diff --git a/-NFLT4oBgHgl3EQfCi71/content/2301.11976v1.pdf b/-NFLT4oBgHgl3EQfCi71/content/2301.11976v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..550523f2fae3e44e8c182c9e061aaf5d77c12b03 --- /dev/null +++ b/-NFLT4oBgHgl3EQfCi71/content/2301.11976v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d3371932b019d9068dc686efabd358690b6a8209c45d030945e5ef9a007ea34d +size 197256 diff --git a/-NFLT4oBgHgl3EQfCi71/vector_store/index.pkl b/-NFLT4oBgHgl3EQfCi71/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..8cc953cc2d0dd2d647003773099ba57168f2e381 --- /dev/null +++ b/-NFLT4oBgHgl3EQfCi71/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8dac84c6544ae11df04bcba403635b745eea6a7102ff532238d3149d97c7b715 +size 102995 diff --git a/.gitattributes b/.gitattributes index 2894d07c40d12afbcca3a9c441dc7261396a628d..a274acb944d77a67e0d54f1d745dc5157cb26c6e 100644 --- a/.gitattributes +++ b/.gitattributes @@ -7839,3 +7839,57 @@ ZNFRT4oBgHgl3EQfPjcf/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -tex LNFRT4oBgHgl3EQf1jiU/content/2301.13657v1.pdf filter=lfs diff=lfs merge=lfs -text Plough[[:space:]]Knowledge[[:space:]]Ocean[[:space:]]Intro/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text m9E_T4oBgHgl3EQf7Ry5/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +n9FKT4oBgHgl3EQfFy0J/content/2301.11721v1.pdf filter=lfs diff=lfs merge=lfs -text +GNAzT4oBgHgl3EQfHPvT/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +ZNE2T4oBgHgl3EQfZQdS/content/2301.03862v1.pdf filter=lfs diff=lfs merge=lfs -text +ttAzT4oBgHgl3EQfBfrk/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +mNE2T4oBgHgl3EQfeQdZ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +ztFJT4oBgHgl3EQfjCwq/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +wtAzT4oBgHgl3EQfCPrg/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +JdE1T4oBgHgl3EQfsAXy/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +mdE2T4oBgHgl3EQfJQZR/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +qtFRT4oBgHgl3EQfezdF/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +5tE4T4oBgHgl3EQf1g0M/content/2301.05290v1.pdf filter=lfs diff=lfs merge=lfs -text +p9E4T4oBgHgl3EQfvg1C/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +5dE1T4oBgHgl3EQfBAJm/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +i9FQT4oBgHgl3EQflDYi/content/2301.13360v1.pdf filter=lfs diff=lfs merge=lfs -text +o9FKT4oBgHgl3EQfGS3Z/content/2301.11725v1.pdf filter=lfs diff=lfs merge=lfs -text +MNFLT4oBgHgl3EQfMy8Y/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +SdAyT4oBgHgl3EQfVPfZ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +udE2T4oBgHgl3EQf2Qhr/content/2301.04159v1.pdf filter=lfs diff=lfs merge=lfs -text +X9E0T4oBgHgl3EQfWQAZ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +i9FQT4oBgHgl3EQflDYi/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +BtAyT4oBgHgl3EQf4PqZ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +l9E4T4oBgHgl3EQftw0Q/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +5tE0T4oBgHgl3EQfvwEb/content/2301.02621v1.pdf filter=lfs diff=lfs merge=lfs -text +6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf filter=lfs diff=lfs merge=lfs -text +zdE3T4oBgHgl3EQfmgot/content/2301.04616v1.pdf filter=lfs diff=lfs merge=lfs -text +udE2T4oBgHgl3EQf2Qhr/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +n9FKT4oBgHgl3EQfFy0J/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +K9A0T4oBgHgl3EQfC_82/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +MNFLT4oBgHgl3EQfMy8Y/content/2301.12017v1.pdf filter=lfs diff=lfs merge=lfs -text +2NE2T4oBgHgl3EQfjAcm/content/2301.03963v1.pdf filter=lfs diff=lfs merge=lfs -text +ZNE2T4oBgHgl3EQfZQdS/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +bNFLT4oBgHgl3EQfXy8v/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +_dFKT4oBgHgl3EQfUy2I/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +otFPT4oBgHgl3EQf6zWU/content/2301.13202v1.pdf filter=lfs diff=lfs merge=lfs -text +5tE4T4oBgHgl3EQf1g0M/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +6dE1T4oBgHgl3EQfTQOX/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +LNFRT4oBgHgl3EQf1jiU/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +odFKT4oBgHgl3EQfFy38/content/2301.11723v1.pdf filter=lfs diff=lfs merge=lfs -text +E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf filter=lfs diff=lfs merge=lfs -text +_dE2T4oBgHgl3EQfmgfK/content/2301.04000v1.pdf filter=lfs diff=lfs merge=lfs -text +FtE0T4oBgHgl3EQfzQIy/content/2301.02669v1.pdf filter=lfs diff=lfs merge=lfs -text +5dE0T4oBgHgl3EQfegDz/content/2301.02393v1.pdf filter=lfs diff=lfs merge=lfs -text +MNE3T4oBgHgl3EQfwQsY/content/2301.04700v1.pdf filter=lfs diff=lfs merge=lfs -text +FtE0T4oBgHgl3EQfzQIy/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +E9E1T4oBgHgl3EQfqgWQ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +R9AzT4oBgHgl3EQf0f4J/content/2301.01783v1.pdf filter=lfs diff=lfs merge=lfs -text +-NFLT4oBgHgl3EQfCi71/content/2301.11976v1.pdf filter=lfs diff=lfs merge=lfs -text +AtAzT4oBgHgl3EQfhv1C/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +JtE5T4oBgHgl3EQfXg-b/content/2301.05567v1.pdf filter=lfs diff=lfs merge=lfs -text +xdFRT4oBgHgl3EQfhzcF/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +JdE1T4oBgHgl3EQfsAXy/content/2301.03362v1.pdf filter=lfs diff=lfs merge=lfs -text +JtE5T4oBgHgl3EQfXg-b/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf filter=lfs diff=lfs merge=lfs -text +AtAzT4oBgHgl3EQfhv1C/content/2301.01488v1.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf b/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b7f9e4b5324f805e01f525d36704dbe79989b50c Binary files /dev/null and b/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf differ diff --git a/09AzT4oBgHgl3EQfRfsM/content/tmp_files/2301.01215v1.pdf.txt b/09AzT4oBgHgl3EQfRfsM/content/tmp_files/2301.01215v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2cc005f7c23735dbbbbe2e51733285ed47f50afb --- /dev/null +++ b/09AzT4oBgHgl3EQfRfsM/content/tmp_files/2301.01215v1.pdf.txt @@ -0,0 +1,241 @@ +arXiv:2301.01215v1 [quant-ph] 3 Jan 2023 +Comment on ’The operational foundations of +PT-symmetric and quasi-Hermitian quantum theory’ +Miloslav Znojil +Nuclear Physics Institute ASCR, Hlavn´ı 130, 250 68 ˇReˇz, Czech Republic +e-mail: znojil@ujf.cas.cz +Abstract +In J. Phys. A: Math. Theor. 55 (2022) 244003, Alase et al wrote that “the constraint of quasi- +Hermiticity on observables” is not “sufficient to extend the standard quantum theory” because +“such a system is equivalent to a standard quantum system.” Three addenda elucidating the +current state of the art are found necessary. The first one concerns the project: In the related +literature the original “aim of extending standard quantum theory” has already been abandoned +shortly after its formulation. The second comment concerns the method, viz., the study in “the +framework of general probabilistic theories” (GPT). It is noticed that a few other, mathematically +consistent GPT-like theories are available. The authors do not mention, in particular, the progress +achieved, under the quasi-Hermiticity constraint, in the approach using the effect algebras. We +add that this approach already found its advanced realistic applications in the quasi-Hermitian +models using the unbounded operators of observables acting in the infinite-dimensional Hilbert +spaces. Thirdly, the “intriguing open question” about “what possible constraints, if any, could +lead to such a meaningful extension” (in the future) is given an immediate tentative answer: The +possibility is advocated that the desirable constraint could really be just the quasi-Hermiticity of +the observables, provided only that one has in mind its recently developed non-stationary version. +1 + +1 +Introduction +As a part of issue “Foundational Structures in Quantum Theory” the paper “The operational +foundations of PT-symmetric and quasi-Hermitian quantum theory” by Abhijeet Alase, Salini +Karuvade and Carlo Maria Scandolo [1] fitted very well the scope of the volume. In a rigorous +mathematical style it offered the readers an interesting material confirming the compatibility +between the three recent conceptual innovations of quantum theory. Still, we believe that the +authors’ coverage of the subject deserves a few addenda, mainly because in loc. cit., the deeply +satisfactory nature of the mathematical analysis seems to be accompanied by a perceivably less +careful presentation of its implications in the context of the theoretical quantum physics. +2 +The absence of extensions of standard quantum theory +Our first addendum is motivated by the last sentence of the abstract in [1]. It states that “our +results show that neither PT-symmetry nor quasi-Hermiticity constraints are sufficient to extend +standard quantum theory consistently”. +Indeed, it is rather unfortunate that this statement +diverts attention from the very interesting main mathematical message of the paper (viz. from the +rigorous confirmation of compatibility between the three alternative versions of quantum theory) +to its much less satisfactory physical contextualization. The impression is further strengthened by +the last paragraph of the whole text where we read that “in conclusion, neither PT-symmetry nor +quasi-Hermiticity of observables leads to an extension of standard quantum mechanics.” Certainly, +non-specialists could be mislead to interpret such a conclusion wrongly, as a disproof of usefulness +of what is usually called PT-symmetric quantum theory (PTQT, an approach which is briefly +reviewed in section 2.1 of loc. cit.) or of the so called quasi-Hermitian quantum theory (QHQT, +cf. its compact review in the subsequent section 2.2 of loc. cit.). The misunderstanding seems +completed by the combination of the very first sentence of the abstract with the very last sentence +of the text: At the beginning of the Abstract we are told that “PT-symmetric quantum theory +was originally proposed with the aim of extending standard quantum theory” (which is not too +relevant at present), while the final question reads “what possible constraints, if any, could lead +to such a meaningful extension” [1]. +The main weakness of such a “theory-extension” motivation and of the “physical” framing of +paper [1] is that the original purpose of “relaxing the Hermiticity constraint on Hamiltonians” +(as proposed, by Bender with Boettcher, in their enormously influential letter [2]) was almost +immediately shown overambitious and unfulfilled (see, e.g., the Mostafazadeh’s 2010 very mathe- +matical and detailed criticism and explanation “that neither PT-symmetry nor quasi-Hermiticity +constraints are sufficient to extend standard quantum theory” [3]). Thus, the authors of [1] only +come with their “aim to answer the question of whether a consistent physical theory with PT- +symmetric observables extends standard quantum theory” too late. For more than twelve years +the answer is known to be negative [4]. +3 +A comment on the method +Naturally, nobody claims that the PTQT itself is not useful. Nobody could also deny the relevance +and the novelty of the mathematical results presented in paper [1]. It is only a pity that its authors +did not better emphasize how well their analysis fits the subject of the special issue, especially +2 + +due to their innovative turn of attention to the so called general probabilistic theories (GPT, cf. +their compact outline in section 2.3 of [1]). +Paradoxically, in the GPT context one immediately identifies the second weakness of the paper. +It lies in a surprisingly short list of the GPT-approach-representing references. In the paper the +list just incorporates the eight newer papers [5] - [12] (all of them published after the year 2000) +plus a single older, Foulis-coauthored 1970 paper [13]. Not quite expectedly, the list of references +does not contain any Gudder’s results – after all, paper [1] is a part of the special issue which is +explicitly declared to honor his contribution to the field. Thus, one would expect, for example, +a reference to his later review papers [14, 15] where he formulated one of the key GPT-related +mathematical theses that “a physical system S under experimental investigation and governed by a +scientific theory (which may be subject to modification in the light of new experimental evidence) +is represented by a CB-effect algebra”. An equally unexpected gap in the references also concerns +the absence of the Foulis’ pioneering, effect-algebras introducing 1994 paper with Bennet [16], or +his comparatively recent review [17]. Indeed, both of these papers sought and offered operational +foundations and gained insight into the GPT-motivating relationship between quantum theory +and classical probability theory (this was emphasized also in [5], etc). +What is an even worse omission is that the list of references does not contain any other subject- +related studies like, e.g., paper [18] in which the predecessors of the present authors considered, +explicitly, the PTQT-GPT relationship, having reconfirmed that “from the standpoint of (gen- +eralized) effect algebra theory both representations of our quantum system coincide”. Similarly, +the QHQT-GPT relationship may be found studied in paper [19] in which the mathematically +fairly advanced analysis incorporated even the fairly realistic quantum models using unbounded +operators. Indeed, the rate of the progress is striking, especially when one recalls just a few years +younger report [14] in which the “separable complex Hilbert space” is assumed to be just “of +dimension 1 or more”. +4 +New and promising non-stationary constraints +At present, it makes sense to accept the fact that in spite of the robust nature of the existing “stan- +dard” formulations of quantum theory and, in particular, of the quantum mechanics of unitary +systems, there still exist differences in the practical applicability of their various specific imple- +mentations. The motivation of the diversity is that ”no [particular] formulation produces a royal +road to quantum mechanics” [20]. In some sense this implies that the concept of the “extension” +of the existing quantum theory is vague. The apparently minor technical differences between the +current alternative formulations of quantum mechanics (as sampled, in [20], on elementary level) +could happen to lead to “decisive extensions” in the future. +A good illustrative example can be provided even within the current stationary forms of QHQT. +Indeed, even in this framework the formalism can really be declared equivalent to its standard +textbook form. Still, the equivalence can be confirmed only under certain fairly detailed and +specific mathematical assumptions (cf. [21]). These assumptions are, even in the abstract context +of functional analysis, far from trivial [22]. Paradoxically, even the popular physical quantum +models of Bender and Boettcher [2] have been later found not to belong to the “admissible”, +QHQT-compatible class (see, e.g., [23, 24] for the corresponding subtle details). Thus, in spite +of their manifest and unbroken PT-symmetry, even these originally proposed benchmark models +still wait for a “meaningful extension” of their fully consistent GPT interpretation. +In our third, last addendum we are now prepared to reopen the vague but important question +3 + +of what the words of “extension” of the “standard” quantum theory could, or do, really mean. +On one side, it is known and widely accepted that the various existing formulations of quantum +theory “differ dramatically in mathematical and conceptual overview, yet each one makes identical +predictions for all experimental results” [20]. On the other side, such a rigidity of the theory is +far from satisfactory. For example, a suitable future amendment of quantum theory would be +necessary for a still absent clarification of the concept of quantum gravity [25]. +For the sake of brevity let us skip here the discussion of the parallel questions concerning +the PT-symmetric quantum models. This being said we believe that even the QHQT formalism +itself did not say its last word yet. Indeed, our optimism concerning its potential “theory exten- +sion status” is based on the recent fundamental clarifications of its scope and structure. First of +all, it became clear that in the QHQT descriptions of unitary systems it is sufficient to distin- +guish just between their representations in the “generalized Schr¨odinger picture” (GSP, stationary +and best presented, by our opinion, in reviews [21, 26] and [3]) and in its non-stationary “non- +Hermitian interaction picture” alternative (NIP, [27, 28]). Using this terminology one immediately +reveals that the QHQT-related considerations of paper [1] just cover the GSP approach. In other +words, the physical inner-product metric (denoted by symbol η) is perceived there as strictly +time-independent. This means that in the GSP language one can easily identify the (stationary) +generator G of the evolution of the wave functions with the (“observable-energy”) Hamiltonian H +(which has real spectrum and is, by assumption, η−quasi-Hermitian). +The situation becomes different after the extension of the QHQT approach to the non-stationary, +NIP dynamical regime. In this case we will denote the inner-product metric by another dedicated +symbol Θ = Θ(t) as introduced in the first description of NIP in [29]. What is important is that +the observable-energy operator H = H(t) will get split in the sum of the two auxiliary operators +G(t) and Σ(t). As long as they are both neither observable nor Θ-quasi-Hermitian in general, +we will exclusively assign the name of the Hamiltonian to the instantaneous energy operator H +(with real spectrum), adding a word of warning that a different, less consequent terminology is +often used by some other authors (see, e.g., [30, 31]). Even though neither the spectrum of G(t) +nor the spectrum of Σ(t) is real in general, the introduction of these operators endows the NIP +formalism with an additional flexibility, capable, as we believe, of opening the new horizons in the +contemporary quantum physics: In the context of relativistic quantum mechanics, for example, +such a hypothetical “theory-extension” possibility has been discussed, in detail, in [27]. For the +purposes of a potentially new approach to the problem of the unitary-evolution models of quantum +phase transitions in many-body context, the formalism has slightly been adapted in [32]. Last but +not least, our very recent paper [28] has been devoted to the possible use of the NIP evolution +equations in a Wheeler-DeWitt-equation-based schematic model of Big Bang in the context of +quantum gravity and cosmology. In this spirit, therefore, certain sufficiently realistic NIP-based +models could easily happen to acquire an “extended quantum mechanics” status, perhaps, in the +nearest future. +5 +Conclusions +The key subject discussed in paper [1] was the question of the possible extension of the scope of +quantum theory in general, and of the realization of such an ambitious project, in the respective +PTQT and QHQT theoretical frameworks, in particular. In our present commentary we reminded +the readers, marginally, of the existence of several older, comparably sceptical conclusions as +available in the related literature (see section 2 for details). In section 3 we then added a few +4 + +similar broader-context-emphasizing remarks on the mathematical, GPT-related aspects of the +results of [1]. Still, the core of our present message (as presented in the longest section 4) concerned +physics. We pointed out that at present, the question of the possible extension of the scope of +the standard quantum theory should be considered open even in the narrower PTQT and QHQT +frameworks. +In support of the latter statement we mentioned that +• even for the stationary and, apparently, most elementary PTQT potentials (sampled, say, by +the most popular V (x) = ix3), the widespread initial optimism and intuitive “nothing new” +understanding of their physical meaning and mathematical background have both recently +been shattered by their more rigorous mathematical analysis; +• one can hardly say “nothing new” even in a mathematically much better understood station- +ary QHQT alias GSP framework where, typically, the use of certain stronger assumptions +enables one to circumvent the obstacles revealed by rigorous mathematics. Indeed, even in +the GSP framework one can search for an entirely new physics. Typically, a non-standard +phenomenology becomes described by the QHQT models in an infinitesimally small vicinity +of the so called exceptional points: Paper [33] offers an illustrative sample of the quantum +systems which cannot be described by the standard quantum theory; +• in fact, our return to optimism and expectation that the QHQT may be a “fundamentally +innovative” theory found its most explicit formulation in section 4. Briefly we exposed there +an enormous growth of the flexibility of the QHQT approach after its ultimate non-stationary +NIP generalization. In some sense, the emphasis put upon the deeply promising conceptual +nature of such a flexibility can be read as the deepest core of our present comment and +message. +Data availability statement +No new data were created or analysed in this study. +ORCID iD +https://orcid.org/0000-0001-6076-0093 +5 + +References +[1] Alase A, Karuvade S and Scandolo C M 2022 J. Phys. A: Math. Theor. 55 244003 +[2] Bender C M and Boettcher S 1998 Phys. Rev. Lett. 80 5243 - 5246 +[3] Mostafazadeh A 2010 Int. J. Geom. Methods Mod. Phys. 07 1191 - 1306 +[4] Znojil M 2015 Non-Self-Adjoint Operators in Quantum Physics: Ideas, People, and Trends +(New York:Wiley) ch 1 pp 7 - 58 +[5] Hardy L 2001 Quantum theory from five reasonable axioms (arXiv:quant-ph/0101012) +[6] Barrett J 2007 Phys Rev A 75 032304 +[7] Chiribella G, D’Ariano G M and Perinotti P 2010 Phys Rev A 81 062348 +[8] Hardy L 2011 Foliable Operational Structures for General Probabilistic Theories (Cambridge: +Cambridge University Press) pp 409 - 442 +[9] Barnum H and Wilce A 2011 Electron. Notes Theor. Comput. Sci. 270 3 - 15 +[10] Janotta P and Hinrichsen H 2014 J. Phys. A: Math. Theor. 47 323001 +[11] Barnum H and Wilce A 2016 Post-Classical Probability Theory (Berlin: Springer) pp 367.420 +[12] Scandolo C M 2018 Information-theoretic foundations of thermodynamics in general proba- +bilistic theories. PhD Thesis, University of Oxford +[13] Randall C H and Foulis D J 1970 Am Math Mon 77 363 - 374 +[14] Gudder S P 2004 Rep Math Phys 54 93 - 114 +[15] Gudder S P 2006 Demonstratio Math 39 43 - 54 +[16] Foulis D J and Bennett M K 1994 Found Phys 24 1331 - 1352 +[17] Foulis D J 2007 Rep Math Phys 60 329 - 346 +[18] Paseka J 2011 Int J Theor Phys 50 1198 - 1205 +[19] Paseka J, Pulmannova S and Riecanova Z 2013 Int J Theor Phys 52 1994 - 2000 +[20] Styer D F et al 2002 Am J Phys 70 288 - 297 +[21] Scholtz F G, Geyer H B and Hahne F J W 1992 Ann Phys 213 74 - 101 +[22] Dieudonne J 1961 Proc Int Symp Lin Spaces (Pergamon, Oxford, 1961), pp. 115 - 122 +[23] Siegl P and Krejˇciˇr´ık D 2012 Phys Rev D 86 121702(R) +Krejˇciˇr´ık D, Siegl P, Tater M and Viola J 2015 J Math Phys 56 103513 +6 + +[24] Guenther U and Frank Stefani F 2019 IR-truncated PT -symmetric ix3 model and its asymp- +totic spectral scaling graph. arXiv 1901.08526 +Semor´adov´a I and Siegl P 2022 SIAM J. Math. Anal. 54 5064 - 5101 +[25] Rovelli C 2004 Quantum Gravity (Cambridge University Press, Cambridge, UK) +[26] Bender C M 2007 Rep Prog Phys 70 947 - 1118 +[27] Znojil M 2017 Ann Phys (NY) 385 162 - 179 +[28] Znojil M 2022 Universe 8 385 +[29] Znojil M 2008 Phys Rev D 78 085003 +Znojil M 2009 SIGMA 5 001 +[30] Fring A and Moussa M H Y 2016 Phys Rev A 93 042114 +[31] B´ıla H 2009 e-print arXiv: 0902.0474 +Gong J-B and Wang Q-H 2013 J Phys A Math Theor 46 485302 +Amaouche N, Sekhri M, Zerimeche R, Maamache M and Liang J-Q 2022 eprint: +arXiv:2207.02477 (quant-ph) +[32] Bishop R F and Znojil M 2020 Eur Phys J Plus 135 374 +[33] Znojil M 2022 Mathematics 10 3721 +7 + diff --git a/09AzT4oBgHgl3EQfRfsM/content/tmp_files/load_file.txt b/09AzT4oBgHgl3EQfRfsM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7c0a9b5ab6c031bc4c6ba8e960a8323c0fa2c4b7 --- /dev/null +++ b/09AzT4oBgHgl3EQfRfsM/content/tmp_files/load_file.txt @@ -0,0 +1,144 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf,len=143 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content='01215v1 [quant-ph] 3 Jan 2023 Comment on ’The operational foundations of PT-symmetric and quasi-Hermitian quantum theory’ Miloslav Znojil Nuclear Physics Institute ASCR, Hlavn´ı 130, 250 68 ˇReˇz, Czech Republic e-mail: znojil@ujf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content='cas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content='cz Abstract In J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' 55 (2022) 244003, Alase et al wrote that “the constraint of quasi- Hermiticity on observables” is not “sufficient to extend the standard quantum theory” because “such a system is equivalent to a standard quantum system.” Three addenda elucidating the current state of the art are found necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' The first one concerns the project: In the related literature the original “aim of extending standard quantum theory” has already been abandoned shortly after its formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' The second comment concerns the method, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=', the study in “the framework of general probabilistic theories” (GPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' It is noticed that a few other, mathematically consistent GPT-like theories are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' The authors do not mention, in particular, the progress achieved, under the quasi-Hermiticity constraint, in the approach using the effect algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' We add that this approach already found its advanced realistic applications in the quasi-Hermitian models using the unbounded operators of observables acting in the infinite-dimensional Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Thirdly, the “intriguing open question” about “what possible constraints, if any, could lead to such a meaningful extension” (in the future) is given an immediate tentative answer: The possibility is advocated that the desirable constraint could really be just the quasi-Hermiticity of the observables, provided only that one has in mind its recently developed non-stationary version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' 1 1 Introduction As a part of issue “Foundational Structures in Quantum Theory” the paper “The operational foundations of PT-symmetric and quasi-Hermitian quantum theory” by Abhijeet Alase, Salini Karuvade and Carlo Maria Scandolo [1] fitted very well the scope of the volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' In a rigorous mathematical style it offered the readers an interesting material confirming the compatibility between the three recent conceptual innovations of quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Still, we believe that the authors’ coverage of the subject deserves a few addenda, mainly because in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=', the deeply satisfactory nature of the mathematical analysis seems to be accompanied by a perceivably less careful presentation of its implications in the context of the theoretical quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' 2 The absence of extensions of standard quantum theory Our first addendum is motivated by the last sentence of the abstract in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' It states that “our results show that neither PT-symmetry nor quasi-Hermiticity constraints are sufficient to extend standard quantum theory consistently”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Indeed, it is rather unfortunate that this statement diverts attention from the very interesting main mathematical message of the paper (viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' from the rigorous confirmation of compatibility between the three alternative versions of quantum theory) to its much less satisfactory physical contextualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' The impression is further strengthened by the last paragraph of the whole text where we read that “in conclusion, neither PT-symmetry nor quasi-Hermiticity of observables leads to an extension of standard quantum mechanics.” Certainly, non-specialists could be mislead to interpret such a conclusion wrongly, as a disproof of usefulness of what is usually called PT-symmetric quantum theory (PTQT, an approach which is briefly reviewed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content='1 of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=') or of the so called quasi-Hermitian quantum theory (QHQT, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' its compact review in the subsequent section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content='2 of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' The misunderstanding seems completed by the combination of the very first sentence of the abstract with the very last sentence of the text: At the beginning of the Abstract we are told that “PT-symmetric quantum theory was originally proposed with the aim of extending standard quantum theory” (which is not too relevant at present), while the final question reads “what possible constraints, if any, could lead to such a meaningful extension” [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' The main weakness of such a “theory-extension” motivation and of the “physical” framing of paper [1] is that the original purpose of “relaxing the Hermiticity constraint on Hamiltonians” (as proposed, by Bender with Boettcher, in their enormously influential letter [2]) was almost immediately shown overambitious and unfulfilled (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=', the Mostafazadeh’s 2010 very mathe- matical and detailed criticism and explanation “that neither PT-symmetry nor quasi-Hermiticity constraints are sufficient to extend standard quantum theory” [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Thus, the authors of [1] only come with their “aim to answer the question of whether a consistent physical theory with PT- symmetric observables extends standard quantum theory” too late.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' For more than twelve years the answer is known to be negative [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' 3 A comment on the method Naturally, nobody claims that the PTQT itself is not useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Nobody could also deny the relevance and the novelty of the mathematical results presented in paper [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' It is only a pity that its authors did not better emphasize how well their analysis fits the subject of the special issue, especially 2 due to their innovative turn of attention to the so called general probabilistic theories (GPT, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' their compact outline in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content='3 of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Paradoxically, in the GPT context one immediately identifies the second weakness of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' It lies in a surprisingly short list of the GPT-approach-representing references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' In the paper the list just incorporates the eight newer papers [5] - [12] (all of them published after the year 2000) plus a single older, Foulis-coauthored 1970 paper [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Not quite expectedly, the list of references does not contain any Gudder’s results – after all, paper [1] is a part of the special issue which is explicitly declared to honor his contribution to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Thus, one would expect, for example, a reference to his later review papers [14, 15] where he formulated one of the key GPT-related mathematical theses that “a physical system S under experimental investigation and governed by a scientific theory (which may be subject to modification in the light of new experimental evidence) is represented by a CB-effect algebra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' An equally unexpected gap in the references also concerns the absence of the Foulis’ pioneering, effect-algebras introducing 1994 paper with Bennet [16], or his comparatively recent review [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Indeed, both of these papers sought and offered operational foundations and gained insight into the GPT-motivating relationship between quantum theory and classical probability theory (this was emphasized also in [5], etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' What is an even worse omission is that the list of references does not contain any other subject- related studies like, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=', paper [18] in which the predecessors of the present authors considered, explicitly, the PTQT-GPT relationship, having reconfirmed that “from the standpoint of (gen- eralized) effect algebra theory both representations of our quantum system coincide”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Similarly, the QHQT-GPT relationship may be found studied in paper [19] in which the mathematically fairly advanced analysis incorporated even the fairly realistic quantum models using unbounded operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Indeed, the rate of the progress is striking, especially when one recalls just a few years younger report [14] in which the “separable complex Hilbert space” is assumed to be just “of dimension 1 or more”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' 4 New and promising non-stationary constraints At present, it makes sense to accept the fact that in spite of the robust nature of the existing “stan- dard” formulations of quantum theory and, in particular, of the quantum mechanics of unitary systems, there still exist differences in the practical applicability of their various specific imple- mentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' The motivation of the diversity is that ”no [particular] formulation produces a royal road to quantum mechanics” [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' In some sense this implies that the concept of the “extension” of the existing quantum theory is vague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' The apparently minor technical differences between the current alternative formulations of quantum mechanics (as sampled, in [20], on elementary level) could happen to lead to “decisive extensions” in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' A good illustrative example can be provided even within the current stationary forms of QHQT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Indeed, even in this framework the formalism can really be declared equivalent to its standard textbook form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Still, the equivalence can be confirmed only under certain fairly detailed and specific mathematical assumptions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' These assumptions are, even in the abstract context of functional analysis, far from trivial [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Paradoxically, even the popular physical quantum models of Bender and Boettcher [2] have been later found not to belong to the “admissible”, QHQT-compatible class (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=', [23, 24] for the corresponding subtle details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Thus, in spite of their manifest and unbroken PT-symmetry, even these originally proposed benchmark models still wait for a “meaningful extension” of their fully consistent GPT interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' In our third, last addendum we are now prepared to reopen the vague but important question 3 of what the words of “extension” of the “standard” quantum theory could, or do, really mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' On one side, it is known and widely accepted that the various existing formulations of quantum theory “differ dramatically in mathematical and conceptual overview, yet each one makes identical predictions for all experimental results” [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' On the other side, such a rigidity of the theory is far from satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' For example, a suitable future amendment of quantum theory would be necessary for a still absent clarification of the concept of quantum gravity [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' For the sake of brevity let us skip here the discussion of the parallel questions concerning the PT-symmetric quantum models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' This being said we believe that even the QHQT formalism itself did not say its last word yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Indeed, our optimism concerning its potential “theory exten- sion status” is based on the recent fundamental clarifications of its scope and structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' First of all, it became clear that in the QHQT descriptions of unitary systems it is sufficient to distin- guish just between their representations in the “generalized Schr¨odinger picture” (GSP, stationary and best presented, by our opinion, in reviews [21, 26] and [3]) and in its non-stationary “non- Hermitian interaction picture” alternative (NIP, [27, 28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Using this terminology one immediately reveals that the QHQT-related considerations of paper [1] just cover the GSP approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' In other words, the physical inner-product metric (denoted by symbol η) is perceived there as strictly time-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' This means that in the GSP language one can easily identify the (stationary) generator G of the evolution of the wave functions with the (“observable-energy”) Hamiltonian H (which has real spectrum and is, by assumption, η−quasi-Hermitian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' The situation becomes different after the extension of the QHQT approach to the non-stationary, NIP dynamical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' In this case we will denote the inner-product metric by another dedicated symbol Θ = Θ(t) as introduced in the first description of NIP in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' What is important is that the observable-energy operator H = H(t) will get split in the sum of the two auxiliary operators G(t) and Σ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' As long as they are both neither observable nor Θ-quasi-Hermitian in general, we will exclusively assign the name of the Hamiltonian to the instantaneous energy operator H (with real spectrum), adding a word of warning that a different, less consequent terminology is often used by some other authors (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=', [30, 31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Even though neither the spectrum of G(t) nor the spectrum of Σ(t) is real in general, the introduction of these operators endows the NIP formalism with an additional flexibility, capable, as we believe, of opening the new horizons in the contemporary quantum physics: In the context of relativistic quantum mechanics, for example, such a hypothetical “theory-extension” possibility has been discussed, in detail, in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' For the purposes of a potentially new approach to the problem of the unitary-evolution models of quantum phase transitions in many-body context, the formalism has slightly been adapted in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Last but not least, our very recent paper [28] has been devoted to the possible use of the NIP evolution equations in a Wheeler-DeWitt-equation-based schematic model of Big Bang in the context of quantum gravity and cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' In this spirit, therefore, certain sufficiently realistic NIP-based models could easily happen to acquire an “extended quantum mechanics” status, perhaps, in the nearest future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' 5 Conclusions The key subject discussed in paper [1] was the question of the possible extension of the scope of quantum theory in general, and of the realization of such an ambitious project, in the respective PTQT and QHQT theoretical frameworks, in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' In our present commentary we reminded the readers, marginally, of the existence of several older, comparably sceptical conclusions as available in the related literature (see section 2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' In section 3 we then added a few 4 similar broader-context-emphasizing remarks on the mathematical, GPT-related aspects of the results of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Still, the core of our present message (as presented in the longest section 4) concerned physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' We pointed out that at present, the question of the possible extension of the scope of the standard quantum theory should be considered open even in the narrower PTQT and QHQT frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' In support of the latter statement we mentioned that even for the stationary and, apparently, most elementary PTQT potentials (sampled, say, by the most popular V (x) = ix3), the widespread initial optimism and intuitive “nothing new” understanding of their physical meaning and mathematical background have both recently been shattered by their more rigorous mathematical analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' one can hardly say “nothing new” even in a mathematically much better understood station- ary QHQT alias GSP framework where, typically, the use of certain stronger assumptions enables one to circumvent the obstacles revealed by rigorous mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Indeed, even in the GSP framework one can search for an entirely new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Typically, a non-standard phenomenology becomes described by the QHQT models in an infinitesimally small vicinity of the so called exceptional points: Paper [33] offers an illustrative sample of the quantum systems which cannot be described by the standard quantum theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' in fact, our return to optimism and expectation that the QHQT may be a “fundamentally innovative” theory found its most explicit formulation in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Briefly we exposed there an enormous growth of the flexibility of the QHQT approach after its ultimate non-stationary NIP generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' In some sense, the emphasis put upon the deeply promising conceptual nature of such a flexibility can be read as the deepest core of our present comment and message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Data availability statement No new data were created or analysed in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' ORCID iD https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content='org/0000-0001-6076-0093 5 References [1] Alase A, Karuvade S and Scandolo C M 2022 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' 55 244003 [2] Bender C M and Boettcher S 1998 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' 80 5243 - 5246 [3] Mostafazadeh A 2010 Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Methods Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' 07 1191 - 1306 [4] Znojil M 2015 Non-Self-Adjoint Operators in Quantum Physics: Ideas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' People,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' and Trends (New York:Wiley) ch 1 pp 7 - 58 [5] Hardy L 2001 Quantum theory from five reasonable axioms (arXiv:quant-ph/0101012) [6] Barrett J 2007 Phys Rev A 75 032304 [7] Chiribella G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' D’Ariano G M and Perinotti P 2010 Phys Rev A 81 062348 [8] Hardy L 2011 Foliable Operational Structures for General Probabilistic Theories (Cambridge: Cambridge University Press) pp 409 - 442 [9] Barnum H and Wilce A 2011 Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Notes Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' 270 3 - 15 [10] Janotta P and Hinrichsen H 2014 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' 47 323001 [11] Barnum H and Wilce A 2016 Post-Classical Probability Theory (Berlin: Springer) pp 367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content='420 [12] Scandolo C M 2018 Information-theoretic foundations of thermodynamics in general proba- bilistic theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' PhD Thesis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' University of Oxford [13] Randall C H and Foulis D J 1970 Am Math Mon 77 363 - 374 [14] Gudder S P 2004 Rep Math Phys 54 93 - 114 [15] Gudder S P 2006 Demonstratio Math 39 43 - 54 [16] Foulis D J and Bennett M K 1994 Found Phys 24 1331 - 1352 [17] Foulis D J 2007 Rep Math Phys 60 329 - 346 [18] Paseka J 2011 Int J Theor Phys 50 1198 - 1205 [19] Paseka J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Pulmannova S and Riecanova Z 2013 Int J Theor Phys 52 1994 - 2000 [20] Styer D F et al 2002 Am J Phys 70 288 - 297 [21] Scholtz F G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Geyer H B and Hahne F J W 1992 Ann Phys 213 74 - 101 [22] Dieudonne J 1961 Proc Int Symp Lin Spaces (Pergamon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' 1961),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' 115 - 122 [23] Siegl P and Krejˇciˇr´ık D 2012 Phys Rev D 86 121702(R) Krejˇciˇr´ık D, Siegl P, Tater M and Viola J 2015 J Math Phys 56 103513 6 [24] Guenther U and Frank Stefani F 2019 IR-truncated PT -symmetric ix3 model and its asymp- totic spectral scaling graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' arXiv 1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content='08526 Semor´adov´a I and Siegl P 2022 SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content=' 54 5064 - 5101 [25] Rovelli C 2004 Quantum Gravity (Cambridge University Press, Cambridge, UK) [26] Bender C M 2007 Rep Prog Phys 70 947 - 1118 [27] Znojil M 2017 Ann Phys (NY) 385 162 - 179 [28] Znojil M 2022 Universe 8 385 [29] Znojil M 2008 Phys Rev D 78 085003 Znojil M 2009 SIGMA 5 001 [30] Fring A and Moussa M H Y 2016 Phys Rev A 93 042114 [31] B´ıla H 2009 e-print arXiv: 0902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content='0474 Gong J-B and Wang Q-H 2013 J Phys A Math Theor 46 485302 Amaouche N, Sekhri M, Zerimeche R, Maamache M and Liang J-Q 2022 eprint: arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} +page_content='02477 (quant-ph) [32] Bishop R F and Znojil M 2020 Eur Phys J Plus 135 374 [33] Znojil M 2022 Mathematics 10 3721 7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfRfsM/content/2301.01215v1.pdf'} diff --git a/29FIT4oBgHgl3EQf5SvC/vector_store/index.pkl b/29FIT4oBgHgl3EQf5SvC/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..8a51bf04f81090ae34cb493ef2251fb83aa60109 --- /dev/null +++ b/29FIT4oBgHgl3EQf5SvC/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ca2d915f19303542eaa6cd7adedbc7854c2ddb7d2ab92fc7f90a62d34baaa7e7 +size 153492 diff --git a/2NE2T4oBgHgl3EQf5ggG/content/tmp_files/2301.04190v1.pdf.txt b/2NE2T4oBgHgl3EQf5ggG/content/tmp_files/2301.04190v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c4b05bf7963ff4389f9b8da72e2c734fbf669a42 --- /dev/null +++ b/2NE2T4oBgHgl3EQf5ggG/content/tmp_files/2301.04190v1.pdf.txt @@ -0,0 +1,2677 @@ +arXiv:2301.04190v1 [math.DG] 10 Jan 2023 +NOTES ON HARMONIC MAPS +GEORGIOS DASKALOPOULOS AND CHIKAKO MESE +This is set of notes prepared for the Summer School on non-Abelian Hodge theory +in Abbaye de Saint-Jacut de la Mer June, 6-19, 2022. +Table of Contents +Lecture 1: Harmonic Maps Between Riemannian Manifolds +p. 2 +Lecture 2: Existence and regularity +p. 9 +Lecture 3: Pluriharmonic Maps and the Siu-Sampson Formula p. 16 +Lecture 4: Donaldson Corlette Theorem +p. 30 +Date: June 2022. +GD supported in part by NSF DMS-2105226, CM supported in part by NSF DMS-2005406. We +would like to thank Yitong Sun for carefully reading this document and making useful suggestions to +improve the exposition of this paper. +1 + +2 +GEORGIOS DASKALOPOULOS AND CHIKAKO MESE +1. Harmonic Maps Between Riemannian manifolds +1.1. Introduction: Basics. In this section, we define energy of maps between Rie- +mannian manifolds, harmonic maps, and the first and second variation formulas after +the pioneering work of Eells-Sampson [ES]. A good reference is also [J]. +1.2. The energy of maps. Let (M, g), (N, h) be Riemannian manifolds. Let f : +M → N be a smooth map which induces a map df : TM → TN +df +� ∂ +∂xα +� ���� +p += ∂f i +∂xα +∂ +∂yi +���� +f(p) +where (xα) (resp. (yi)) are the local coordinates of M (resp. N). +The map f also induces a vector bundle f ∗TN over M. Let ∇ be a connection on +f ∗TN inherited from the Levi-Civita connection on TN. Then ∇ induces an exterior +derivative +d∇ : C∞((ΛpT ∗M) ⊗ f ∗TN) → C∞((Λp+1T ∗M) ⊗ f ∗TN). +We view df as a section +df ∈ C∞(T ∗M ⊗ f ∗TN) = Ω1(f ∗TN). +Using the notation +∂ +∂f i := ∂ +∂yi ◦ f ∈ C∞ +loc(M, f ∗TN), +we have +df = ∂f i +∂xα dxα ⊗ ∂ +∂f i . +Let (gαβ) (resp. (hij)) be the expression of the Riemannian metric g of M (resp. h +of N) with respect to local coordinates (xα) (resp. (yi)). +Definition 1.1. Set +e(f) := 1 +2|df|2 = 1 +2 +∂f i +∂xα +∂f j +∂xβ gαβhij ◦ f. +The energy of f is +E(f) := +� +M +e(f) ⋆ 1 = 1 +2 +� +M +gαβ(x)hij(f(x)) ∂f i +∂xα +∂f j +∂xβ +� +g(x) dx1 ∧ · · · ∧ dxn. +Here, recall that the Hodge star operator ⋆ : ΛkTM → Λn−kTM, is the unique linear +operator such that for all α, β ∈ ΛkV , +α ∧ ⋆β = g(α, β) ⋆ 1. + +NOTES ON HARMONIC MAPS +3 +Lemma 1.2. Let f = (ft) be a smooth one-parameter family of C∞ maps +f : M × (−ǫ, ǫ) → N, +f(x, t) = ft(x). +Then +∇∂f +∂t = ∇∂/∂tdf +where f = ft and +∂f +∂t = ∂f i +∂t +∂ +∂f i ∈ C∞(f ∗TN). +Proof. Both ∇ ∂f +∂t = ∇∂/∂xα ∂f +∂t dxα and ∇∂/∂tdf = ∇∂/∂t +∂f +∂xαdxα are 1-forms with values +in f ∗TN. Here, +∂f +∂xα = ∂f j +∂xα +∂ +∂f j ∈ C∞(f ∗TN). +Consider f as a map f : M × (−ǫ, ǫ) → N. Since +� +f∗ +� ∂ +∂t +� +, f∗ +� ∂ +∂xα +�� += f∗ +� ∂ +∂t, +∂ +∂xα +� += 0 +and ∇ is torsion-free, +∇∂/∂xα ∂f +∂t = +� +∇f∗( +∂ +∂xα)f∗ +� ∂ +∂t +�� +◦ f = +� +∇f∗( ∂f +∂t )f∗ +� ∂ +∂xα +�� +◦ f = ∇∂/∂tdf( ∂ +∂xα) +which proves the equality. +□ +Corollary 1.3 (First Variation Formula). For (ft) as above, +d +dtE(ft) = +� +M +� +∇∂f +∂t , df +� +⋆ 1. +Proof. We compute +d +dtE(ft) = 1 +2 +� +M +d +dt ⟨df, df⟩ ⋆ 1 += +� +M +� +∇∂/∂tdf, df +� +⋆ 1 += +� +M +� +∇∂f +∂t , df +� +⋆ 1. +□ +Corollary 1.4. The critical points f of the functional E satisfy +� +M +⟨∇ψ, df⟩ ⋆ 1 = 0, +∀ψ ∈ C∞(f ∗TN). +(1.1) +By taking ψ compactly supported away from ∂M, we obtain the Euler Lagrange equation +of E, +τ(f) := −d⋆ +∇df = ⋆d∇ ⋆ df = 0. +(1.2) + +4 +GEORGIOS DASKALOPOULOS AND CHIKAKO MESE +Here, ∇ is the pullback of the Levi-Civita connection on f ∗(TN). +Proof. Let ft be a family of maps with +d +dt +���� +t=0 +ft = ψ ∈ C∞(f ∗TN). +Taking ψ compactly supported and integrating by parts, +d +dt +���� +t=0 +E(ft) = +� +M +⟨ψ, −τf⟩ ⋆ 1 = 0 +which holds for every ψ ∈ C∞ +c (f ∗TM) iff τf = 0. +□ +Definition 1.5. A smooth map f : M → N satisfying d⋆ +∇df = 0 is called a harmonic +map. +1.3. Harmonic map equations in local coordinates. First define +ω +� ∂ +∂yi +� +:= Γi +jkdyk ⊗ ∂ +∂yj , +where Γi +jk are Christoffel symbols on N and set +˜ω := ω ◦ f = +� +Γk +ijdyk ⊗ ∂ +∂yj +� +◦ f = (Γj +ik ◦ f)∂f k +∂xβ dxβ ⊗ ∂ +∂f j . +Then d∇ = d + ˜ω and +d⋆ +∇df = − ⋆ d∇ ⋆ df += − ⋆ (d + ˜ω) +� +⋆df i ⊗ ∂ +∂f i +� += −(⋆d ⋆ df i) ∂ +∂f i − (−1)m−1 ⋆ +� +⋆df i ⊗ ˜ω +� ∂ +∂f i +�� += −∆f i +∂ +∂f i − (−1)m−1 ⋆ +� ∂f i +∂xα ⋆ dxα ∧ (Γj +ik ◦ f)∂f k +∂xβ dxβ ⊗ ∂ +∂f j +� += −∆f i +∂ +∂f i − (−1)m−1 +� ∂f i +∂xα +∂f k +∂xβ Γj +ik ◦ f ⋆ (⋆dxα ∧ dxβ) ⊗ +∂ +∂f j +� += − +� +∆f k + gαβ ∂f i +∂xα +∂f j +∂xβ Γk +ij ◦ f +� ∂ +∂f k . +Thus the harmonic map equation is +∆f k + gαβ ∂f i +∂xα +∂f j +∂xβ Γk +ij ◦ f = 0. +(1.3) +Examples 1.6. +(1) Suppose N = R, then the harmonic map equation reduces to ∆f = 0; i.e. f is a +harmonic function on M. + +NOTES ON HARMONIC MAPS +5 +(2) Suppose M = S1. Then +E(f) = 1 +2 +� 2π +0 +| ˙f(t)|dt +and the critical points of E(f) are geodesics. We can also see this from the harmonic +maps equation. Since S1 is 1-dimensional we can take gαβ = δαβ. Then +∂2f k +∂t2 + Γk +ij +∂f i +∂xα +∂f j +∂xβ = 0. +This is the geodesic equation. +(3) We’ll show later that holomorphic maps between K¨ahler manifolds are harmonic +(cf. Remark 3.3). +1.4. The Dirichlet and Neumann problems. If M has non-empty boundary (N +is without boundary) there are two different boundary value problems to consider: +• Dirichlet problem: Minimize E in a fixed homotopy class of maps from M to +N relative to the boundary of M. This is equivalent to considering compactly +supported variations ψ. +• Neumann problem: Minimize E in a fixed free homotopy class of maps from +M to N, in other words no restriction on the type of variations. +1.5. The second variation. The Riemannian tensor +RN : TN × TN × TN → TN +induces and operator +RN : f ∗TN × f ∗TN × f ∗TN → f ∗TN +in the natural way. +Lemma 1.7. Let ft : M → N, and let V be a vector field along ft. Then +∇∂/∂t∇V = ∇∇∂/∂tV − RN +� +df, ∂f +∂t +� +V. +Proof. Both +∇∂/∂t∇V − ∇∇∂/∂tV += +� +∇∂/∂t∇∂/∂xαV − ∇∂/∂xα∇∂/∂tV +� +dxα += +�� +∇f∗(∂/∂t)∇f∗(∂/∂xα)V − ∇f∗(∂/∂xα)∇f∗(∂/∂t) +� +f∗V +� +◦ f dxα +and +RN +� +df, ∂f +∂t +� +V = +� +RN +� +f∗ +� ∂ +∂xα +� +, f∗ +� ∂ +∂t +�� +f∗(V ) +� +◦ f dxα +are 1−forms on M with values in f ∗TN. Thus, assertion follows from the definition +of the Riemannian tensor RN. +□ + +6 +GEORGIOS DASKALOPOULOS AND CHIKAKO MESE +Theorem 1.8 (Second Variation Formula). One has +d2 +dt2E(ft) = ||∇∂f +∂t ||2 − +� +M +� +RN +� +df, ∂f +∂t +� ∂f +∂t , df +� +⋆ 1 + +� +M +� +∇∂/∂t +∂f +∂t , τf +� +⋆ 1. +Proof. We compute +d2 +dt2E(ft) = +� +M +d +dt +� +∇∂f +∂t , df +� +⋆ 1 += +� +M +�� +∇∂/∂t∇∂f +∂t , df +� ++ +� +∇∂f +∂t , ∇∂/∂tdf +�� +⋆ 1 += +� +M +�� +∇∇∂/∂t +∂f +∂t , df +� +− +� +RN +� +df, ∂f +∂t +� ∂f +∂t , df +� ++ ||∇∂f +∂t ||2 +� +⋆ 1. +□ +2. Existence and regularity +2.1. Introduction: Non-positive curvature. In this section, we examine the role +of non-positive curvature of the target metric on harmonic maps. We show uniqueness +and discuss regularity. We also study the equivariant problem and prove existence of +equivariant harmonic maps into non-positively curved metric spaces. Some references +are [S], [KS1], [KS2] and [GS]. +2.2. Second variation formula and non-positive curvature. The following are +corollaries of Theorem 1.8. +Corollary 2.1. If N has ≤ 0 sectional curvature and ft is a geodesic interpolation, +then E(ft) is convex. +Proof. In the second variation formula the last term vanishes and the others are ≥ +0. +□ +Corollary 2.2. Let f, φ : M → N be homotopic with f|∂M = φ|∂M. If N has ≤ 0 +sectional curvature and f is harmonic, then +E(f) ≤ E(φ). +Proof. Let ft be a geodesic homotopy between f, φ, thus f0 = f, f1 = φ. +Then +E(t) = E(ft) is convex, and E′(0) = 0. So E(1) ≥ E(0), hence E(φ) ≥ E(f). +□ +Corollary 2.3. If f0, f1 : M → N are homotopic harmonic maps with f0|∂M = f1|∂M +and N has ≤ 0 sectional curvature, then: +(1) If ∂M is nonempty, then f0 = f1. +(2) If ∂M is empty, F is a geodesic homotopy between f0, f1 and N has sectional +curvature < 0 at one point p in the image of F, then either f0 = f1 or the rank +of f0 is ≤ 1. + +NOTES ON HARMONIC MAPS +7 +Proof. (1) Let ft be a geodesic homotopy between f0, f1, E(t) = E(ft). Then E is +convex. Since E′(0) = E′(1) = 0, we conclude E′(t) = 0 = E′′(t). By Theorem 1.8, +∇∂F +∂t = 0 +and +� +RN +� +df, ∂f +∂t +� ∂f +∂t , df +� += 0. +Thus, +∂ +∂xα||∂F +∂t ||2 = 2 +� +∇∂/∂xα ∂F +∂t , ∂F +∂t +� += 0 +which implies that ||∂F/∂t|| is constant. +But ∂F/∂t = 0 on ∂M, so ∂F/∂t = 0 +everywhere if ∂M is nonempty and hence f0 = f1. +(2) If ||∂F/∂t|| = 0, then f0 = f1. Otherwise, ∂F/∂t ̸= 0 for every x, t. The negative +sectional curvature at p implies df is parallel to ∂F/∂t at p and therefore everywhere. +Thus, the image of df has dimension ≤ 1. +□ +2.3. The Weitzenb¨ock formula. +Theorem 2.4. Let f : M → N be a harmonic map and (eα) an orthonormal frame +for TM. Then +∆e(f) = |∇df|2 + 1 +2 +� +df(RicM(eα)), df(eα) +� +− 1 +2 +� +RN(df(eα), df(eβ))df(eβ), df(eα) +� +. +Proof. Expanding out the Laplacian with respect to local coordinates, the harmonic +map equation (1.3) is +gαβf i +/αβ − gαβ MΓη +αβf i +/η + gαβ NΓi +kℓ ◦ ff k +/αf ℓ +/β = 0. +We use normal coordinates at x ∈ M and f(x) = y. Thus, the metric tensors (gαβ) +and (hij) are Euclidean up to first order at x and y respectively. Differentiating, +f i +/ααε = MΓη +αα/εf i +/η − NΓi +kℓ/mf m +/εf k +/αf ℓ +/α += 1 +2(gαη/αε + gαη/αε − gαα/ηε)f i +/η +− 1 +2(hki/ℓm + hℓi/km − hkℓ/im)f m +/εf k +/αf ℓ +/α. +Furthermore, +gαβ +/ǫǫ = −gαβ/ǫǫ +and +△hij(f(x)) = hij/klf k +/ǫf k +/ǫ. + +8 +GEORGIOS DASKALOPOULOS AND CHIKAKO MESE +Thus, +∆ +�1 +2gαβhij ◦ ff i +/αf j +/β +� += +1 +√g +∂ +∂xσ +�√ggστ ∂ +∂xτ +�1 +2gαβhij ◦ ff i +/αf j +/β +�� += f i +/ασf i +/ασ − 1 +2(gαβ/σσ + gσσ/αβ − gσα/σβ − gσα/σβ)f i +/αf i +/β ++ 1 +2(hij/kℓ + hkℓ/ji − hik/jℓ − hjℓ/ik)f i +/αf j +/αf k +/σf ℓ +/σ += f i +/ασf i +/ασ + 1 +2RicM +αβf i +/αf j +/β − 1 +2RN +ikjℓf i +/αf j +/αf k +/σf ℓ +/σ. +□ +Here, +RicM +αβ = gδǫRM +αδβǫ +is the Ricci tensor. +2.4. Regularity. Assume N has ≤ 0 sectional curvature. Then from the Weitzenb¨ock +formula, +∆e(f) ≥ −Ce(f) +(2.1) +where C depends only on the geometry of M. +Theorem 2.5. If f : M → N is harmonic, and N has ≤ 0 sectional curvature, then +|f|C2+α +loc ≤ c +where c > 0 depends on E(f) and the geometries of M, N. +Proof. By (2.1) and Moser iteration, +sup +Br(p) +e(f) ≤ c +� +B2r(p) +e(f) ⋆ 1 = E(f) +(2.2) +where c only depends on the geometry and r. Now the right-hand side of (2.1) is +C0-bounded. So by elliptic regularity, fi is C1+α-bounded. But then the right-hand +side of (2.1) is Cα-bounded, so fi is C2+α-bounded. +□ +Corollary 2.6. If f : M → N is harmonic, and N has ≤ 0 sectional curvature, then +f ∈ C∞(M, N). +Proof. Keep bootstrapping with (2.1). +□ +Theorem 2.7. If f : M → N is a harmonic map, M is compact with Ricci curvature +≥ 0 and N has sectional curvature ≤ 0, then f is totally geodesic. + +NOTES ON HARMONIC MAPS +9 +Proof. Since +0 = +� +M +△e(f) ⋆ 1 = +� +M +� +|∇df|2 + 1 +2 +� +df(RicM(eα)), df(eα) +� +−1 +2 +� +RN(df(eα), df(eβ))df(eβ), df(eα) +�� +⋆ 1, +and each of the terms on the right hand side is non-negative, we have +∇df = 0. +□ +2.5. Non-positive curvature in a metric space. A complete metric space (X, d) +is called an NPC space if the following conditions are satisfied: +(i) The space (X, d) is a length (or geodesic) space. That is, for any two points P +and Q in X, there exists a rectifiable curve c so that the length of c is equal to d(P, Q) +(which we will sometimes denote by dP Q for simplicity). We call such distance realizing +curves geodesics. +(ii) For any geodesic triangle with vertices P, R, Q ∈ X, let c : [0, l] → X be the +arclength parameterized geodesic from Q to R and let Qt = c(tl). Then +d2 +P Qt ≤ (1 − t)d2 +P Q + td2 +P R − t(1 − t)d2 +QR. +(2.3) +(iii) Condition (ii) implies the quadralateral comparison inequalities (cf. [KS1, Corol- +lary 2.1.3]) +d2 +PtQt ≤ (1 − t)d2 +P Q + td2 +RS − t(1 − t)(dSP − dQR)2 +(2.4) +d2 +QtP + d2 +Q1−tS ≤ d2 +P Q + d2 +RS − td2 +QR − 2tdSPdQR + 2t2d2 +QR +(2.5) +Example 2.8. The main examples we will be considering are Riemannian manifolds +of non-positive curvature and (locally compact) Euclidean buildings. +Example 2.9. Let (X, d) be an NPC space, P ∈ X and M be a compact Riemannian +manifold. Let Y = L2(M, X) be a set of maps f : M → X such that +� +M +d2(f, P) ⋆ 1 < ∞. +Define a distance function dY on Y by setting +d2 +Y (f0, f1) = +� +M +d2(f0(x), f1(x)) ⋆ 1. +Then (Y, dY ) is an NPC space (cf. [KS1, Lemma 2.1.2]) where the geodesic between +f0 and f1 is the geodesic interpolation map ft(x) = (1 − t)f0(x) + tf1(x). + +10 +GEORGIOS DASKALOPOULOS AND CHIKAKO MESE +2.6. Local existence. We solve the Dirichlet problem for a smooth Riemannian do- +main B ⊂ M. We will motivate the construction by first considering the case X = R +(cf. [KS1, Section 2.2]). Fix φ ∈ H1(B, X) and consider the space +H1 +φ(B, X) = {f ∈ H1(B, X) : f − φ ∈ H1 +0(B, X)} +Let +E0 = inf{E(f) : f ∈ H1 +φ(B, X)}. +By the parallelogram identity +2 +� +B +|df + v +2 +|2 ⋆ 1 + 2 +� +B +|df − v +2 +|2 ⋆ 1 = +� +B +|df|2 ⋆ 1 + +� +B +|dv|2 ⋆ 1 +Take a minimizing sequence fi and apply the previous equality for f = fi, v = fj. +This implies that +2 +� +B +|dfi − fj +2 +|2 ⋆ 1 += +� +B +|dfi|2 ⋆ 1 + +� +B +|dfj|2 ⋆ 1 − 2 +� +B +|dfi + fj +2 +|2 ⋆ 1 +≤ +2E0 + 2ǫi − 2E0 = 2ǫi. +Hence +lim +� +B +|dfi − fj +2 +|2 ⋆ 1 = 0. +(2.6) +By the Poincare inequality +lim +� +B +|fi − fj +2 +|2 ⋆ 1 = 0 +(2.7) +hence +lim +i→∞ fi = f in H1 +φ(B, X) and E(f) = E0. +Now assume X is an NPC space. Korevaar-Schoen [KS1] showed that the energy +density makes sense by taking difference quotients. For the purpose of these lectures, +if X is a locally finite Euclidean building, then we can locally isometricaly embed it +in a Euclidean space of high dimension. Then, we can define the energy density of the +map to the building equal to the energy density of the map considered as a map to the +Euclidean space. In fact, this was the original point of view taken in [GS]. The more +general theory developed later in [KS1] and [KS2]. +With this, we argue as above replacing the parallelogram identity by the quadrilat- +eral inequality. Indeed, for f, v ∈ H1 +φ(B, X), define w(x) = (1 − t)f(x) + tv(x). Then +(2.4) with t = 1 +2 implies +2d2(w(x), w(y)) ≤ d2(f(x), f(y)) + d2(v(x), v(y)) − 1 +2(d(f(y), v(y)) − d(f(x), v(x))2 +which then implies +2Ew ≤ Ef + Ev − 1 +2 +� +B +|∇d(f, v)|2 ⋆ 1 + +NOTES ON HARMONIC MAPS +11 +Take a minimizing sequence fi and apply the previous inequality with f = fi and +v = vi to conclude (cf. (2.6)) +lim +i,j→∞ +� +B +|∇d(fi, fj)|2 ⋆ 1 → 0. +By the Poincare inequality, fi is a Cauchy sequence in (Y, dY ) and converges to a map +which is minimizing by the lower semicontinuity of energy [KS1, Theorem 1.6.1]. +2.7. Basic Regularity result of Gromov-Schoen and Korevaar-Schoen. This +is the analogue of (2.2) without using the PDE. +Theorem 2.10. If f ∈ H1(B, X) is a harmonic map, then f is locally Lipschitz. More +precisely, for any B′ ⊂⊂ B, there exists a constant C only depending on the metric on +B′ and the distance of B′ to ∂B such that +sup +B′ |df|2 ≤ c +� +B +|df|2 ⋆ 1. +2.8. Equivariant maps. Let ρ : π1(M) → Isom(X) be a homomorphism. A map +v : ˜ +M → X +is called a ρ-equivariant map, if +v(γx) = ρ(γ)v(x). +Since |dv|2 is π1(M)-invariant, it descends to a function on M. Define: +E(v) = +� +M +|dv|2 ⋆ 1. +If v descends to a map to M/ρ(π1(M)) this agrees with our previous definition. +2.9. Existence of ρ-equivariant locally Lipschitz maps. Let (M, ν) be a proba- +bility space, X an NPC-space and f ∈ L2(M, X). +Lemma 2.11. There exists a unique point Qf,ν that minimizes the integral +If,ν(Q) := +� +M +d2(f(m), Q)dν(m) ∀Q ∈ X. +We call Qf,ν the center of mass. +Proof. Let {Qi} be a minimizing sequence and let Qij = 1 +2Qi + 1 +2Qj. By (2.3) with +t = 1 +2, +d2(f(x), Qij) ≤ 1 +2d2(f(x), Qi) + 1 +2d2(f(x), Qj) − 1 +4d2(Qi, Qj). +Integrating, we obtain +If,ν(Qij) ≤ 1 +2If,ν(Qi) + 1 +2If,ν(Qj) − 1 +2d2(Qi, Qj). + +12 +GEORGIOS DASKALOPOULOS AND CHIKAKO MESE +Thus, d2(Qi, Qj) is a Cauchy sequence. We conclude that any minimizing sequence is +a Cauchy sequence and converges to a minimizing element. +□ +Lemma 2.12. There exists a locally Lipschitz ρ-equivariant map ˜f : ˜ +M → X. If X is +smooth, then ˜f can be chosen to be smooth. +Proof. Let Q0 := Qf,µ0 (resp. Q1 := Qf,µ1) be the center of mass for the function f ∈ +L2(M, X) and the probability space (M, µ0) (resp. (M, µ1)). Let Qt = (1−t)Q0+tQ1. +By the minimizing property of Q0 and Q1, +� +d2(f, Q0) + d2(f, Q1) dµ0 += +� +d2(f, Q0)dµ0 + +� +d2(f, Q1)dµ1 + +� +d2(f, Q1)(dµ0 − dµ1) +≤ +2 +� +d2(f, Q1/2) dµ0 + +� � +d2(f, Q1) − d2(f, Q1/2) +� +(dµ0 − dµ1) +≤ +� +d2(f, Q0) + d2(f, Q1) − 1 +4d2(Q0, Q1) dµ0 ++ +� � +d2(f, Q1) − d2(f, Q1/2) +� +(dµ0 − dµ1). +The last inequality is by triangle comparison. Consequently, +d2(Q0, Q1) ≤ 4 +� � +d2(f, Q1) − d2(f, Q 1 +2) +� +(dµ0 − dµ1). +(2.8) +For each x ∈ ˜ +M, let +dµx = dvol +B1(x) +V (x) +, +V (x) = vol(B1(x)). +where vol = ⋆1 is the volume form of ˜ +M. Since µx is only dependent on the metric of +M, x �→ µx is invariant under the isometric action of π1(M). Furthermore, +� +|dµx0 − dµx1| = +� ���� +1 +V (x0)χB1(x0) − +1 +V (x0)χB1(x0) +���� ⋆ 1 +≤ Cρ(x0, x1) +where ρ denotes the distance function on M. +Let M0 be a fundamental domain. Let f(M0) = P and extend equivariantly to +f : ˜ +M → X. For simplicity, assume M0 is compact. Then there exists a constant L +such that +d(f(x0), f(x1)) ≤ L whenever ρ(x0, x1) < 2. +Thus, for ρ(x0, x1) < 1 and in the support of |dµx0 − dµx1|, +d2(f, Q1) − d2(f, Q1/2) ≤ d2(f, Q1) + d2(f, Qt) ≤ 2L2. + +NOTES ON HARMONIC MAPS +13 +Define +˜f : ˜ +M → X, +˜f(x) = Qf,µx +The π1(M)-invariance of µx and the ρ-equivariance of f imply the ρ-equivariance of ˜f. +Apply (2.8) with M = M, µ0 = µx0 and µ1 = µx1 to obtain +d2( ˜f(x0), ˜f(x1)) ≤ 2L2 +� +|dµx0 − dµx1| ≤ 2L2Cρ(x0, x1). +□ +2.10. The boundary at infinity. A good reference is [BH]. Suppose X is an NPC- +space. Two geodesic rays c, c′ : [0, ∞) → X are said to be asymptotic if there exists a +constant K such that d(c(t), c′(t)) < K for all t > 0. The set ∂X of boundary points +of X (which we shall also call the points at infinity) is the set of equivalence classes +of geodesic rays, two geodesic rays being equivalent if and only if they are asymptotic. +We denote ¯X = X ∪ ∂X. Notice that the images of two asymptotic geodesic rays +under any isometry of X are again asymptotic geodesic rays, and hence any isometry +extends to give a bijection of ¯X. The next proposition is [BH, Proposition 8.2]. +Proposition 2.13. If X is an NPC-space and c : [0, ∞) → X is a geodesic ray starting +from P, then for every point P1 ∈ X there is a unique geodesic ray which starts from +P1 and is asymptotic to c. +The topology of ¯X is defined as follows: A sequence of points Pi converges to a point +P ∗ ∈ ∂X if and only if the geodesics joining P0 to Pi converge (uniformly on compact +subsets) to the geodesic ray that starts from P0 and belongs to the class of P ∗. +Example 2.14. If X is a complete n-dimensional Riemannian manifold of non-positive +sectional curvature, then ∂X is homeomorphic to Sn−1. Indeed, given a base point +P0, we can obtain a homeomorphism by considering the map which associates to each +unit vector V tangent to X at P0 the class of the geodesic ray c starting at P0 with +velocity vector V. In particular, if X is the n-dimensional hyperbolic space, then ¯X is +homeomorphic to the n-dimensional ball in Rn. If X is a locally compact Euclidean +building, then ∂X is a compact spherical building (cf. [KL, Proposition 4.2.1]). +Lemma 2.15. If Pi is a sequence in X with lim Pi = P ∗ ∈ ∂X and if Qi is another +sequence in X with d(Pi, Qi) ≤ C independently of i, then lim Qi = P ∗. +Proof. Fix P0 ∈ X. Let γ : [0, ∞) → ∞ be an arclength parameterized geodesic ray +in the equivalence class P ∗ with γ(0) = P0. Let ti = d(P0, Pi) (resp. τi = d(P0, Qi)) +and let γi : [0, ti] → X (resp. ˆγi : [0, τi] → X) be the arclength parameterized geodesic +segment connecting P0 and Pi (resp. Qi). By the triangle inequality, +d(ˆγi(ti), ˆγi(τi)) = |ti − τi| = |d(P0, Pi) − d(P0, Qi)| ≤ d(Pi, Qi) ≤ C. + +14 +GEORGIOS DASKALOPOULOS AND CHIKAKO MESE +Thus, assuming t ≤ ti ≤ τi, the NPC condition implies +d(ˆγi(t), γi(t)) ≤ t +ti +d(ˆγi(ti), γi(ti)) ≤ t +ti +� +d(ˆγi(ti), ˆγi(τi)) + d(ˆγi(τi), γi(ti)) +� +≤ 2Ct +ti +. +Similarly, assuming t ≤ τi < ti, +d(ˆγi(t), γi(t)) ≤ 2Ct +τi +. +Thus, for t ≤ min{ti, τi}, +d(ˆγi(t), γ(t)) ≤ d(ˆγi(t), γi(t)) + d(γi(t), γ(t)) ≤ +2Ct +max{ti, τi} + d(γi(t), γ(t)). +Fix T0 > ∞. +The assumption that lim Pi = P ∗ implies that ti, τi → ∞ and the +geodesics γi converge uniformly to γ in [0, T0]. Thus, ˆγi also converge uniformly to γ +in [0, T0]. +□ +Lemma 2.16. The stabilizer of a point at infinity is contained in a parabolic subgroup. +So if the image of ρ is Zariski dense it cannot fix a point at infinity. +2.11. Global existence result. We prove existence of equivariant harmonic maps +[GS, Theorem 7.1]. +Theorem 2.17. Let X be a locally compact NPC space. Assume that the image of ρ +doesn’t fix a point in ∂X and that there exists a Lipschitz equivariant map v : ˜ +M → X +with finite energy. Then there is a Lipschitz equivariant map f of least energy and the +restriction of f to a small ball about any point is minimizing. +Proof. Let E0 denote the infimum of the energy taken over all Lipschitz equivariant +maps. Let vi be a sequence of Lipschitz equivariant maps with E(vi) → E0. Let B be +a ball in ˜ +M such that γ(B) ∩ B = ∅ for all γ ∈ π1(M). We may then construct a new +minimizing sequence ¯vi, by replacing vi with the solution to the Dirichlet problem on +each γ(B). Clearly ¯vi is also a minimizing sequence. +On a compact subset of B, the sequence ¯vi is uniformly Lipschitz by Theorem 2.10. +It follows that a subsequence of ¯vi converges uniformly on compact subsets of B to +a map into ¯X which either maps into X or maps to a single point P ∗ ∈ ∂X. We +exclude the second possibility as follows. Let x0 ∈ +˜ +M be the center of the chosen +ball B. +Let C be any smooth embedded curve from x0 to γ(x0). +An elementary +argument using Fubini’s theorem shows that C may be chosen so that the energy of +the restriction of each map ¯vi to C is uniformly bounded. Therefore the length of +the curve ¯vi(C) is uniformly bounded, and in particular d(vi(x0), ρ(γ)vi((x0)) ≤ C. +Lemma 2.15 implies lim ρ(γ)vi(x0) = P ∗, and hence ρ(γ)P ∗ = P ∗ for all γ. This is +a contradiction. Therefore we may assume that ¯vi converges uniformly on compact +subsets of B. + +NOTES ON HARMONIC MAPS +15 +From (2.6) as before, we have� +K +|∇d(¯vi, ¯vj)|2 ⋆ 1 → 0 +for any compact set K ⊂ ˜ +M. Since ¯vi converges uniformly on compact subsets of B, +the function d(¯vi, v) is uniformly bounded there. It then follows from Poincare type +inequalities that +� +K +d(¯vi, ¯vj)2 ⋆ 1 → 0. +In particular, the sequence ¯vi → f which is a minimizer by lower semicontinuity of +energy. The local minimizing property of f follows. This completes the proof. +□ + +16 +GEORGIOS DASKALOPOULOS AND CHIKAKO MESE +3. Pluriharmonic maps and the Siu-Sampson Formula +3.1. Introduction: Bochner methods for harmonic maps. In this section, we +discuss the Bochner formulas of Siu [Siu] and Sampson [Sa]. Our exposition closely +follows the approach of [LY]. We also present a variation of these formulas inspired +by the work of Mochizuki [M]. Lastly, we sketch the existence of pluriharmonic maps +into Euclidean buildings. +3.2. Pluriharmonic maps from K¨ahler manifolds to Riemannian manifolds. +Let (M, ω, J) be a K¨ahler manifold along with its K¨ahler form and complex structure. +Let +TM ⊗ C = T (1,0)M ⊕ T (0,1)M +be its complexified tangent bundle decomposed into the ±√−1-eigenspaces of J. We +can decompose v ∈ TM ⊗ C into +v = v1,0 + v0,1 where v1,0 = 1 +2(v − +√ +−1Jv), v0,1 = 1 +2(v + +√ +−1Jv). +The cotangent space T ∗M has a complex structure still denoted J defined by Jα = +α ◦ J. Accordingly, we have an analogous decomposition +T ∗M ⊗ C = T ∗(1,0)M ⊕ T ∗(0,1)M. +Let (N, h) be a Riemannian manifold and TN ⊗ C its complexified tangent bundle. +For a smooth map f : M → N, let +E := f ∗(TN ⊗ C). +(3.1) +Extending complex linearly, df : TM → TN gives rise to a map df : TM ⊗ C → +TN ⊗ C. Denote by Ωp,q(E), the space of E-valued (p, q)-forms. Define +d′f := 1 +2(df − +√ +−1 df ◦ J) ∈ Ω1,0(E), +d′′f := 1 +2(df + +√ +−1 df ◦ J) ∈ Ω0,1(E). +We have that +df = d′f + d′′f +Jdf = df ◦ J = − +√ +−1 (d′f − d′′f). +For local coordinates (yi) of N, let +∂ +∂fi = +∂ +∂yi ◦ f. Then +d′f = d′f i ∂ +∂f i +d′′f = d′′f i ∂ +∂f i +d′f = d′′f +d′′f = d′f. +Similarly, we can decompose the pullback of the Levi-Civita connection (cf. Section 1) +as +∇ = ∇′ + ∇′′ +where +∇′ : C∞(E) → Ω1,0(E), +∇′′ : C∞(E) → Ω0,1(E). + +NOTES ON HARMONIC MAPS +17 +In turn, ∇′ and ∇′′ induce differential operators +d′ +E : Ωp,q(E) → Ωp+1,q(E), +d′′ +E : Ωp,q(E) → Ωp,q+1(E) +where +d′ +E(φ ⊗ s) += +d′φ ⊗ s + (−1)p+qφ ⊗ ∇′ +Es +d′′ +E(φ ⊗ s) += +d′′φ ⊗ s + (−1)p+qφ ⊗ ∇′′ +Es. +A straightforward calculation implies that +d′ +Ed′′f = −d′′ +Ed′f, +d′ +Ed′f = 0, +d′′ +Ed′′f = 0. +(3.2) +Lemma 3.1. +τ(f) = 2i ⋆ +� ωn−1 +(n − 1)! ∧ d′ +Ed′′f +� +. +Proof. We claim +⋆α = +ωn−1 +(n − 1)! ∧ Jα, +∀α ∈ Ω1(M, R). +To check the claim, use normal coordinates (zi = xi + √−1yi) at a point x ∈ M. For +α = dxi or α = dyi, we have +dxi ∧ +ωn−1 +(n − 1)! ∧ Jdxi = dxi ∧ dyi ∧ +ωn−1 +(n − 1)! = +√−1 +2 +dzi ∧ d¯zi ∧ +ωn−1 +(n − 1)! = ωn +n! +and +dyi ∧ +ωn−1 +(n − 1)! ∧ Jdyi = −dyi ∧ dxi ∧ +ωn−1 +(n − 1)! = +√−1 +2 +dzi ∧ d¯zi ∧ +ωn−1 +(n − 1)! = ωn +n! . +The claim follows by linearity. +Next, note that +Jd′f += +1 +2(df ◦ J + +√ +−1df) = +√−1 +2 +(df − +√ +−1df ◦ J) = +√ +−1d′f +Jd′′f += +1 +2(df ◦ J − +√ +−1df) = +√−1 +2 +(df + +√ +−1df ◦ J) = − +√ +−1d′′f, + +18 +GEORGIOS DASKALOPOULOS AND CHIKAKO MESE +which implies Jdf = Jd′f + Jd′′f = √−1(d′f − d′′f). Applying the claim for α = df, +we use the fact that dω = 0 to obtain +τ(f) += +−d⋆ +∇df += +⋆d∇(⋆df) += +⋆d∇ +� ωn−1 +(n − 1)! ∧ Jdf +� += +⋆d∇ +� ωn−1 +(n − 1)! ∧ ( +√ +−1(d′f − d′′f)) +� += +− +√ +−1 ⋆ +� ωn−1 +(n − 1)! ∧ (d′ +Ed′′f − d′′ +Ed′f) +� += +−2 +√ +−1 ⋆ +� ωn−1 +(n − 1)! ∧ d′ +Ed′′f +� +. +□ +Definition 3.2. f is called pluriharmonic d′ +Ed′′f = 0. +Remark 3.3. Lemma 3.1 implies +pluriharmonic =⇒ harmonic. +Note that holomorphic maps between K¨ahler manifolds are pluriharmonic, and thus +harmonic. +3.3. Sampson’s Bochner formula. +Theorem 3.4 (Sampson’s Bochner formula, [Sa]). For a harmonic map f : M → N +from a K¨ahler manifold (M, g) to a Riemannian manifold (N, h), +d′d′′{d′′f, d′′f} ∧ +ωn−2 +(n − 2)! += +4 +� +|d′ +Ed′′f|2 + Q0 +� ωn +n! +where {·, ·} is given in Definition 3.5 below and +Q0 = −2gα¯δgγ ¯βRijkl +∂f i +∂zα +∂f k +∂¯zβ +∂f j +∂zγ +∂f l +∂¯zδ +in local coordinates (zα) of M and (yi) of N. +Proof. Combine Lemma 3.6, Lemma 3.8 and Lemma 3.20 below. +□ +Definition 3.5. Let {si} be a local frame of E. For +ψ = ψi ⊗ si ∈ Ωp,q(E) and ξ = ξi ⊗ si ∈ Ωp′,q′(E) +we set +{ψ, ξ} = ⟨si, sj⟩ψi ∧ ¯ξj ∈ Ωp+q′,q+p′ +where ⟨·, ·⟩ is the complex-linear extention of the Riemannian metric on E. + +NOTES ON HARMONIC MAPS +19 +Lemma 3.6. For any smooth map f : M → N from a K¨ahler manifold to a Riemann- +ian manifold, we have +d′d′′{d′′f, d′′f} ∧ +ωn−2 +(n − 2)! = +� +−{d′ +Ed′′f, d′ +Ed′′f} + {d′′f, R(1,1) +E +(d′′f)} +� +∧ +ωn−2 +(n − 2)! +where +R(1,1) +E += (d′ +Ed′′ +E + d′′ +Ed′ +E) +is the (1, 1)-part of the curvature RE = d2 +E. +Proof. Repeatedly using the fact that d′′ +Ed′′f = 0 (cf. (3.2)), +d′d′′{d′′f, d′′f} += +−{d′ +Ed′′f, d′ +Ed′′f} + {d′′f, d′′ +Ed′ +Ed′′f} += +−{d′ +Ed′′f, d′ +Ed′′f} + {d′′f, (d′′ +Ed′ +E + d′ +Ed′′ +E + d′′ +E +2)d′′f}. +Since {d′′f, d′ +Ed′ +Ed′′f} ∧ ωn−2 +(n−2)! is an (n − 1, n + 1)-form and hence zero for dimensional +reasons, we can complete the square to obtain +d′d′′{d′′f, d′′f} ∧ +ωn−2 +(n − 2)! += +� +−{d′ +Ed′′f, d′ +Ed′′f} + {d′′f, (d′ +E + d′′ +E)2d′′f} +� +∧ +ωn−2 +(n − 2)! += +� +−{d′ +Ed′′f, d′ +Ed′′f} + {d′′f, R(1,1) +E +(d′′f)} +� +∧ +ωn−2 +(n − 2)! +which proves the first equation. +□ +Lemma 3.7. For any E-valued (1, 1)-form φ on M, +−{φ, φ} ∧ +ωn−2 +(n − 2)! = 4(|φ|2 − |Traceωφ|2)ωn +n! . +Proof. Let (zp) be normal coordinates at x ∈ M and let φp¯qdzp ∧ d¯zq. At x, +ωn +n! = +�√−1 +2 +�2 �� +p +dzp ∧ d¯zp +�2 +∧ +ωn−2 +(n − 2)!. +For p, q such that p ̸= q, +s ̸= p or t ̸= q ⇒ dzp ∧ d¯zq ∧ d¯zs ∧ dzt ∧ +�� +j +dzj ∧ d¯zj +�n−2 += 0. +For p = q, +s ̸= t ⇒ dzp ∧ d¯zq ∧ d¯zs ∧ dzt ∧ +�� +j +dzj ∧ d¯zj +�n−2 += 0. + +20 +GEORGIOS DASKALOPOULOS AND CHIKAKO MESE +Furthermore, +φp¯qφp¯qdzp ∧ d¯zq ∧ d¯zp ∧ dzq += +φp¯qφp¯qdzp ∧ d¯zp ∧ dzq ∧ d¯zq +φp¯pφq¯qdzp ∧ d¯zp ∧ d¯zq ∧ dzq += +−φp¯pφq¯qdzp ∧ d¯zp ∧ dzq ∧ d¯zq. +Thus, +�√−1 +2 +�2 +{φ, φ} ∧ +ωn−2 +(n − 2)! += +�√−1 +2 +�2 � � +p,q,s,t +φp¯qφs¯tdzp ∧ d¯zq ∧ d¯zs ∧ dzt +� +∧ +ωn−2 +(n − 2)! += +� +p̸=q +� +|φp¯q|2 − φp¯pφq¯q +� ωn +n! += +� +p,q +� +|φp¯q|2 − φp¯pφq¯q +� ωn +n! += +� +|φ|2 − |traceωφ|2� ωn +n! . +□ +Lemma 3.8. For any harmonic map f : M → N from a K¨ahler manifold to a +Riemannian manifold, we have +−{d′ +Ed′′f, d′ +Ed′′f} ∧ +ωn−2 +(n − 2)! += +4 |d′ +Ed′′f|2 ωn +n! . +Proof. We apply Lemma 3.7 with φ = d′ +Ed′′f. Since f is harmonic, Trωd′ +Ed′′f = 0 by +Lemma 3.1. +□ +Lemma 3.9. For a harmonic map f : M → N from a K¨ahler manifold to a Hermitian- +negative Riemannian manifold, we have +{d′′f, R(1,1) +E +(d′′f)} ∧ +ωn−2 +(n − 2)! = −2 Rijkld′f i ∧ d′′f k ∧ d′f j ∧ d′′f l ∧ ωn +n! +where R(1,1) +E +is defined in Lemma 3.6. +Proof. Let (zα) (resp. (yi)) be normal coordinates at a point x ∈ M (resp. f(x) ∈ N). +Then +∇ +∂ +∂¯zγ ∇ +∂ +∂zβ +∂ +∂f j += +∇ +∂ +∂¯zγ +�∂f k +∂zβ ∇ +∂ +∂fk +∂ +∂f j +� += +∂f k +∂zβ +∂f l +∂¯zγ ∇ +∂ +∂fl ∇ +∂ +∂fk +∂ +∂f j + +NOTES ON HARMONIC MAPS +21 +and +d′′ +Ed′ +Ed′′f += +d′′ +Ed′ +E +�∂f j +∂¯zα d¯zα ⊗ ∂ +∂f j +� += +d′′d′ +�∂f j +∂¯zα d¯zα +� +⊗ ∂ +∂f j − ∂f j +∂¯zα d¯zα ∧ d¯zγ ∧ dzβ ⊗ ∇ +∂ +∂¯zγ ∇ +∂ +∂zβ +∂ +∂f j += +d′′d′ +�∂f j +∂¯zα d¯zα +� +⊗ ∂ +∂f j + ∂f j +∂¯zα +∂f k +∂zβ +∂f l +∂¯zγ d¯zα ∧ dzβ ∧ d¯zγ ⊗ ∇ +∂ +∂fl ∇ +∂ +∂fk +∂ +∂f j . +Similarly, +d′ +Ed′′ +Ed′′f += +d′d′′ +�∂f j +∂¯zα d¯zα +� +⊗ ∂ +∂f j − ∂f j +∂¯zα +∂f k +∂zβ +∂f l +∂¯zγ d¯zα ∧ dzβ ∧ d¯zγ ⊗ ∇ +∂ +∂fk ∇ +∂ +∂fl +∂ +∂f j . +Combining the above two equalities, +R(1,1) +E +(d′′f) += +∂f j +∂¯zα +∂f k +∂zβ +∂f l +∂¯zγ d¯zα ∧ dzβ ∧ d¯zγ ⊗ Rs +jkl +∂ +∂f s. +We compute +{d′′f, R(1,1) +E +(d′′f)} += +Rijkl +∂f i +∂¯zδ +∂f j +∂zα +∂f k +∂¯zβ +∂f l +∂zγ d¯zδ ∧ dzα ∧ d¯zβ ∧ dzγ += +Rjilk +∂f j +∂zα +∂f i +∂¯zδ +∂f l +∂zγ +∂f k +∂¯zβ dzα ∧ d¯zδ ∧ dzγ ∧ d¯zβ += +(−Rjlki + Rjkli)∂f j +∂zα +∂f i +∂¯zδ +∂f l +∂zγ +∂f k +∂¯zβ dzα ∧ d¯zδ ∧ dzγ ∧ d¯zβ. +Since +Rjkli +∂f j +∂zα +∂f i +∂¯zδ +∂f l +∂zγ +∂f k +∂¯zβ dzα ∧ d¯zδ ∧ dzγ ∧ d¯zβ ∧ +ωn−2 +(n − 2)! += +Rjkli +∂f j +∂zα +∂f i +∂¯zδ +∂f l +∂zγ +∂f k +∂¯zβ dzα ∧ d¯zα ∧ dzβ ∧ d¯zβ ∧ +ωn−2 +(n − 2)! ++Rjkli +∂f j +∂zα +∂f i +∂¯zδ +∂f l +∂zγ +∂f k +∂¯zβ dzα ∧ d¯zβ ∧ dzβ ∧ d¯zα ∧ +ωn−2 +(n − 2)! += +0, + +22 +GEORGIOS DASKALOPOULOS AND CHIKAKO MESE +we obtain +{d′′f, R(1,1) +E +(d′′f)} ∧ +ωn−2 +(n − 2)! += +−Rjlki +∂f j +∂zα +∂f i +∂¯zα +∂f l +∂zβ +∂f k +∂¯zβ dzα ∧ d¯zα ∧ dzβ ∧ d¯zβ ∧ +ωn−2 +(n − 2)! +−Rjlki +∂f j +∂zα +∂f i +∂¯zβ +∂f l +∂zβ +∂f k +∂¯zα dzα ∧ d¯zβ ∧ dzβ ∧ d¯zα ∧ +ωn−2 +(n − 2)! += +−2Rjlki +∂f j +∂zα +∂f i +∂¯zα +∂f l +∂zβ +∂f k +∂¯zβ dzα ∧ d¯zα ∧ dzβ ∧ d¯zβ ∧ +ωn−2 +(n − 2)! += +−2Rjlkid′f j ∧ d′′f i ∧ d′f l ∧ d′′f k ∧ +ωn−2 +(n − 2)!. +□ +Definition 3.10. A Riemannian manifold N is said to be Hermitian-negative (resp. +strongly Hermitian negative) if +RijklAi¯lAj¯k ≤ 0 (resp. < 0) +for any Hermitian semi-positive matrix A = +� +Ai¯l� +. +Remark 3.11. Locally symmetric spaces whose irreducible local factors are all non- +compact or Euclidean type are Hermitian negative (cf. [Sa, Theorem 2]). +Theorem 3.12 (Sampson). If f : M → N is a harmonic map from a K¨ahler manifold +into a Hermitian negative Riemannian manifold, then f is pluriharmonic. +Proof. Integrate Sampson’s Bochner formula over M. Applying Stoke’s theorem results +in the left hand side being 0. The two terms on the right hand side are non-negative +pointwise, hence they must be identically equal to 0. In particular, d′ +Ed′′f = 0; i.e. f +is is pluriharmonic. +□ +3.4. Maps between K¨ahler manifolds. Let f : M → N be a smooth map between +K¨ahler manifolds. By decomposing +TN ⊗ C = T (1,0)N ⊕ T (0,1)N +we get the decomposition of E := f −1(TN ⊗ C) as +E = E′ ⊕ E′′ where E′ := f −1(T (1,0)N), +E′′ := f −1(T (0,1)N). +Denote by Ωp,q(E), Ωp,q(E′) and Ωp,q(E′′) the space of E-, E′- and E′′-valued (p, q)- +forms respectively. If (wi) are local holomorphic coordinates in N, then { ∂ +∂fi := +∂ +∂wi ◦ +f, +∂ +∂ ¯fi := +∂ +∂ ¯wi ◦ f} is a local frame of E. If d′f, d′f ′ are as in Section 3.2, then +d′f = ∂f + ∂ ¯f, +d′′f = ¯∂f + ¯∂ ¯f, +df = d′f + d′′f = ∂f + ∂ ¯f + ¯∂f + ¯∂ ¯f. + +NOTES ON HARMONIC MAPS +23 +where +∂f = ∂f i ∂ +∂f i +¯∂f = ¯∂f i ∂ +∂f i +∂ ¯f = ∂ ¯f i ∂ +∂ ¯f i +¯∂ ¯f = ¯∂ ¯f i ∂ +∂ ¯f i +∂f = ¯∂ ¯f +¯∂f = ∂ ¯f. +Analogously, d∇ = d′ +E + d′′ +E is decomposed into the induced operators ∂E′, ¯∂E′, ∂E′′, +¯∂E′′. +A straightforward calculation yields +∂E′ ¯∂f = −¯∂E′∂f +∂E′′ ¯∂ ¯f = −¯∂E′′∂ ¯f +(3.3) +∂E′∂f = 0 +¯∂E′ ¯∂f = 0 +∂E′′∂ ¯f = 0 +¯∂E′′ ¯∂ ¯f = 0. +(3.4) +For any map f : M → N between K¨ahler manifolds, we have +��∂E′′ ¯∂ ¯f +��2 = +��¯∂E′′∂ ¯f +��2 = +��∂E′ ¯∂f +��2 . +(3.5) +Indeed, the left equality follows from (3.3) and the right from the fact that conjugation +is an isometry. +3.5. Siu’s curvature. +Definition 3.13. Let N be a K¨ahler manifold and R its complexified curvature tensor. +We say N has negative (resp. non-positive) complex sectional curvature, if +R(V, ¯W, W, ¯V ) < 0 (resp. ≤ 0) ∀V, W ∈ TNC. +In [Siu], Siu introduced the following notion of negative curvature. Recall that for +local holomorphic coordinates (wi) of a K¨ahler manifold N, the curvature tensor is of +type (1,1) and is given explicitly by +Ri¯jk¯l = − ∂2hi¯j +∂wk∂ ¯wl + hp¯q ∂hk¯q +∂wi +∂hp¯l +∂ ¯wj +where h is the K¨ahler metric on N. We say N has strongly negative (resp. strongly +semi-negative) curvature if +Ri¯jk¯l(AiBj − CiDj)(AlBk − ClDk) < 0 (resp. ≤ 0). +for arbitrary complex numbers Ai, Bi, Ci, Di when AiBj − CiDj ̸= 0 for at least one +pair of indices (i, j). +Remark 3.14. A K¨ahler manifold N is strongly semi-negative if and only if it has +non-positive complex sectional curvature (cf. [LSY, Theorem 4.4]). + +24 +GEORGIOS DASKALOPOULOS AND CHIKAKO MESE +Lemma 3.15. Let N be a K¨ahler manifold with K¨ahler form ω and of strongly semi- +negative curvature. Let M be another K¨ahler manifold and f : M → N be a smooth +map. If Q : M → R is defined by setting +Qωn +n! = −Ri¯jk¯l ¯∂f i ∧ ∂ ¯f j ∧ ∂f k ∧ ¯∂ ¯f l ∧ +ωn−2 +(n − 2)!, +then Q ≥ 0. +Proof. At a point with normal coordinates in the domain +Ri¯jk¯l ¯∂f i ∧ ∂f j ∧ ∂f k ∧ ¯∂ f l ∧ +ωn−2 +(n − 2)! += +� +α,β +( +√ +−1)n−2Ri¯jk¯l +� +−∂¯αf i∂αf j∂βf k∂¯βf l + ∂¯αf i∂βf j∂αf k∂¯βf l ++∂¯βf i∂αf j∂βf k∂¯αf l − ∂¯βf i∂βf j∂αf k∂¯αf l +� +∧ (∧γ(dzγ ∧ d¯zγ)) += +4 +� +α,β +Ri¯jk¯l +� +(∂¯αf i)(∂βf l) − (∂¯βf i)(∂αf l) +� � +(∂¯αf j)(∂βf k) − (∂¯βf j)(∂αf k) +�ωn +n! += +4 +� +α,β +Ri¯jk¯l +� +(∂¯αf i)(∂βf j) − (∂¯βf i)(∂αf j) +� � +(∂¯αf l)(∂βf k) − (∂¯βf l)(∂αf k) +�ωn +n! +≤ +0. +The last equality is because Rı¯jk¯l = Ri¯lk¯l, and the last inequality is because of the +assumption that N has strong semi-negative curvature. +□ +3.6. Siu’s Bochner Formula. +Theorem 3.16 (Siu-Bochner formula, [Siu] Proposition 2). For a harmonic map f : +M → N between K¨ahler manifolds, +∂ ¯∂{¯∂f, ¯∂f} ∧ +ωn−2 +(n − 2)! += +� +4 +��∂E′ ¯∂f +��2 + Q +� ωn +n! . +Proof. Combine Lemma 3.15, Lemma 3.17, Lemma 3.18 and Corollary 3.20 below. +□ +The curvature operators of E′ and E′′ are RE′ = −(∂E′ + ¯∂E′)2 and RE′′ = −(∂E′′ + +¯∂E′′)2 respectively. +Lemma 3.17. For any smooth map f : M → N between K¨ahler manifolds, we have +∂ ¯∂{¯∂f, ¯∂f} ∧ +ωn−2 +(n − 2)! = +� +−{∂E′ ¯∂f, ∂E′ ¯∂f} − {¯∂f, RE′(¯∂f)} +� +∧ +ωn−2 +(n − 2)! +and +∂ ¯∂{¯∂ ¯f, ¯∂ ¯f} ∧ +ωn−2 +(n − 2)! = +� +−{∂E′′ ¯∂ ¯f, ∂E′′ ¯∂ ¯f} − {¯∂ ¯f, RE′′(¯∂ ¯f)} +� +∧ +ωn−2 +(n − 2)!. + +NOTES ON HARMONIC MAPS +25 +Proof. By setting ψ = ξ = ¯∂f ∈ Ω0,1(E′) in (3.5) and repeatedly using the fact that +¯∂E′ ¯∂f = 0 (cf. (3.3)), +∂ ¯∂{¯∂f, ¯∂f} += +−{∂E′ ¯∂f, ∂E′ ¯∂f} + {¯∂f, ¯∂E′∂E′ ¯∂f} += +−{∂E′ ¯∂f, ∂E′ ¯∂f} + {¯∂f, (¯∂E′∂E′ + ∂E′ ¯∂E′ + ¯∂2 +E′)¯∂f}. +Since {¯∂f, ∂2 +E′ ¯∂f} ∧ +ωn−2 +(n−2)! is an (n − 1, n + 1)-form and hence zero for dimensional +reasons, we can complete the square to obtain +∂ ¯∂{¯∂f, ¯∂f} ∧ +ωn−2 +(n − 2)! += +� +−{∂E′ ¯∂f, ∂E′ ¯∂f} + {¯∂f, (∂E′ + ¯∂E′)2 ¯∂f} +� +∧ +ωn−2 +(n − 2)! += +� +−{∂E′ ¯∂f, ∂E′ ¯∂f} − {¯∂f, RE′(¯∂f)} +� +∧ +ωn−2 +(n − 2)! +which proves the first equation. The second equation follows by setting ψ = ξ = ¯∂ ¯f ∈ +Ω0,1(E′′) in (3.5) and following exactly the same computation. +□ +Lemma 3.18. For any harmonic map f : M → N between K¨ahler manifolds, we have +−{∂E′ ¯∂f, ∂E′ ¯∂f} ∧ +ωn−2 +(n − 2)! += +4 +��∂E′ ¯∂f +��2 ωn +n! +−{∂E′′ ¯∂ ¯f, ∂E′′ ¯∂ ¯f} ∧ +ωn−2 +(n − 2)! += +4 +��∂E′′ ¯∂ ¯f +��2 ωn +n! . +Proof. Apply Lemma 3.7 with φ = ∂E′ ¯∂f (resp. φ = ∂E′′ ¯∂ ¯f). Since f is harmonic, +Trω∂E′ ¯∂f = 0 and Trω∂E′′ ¯∂ ¯f = 0 by Lemma 3.1. +□ +Lemma 3.19. For any smooth map f : M → N between K¨ahler manifolds, we have +{¯∂f, RE′(¯∂f)} = Ri¯jk¯l ¯∂f i ∧ ∂ ¯f j ∧ ∂f k ∧ ¯∂ ¯f l = {¯∂ ¯f, RE′′(¯∂ ¯f)}. +Proof. Using normal coordinates, we compute +{¯∂f, RE′(¯∂f)} += +{¯∂f i ∂ +∂f i , RE′(¯∂f j ∂ +∂f j )} += +{¯∂f i ∂ +∂f i , ¯∂f j ∧ RE′( ∂ +∂f j )} += +{¯∂f i ∂ +∂f i , ¯∂f j ∧ Rs +jk¯l∂f k ∧ ∂f l ∂ +∂f s } += +Ri¯j¯kl ¯∂f i ∧ ∂ ¯f j ∧ ¯∂ ¯f k ∧ ∂f l += +Ri¯jk¯l ¯∂f i ∧ ∂ ¯f j ∧ ∂f k ∧ ¯∂ ¯f l + +26 +GEORGIOS DASKALOPOULOS AND CHIKAKO MESE +which proves the first equality. The second equality is proved similarly: +{¯∂ ¯f, RE′′(¯∂ ¯f)} += +{¯∂ ¯f i ∂ +∂ ¯f i , RE′′(¯∂ ¯f j ∂ +∂ ¯f j )} += +{¯∂ ¯f i ∂ +∂ ¯f i , ¯∂ ¯f j ∧ RE′′( ∂ +∂ ¯f j )} += +{¯∂ ¯f i ∂ +∂ ¯f i , ¯∂ ¯f j ∧ R¯s +¯j¯kl∂ ¯f k ∧ ¯∂f l ∂ +∂ ¯f s } += +R¯ijk¯l ¯∂ ¯f i ∧ ∂f j ∧ ¯∂f k ∧ ∂ ¯f l += +Rj¯ik¯l∂f j ∧ ¯∂ ¯f i ∧ ¯∂f k ∧ ∂ ¯f l += +Ri¯jk¯l∂f i ∧ ¯∂ ¯f j ∧ ¯∂f k ∧ ∂ ¯f l += +Ri¯jk¯l ¯∂f i ∧ ∂ ¯f j ∧ ∂f k ∧ ¯∂ ¯f l. +□ +Corollary 3.20. For any smooth map f : M → N between K¨ahler manifolds, we have +−{¯∂f, RE′(¯∂f)} ∧ +ωn−2 +(n − 2)! = Qωn +n! . +Proof. Combine Lemma 3.19 with the definition of Q given in Lemma 3.15. +□ +Theorem 3.21. Suppose M and N are compact K¨ahler manifolds and the curvature +of N is strongly semi-negative. If f : M → N is a harmonic map, then f is plurihar- +monic. If, in addition, the curvature of N is strongly negative and the rankRdf ≥ 3 at +some point of M, then f is either holomorphic or conjugate holomorphic. +Proof. Integrate Siu’s Bochner formula over M. Applying Stoke’s theorem results in +the left hand side being 0. The two terms on the right hand side are non-negative +pointwise, hence they must be identically equal to 0. In particular, ∂E′ ¯∂f = 0; i.e. f +is is pluriharmonic. If the rank is ≥ 3 at some point x, ¯∂f = 0 in some neighborhood +of x by the definition of Q. Hence ¯∂f = 0 in all of M. +□ +3.7. Variations of the Siu and Sampson Formulas. The following is a variation +of the Sampson’s Bochner Formula. For harmonic metrics, this is due to Mochizuki +(cf. [M, Proposition 21.42]). +Theorem 3.22. For a harmonic map f : M → N from a K¨ahler manifold to a +Riemannian manifold, +d{d′ +Ed′f, d′′f − d′f} ∧ +ωn−2 +(n − 2)! += +8 +� +|d′ +Ed′′f|2 + Q0 +� +∧ ωn +n! . +Proof. The key observation is that, since d′{d′ +Ed′′f, d′′f} ∧ +ωn−2 +(n−2)! is an (n + 1, n − 1)- +form and d′′{d′ +Ed′′f, d′f} ∧ +ωn−2 +(n−2)! is an (n − 1, n + 1)-form, these two forms are both + +NOTES ON HARMONIC MAPS +27 +identically equal to zero. Thus, +d′{d′ +Ed′′f, d′f − d′′f} ∧ +ωn−2 +(n − 2)! += +d′{d′ +Ed′′f, d′f} ∧ +ωn−2 +(n − 2)! += +−d′{d′′ +Ed′f, d′f} ∧ +ωn−2 +(n − 2)! +(by (3.2)). += +−d′d′′{d′f, d′f} ∧ +ωn−2 +(n − 2)! += +d′d′′{d′′f, d′′f} ∧ +ωn−2 +(n − 2)! +(3.6) +d′′{d′ +Ed′′f, d′f − d′′f} ∧ +ωn−2 +(n − 2)! += +−d′′{d′ +Ed′′f, d′′f} ∧ +ωn−2 +(n − 2)! += +−d′′d′{d′′f, d′′f} ∧ +ωn−2 +(n − 2)! += +d′d′′{d′′f, d′′f} ∧ +ωn−2 +(n − 2)!. +(3.7) +Thus, +d{d′′ +Ed′f, d′′f − d′f} ∧ +ωn−2 +(n − 2)! += +d{d′ +Ed′′f, d′f − d′′f} ∧ +ωn−2 +(n − 2)! +(by (3.2)) += +(d′ + d′′){d′ +Ed′′f, d′f − d′′f} ∧ +ωn−2 +(n − 2)! += +2d′d′′{d′′f, d′′f} ∧ +ωn−2 +(n − 2)! (by (3.6) and (3.7)). +Thus, the asserted identity follows from Theorem 3.4. +□ +By applying a similar proof as Theorem 3.23, we obtain a variation of the Siu’s +Bochner formula. +Theorem 3.23. For a harmonic map f : M → X between K¨ahler manifolds, +d{¯∂E′∂f, ¯∂f − ∂f} ∧ +ωn−2 +(n − 2)! += +� +8 +��∂E′ ¯∂f +��2 + 2Q +� +∧ ωn +n! . +Proof. As in the proof of Theorem 3.22, ∂{∂E′ ¯∂f, ¯∂f} ∧ +ωn−2 +(n−2)! = 0 = ¯∂{∂E′ ¯∂f, ∂f} ∧ +ωn−2 +(n−2)! and hence +∂{∂E′ ¯∂f, ∂f − ¯∂f} ∧ +ωn−2 +(n − 2)! += +∂ ¯∂{¯∂ ¯f, ¯∂ ¯f} ∧ +ωn−2 +(n − 2)!, +¯∂{∂E′ ¯∂f, ∂f − ¯∂f} ∧ +ωn−2 +(n − 2)! += +∂ ¯∂{¯∂f, ¯∂f} ∧ +ωn−2 +(n − 2)!. + +28 +GEORGIOS DASKALOPOULOS AND CHIKAKO MESE +Consequently, +d{¯∂E′∂f, ¯∂f − ∂f} ∧ +ωn−2 +(n − 2)! += +d{∂E′ ¯∂f, ∂f − ¯∂f} ∧ +ωn−2 +(n − 2)! += +(∂ + ¯∂){∂E′ ¯∂f, ∂f − ¯∂f} ∧ +ωn−2 +(n − 2)! += +� +∂ ¯∂{¯∂f, ¯∂f} + ∂ ¯∂{¯∂ ¯f, ¯∂ ¯f} +� +∧ +ωn−2 +(n − 2)!. +The asserted identity follows from Theorem 3.16. +□ +3.8. Pluriharmonic maps into Euclidean buildings. +Theorem 3.24. Let M be a compact K¨ahler manifold and ∆(G) be the Bruhat-Tits +building associated to a semisimple algebraic group G defined over a non-Archimedean +local field K. For any Zariski dense representation of ρ : π1(X) → G(K), there exists +a ρ-equivariant, locally Lipschitz pluriharmonic map f : ˜ +M → ∆(G) from the universal +cover ˜ +M. +Definition 3.25. A Euclidean building of dimension n is a piecewise Euclidean sim- +plicial complex ∆ such that: +• ∆ is the union of a collection A of subcomplexes A, called apartments, such +that the intrinsic metric dA on A makes (A, dA) isometric to the Euclidean +space Rn and induces the given Euclidean metric on each simplex. +• Given two apartments A and A′ containing both simplices S and S′, there +is a simplicial isometry from (A, dA) to (A′, dA′) which leaves both S and S′ +pointwise fixed. +• ∆ is locally finite. +Definition 3.26. A point x0 is said to be a regular point of a harmonic map f, if there +exists r > 0 such that f(Br(x0)) of x is contained in an apartment of ∆. A singular +point of f is a point of Ω that is not a regular point. The regular (resp. singular) set +R(f) (resp. S(u)) of f is the set of all regular (resp. singular) points of f. +Example 3.27. Consider a measured foliation defined by the quadratic differential +zdz2 on C. The leaves of the horizontal foliation define a 3-pod T and the transverse +measure gives T a distance function d making (T, d) into a NPC space. The projection +along the vertical foliation u : C → T is a harmonic map. +The leaf containing 0 +is a non-manifold point of T. Let K = u−1(0). Then K is also a 3-pod. On the +other hand, every point of K besides 0 has a neighborhood mapping into an isometric +copy of R and S(0) = {0}. In particular, the singular set is of Hausdorff codimension +2. Similarly one can construct harmonic maps to other homogeneous trees by taking +quadratic differentials of higher order. + +NOTES ON HARMONIC MAPS +29 +The next two theorems are proved in [GS]. +Theorem 3.28. The singular set S(f) of a harmonic map f : M → ∆ is a closed set +of Hausdorff codimension ≥ 2. +Theorem 3.29. Let f : M → ∆ be as in Theorem 3.28. There exists a sequence of +smooth functions ψi with ψi ≡ 0 in a neighborhood of S(u), 0 ≤ ψi ≤ 1 and ψi(x) → 1 +for all x ∈ S(u) such that +lim +i→∞ +� +M +|∇∇u||∇ψi| dµ = 0. +By Theorem 3.28, Siu’s or Sampson’s Bochner formula holds at a.e. x ∈ ˜ +M. We +now follow the proof of Theorem 3.21 where integration by parts can be justified using +Theorem 3.28 and Theorem 3.29. + +30 +GEORGIOS DASKALOPOULOS AND CHIKAKO MESE +4. Donaldson Corlette theorem +4.1. Introduction: Higgs bundles via harmonic maps. In this lecture, we prove +the theorem of Donaldson and Corlette relating harmonic maps to symmetric spaces +of non-compact type and flat connections. +We do it explicitly for SL(n, C). +This +correspondence is very well known and there are many excellent references to consult. +Given the interest of the audience in this subject, we decided to give all the details of +the proof explicitly. See also [Do], [Co] and the expositional paper [Li]. +4.2. The flat vector bundle associated to a representation. Let ρ : π1(M) → +G = SL(n, C) be a homomorphism and +E = ˜ +M ×ρ Cn → M +be the associated flat vector bundle. Let H denote the space of positive definite self- +adjoint matrices of determinant one. For g ∈ SL(n, C), define an action Ag on the +space of (n × n)-matrices Mn×n(C) by +Ag(h) = g−1∗hg−1. +(4.1) +Note that H is invariant under Ag and hence it defines an action on H. +A ρ-equivariant map h : ˜ +M → H defines a hermitian metric on E by first defining +H(s, t) = ¯stht +(4.2) +on the universal cover ˜ +M × Cn and descending to a metric on E by equivariance. +Given the flat vector bundle (E, d) defined by ρ and the Hermitian metric H defined +by a ρ-equivariant map, we define θ ∈ Ω1(M, End(E)) by the formula +H(θs, t) = 1/2 (H(ds, t) + H(s, dt) − dH(s, t)) +(4.3) +and D by the formula +d = D + θ. +(4.4) +Formulas (4.3) and (4.4) immediately imply that +H(θs, t) = H(s, θt) +(4.5) +and +H(Ds, t) + H(s, Dt) += +H(ds, t) − H(θs, t) + H(s, dt) − H(s, θt) += +dH(s, t). +(4.6) +In other words, D is a Hermitian connection on (E, H). +We claim +θ = −1 +2h−1dh. +(4.7) + +NOTES ON HARMONIC MAPS +31 +To see (4.7), compute +dH(s, t) += +d¯sth t + ¯stdh t + ¯sth dt += +H(ds, t) + ¯stdh t + H(s, dt) += +H(Ds, t) + H(θs, t) + ¯stdh t + H(s, Dt) + H(s, θt) += +dH(s, t) + H(θs, t) + H(s, h−1dh t) + H(s, θt). +Thus, +H(θs, t) = H(s, θt) = −1/2H(s, h−1dh t) +and (4.7) follows. +Let End0(E) denote the space of trace-less endomorphisms of E. We claim that D +is a SL(n, C)-connection and θ ∈ End0(E). By (4.4) and since d is traceless, it suffices +to show that θ is traceless. Indeed, since G/K is a Cartan-Hadamard space, we can +write h = eu over a simply connected region U in M (or passing to the universal cover) +where u(x) ∈ p for all x ∈ U. Thus, +θ = h−1dh = du +is traceless since u is traceless. +As connections on End0(E), +D = d + 1 +2 +� +h−1dh, · +� +. +(4.8) +We apply harmonic map theory to prove: +Theorem 4.1. Given an irreducible representation ρ : π1(M) → SL(n, C), there exists +a ρ-equivariant map h : ˜ +M → H such that for the Hermitian metric H, Hermitian +connection D on End0(E) and θ ∈ Ω1(M, End0(E)) defined by (4.2), (4.3) and (4.4) +respectively, +d⋆ +Dθ = 0. +(4.9) +The proof of Theorem 4.1 is given several steps: (1) Choose h to be a harmonic map +(cf. Section 4.3). (2) Show that the Hermitian connection D is related to the Levi- +Civita connection on H (cf. Section 4.4). (3) Show that the harmonic map equation +for h is equivalent to (4.9) (cf. Section 4.5). +4.3. The equivariant map h is harmonic. The first step in the proof of Theorem 4.1 +is to choose the map h of Theorem 4.1 as a harmonic map into (H, gH) where the metric +is given by +gH(X, Y ) = n +2 trace(h−1Xh−1Y ) for X, Y ∈ ThH. +Definition 4.2. We call h or H a harmonic metric. + +32 +GEORGIOS DASKALOPOULOS AND CHIKAKO MESE +For G = SL(n, C), K = SU(n), let sl(n) = k ⊕ p be the Cartan decomposition and +B(X, Y ) = 2n trace(XY ) +be the Killing form on sl(n). The inner product is positive definite on p. +Let Lg : G/K → G/K be left multiplication and define a metric gG/K on G/K by +metrically identifying +dLg−1 : TgKG/K → TeKG/K = p. +(4.10) +This defines (G/K, gG/K) as a symmetric space of non-compact type. +Lemma 4.3. The map +Ψ : G/K +�→ +H +gK +�→ +g−1∗g−1 = h +identifies (G/K, gG/K) isometrically with (H, gH) as G-spaces. +Proof. First, Ψ is equivariant with respect to the action Lg on G/K and the action Ag +on H. Indeed, +Ψ ◦ Lg(g1K) = Ψ(gg1K) = (gg1)−1∗(gg1)−1 = g−1∗(g−1∗ +1 +g−1 +1 )g−1 += g−1∗Ψ(g1K)g−1 = Ag ◦ Ψ(g1K). +Second, Ψ is an isometry. Since the metric gG/K is defined by (4.10), we need to show +that with h = g−1∗g−1 ∈ H, +d(Lg−1)gK ◦ d(Ψ−1)h = d +� +(Ψ ◦ Lg)−1� +h : ThH → TeKG/K = p +is an isometry. This is a straightforward calculation: Let t �→ gt be a path in G/K with +g0 = eK and ˙g0 ∈ TeKG/K (where dot indicates the t-derivative). For ˙g ∈ TeKG/K, +since ˙g0 is self-adjoint, +(dΨ)e(˙g0) = d +dt +��� +t=0(g−1∗ +t +g−1 +t ) = −˙g∗ +0 − ˙g0 = −2˙g0. +(4.11) +For X ∈ ThH, +d +� +(Ψ ◦ Lg)−1� +h(X) += +d +� +(Ag ◦ Ψ)−1� +h(X) = d(Ψ−1 ◦ Ag−1)h(X) += +d(Ψ−1 ◦ Ag−1)g−1∗g−1(X) = (dΨe)−1 ◦ (dAg−1)g−1∗g−1(X) += +(dΨ−1)e(g∗Xg) = −1 +2g∗Xg += +−1 +2Adg−1(gg∗X) = −1 +2Adg−1(h−1X). + +NOTES ON HARMONIC MAPS +33 +Here we used (4.11) in the third to last equality. +Using this formula and the Ad- +invariance of the Killing form, we have for X, Y ∈ ThH +B +� +d +� +(Ψ ◦ Lg)−1� +h(X), d +� +(Ψ ◦ Lg)−1� +h(Y ) +� += 1 +4B(h−1X, h−1Y ) += n +2trace(h−1Xh−1Y ) += gH(X, Y ). +□ +By Theorem 2.17, there exists a ρ-equivariant harmonic map f : ˜ +M → G/K. In +view of the Lemma 4.3, we identify G/K with H and obtain a ρ-equivariant harmonic +map +h = f −1∗f −1 : ˜ +M → H, +d⋆ +∇dh = 0 +where ∇ is the pullback to h∗TH of the Levi-Civita connection of (H, gH). +4.4. The hermitian connection D and the Levi-Civita connection on (H, gH). +Recall that H ⊂ SL(n, C) is the space of positive definite, self-adjoint matrices of +determinant one and consider the map +Ph : ThH → Ph(ThH) ⊂ sl(n), +X �→ h−1X +whose image Ph(ThH) consists of matrices self-adjoint with respect to h. Indeed, +(h−1X)∗h = h−1(h−1X)∗h = h−1X. +Extending this map complex linearly induces an isomorphism +P C +h : ThHC +≃−→ sl(n) +which defines a global isomorphism +P C : THC ≃−→ H × sl(n). +(4.12) +The trivial connection d on H × sl(n) pulls back by the isomorphism P C to a flat +connection ¯∇ on THC; i.e. +¯∇XY +��� +h = P C−1 ◦ dX ◦ P C(Y ) +��� +h. +We next compute the formula for ¯∇ with respect to the coordinates that identify the +space of (n × n)-matrices Mn×n(C) with Rn2. Let t �→ ht be a curve in H with h0 = h +and ˙h0 = X(h). We have +dX(P C(Y )) = d +dt +��� +t=0 +� +h−1 +t +Y (ht) +� += h−1 d +dt +��� +t=0Y (ht) − h−1 ˙h0h−1 Y (h0). + +34 +GEORGIOS DASKALOPOULOS AND CHIKAKO MESE +Using the embedding H ֒→ Mn×n(C), we express ht = (hij +t ). Furthermore, we can ex- +press X = Xij∂ij and Y = Y kl∂kl with respect to the coordinate basis (∂ij). Extending +Y = Y kl∂kl as a vector field on Mn×n(C), we apply the chain rule to obtain +d +dt +��� +t=0Y (ht) = d +dt +��� +t=0Y kl(ht)∂kl = +� +∂ijY kl��� +h +˙hij +0 +� +∂kl = +� +Xij∂ijY kl� ��� +h ∂kl = ∂Y +∂X +��� +h. +(4.13) +Thus, the formula for the flat connection at the point h ∈ H is +¯∇XY = ∂Y +∂X − Xh−1Y. +The Levi-Civita connection on TH, denoted by ∇ and extended complex linearly to +THC, is given at h ∈ H by the formula +∇XY = ∂Y +∂X − 1 +2 +� +Xh−1Y + Y h−1X +� +. +Indeed: +(i) ∇ is torsion free: First, for a function f defined near h, +� ∂Y +∂X − ∂X +∂Y +� +f += +(Xij∂ijY kl)∂klf − (Y kl∂klXij)∂ijf += +(Xij∂ijY kl)∂klf + XijY kl∂ij∂klf − (Y kl∂klXij)∂ijf − Y klXij∂kl∂ijf += +X(Y f) − Y (Xf) = [X, Y ]f. +Thus, +∇XY − ∇Y X += +� ∂Y +∂X − 1 +2(Xh−1Y + Y h−1X) +� +− +�∂X +∂Y − 1 +2(Y h−1X + Xh−1Y ) +� += +∂Y +∂X − ∂X +∂Y = [X, Y ]. +(ii) ∇ is metric compatible: Using the path t �→ ht given above and using (4.13), +XgH(Y, Z) += +n +2trace +� ∂ +∂t +��� +t=0 +� +h−1 +t Y (ht)h−1 +t Z(ht) +�� += +n +2trace +�� +h−1 ∂Y +∂X − h−1Xh−1Y +� +h−1Z + h−1Y +� +h−1 ∂Z +∂X − h−1Xh−1Z +�� += +n +2trace +� +h−1 +� ∂Y +∂X − 1 +2 +� +Xh−1Y + Y h−1X +�� +h−1Z +� ++n +2trace +� +h−1Y h−1 +� ∂Z +∂X − 1 +2(Xh−1Z + Zh−1X) +�� += +gH(∇XY, Z) + gH(Y, ∇XZ). + +NOTES ON HARMONIC MAPS +35 +The difference of the flat connection ¯∇ and the Levi-Civita connection ∇ on THC is +� ¯∇XY − ∇XY +� += 1 +2 +� +Y h−1X − Xh−1Y +� += −1 +2h +� +h−1X, h−1Y +� +. +(4.14) +Let ˆ∇ = P C◦∇◦P C−1 denote the pullback to H×sl(n) of the Levi-Civita connection +∇ on THC via (4.12). Then the corresponding formula to (4.14) for the difference +between the flat connection d and ˆ∇ on H × sl(n) → H is +dX − ˆ∇X = −1 +2 +� +h−1X, · +� +. +(4.15) +The bundle H × sl(n) pulls back by h : ˜ +M → H to the trivial SL(n, C)-bundle h∗(H × +sl(n)) on the universal cover ˜ +M. +h∗(H × sl(n)) +H × sl(n) +˜ +M +H +h +From (4.15), the difference of the flat connection d and the pullback h∗ ˆ∇ is given by +the formula +dV − (h∗ ˆ∇)V = −1 +2 +� +h−1dh(V ), · +� +. +(4.16) +Next, the pullback to ˜ +M of the endormorphism bundle End0(E) is isomorphic to +the trivial bundle. +Taking the quotient by the induced action from ρ, End0(E) ≃ +h∗(H ×ρ sl(n)) → M and the connection h∗ ˆ∇ induces a connection on End0(E) (which +we also call ˆ∇). +From (4.16), we have +ˆ∇ = d + 1 +2 +� +h−1dh, · +� +. +Hence, +ˆ∇ = D +by (4.8). In other words, D is the connection on End0(E) induced by the Levi-Civita +connection on T CH. +4.5. Completion of the proof of Theorem 4.1. The bundle isomorphism P C−1 of +(4.12) induces a bundle isomorphism (still denoted by P C−1) +h∗(H ×ρ sl(n)) +≃ +h∗(THC) → M +φ +�→ +hφ. +Also, +ˆ∇ = P C ◦ ∇ ◦ P C−1. +(4.17) + +36 +GEORGIOS DASKALOPOULOS AND CHIKAKO MESE +In particular, since +θ = −1 +2h−1dh ∈ Ω1(M, End0(E)) ≃ Ω1(M, h∗(H ×ρ sl(n))), +we have +P C−1θ = hθ = −1 +2dh ∈ Ω1(M, h∗(THC)). +(4.18) +Theorem 4.1 follows from the the following implications: +h is harmonic +⇒ +0 = −1 +2d∗ +∇dh = d∗ +∇hθ = d∗ +∇P C−1θ +by (4.18) +⇒ +0 = P Cd∗ +∇P C−1θ = d∗ +ˆ∇θ = d∗ +Dθ +by (4.4) and (4.17). +References +[BH] +M. R. Bridson and A. Haefliger. Metric Spaces of Non-Positive Curvature. Springer-Verlag, +Berlin (1999). +[Co] +K. Corlette. Flat G-bundles with canonical metrics. J. Differential Geom. 28 (1988) 361-382. +[Do] +S. Donaldson. Twisted harmonic maps and the self-duality equations. Proc. London +Math. Soc. 55 (1987) 127-131. +[ES] +J. Eells and J. H. Sampson. Harmonic mappings of Riemannian manifolds. Amer. J. Math. +86 (1964) 109-160. +[GS] +M. Gromov and R. Schoen. Harmonic maps into singular spaces and p-adic superrigidity for +lattices in groups of rank one. Publ. Math. IHES 76 (1992) 165-246. +[J] +J. Jost. Nonlinear Methods in Riemannian and K¨ahlerian Geometry. Birkh¨auser Verlag 1988. +[KL] +B. Kleiner and B. Lieb. Rigidity of quasi-isometries for symmetric spaces and Euclidean build- +ings. Publications mathematiques de I.H.E.S, tome 86 (1997), 115-197. +[KS1] +N. Korevaar and R. Schoen. Global existence theorems for harmonic maps to non-locally com- +pact spaces. Comm. Anal. Geom. 5 (1997) 213-266. +[KS2] +N. Korevaar and R. Schoen. Global existence theorem for harmonic maps to non-locally com- +pact spaces. Comm. Anal. Geom. 5 (1997), 333-387. +[Li] +Q. Li. An Introduction to Higgs Bundles via Harmonic Maps. SIGMA 15 (2019). +[LSY] K. Liu, X. Sun, X. Yang and ST. Yau.Curvatures of moduli space of curves and applications. +Asian J. of Math. Vol. 21, No. 5, (2017) 841-854. +[LY] +K. Liu and X. Yang. Hermitian harmonic maps and non-degenerate curvatures. Mathematical +Research Letters 21 (2014) 831-862. +[M] +T. Mochizuki. Asymptotic behaviour of tame harmonic bundles and an application to pure +twistor D-modules. Memoirs of the AMS 185 (2007). +[Sa] +J. H. Sampson. Harmonic maps in K¨ahler geometry. Harmonic mappings and minimal im- +mersions, 193–205, Lecture Notes in Math., 1161, Springer, Berlin, 1985. +[S] +R. Schoen. Analytic Aspects of the Harmonic Map Problem. Seminar on Nonlinear Partial +Differential Equations, 1984, MSRI Publications book series, Volume 2. +[Siu] +Y.-T. Siu. The complex analyticity of harmonic maps and the strong rigidity of compact K¨ahler +manifolds. Ann. of Math. 112 (1980) 73-111. + diff --git a/2NE2T4oBgHgl3EQf5ggG/content/tmp_files/load_file.txt b/2NE2T4oBgHgl3EQf5ggG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3554752c99e86f0e29ee2e176ea17b3644d1ffec --- /dev/null +++ b/2NE2T4oBgHgl3EQf5ggG/content/tmp_files/load_file.txt @@ -0,0 +1,1042 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf,len=1041 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='04190v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='DG] 10 Jan 2023 NOTES ON HARMONIC MAPS GEORGIOS DASKALOPOULOS AND CHIKAKO MESE This is set of notes prepared for the Summer School on non-Abelian Hodge theory in Abbaye de Saint-Jacut de la Mer June, 6-19, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Table of Contents Lecture 1: Harmonic Maps Between Riemannian Manifolds p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 2 Lecture 2: Existence and regularity p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 9 Lecture 3: Pluriharmonic Maps and the Siu-Sampson Formula p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 16 Lecture 4: Donaldson Corlette Theorem p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 30 Date: June 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' GD supported in part by NSF DMS-2105226, CM supported in part by NSF DMS-2005406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We would like to thank Yitong Sun for carefully reading this document and making useful suggestions to improve the exposition of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 1 2 GEORGIOS DASKALOPOULOS AND CHIKAKO MESE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Harmonic Maps Between Riemannian manifolds 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Introduction: Basics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' In this section, we define energy of maps between Rie- mannian manifolds, harmonic maps, and the first and second variation formulas after the pioneering work of Eells-Sampson [ES].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' A good reference is also [J].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The energy of maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let (M, g), (N, h) be Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let f : M → N be a smooth map which induces a map df : TM → TN df � ∂ ∂xα � ���� p = ∂f i ∂xα ∂ ∂yi ���� f(p) where (xα) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (yi)) are the local coordinates of M (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The map f also induces a vector bundle f ∗TN over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let ∇ be a connection on f ∗TN inherited from the Levi-Civita connection on TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then ∇ induces an exterior derivative d∇ : C∞((ΛpT ∗M) ⊗ f ∗TN) → C∞((Λp+1T ∗M) ⊗ f ∗TN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We view df as a section df ∈ C∞(T ∗M ⊗ f ∗TN) = Ω1(f ∗TN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Using the notation ∂ ∂f i := ∂ ∂yi ◦ f ∈ C∞ loc(M, f ∗TN), we have df = ∂f i ∂xα dxα ⊗ ∂ ∂f i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let (gαβ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (hij)) be the expression of the Riemannian metric g of M (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' h of N) with respect to local coordinates (xα) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (yi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Set e(f) := 1 2|df|2 = 1 2 ∂f i ∂xα ∂f j ∂xβ gαβhij ◦ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The energy of f is E(f) := � M e(f) ⋆ 1 = 1 2 � M gαβ(x)hij(f(x)) ∂f i ∂xα ∂f j ∂xβ � g(x) dx1 ∧ · · · ∧ dxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Here, recall that the Hodge star operator ⋆ : ΛkTM → Λn−kTM, is the unique linear operator such that for all α, β ∈ ΛkV , α ∧ ⋆β = g(α, β) ⋆ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' NOTES ON HARMONIC MAPS 3 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let f = (ft) be a smooth one-parameter family of C∞ maps f : M × (−ǫ, ǫ) → N, f(x, t) = ft(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then ∇∂f ∂t = ∇∂/∂tdf where f = ft and ∂f ∂t = ∂f i ∂t ∂ ∂f i ∈ C∞(f ∗TN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Both ∇ ∂f ∂t = ∇∂/∂xα ∂f ∂t dxα and ∇∂/∂tdf = ∇∂/∂t ∂f ∂xαdxα are 1-forms with values in f ∗TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Here, ∂f ∂xα = ∂f j ∂xα ∂ ∂f j ∈ C∞(f ∗TN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Consider f as a map f : M × (−ǫ, ǫ) → N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Since � f∗ � ∂ ∂t � , f∗ � ∂ ∂xα �� = f∗ � ∂ ∂t, ∂ ∂xα � = 0 and ∇ is torsion-free, ∇∂/∂xα ∂f ∂t = � ∇f∗( ∂ ∂xα)f∗ � ∂ ∂t �� f = � ∇f∗( ∂f ∂t )f∗ � ∂ ∂xα �� f = ∇∂/∂tdf( ∂ ∂xα) which proves the equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3 (First Variation Formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For (ft) as above, d dtE(ft) = � M � ∇∂f ∂t , df � ⋆ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We compute d dtE(ft) = 1 2 � M d dt ⟨df, df⟩ ⋆ 1 = � M � ∇∂/∂tdf, df � ⋆ 1 = � M � ∇∂f ∂t , df � ⋆ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The critical points f of the functional E satisfy � M ⟨∇ψ, df⟩ ⋆ 1 = 0, ∀ψ ∈ C∞(f ∗TN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1) By taking ψ compactly supported away from ∂M, we obtain the Euler Lagrange equation of E, τ(f) := −d⋆ ∇df = ⋆d∇ ⋆ df = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2) 4 GEORGIOS DASKALOPOULOS AND CHIKAKO MESE Here, ∇ is the pullback of the Levi-Civita connection on f ∗(TN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let ft be a family of maps with d dt ���� t=0 ft = ψ ∈ C∞(f ∗TN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Taking ψ compactly supported and integrating by parts, d dt ���� t=0 E(ft) = � M ⟨ψ, −τf⟩ ⋆ 1 = 0 which holds for every ψ ∈ C∞ c (f ∗TM) iff τf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' A smooth map f : M → N satisfying d⋆ ∇df = 0 is called a harmonic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Harmonic map equations in local coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' First define ω � ∂ ∂yi � := Γi jkdyk ⊗ ∂ ∂yj , where Γi jk are Christoffel symbols on N and set ˜ω := ω ◦ f = � Γk ijdyk ⊗ ∂ ∂yj � f = (Γj ik ◦ f)∂f k ∂xβ dxβ ⊗ ∂ ∂f j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then d∇ = d + ˜ω and d⋆ ∇df = − ⋆ d∇ ⋆ df = − ⋆ (d + ˜ω) � ⋆df i ⊗ ∂ ∂f i � = −(⋆d ⋆ df i) ∂ ∂f i − (−1)m−1 ⋆ � ⋆df i ⊗ ˜ω � ∂ ∂f i �� = −∆f i ∂ ∂f i − (−1)m−1 ⋆ � ∂f i ∂xα ⋆ dxα ∧ (Γj ik ◦ f)∂f k ∂xβ dxβ ⊗ ∂ ∂f j � = −∆f i ∂ ∂f i − (−1)m−1 � ∂f i ∂xα ∂f k ∂xβ Γj ik ◦ f ⋆ (⋆dxα ∧ dxβ) ⊗ ∂ ∂f j � = − � ∆f k + gαβ ∂f i ∂xα ∂f j ∂xβ Γk ij ◦ f � ∂ ∂f k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Thus the harmonic map equation is ∆f k + gαβ ∂f i ∂xα ∂f j ∂xβ Γk ij ◦ f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3) Examples 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (1) Suppose N = R, then the harmonic map equation reduces to ∆f = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' f is a harmonic function on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' NOTES ON HARMONIC MAPS 5 (2) Suppose M = S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then E(f) = 1 2 � 2π 0 | ˙f(t)|dt and the critical points of E(f) are geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We can also see this from the harmonic maps equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Since S1 is 1-dimensional we can take gαβ = δαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then ∂2f k ∂t2 + Γk ij ∂f i ∂xα ∂f j ∂xβ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' This is the geodesic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (3) We’ll show later that holomorphic maps between K¨ahler manifolds are harmonic (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The Dirichlet and Neumann problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If M has non-empty boundary (N is without boundary) there are two different boundary value problems to consider: Dirichlet problem: Minimize E in a fixed homotopy class of maps from M to N relative to the boundary of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' This is equivalent to considering compactly supported variations ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Neumann problem: Minimize E in a fixed free homotopy class of maps from M to N, in other words no restriction on the type of variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The second variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The Riemannian tensor RN : TN × TN × TN → TN induces and operator RN : f ∗TN × f ∗TN × f ∗TN → f ∗TN in the natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let ft : M → N, and let V be a vector field along ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then ∇∂/∂t∇V = ∇∇∂/∂tV − RN � df, ∂f ∂t � V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Both ∇∂/∂t∇V − ∇∇∂/∂tV = � ∇∂/∂t∇∂/∂xαV − ∇∂/∂xα∇∂/∂tV � dxα = �� ∇f∗(∂/∂t)∇f∗(∂/∂xα)V − ∇f∗(∂/∂xα)∇f∗(∂/∂t) � f∗V � f dxα and RN � df, ∂f ∂t � V = � RN � f∗ � ∂ ∂xα � , f∗ � ∂ ∂t �� f∗(V ) � f dxα are 1−forms on M with values in f ∗TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Thus, assertion follows from the definition of the Riemannian tensor RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ 6 GEORGIOS DASKALOPOULOS AND CHIKAKO MESE Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='8 (Second Variation Formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' One has d2 dt2E(ft) = ||∇∂f ∂t ||2 − � M � RN � df, ∂f ∂t � ∂f ∂t , df � ⋆ 1 + � M � ∇∂/∂t ∂f ∂t , τf � ⋆ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We compute d2 dt2E(ft) = � M d dt � ∇∂f ∂t , df � ⋆ 1 = � M �� ∇∂/∂t∇∂f ∂t , df � + � ∇∂f ∂t , ∇∂/∂tdf �� ⋆ 1 = � M �� ∇∇∂/∂t ∂f ∂t , df � − � RN � df, ∂f ∂t � ∂f ∂t , df � + ||∇∂f ∂t ||2 � ⋆ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Existence and regularity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Introduction: Non-positive curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' In this section, we examine the role of non-positive curvature of the target metric on harmonic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We show uniqueness and discuss regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We also study the equivariant problem and prove existence of equivariant harmonic maps into non-positively curved metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Some references are [S], [KS1], [KS2] and [GS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Second variation formula and non-positive curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The following are corollaries of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If N has ≤ 0 sectional curvature and ft is a geodesic interpolation, then E(ft) is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' In the second variation formula the last term vanishes and the others are ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let f, φ : M → N be homotopic with f|∂M = φ|∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If N has ≤ 0 sectional curvature and f is harmonic, then E(f) ≤ E(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let ft be a geodesic homotopy between f, φ, thus f0 = f, f1 = φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then E(t) = E(ft) is convex, and E′(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' So E(1) ≥ E(0), hence E(φ) ≥ E(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If f0, f1 : M → N are homotopic harmonic maps with f0|∂M = f1|∂M and N has ≤ 0 sectional curvature, then: (1) If ∂M is nonempty, then f0 = f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (2) If ∂M is empty, F is a geodesic homotopy between f0, f1 and N has sectional curvature < 0 at one point p in the image of F, then either f0 = f1 or the rank of f0 is ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' NOTES ON HARMONIC MAPS 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (1) Let ft be a geodesic homotopy between f0, f1, E(t) = E(ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then E is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Since E′(0) = E′(1) = 0, we conclude E′(t) = 0 = E′′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='8, ∇∂F ∂t = 0 and � RN � df, ∂f ∂t � ∂f ∂t , df � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Thus, ∂ ∂xα||∂F ∂t ||2 = 2 � ∇∂/∂xα ∂F ∂t , ∂F ∂t � = 0 which implies that ||∂F/∂t|| is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' But ∂F/∂t = 0 on ∂M, so ∂F/∂t = 0 everywhere if ∂M is nonempty and hence f0 = f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (2) If ||∂F/∂t|| = 0, then f0 = f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Otherwise, ∂F/∂t ̸= 0 for every x, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The negative sectional curvature at p implies df is parallel to ∂F/∂t at p and therefore everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Thus, the image of df has dimension ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The Weitzenb¨ock formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let f : M → N be a harmonic map and (eα) an orthonormal frame for TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then ∆e(f) = |∇df|2 + 1 2 � df(RicM(eα)), df(eα) � − 1 2 � RN(df(eα), df(eβ))df(eβ), df(eα) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Expanding out the Laplacian with respect to local coordinates, the harmonic map equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3) is gαβf i /αβ − gαβ MΓη αβf i /η + gαβ NΓi kℓ ◦ ff k /αf ℓ /β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We use normal coordinates at x ∈ M and f(x) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Thus, the metric tensors (gαβ) and (hij) are Euclidean up to first order at x and y respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Differentiating, f i /ααε = MΓη αα/εf i /η − NΓi kℓ/mf m /εf k /αf ℓ /α = 1 2(gαη/αε + gαη/αε − gαα/ηε)f i /η − 1 2(hki/ℓm + hℓi/km − hkℓ/im)f m /εf k /αf ℓ /α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Furthermore, gαβ /ǫǫ = −gαβ/ǫǫ and △hij(f(x)) = hij/klf k /ǫf k /ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 8 GEORGIOS DASKALOPOULOS AND CHIKAKO MESE Thus, ∆ �1 2gαβhij ◦ ff i /αf j /β � = 1 √g ∂ ∂xσ �√ggστ ∂ ∂xτ �1 2gαβhij ◦ ff i /αf j /β �� = f i /ασf i /ασ − 1 2(gαβ/σσ + gσσ/αβ − gσα/σβ − gσα/σβ)f i /αf i /β + 1 2(hij/kℓ + hkℓ/ji − hik/jℓ − hjℓ/ik)f i /αf j /αf k /σf ℓ /σ = f i /ασf i /ασ + 1 2RicM αβf i /αf j /β − 1 2RN ikjℓf i /αf j /αf k /σf ℓ /σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ Here, RicM αβ = gδǫRM αδβǫ is the Ricci tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Assume N has ≤ 0 sectional curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then from the Weitzenb¨ock formula, ∆e(f) ≥ −Ce(f) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1) where C depends only on the geometry of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If f : M → N is harmonic, and N has ≤ 0 sectional curvature, then |f|C2+α loc ≤ c where c > 0 depends on E(f) and the geometries of M, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1) and Moser iteration, sup Br(p) e(f) ≤ c � B2r(p) e(f) ⋆ 1 = E(f) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2) where c only depends on the geometry and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Now the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1) is C0-bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' So by elliptic regularity, fi is C1+α-bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' But then the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1) is Cα-bounded, so fi is C2+α-bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If f : M → N is harmonic, and N has ≤ 0 sectional curvature, then f ∈ C∞(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Keep bootstrapping with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If f : M → N is a harmonic map, M is compact with Ricci curvature ≥ 0 and N has sectional curvature ≤ 0, then f is totally geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' NOTES ON HARMONIC MAPS 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Since 0 = � M △e(f) ⋆ 1 = � M � |∇df|2 + 1 2 � df(RicM(eα)), df(eα) � −1 2 � RN(df(eα), df(eβ))df(eβ), df(eα) �� ⋆ 1, and each of the terms on the right hand side is non-negative, we have ∇df = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Non-positive curvature in a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' A complete metric space (X, d) is called an NPC space if the following conditions are satisfied: (i) The space (X, d) is a length (or geodesic) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' That is, for any two points P and Q in X, there exists a rectifiable curve c so that the length of c is equal to d(P, Q) (which we will sometimes denote by dP Q for simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We call such distance realizing curves geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (ii) For any geodesic triangle with vertices P, R, Q ∈ X, let c : [0, l] → X be the arclength parameterized geodesic from Q to R and let Qt = c(tl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then d2 P Qt ≤ (1 − t)d2 P Q + td2 P R − t(1 − t)d2 QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3) (iii) Condition (ii) implies the quadralateral comparison inequalities (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [KS1, Corol- lary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3]) d2 PtQt ≤ (1 − t)d2 P Q + td2 RS − t(1 − t)(dSP − dQR)2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='4) d2 QtP + d2 Q1−tS ≤ d2 P Q + d2 RS − td2 QR − 2tdSPdQR + 2t2d2 QR (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='5) Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The main examples we will be considering are Riemannian manifolds of non-positive curvature and (locally compact) Euclidean buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let (X, d) be an NPC space, P ∈ X and M be a compact Riemannian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let Y = L2(M, X) be a set of maps f : M → X such that � M d2(f, P) ⋆ 1 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Define a distance function dY on Y by setting d2 Y (f0, f1) = � M d2(f0(x), f1(x)) ⋆ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then (Y, dY ) is an NPC space (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [KS1, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2]) where the geodesic between f0 and f1 is the geodesic interpolation map ft(x) = (1 − t)f0(x) + tf1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 10 GEORGIOS DASKALOPOULOS AND CHIKAKO MESE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Local existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We solve the Dirichlet problem for a smooth Riemannian do- main B ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We will motivate the construction by first considering the case X = R (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [KS1, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Fix φ ∈ H1(B, X) and consider the space H1 φ(B, X) = {f ∈ H1(B, X) : f − φ ∈ H1 0(B, X)} Let E0 = inf{E(f) : f ∈ H1 φ(B, X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' By the parallelogram identity 2 � B |df + v 2 |2 ⋆ 1 + 2 � B |df − v 2 |2 ⋆ 1 = � B |df|2 ⋆ 1 + � B |dv|2 ⋆ 1 Take a minimizing sequence fi and apply the previous equality for f = fi, v = fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' This implies that 2 � B |dfi − fj 2 |2 ⋆ 1 = � B |dfi|2 ⋆ 1 + � B |dfj|2 ⋆ 1 − 2 � B |dfi + fj 2 |2 ⋆ 1 ≤ 2E0 + 2ǫi − 2E0 = 2ǫi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Hence lim � B |dfi − fj 2 |2 ⋆ 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='6) By the Poincare inequality lim � B |fi − fj 2 |2 ⋆ 1 = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='7) hence lim i→∞ fi = f in H1 φ(B, X) and E(f) = E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Now assume X is an NPC space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Korevaar-Schoen [KS1] showed that the energy density makes sense by taking difference quotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For the purpose of these lectures, if X is a locally finite Euclidean building, then we can locally isometricaly embed it in a Euclidean space of high dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then, we can define the energy density of the map to the building equal to the energy density of the map considered as a map to the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' In fact, this was the original point of view taken in [GS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The more general theory developed later in [KS1] and [KS2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' With this, we argue as above replacing the parallelogram identity by the quadrilat- eral inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Indeed, for f, v ∈ H1 φ(B, X), define w(x) = (1 − t)f(x) + tv(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='4) with t = 1 2 implies 2d2(w(x), w(y)) ≤ d2(f(x), f(y)) + d2(v(x), v(y)) − 1 2(d(f(y), v(y)) − d(f(x), v(x))2 which then implies 2Ew ≤ Ef + Ev − 1 2 � B |∇d(f, v)|2 ⋆ 1 NOTES ON HARMONIC MAPS 11 Take a minimizing sequence fi and apply the previous inequality with f = fi and v = vi to conclude (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='6)) lim i,j→∞ � B |∇d(fi, fj)|2 ⋆ 1 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' By the Poincare inequality, fi is a Cauchy sequence in (Y, dY ) and converges to a map which is minimizing by the lower semicontinuity of energy [KS1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Basic Regularity result of Gromov-Schoen and Korevaar-Schoen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' This is the analogue of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2) without using the PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If f ∈ H1(B, X) is a harmonic map, then f is locally Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' More precisely, for any B′ ⊂⊂ B, there exists a constant C only depending on the metric on B′ and the distance of B′ to ∂B such that sup B′ |df|2 ≤ c � B |df|2 ⋆ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Equivariant maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let ρ : π1(M) → Isom(X) be a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' A map v : ˜ M → X is called a ρ-equivariant map, if v(γx) = ρ(γ)v(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Since |dv|2 is π1(M)-invariant, it descends to a function on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Define: E(v) = � M |dv|2 ⋆ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If v descends to a map to M/ρ(π1(M)) this agrees with our previous definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Existence of ρ-equivariant locally Lipschitz maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let (M, ν) be a proba- bility space, X an NPC-space and f ∈ L2(M, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' There exists a unique point Qf,ν that minimizes the integral If,ν(Q) := � M d2(f(m), Q)dν(m) ∀Q ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We call Qf,ν the center of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let {Qi} be a minimizing sequence and let Qij = 1 2Qi + 1 2Qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3) with t = 1 2, d2(f(x), Qij) ≤ 1 2d2(f(x), Qi) + 1 2d2(f(x), Qj) − 1 4d2(Qi, Qj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Integrating, we obtain If,ν(Qij) ≤ 1 2If,ν(Qi) + 1 2If,ν(Qj) − 1 2d2(Qi, Qj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 12 GEORGIOS DASKALOPOULOS AND CHIKAKO MESE Thus, d2(Qi, Qj) is a Cauchy sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We conclude that any minimizing sequence is a Cauchy sequence and converges to a minimizing element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' There exists a locally Lipschitz ρ-equivariant map ˜f : ˜ M → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If X is smooth, then ˜f can be chosen to be smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let Q0 := Qf,µ0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Q1 := Qf,µ1) be the center of mass for the function f ∈ L2(M, X) and the probability space (M, µ0) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (M, µ1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let Qt = (1−t)Q0+tQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' By the minimizing property of Q0 and Q1, � d2(f, Q0) + d2(f, Q1) dµ0 = � d2(f, Q0)dµ0 + � d2(f, Q1)dµ1 + � d2(f, Q1)(dµ0 − dµ1) ≤ 2 � d2(f, Q1/2) dµ0 + � � d2(f, Q1) − d2(f, Q1/2) � (dµ0 − dµ1) ≤ � d2(f, Q0) + d2(f, Q1) − 1 4d2(Q0, Q1) dµ0 + � � d2(f, Q1) − d2(f, Q1/2) � (dµ0 − dµ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The last inequality is by triangle comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Consequently, d2(Q0, Q1) ≤ 4 � � d2(f, Q1) − d2(f, Q 1 2) � (dµ0 − dµ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='8) For each x ∈ ˜ M, let dµx = dvol B1(x) V (x) , V (x) = vol(B1(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' where vol = ⋆1 is the volume form of ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Since µx is only dependent on the metric of M, x �→ µx is invariant under the isometric action of π1(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Furthermore, � |dµx0 − dµx1| = � ���� 1 V (x0)χB1(x0) − 1 V (x0)χB1(x0) ���� ⋆ 1 ≤ Cρ(x0, x1) where ρ denotes the distance function on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let M0 be a fundamental domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let f(M0) = P and extend equivariantly to f : ˜ M → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For simplicity, assume M0 is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then there exists a constant L such that d(f(x0), f(x1)) ≤ L whenever ρ(x0, x1) < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Thus, for ρ(x0, x1) < 1 and in the support of |dµx0 − dµx1|, d2(f, Q1) − d2(f, Q1/2) ≤ d2(f, Q1) + d2(f, Qt) ≤ 2L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' NOTES ON HARMONIC MAPS 13 Define ˜f : ˜ M → X, ˜f(x) = Qf,µx The π1(M)-invariance of µx and the ρ-equivariance of f imply the ρ-equivariance of ˜f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Apply (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='8) with M = M, µ0 = µx0 and µ1 = µx1 to obtain d2( ˜f(x0), ˜f(x1)) ≤ 2L2 � |dµx0 − dµx1| ≤ 2L2Cρ(x0, x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The boundary at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' A good reference is [BH].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Suppose X is an NPC- space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Two geodesic rays c, c′ : [0, ∞) → X are said to be asymptotic if there exists a constant K such that d(c(t), c′(t)) < K for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The set ∂X of boundary points of X (which we shall also call the points at infinity) is the set of equivalence classes of geodesic rays, two geodesic rays being equivalent if and only if they are asymptotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We denote ¯X = X ∪ ∂X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Notice that the images of two asymptotic geodesic rays under any isometry of X are again asymptotic geodesic rays, and hence any isometry extends to give a bijection of ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The next proposition is [BH, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If X is an NPC-space and c : [0, ∞) → X is a geodesic ray starting from P, then for every point P1 ∈ X there is a unique geodesic ray which starts from P1 and is asymptotic to c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The topology of ¯X is defined as follows: A sequence of points Pi converges to a point P ∗ ∈ ∂X if and only if the geodesics joining P0 to Pi converge (uniformly on compact subsets) to the geodesic ray that starts from P0 and belongs to the class of P ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If X is a complete n-dimensional Riemannian manifold of non-positive sectional curvature, then ∂X is homeomorphic to Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Indeed, given a base point P0, we can obtain a homeomorphism by considering the map which associates to each unit vector V tangent to X at P0 the class of the geodesic ray c starting at P0 with velocity vector V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' In particular, if X is the n-dimensional hyperbolic space, then ¯X is homeomorphic to the n-dimensional ball in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If X is a locally compact Euclidean building, then ∂X is a compact spherical building (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [KL, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If Pi is a sequence in X with lim Pi = P ∗ ∈ ∂X and if Qi is another sequence in X with d(Pi, Qi) ≤ C independently of i, then lim Qi = P ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Fix P0 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let γ : [0, ∞) → ∞ be an arclength parameterized geodesic ray in the equivalence class P ∗ with γ(0) = P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let ti = d(P0, Pi) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' τi = d(P0, Qi)) and let γi : [0, ti] → X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' ˆγi : [0, τi] → X) be the arclength parameterized geodesic segment connecting P0 and Pi (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' By the triangle inequality, d(ˆγi(ti), ˆγi(τi)) = |ti − τi| = |d(P0, Pi) − d(P0, Qi)| ≤ d(Pi, Qi) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 14 GEORGIOS DASKALOPOULOS AND CHIKAKO MESE Thus, assuming t ≤ ti ≤ τi, the NPC condition implies d(ˆγi(t), γi(t)) ≤ t ti d(ˆγi(ti), γi(ti)) ≤ t ti � d(ˆγi(ti), ˆγi(τi)) + d(ˆγi(τi), γi(ti)) � ≤ 2Ct ti .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Similarly, assuming t ≤ τi < ti, d(ˆγi(t), γi(t)) ≤ 2Ct τi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Thus, for t ≤ min{ti, τi}, d(ˆγi(t), γ(t)) ≤ d(ˆγi(t), γi(t)) + d(γi(t), γ(t)) ≤ 2Ct max{ti, τi} + d(γi(t), γ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Fix T0 > ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The assumption that lim Pi = P ∗ implies that ti, τi → ∞ and the geodesics γi converge uniformly to γ in [0, T0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Thus, ˆγi also converge uniformly to γ in [0, T0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The stabilizer of a point at infinity is contained in a parabolic subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' So if the image of ρ is Zariski dense it cannot fix a point at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Global existence result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We prove existence of equivariant harmonic maps [GS, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let X be a locally compact NPC space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Assume that the image of ρ doesn’t fix a point in ∂X and that there exists a Lipschitz equivariant map v : ˜ M → X with finite energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then there is a Lipschitz equivariant map f of least energy and the restriction of f to a small ball about any point is minimizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let E0 denote the infimum of the energy taken over all Lipschitz equivariant maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let vi be a sequence of Lipschitz equivariant maps with E(vi) → E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let B be a ball in ˜ M such that γ(B) ∩ B = ∅ for all γ ∈ π1(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We may then construct a new minimizing sequence ¯vi, by replacing vi with the solution to the Dirichlet problem on each γ(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Clearly ¯vi is also a minimizing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' On a compact subset of B, the sequence ¯vi is uniformly Lipschitz by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' It follows that a subsequence of ¯vi converges uniformly on compact subsets of B to a map into ¯X which either maps into X or maps to a single point P ∗ ∈ ∂X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We exclude the second possibility as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let x0 ∈ ˜ M be the center of the chosen ball B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let C be any smooth embedded curve from x0 to γ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' An elementary argument using Fubini’s theorem shows that C may be chosen so that the energy of the restriction of each map ¯vi to C is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Therefore the length of the curve ¯vi(C) is uniformly bounded, and in particular d(vi(x0), ρ(γ)vi((x0)) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='15 implies lim ρ(γ)vi(x0) = P ∗, and hence ρ(γ)P ∗ = P ∗ for all γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Therefore we may assume that ¯vi converges uniformly on compact subsets of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' NOTES ON HARMONIC MAPS 15 From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='6) as before, we have� K |∇d(¯vi, ¯vj)|2 ⋆ 1 → 0 for any compact set K ⊂ ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Since ¯vi converges uniformly on compact subsets of B, the function d(¯vi, v) is uniformly bounded there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' It then follows from Poincare type inequalities that � K d(¯vi, ¯vj)2 ⋆ 1 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' In particular, the sequence ¯vi → f which is a minimizer by lower semicontinuity of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The local minimizing property of f follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ 16 GEORGIOS DASKALOPOULOS AND CHIKAKO MESE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Pluriharmonic maps and the Siu-Sampson Formula 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Introduction: Bochner methods for harmonic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' In this section, we discuss the Bochner formulas of Siu [Siu] and Sampson [Sa].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Our exposition closely follows the approach of [LY].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We also present a variation of these formulas inspired by the work of Mochizuki [M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Lastly, we sketch the existence of pluriharmonic maps into Euclidean buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Pluriharmonic maps from K¨ahler manifolds to Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let (M, ω, J) be a K¨ahler manifold along with its K¨ahler form and complex structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let TM ⊗ C = T (1,0)M ⊕ T (0,1)M be its complexified tangent bundle decomposed into the ±√−1-eigenspaces of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We can decompose v ∈ TM ⊗ C into v = v1,0 + v0,1 where v1,0 = 1 2(v − √ −1Jv), v0,1 = 1 2(v + √ −1Jv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The cotangent space T ∗M has a complex structure still denoted J defined by Jα = α ◦ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Accordingly, we have an analogous decomposition T ∗M ⊗ C = T ∗(1,0)M ⊕ T ∗(0,1)M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let (N, h) be a Riemannian manifold and TN ⊗ C its complexified tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For a smooth map f : M → N, let E := f ∗(TN ⊗ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1) Extending complex linearly, df : TM → TN gives rise to a map df : TM ⊗ C → TN ⊗ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Denote by Ωp,q(E), the space of E-valued (p, q)-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Define d′f := 1 2(df − √ −1 df ◦ J) ∈ Ω1,0(E), d′′f := 1 2(df + √ −1 df ◦ J) ∈ Ω0,1(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We have that df = d′f + d′′f Jdf = df ◦ J = − √ −1 (d′f − d′′f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For local coordinates (yi) of N, let ∂ ∂fi = ∂ ∂yi ◦ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then d′f = d′f i ∂ ∂f i d′′f = d′′f i ∂ ∂f i d′f = d′′f d′′f = d′f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Similarly, we can decompose the pullback of the Levi-Civita connection (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Section 1) as ∇ = ∇′ + ∇′′ where ∇′ : C∞(E) → Ω1,0(E), ∇′′ : C∞(E) → Ω0,1(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' NOTES ON HARMONIC MAPS 17 In turn, ∇′ and ∇′′ induce differential operators d′ E : Ωp,q(E) → Ωp+1,q(E), d′′ E : Ωp,q(E) → Ωp,q+1(E) where d′ E(φ ⊗ s) = d′φ ⊗ s + (−1)p+qφ ⊗ ∇′ Es d′′ E(φ ⊗ s) = d′′φ ⊗ s + (−1)p+qφ ⊗ ∇′′ Es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' A straightforward calculation implies that d′ Ed′′f = −d′′ Ed′f, d′ Ed′f = 0, d′′ Ed′′f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2) Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' τ(f) = 2i ⋆ � ωn−1 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' ∧ d′ Ed′′f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We claim ⋆α = ωn−1 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' ∧ Jα, ∀α ∈ Ω1(M, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' To check the claim, use normal coordinates (zi = xi + √−1yi) at a point x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For α = dxi or α = dyi, we have dxi ∧ ωn−1 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' ∧ Jdxi = dxi ∧ dyi ∧ ωn−1 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = √−1 2 dzi ∧ d¯zi ∧ ωn−1 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = ωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' and dyi ∧ ωn−1 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' ∧ Jdyi = −dyi ∧ dxi ∧ ωn−1 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = √−1 2 dzi ∧ d¯zi ∧ ωn−1 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = ωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The claim follows by linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Next, note that Jd′f = 1 2(df ◦ J + √ −1df) = √−1 2 (df − √ −1df ◦ J) = √ −1d′f Jd′′f = 1 2(df ◦ J − √ −1df) = √−1 2 (df + √ −1df ◦ J) = − √ −1d′′f, 18 GEORGIOS DASKALOPOULOS AND CHIKAKO MESE which implies Jdf = Jd′f + Jd′′f = √−1(d′f − d′′f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Applying the claim for α = df, we use the fact that dω = 0 to obtain τ(f) = −d⋆ ∇df = ⋆d∇(⋆df) = ⋆d∇ � ωn−1 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' ∧ Jdf � = ⋆d∇ � ωn−1 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' ∧ ( √ −1(d′f − d′′f)) � = − √ −1 ⋆ � ωn−1 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' ∧ (d′ Ed′′f − d′′ Ed′f) � = −2 √ −1 ⋆ � ωn−1 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' ∧ d′ Ed′′f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' f is called pluriharmonic d′ Ed′′f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1 implies pluriharmonic =⇒ harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Note that holomorphic maps between K¨ahler manifolds are pluriharmonic, and thus harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Sampson’s Bochner formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='4 (Sampson’s Bochner formula, [Sa]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For a harmonic map f : M → N from a K¨ahler manifold (M, g) to a Riemannian manifold (N, h), d′d′′{d′′f, d′′f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = 4 � |d′ Ed′′f|2 + Q0 � ωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' where {·, ·} is given in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='5 below and Q0 = −2gα¯δgγ ¯βRijkl ∂f i ∂zα ∂f k ∂¯zβ ∂f j ∂zγ ∂f l ∂¯zδ in local coordinates (zα) of M and (yi) of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Combine Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='6, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='8 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='20 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let {si} be a local frame of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For ψ = ψi ⊗ si ∈ Ωp,q(E) and ξ = ξi ⊗ si ∈ Ωp′,q′(E) we set {ψ, ξ} = ⟨si, sj⟩ψi ∧ ¯ξj ∈ Ωp+q′,q+p′ where ⟨·, ·⟩ is the complex-linear extention of the Riemannian metric on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' NOTES ON HARMONIC MAPS 19 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For any smooth map f : M → N from a K¨ahler manifold to a Riemann- ian manifold, we have d′d′′{d′′f, d′′f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = � −{d′ Ed′′f, d′ Ed′′f} + {d′′f, R(1,1) E (d′′f)} � ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' where R(1,1) E = (d′ Ed′′ E + d′′ Ed′ E) is the (1, 1)-part of the curvature RE = d2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Repeatedly using the fact that d′′ Ed′′f = 0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2)), d′d′′{d′′f, d′′f} = −{d′ Ed′′f, d′ Ed′′f} + {d′′f, d′′ Ed′ Ed′′f} = −{d′ Ed′′f, d′ Ed′′f} + {d′′f, (d′′ Ed′ E + d′ Ed′′ E + d′′ E 2)d′′f}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Since {d′′f, d′ Ed′ Ed′′f} ∧ ωn−2 (n−2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' is an (n − 1, n + 1)-form and hence zero for dimensional reasons, we can complete the square to obtain d′d′′{d′′f, d′′f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = � −{d′ Ed′′f, d′ Ed′′f} + {d′′f, (d′ E + d′′ E)2d′′f} � ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = � −{d′ Ed′′f, d′ Ed′′f} + {d′′f, R(1,1) E (d′′f)} � ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' which proves the first equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For any E-valued (1, 1)-form φ on M, −{φ, φ} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = 4(|φ|2 − |Traceωφ|2)ωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let (zp) be normal coordinates at x ∈ M and let φp¯qdzp ∧ d¯zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' At x, ωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = �√−1 2 �2 �� p dzp ∧ d¯zp �2 ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='. For p, q such that p ̸= q, s ̸= p or t ̸= q ⇒ dzp ∧ d¯zq ∧ d¯zs ∧ dzt ∧ �� j dzj ∧ d¯zj �n−2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For p = q, s ̸= t ⇒ dzp ∧ d¯zq ∧ d¯zs ∧ dzt ∧ �� j dzj ∧ d¯zj �n−2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 20 GEORGIOS DASKALOPOULOS AND CHIKAKO MESE Furthermore, φp¯qφp¯qdzp ∧ d¯zq ∧ d¯zp ∧ dzq = φp¯qφp¯qdzp ∧ d¯zp ∧ dzq ∧ d¯zq φp¯pφq¯qdzp ∧ d¯zp ∧ d¯zq ∧ dzq = −φp¯pφq¯qdzp ∧ d¯zp ∧ dzq ∧ d¯zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Thus, �√−1 2 �2 {φ, φ} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = �√−1 2 �2 � � p,q,s,t φp¯qφs¯tdzp ∧ d¯zq ∧ d¯zs ∧ dzt � ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = � p̸=q � |φp¯q|2 − φp¯pφq¯q � ωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = � p,q � |φp¯q|2 − φp¯pφq¯q � ωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = � |φ|2 − |traceωφ|2� ωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For any harmonic map f : M → N from a K¨ahler manifold to a Riemannian manifold, we have −{d′ Ed′′f, d′ Ed′′f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = 4 |d′ Ed′′f|2 ωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='7 with φ = d′ Ed′′f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Since f is harmonic, Trωd′ Ed′′f = 0 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For a harmonic map f : M → N from a K¨ahler manifold to a Hermitian- negative Riemannian manifold, we have {d′′f, R(1,1) E (d′′f)} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = −2 Rijkld′f i ∧ d′′f k ∧ d′f j ∧ d′′f l ∧ ωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' where R(1,1) E is defined in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let (zα) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (yi)) be normal coordinates at a point x ∈ M (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' f(x) ∈ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then ∇ ∂ ∂¯zγ ∇ ∂ ∂zβ ∂ ∂f j = ∇ ∂ ∂¯zγ �∂f k ∂zβ ∇ ∂ ∂fk ∂ ∂f j � = ∂f k ∂zβ ∂f l ∂¯zγ ∇ ∂ ∂fl ∇ ∂ ∂fk ∂ ∂f j NOTES ON HARMONIC MAPS 21 and d′′ Ed′ Ed′′f = d′′ Ed′ E �∂f j ∂¯zα d¯zα ⊗ ∂ ∂f j � = d′′d′ �∂f j ∂¯zα d¯zα � ⊗ ∂ ∂f j − ∂f j ∂¯zα d¯zα ∧ d¯zγ ∧ dzβ ⊗ ∇ ∂ ∂¯zγ ∇ ∂ ∂zβ ∂ ∂f j = d′′d′ �∂f j ∂¯zα d¯zα � ⊗ ∂ ∂f j + ∂f j ∂¯zα ∂f k ∂zβ ∂f l ∂¯zγ d¯zα ∧ dzβ ∧ d¯zγ ⊗ ∇ ∂ ∂fl ∇ ∂ ∂fk ∂ ∂f j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Similarly, d′ Ed′′ Ed′′f = d′d′′ �∂f j ∂¯zα d¯zα � ⊗ ∂ ∂f j − ∂f j ∂¯zα ∂f k ∂zβ ∂f l ∂¯zγ d¯zα ∧ dzβ ∧ d¯zγ ⊗ ∇ ∂ ∂fk ∇ ∂ ∂fl ∂ ∂f j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Combining the above two equalities, R(1,1) E (d′′f) = ∂f j ∂¯zα ∂f k ∂zβ ∂f l ∂¯zγ d¯zα ∧ dzβ ∧ d¯zγ ⊗ Rs jkl ∂ ∂f s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We compute {d′′f, R(1,1) E (d′′f)} = Rijkl ∂f i ∂¯zδ ∂f j ∂zα ∂f k ∂¯zβ ∂f l ∂zγ d¯zδ ∧ dzα ∧ d¯zβ ∧ dzγ = Rjilk ∂f j ∂zα ∂f i ∂¯zδ ∂f l ∂zγ ∂f k ∂¯zβ dzα ∧ d¯zδ ∧ dzγ ∧ d¯zβ = (−Rjlki + Rjkli)∂f j ∂zα ∂f i ∂¯zδ ∂f l ∂zγ ∂f k ∂¯zβ dzα ∧ d¯zδ ∧ dzγ ∧ d¯zβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Since Rjkli ∂f j ∂zα ∂f i ∂¯zδ ∂f l ∂zγ ∂f k ∂¯zβ dzα ∧ d¯zδ ∧ dzγ ∧ d¯zβ ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = Rjkli ∂f j ∂zα ∂f i ∂¯zδ ∂f l ∂zγ ∂f k ∂¯zβ dzα ∧ d¯zα ∧ dzβ ∧ d¯zβ ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' +Rjkli ∂f j ∂zα ∂f i ∂¯zδ ∂f l ∂zγ ∂f k ∂¯zβ dzα ∧ d¯zβ ∧ dzβ ∧ d¯zα ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = 0, 22 GEORGIOS DASKALOPOULOS AND CHIKAKO MESE we obtain {d′′f, R(1,1) E (d′′f)} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = −Rjlki ∂f j ∂zα ∂f i ∂¯zα ∂f l ∂zβ ∂f k ∂¯zβ dzα ∧ d¯zα ∧ dzβ ∧ d¯zβ ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' −Rjlki ∂f j ∂zα ∂f i ∂¯zβ ∂f l ∂zβ ∂f k ∂¯zα dzα ∧ d¯zβ ∧ dzβ ∧ d¯zα ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = −2Rjlki ∂f j ∂zα ∂f i ∂¯zα ∂f l ∂zβ ∂f k ∂¯zβ dzα ∧ d¯zα ∧ dzβ ∧ d¯zβ ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = −2Rjlkid′f j ∧ d′′f i ∧ d′f l ∧ d′′f k ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='. □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' A Riemannian manifold N is said to be Hermitian-negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' strongly Hermitian negative) if RijklAi¯lAj¯k ≤ 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' < 0) for any Hermitian semi-positive matrix A = � Ai¯l� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Locally symmetric spaces whose irreducible local factors are all non- compact or Euclidean type are Hermitian negative (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [Sa, Theorem 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='12 (Sampson).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If f : M → N is a harmonic map from a K¨ahler manifold into a Hermitian negative Riemannian manifold, then f is pluriharmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Integrate Sampson’s Bochner formula over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Applying Stoke’s theorem results in the left hand side being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The two terms on the right hand side are non-negative pointwise, hence they must be identically equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' In particular, d′ Ed′′f = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' f is is pluriharmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Maps between K¨ahler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let f : M → N be a smooth map between K¨ahler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' By decomposing TN ⊗ C = T (1,0)N ⊕ T (0,1)N we get the decomposition of E := f −1(TN ⊗ C) as E = E′ ⊕ E′′ where E′ := f −1(T (1,0)N), E′′ := f −1(T (0,1)N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Denote by Ωp,q(E), Ωp,q(E′) and Ωp,q(E′′) the space of E-, E′- and E′′-valued (p, q)- forms respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If (wi) are local holomorphic coordinates in N, then { ∂ ∂fi := ∂ ∂wi ◦ f, ∂ ∂ ¯fi := ∂ ∂ ¯wi ◦ f} is a local frame of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If d′f, d′f ′ are as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2, then d′f = ∂f + ∂ ¯f, d′′f = ¯∂f + ¯∂ ¯f, df = d′f + d′′f = ∂f + ∂ ¯f + ¯∂f + ¯∂ ¯f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' NOTES ON HARMONIC MAPS 23 where ∂f = ∂f i ∂ ∂f i ¯∂f = ¯∂f i ∂ ∂f i ∂ ¯f = ∂ ¯f i ∂ ∂ ¯f i ¯∂ ¯f = ¯∂ ¯f i ∂ ∂ ¯f i ∂f = ¯∂ ¯f ¯∂f = ∂ ¯f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Analogously, d∇ = d′ E + d′′ E is decomposed into the induced operators ∂E′, ¯∂E′, ∂E′′, ¯∂E′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' A straightforward calculation yields ∂E′ ¯∂f = −¯∂E′∂f ∂E′′ ¯∂ ¯f = −¯∂E′′∂ ¯f (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3) ∂E′∂f = 0 ¯∂E′ ¯∂f = 0 ∂E′′∂ ¯f = 0 ¯∂E′′ ¯∂ ¯f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='4) For any map f : M → N between K¨ahler manifolds, we have ��∂E′′ ¯∂ ¯f ��2 = ��¯∂E′′∂ ¯f ��2 = ��∂E′ ¯∂f ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='5) Indeed, the left equality follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3) and the right from the fact that conjugation is an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Siu’s curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let N be a K¨ahler manifold and R its complexified curvature tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We say N has negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' non-positive) complex sectional curvature, if R(V, ¯W, W, ¯V ) < 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' ≤ 0) ∀V, W ∈ TNC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' In [Siu], Siu introduced the following notion of negative curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Recall that for local holomorphic coordinates (wi) of a K¨ahler manifold N, the curvature tensor is of type (1,1) and is given explicitly by Ri¯jk¯l = − ∂2hi¯j ∂wk∂ ¯wl + hp¯q ∂hk¯q ∂wi ∂hp¯l ∂ ¯wj where h is the K¨ahler metric on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We say N has strongly negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' strongly semi-negative) curvature if Ri¯jk¯l(AiBj − CiDj)(AlBk − ClDk) < 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' ≤ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' for arbitrary complex numbers Ai, Bi, Ci, Di when AiBj − CiDj ̸= 0 for at least one pair of indices (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' A K¨ahler manifold N is strongly semi-negative if and only if it has non-positive complex sectional curvature (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [LSY, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 24 GEORGIOS DASKALOPOULOS AND CHIKAKO MESE Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let N be a K¨ahler manifold with K¨ahler form ω and of strongly semi- negative curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let M be another K¨ahler manifold and f : M → N be a smooth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If Q : M → R is defined by setting Qωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = −Ri¯jk¯l ¯∂f i ∧ ∂ ¯f j ∧ ∂f k ∧ ¯∂ ¯f l ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=', then Q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' At a point with normal coordinates in the domain Ri¯jk¯l ¯∂f i ∧ ∂f j ∧ ∂f k ∧ ¯∂ f l ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = � α,β ( √ −1)n−2Ri¯jk¯l � −∂¯αf i∂αf j∂βf k∂¯βf l + ∂¯αf i∂βf j∂αf k∂¯βf l +∂¯βf i∂αf j∂βf k∂¯αf l − ∂¯βf i∂βf j∂αf k∂¯αf l � ∧ (∧γ(dzγ ∧ d¯zγ)) = 4 � α,β Ri¯jk¯l � (∂¯αf i)(∂βf l) − (∂¯βf i)(∂αf l) � � (∂¯αf j)(∂βf k) − (∂¯βf j)(∂αf k) �ωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = 4 � α,β Ri¯jk¯l � (∂¯αf i)(∂βf j) − (∂¯βf i)(∂αf j) � � (∂¯αf l)(∂βf k) − (∂¯βf l)(∂αf k) �ωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The last equality is because Rı¯jk¯l = Ri¯lk¯l, and the last inequality is because of the assumption that N has strong semi-negative curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Siu’s Bochner Formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='16 (Siu-Bochner formula, [Siu] Proposition 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For a harmonic map f : M → N between K¨ahler manifolds, ∂ ¯∂{¯∂f, ¯∂f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = � 4 ��∂E′ ¯∂f ��2 + Q � ωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Combine Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='15, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='17, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='18 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='20 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ The curvature operators of E′ and E′′ are RE′ = −(∂E′ + ¯∂E′)2 and RE′′ = −(∂E′′ + ¯∂E′′)2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For any smooth map f : M → N between K¨ahler manifolds, we have ∂ ¯∂{¯∂f, ¯∂f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = � −{∂E′ ¯∂f, ∂E′ ¯∂f} − {¯∂f, RE′(¯∂f)} � ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' and ∂ ¯∂{¯∂ ¯f, ¯∂ ¯f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = � −{∂E′′ ¯∂ ¯f, ∂E′′ ¯∂ ¯f} − {¯∂ ¯f, RE′′(¯∂ ¯f)} � ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='. NOTES ON HARMONIC MAPS 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' By setting ψ = ξ = ¯∂f ∈ Ω0,1(E′) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='5) and repeatedly using the fact that ¯∂E′ ¯∂f = 0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3)), ∂ ¯∂{¯∂f, ¯∂f} = −{∂E′ ¯∂f, ∂E′ ¯∂f} + {¯∂f, ¯∂E′∂E′ ¯∂f} = −{∂E′ ¯∂f, ∂E′ ¯∂f} + {¯∂f, (¯∂E′∂E′ + ∂E′ ¯∂E′ + ¯∂2 E′)¯∂f}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Since {¯∂f, ∂2 E′ ¯∂f} ∧ ωn−2 (n−2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' is an (n − 1, n + 1)-form and hence zero for dimensional reasons, we can complete the square to obtain ∂ ¯∂{¯∂f, ¯∂f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = � −{∂E′ ¯∂f, ∂E′ ¯∂f} + {¯∂f, (∂E′ + ¯∂E′)2 ¯∂f} � ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = � −{∂E′ ¯∂f, ∂E′ ¯∂f} − {¯∂f, RE′(¯∂f)} � ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' which proves the first equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The second equation follows by setting ψ = ξ = ¯∂ ¯f ∈ Ω0,1(E′′) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='5) and following exactly the same computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For any harmonic map f : M → N between K¨ahler manifolds, we have −{∂E′ ¯∂f, ∂E′ ¯∂f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = 4 ��∂E′ ¯∂f ��2 ωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' −{∂E′′ ¯∂ ¯f, ∂E′′ ¯∂ ¯f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = 4 ��∂E′′ ¯∂ ¯f ��2 ωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='7 with φ = ∂E′ ¯∂f (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' φ = ∂E′′ ¯∂ ¯f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Since f is harmonic, Trω∂E′ ¯∂f = 0 and Trω∂E′′ ¯∂ ¯f = 0 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For any smooth map f : M → N between K¨ahler manifolds, we have {¯∂f, RE′(¯∂f)} = Ri¯jk¯l ¯∂f i ∧ ∂ ¯f j ∧ ∂f k ∧ ¯∂ ¯f l = {¯∂ ¯f, RE′′(¯∂ ¯f)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Using normal coordinates, we compute {¯∂f, RE′(¯∂f)} = {¯∂f i ∂ ∂f i , RE′(¯∂f j ∂ ∂f j )} = {¯∂f i ∂ ∂f i , ¯∂f j ∧ RE′( ∂ ∂f j )} = {¯∂f i ∂ ∂f i , ¯∂f j ∧ Rs jk¯l∂f k ∧ ∂f l ∂ ∂f s } = Ri¯j¯kl ¯∂f i ∧ ∂ ¯f j ∧ ¯∂ ¯f k ∧ ∂f l = Ri¯jk¯l ¯∂f i ∧ ∂ ¯f j ∧ ∂f k ∧ ¯∂ ¯f l 26 GEORGIOS DASKALOPOULOS AND CHIKAKO MESE which proves the first equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The second equality is proved similarly: {¯∂ ¯f, RE′′(¯∂ ¯f)} = {¯∂ ¯f i ∂ ∂ ¯f i , RE′′(¯∂ ¯f j ∂ ∂ ¯f j )} = {¯∂ ¯f i ∂ ∂ ¯f i , ¯∂ ¯f j ∧ RE′′( ∂ ∂ ¯f j )} = {¯∂ ¯f i ∂ ∂ ¯f i , ¯∂ ¯f j ∧ R¯s ¯j¯kl∂ ¯f k ∧ ¯∂f l ∂ ∂ ¯f s } = R¯ijk¯l ¯∂ ¯f i ∧ ∂f j ∧ ¯∂f k ∧ ∂ ¯f l = Rj¯ik¯l∂f j ∧ ¯∂ ¯f i ∧ ¯∂f k ∧ ∂ ¯f l = Ri¯jk¯l∂f i ∧ ¯∂ ¯f j ∧ ¯∂f k ∧ ∂ ¯f l = Ri¯jk¯l ¯∂f i ∧ ∂ ¯f j ∧ ∂f k ∧ ¯∂ ¯f l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For any smooth map f : M → N between K¨ahler manifolds, we have −{¯∂f, RE′(¯∂f)} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = Qωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Combine Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='19 with the definition of Q given in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Suppose M and N are compact K¨ahler manifolds and the curvature of N is strongly semi-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If f : M → N is a harmonic map, then f is plurihar- monic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If, in addition, the curvature of N is strongly negative and the rankRdf ≥ 3 at some point of M, then f is either holomorphic or conjugate holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Integrate Siu’s Bochner formula over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Applying Stoke’s theorem results in the left hand side being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The two terms on the right hand side are non-negative pointwise, hence they must be identically equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' In particular, ∂E′ ¯∂f = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' f is is pluriharmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' If the rank is ≥ 3 at some point x, ¯∂f = 0 in some neighborhood of x by the definition of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Hence ¯∂f = 0 in all of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Variations of the Siu and Sampson Formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The following is a variation of the Sampson’s Bochner Formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For harmonic metrics, this is due to Mochizuki (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [M, Proposition 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For a harmonic map f : M → N from a K¨ahler manifold to a Riemannian manifold, d{d′ Ed′f, d′′f − d′f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = 8 � |d′ Ed′′f|2 + Q0 � ∧ ωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The key observation is that, since d′{d′ Ed′′f, d′′f} ∧ ωn−2 (n−2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' is an (n + 1, n − 1)- form and d′′{d′ Ed′′f, d′f} ∧ ωn−2 (n−2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' is an (n − 1, n + 1)-form, these two forms are both NOTES ON HARMONIC MAPS 27 identically equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Thus, d′{d′ Ed′′f, d′f − d′′f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = d′{d′ Ed′′f, d′f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = −d′{d′′ Ed′f, d′f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = −d′d′′{d′f, d′f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = d′d′′{d′′f, d′′f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='6) d′′{d′ Ed′′f, d′f − d′′f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = −d′′{d′ Ed′′f, d′′f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = −d′′d′{d′′f, d′′f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = d′d′′{d′′f, d′′f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='. (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='7) Thus, d{d′′ Ed′f, d′′f − d′f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = d{d′ Ed′′f, d′f − d′′f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2)) = (d′ + d′′){d′ Ed′′f, d′f − d′′f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = 2d′d′′{d′′f, d′′f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Thus, the asserted identity follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ By applying a similar proof as Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='23, we obtain a variation of the Siu’s Bochner formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For a harmonic map f : M → X between K¨ahler manifolds, d{¯∂E′∂f, ¯∂f − ∂f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = � 8 ��∂E′ ¯∂f ��2 + 2Q � ∧ ωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' As in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='22, ∂{∂E′ ¯∂f, ¯∂f} ∧ ωn−2 (n−2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = 0 = ¯∂{∂E′ ¯∂f, ∂f} ∧ ωn−2 (n−2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' and hence ∂{∂E′ ¯∂f, ∂f − ¯∂f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = ∂ ¯∂{¯∂ ¯f, ¯∂ ¯f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=', ¯∂{∂E′ ¯∂f, ∂f − ¯∂f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = ∂ ¯∂{¯∂f, ¯∂f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='. 28 GEORGIOS DASKALOPOULOS AND CHIKAKO MESE Consequently, d{¯∂E′∂f, ¯∂f − ∂f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = d{∂E′ ¯∂f, ∂f − ¯∂f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = (∂ + ¯∂){∂E′ ¯∂f, ∂f − ¯∂f} ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' = � ∂ ¯∂{¯∂f, ¯∂f} + ∂ ¯∂{¯∂ ¯f, ¯∂ ¯f} � ∧ ωn−2 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='. The asserted identity follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Pluriharmonic maps into Euclidean buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let M be a compact K¨ahler manifold and ∆(G) be the Bruhat-Tits building associated to a semisimple algebraic group G defined over a non-Archimedean local field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For any Zariski dense representation of ρ : π1(X) → G(K), there exists a ρ-equivariant, locally Lipschitz pluriharmonic map f : ˜ M → ∆(G) from the universal cover ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' A Euclidean building of dimension n is a piecewise Euclidean sim- plicial complex ∆ such that: ∆ is the union of a collection A of subcomplexes A, called apartments, such that the intrinsic metric dA on A makes (A, dA) isometric to the Euclidean space Rn and induces the given Euclidean metric on each simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Given two apartments A and A′ containing both simplices S and S′, there is a simplicial isometry from (A, dA) to (A′, dA′) which leaves both S and S′ pointwise fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' ∆ is locally finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' A point x0 is said to be a regular point of a harmonic map f, if there exists r > 0 such that f(Br(x0)) of x is contained in an apartment of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' A singular point of f is a point of Ω that is not a regular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The regular (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' singular) set R(f) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' S(u)) of f is the set of all regular (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' singular) points of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Consider a measured foliation defined by the quadratic differential zdz2 on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The leaves of the horizontal foliation define a 3-pod T and the transverse measure gives T a distance function d making (T, d) into a NPC space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The projection along the vertical foliation u : C → T is a harmonic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The leaf containing 0 is a non-manifold point of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let K = u−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then K is also a 3-pod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' On the other hand, every point of K besides 0 has a neighborhood mapping into an isometric copy of R and S(0) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' In particular, the singular set is of Hausdorff codimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Similarly one can construct harmonic maps to other homogeneous trees by taking quadratic differentials of higher order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' NOTES ON HARMONIC MAPS 29 The next two theorems are proved in [GS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The singular set S(f) of a harmonic map f : M → ∆ is a closed set of Hausdorff codimension ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let f : M → ∆ be as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' There exists a sequence of smooth functions ψi with ψi ≡ 0 in a neighborhood of S(u), 0 ≤ ψi ≤ 1 and ψi(x) → 1 for all x ∈ S(u) such that lim i→∞ � M |∇∇u||∇ψi| dµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='28, Siu’s or Sampson’s Bochner formula holds at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' x ∈ ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We now follow the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='21 where integration by parts can be justified using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='28 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 30 GEORGIOS DASKALOPOULOS AND CHIKAKO MESE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Donaldson Corlette theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Introduction: Higgs bundles via harmonic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' In this lecture, we prove the theorem of Donaldson and Corlette relating harmonic maps to symmetric spaces of non-compact type and flat connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We do it explicitly for SL(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' This correspondence is very well known and there are many excellent references to consult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Given the interest of the audience in this subject, we decided to give all the details of the proof explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' See also [Do], [Co] and the expositional paper [Li].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The flat vector bundle associated to a representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let ρ : π1(M) → G = SL(n, C) be a homomorphism and E = ˜ M ×ρ Cn → M be the associated flat vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let H denote the space of positive definite self- adjoint matrices of determinant one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For g ∈ SL(n, C), define an action Ag on the space of (n × n)-matrices Mn×n(C) by Ag(h) = g−1∗hg−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1) Note that H is invariant under Ag and hence it defines an action on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' A ρ-equivariant map h : ˜ M → H defines a hermitian metric on E by first defining H(s, t) = ¯stht (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2) on the universal cover ˜ M × Cn and descending to a metric on E by equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Given the flat vector bundle (E, d) defined by ρ and the Hermitian metric H defined by a ρ-equivariant map, we define θ ∈ Ω1(M, End(E)) by the formula H(θs, t) = 1/2 (H(ds, t) + H(s, dt) − dH(s, t)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3) and D by the formula d = D + θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='4) Formulas (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='4) immediately imply that H(θs, t) = H(s, θt) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='5) and H(Ds, t) + H(s, Dt) = H(ds, t) − H(θs, t) + H(s, dt) − H(s, θt) = dH(s, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='6) In other words, D is a Hermitian connection on (E, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We claim θ = −1 2h−1dh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='7) NOTES ON HARMONIC MAPS 31 To see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='7), compute dH(s, t) = d¯sth t + ¯stdh t + ¯sth dt = H(ds, t) + ¯stdh t + H(s, dt) = H(Ds, t) + H(θs, t) + ¯stdh t + H(s, Dt) + H(s, θt) = dH(s, t) + H(θs, t) + H(s, h−1dh t) + H(s, θt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Thus, H(θs, t) = H(s, θt) = −1/2H(s, h−1dh t) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='7) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let End0(E) denote the space of trace-less endomorphisms of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We claim that D is a SL(n, C)-connection and θ ∈ End0(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='4) and since d is traceless, it suffices to show that θ is traceless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Indeed, since G/K is a Cartan-Hadamard space, we can write h = eu over a simply connected region U in M (or passing to the universal cover) where u(x) ∈ p for all x ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Thus, θ = h−1dh = du is traceless since u is traceless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' As connections on End0(E), D = d + 1 2 � h−1dh, · � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='8) We apply harmonic map theory to prove: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Given an irreducible representation ρ : π1(M) → SL(n, C), there exists a ρ-equivariant map h : ˜ M → H such that for the Hermitian metric H, Hermitian connection D on End0(E) and θ ∈ Ω1(M, End0(E)) defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='4) respectively, d⋆ Dθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='9) The proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1 is given several steps: (1) Choose h to be a harmonic map (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (2) Show that the Hermitian connection D is related to the Levi- Civita connection on H (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (3) Show that the harmonic map equation for h is equivalent to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='9) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The equivariant map h is harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The first step in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1 is to choose the map h of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1 as a harmonic map into (H, gH) where the metric is given by gH(X, Y ) = n 2 trace(h−1Xh−1Y ) for X, Y ∈ ThH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We call h or H a harmonic metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 32 GEORGIOS DASKALOPOULOS AND CHIKAKO MESE For G = SL(n, C), K = SU(n), let sl(n) = k ⊕ p be the Cartan decomposition and B(X, Y ) = 2n trace(XY ) be the Killing form on sl(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The inner product is positive definite on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let Lg : G/K → G/K be left multiplication and define a metric gG/K on G/K by metrically identifying dLg−1 : TgKG/K → TeKG/K = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='10) This defines (G/K, gG/K) as a symmetric space of non-compact type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The map Ψ : G/K �→ H gK �→ g−1∗g−1 = h identifies (G/K, gG/K) isometrically with (H, gH) as G-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' First, Ψ is equivariant with respect to the action Lg on G/K and the action Ag on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Indeed, Ψ ◦ Lg(g1K) = Ψ(gg1K) = (gg1)−1∗(gg1)−1 = g−1∗(g−1∗ 1 g−1 1 )g−1 = g−1∗Ψ(g1K)g−1 = Ag ◦ Ψ(g1K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Second, Ψ is an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Since the metric gG/K is defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='10), we need to show that with h = g−1∗g−1 ∈ H, d(Lg−1)gK ◦ d(Ψ−1)h = d � (Ψ ◦ Lg)−1� h : ThH → TeKG/K = p is an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' This is a straightforward calculation: Let t �→ gt be a path in G/K with g0 = eK and ˙g0 ∈ TeKG/K (where dot indicates the t-derivative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' For ˙g ∈ TeKG/K, since ˙g0 is self-adjoint, (dΨ)e(˙g0) = d dt ��� t=0(g−1∗ t g−1 t ) = −˙g∗ 0 − ˙g0 = −2˙g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='11) For X ∈ ThH, d � (Ψ ◦ Lg)−1� h(X) = d � (Ag ◦ Ψ)−1� h(X) = d(Ψ−1 ◦ Ag−1)h(X) = d(Ψ−1 ◦ Ag−1)g−1∗g−1(X) = (dΨe)−1 ◦ (dAg−1)g−1∗g−1(X) = (dΨ−1)e(g∗Xg) = −1 2g∗Xg = −1 2Adg−1(gg∗X) = −1 2Adg−1(h−1X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' NOTES ON HARMONIC MAPS 33 Here we used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='11) in the third to last equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Using this formula and the Ad- invariance of the Killing form, we have for X, Y ∈ ThH B � d � (Ψ ◦ Lg)−1� h(X), d � (Ψ ◦ Lg)−1� h(Y ) � = 1 4B(h−1X, h−1Y ) = n 2trace(h−1Xh−1Y ) = gH(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' □ By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='17, there exists a ρ-equivariant harmonic map f : ˜ M → G/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' In view of the Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='3, we identify G/K with H and obtain a ρ-equivariant harmonic map h = f −1∗f −1 : ˜ M → H, d⋆ ∇dh = 0 where ∇ is the pullback to h∗TH of the Levi-Civita connection of (H, gH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The hermitian connection D and the Levi-Civita connection on (H, gH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Recall that H ⊂ SL(n, C) is the space of positive definite, self-adjoint matrices of determinant one and consider the map Ph : ThH → Ph(ThH) ⊂ sl(n), X �→ h−1X whose image Ph(ThH) consists of matrices self-adjoint with respect to h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Indeed, (h−1X)∗h = h−1(h−1X)∗h = h−1X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Extending this map complex linearly induces an isomorphism P C h : ThHC ≃−→ sl(n) which defines a global isomorphism P C : THC ≃−→ H × sl(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='12) The trivial connection d on H × sl(n) pulls back by the isomorphism P C to a flat connection ¯∇ on THC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' ¯∇XY ��� h = P C−1 ◦ dX ◦ P C(Y ) ��� h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We next compute the formula for ¯∇ with respect to the coordinates that identify the space of (n × n)-matrices Mn×n(C) with Rn2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Let t �→ ht be a curve in H with h0 = h and ˙h0 = X(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' We have dX(P C(Y )) = d dt ��� t=0 � h−1 t Y (ht) � = h−1 d dt ��� t=0Y (ht) − h−1 ˙h0h−1 Y (h0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 34 GEORGIOS DASKALOPOULOS AND CHIKAKO MESE Using the embedding H ֒→ Mn×n(C), we express ht = (hij t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Furthermore, we can ex- press X = Xij∂ij and Y = Y kl∂kl with respect to the coordinate basis (∂ij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Extending Y = Y kl∂kl as a vector field on Mn×n(C), we apply the chain rule to obtain d dt ��� t=0Y (ht) = d dt ��� t=0Y kl(ht)∂kl = � ∂ijY kl��� h ˙hij 0 � ∂kl = � Xij∂ijY kl� ��� h ∂kl = ∂Y ∂X ��� h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='13) Thus, the formula for the flat connection at the point h ∈ H is ¯∇XY = ∂Y ∂X − Xh−1Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The Levi-Civita connection on TH, denoted by ∇ and extended complex linearly to THC, is given at h ∈ H by the formula ∇XY = ∂Y ∂X − 1 2 � Xh−1Y + Y h−1X � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Indeed: (i) ∇ is torsion free: First, for a function f defined near h, � ∂Y ∂X − ∂X ∂Y � f = (Xij∂ijY kl)∂klf − (Y kl∂klXij)∂ijf = (Xij∂ijY kl)∂klf + XijY kl∂ij∂klf − (Y kl∂klXij)∂ijf − Y klXij∂kl∂ijf = X(Y f) − Y (Xf) = [X, Y ]f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Thus, ∇XY − ∇Y X = � ∂Y ∂X − 1 2(Xh−1Y + Y h−1X) � − �∂X ∂Y − 1 2(Y h−1X + Xh−1Y ) � = ∂Y ∂X − ∂X ∂Y = [X, Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (ii) ∇ is metric compatible: Using the path t �→ ht given above and using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='13), XgH(Y, Z) = n 2trace � ∂ ∂t ��� t=0 � h−1 t Y (ht)h−1 t Z(ht) �� = n 2trace �� h−1 ∂Y ∂X − h−1Xh−1Y � h−1Z + h−1Y � h−1 ∂Z ∂X − h−1Xh−1Z �� = n 2trace � h−1 � ∂Y ∂X − 1 2 � Xh−1Y + Y h−1X �� h−1Z � +n 2trace � h−1Y h−1 � ∂Z ∂X − 1 2(Xh−1Z + Zh−1X) �� = gH(∇XY, Z) + gH(Y, ∇XZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' NOTES ON HARMONIC MAPS 35 The difference of the flat connection ¯∇ and the Levi-Civita connection ∇ on THC is � ¯∇XY − ∇XY � = 1 2 � Y h−1X − Xh−1Y � = −1 2h � h−1X, h−1Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='14) Let ˆ∇ = P C◦∇◦P C−1 denote the pullback to H×sl(n) of the Levi-Civita connection ∇ on THC via (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Then the corresponding formula to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='14) for the difference between the flat connection d and ˆ∇ on H × sl(n) → H is dX − ˆ∇X = −1 2 � h−1X, · � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='15) The bundle H × sl(n) pulls back by h : ˜ M → H to the trivial SL(n, C)-bundle h∗(H × sl(n)) on the universal cover ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' h∗(H × sl(n)) H × sl(n) ˜ M H h From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='15), the difference of the flat connection d and the pullback h∗ ˆ∇ is given by the formula dV − (h∗ ˆ∇)V = −1 2 � h−1dh(V ), · � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='16) Next, the pullback to ˜ M of the endormorphism bundle End0(E) is isomorphic to the trivial bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Taking the quotient by the induced action from ρ, End0(E) ≃ h∗(H ×ρ sl(n)) → M and the connection h∗ ˆ∇ induces a connection on End0(E) (which we also call ˆ∇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='16), we have ˆ∇ = d + 1 2 � h−1dh, · � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Hence, ˆ∇ = D by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' In other words, D is the connection on End0(E) induced by the Levi-Civita connection on T CH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Completion of the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The bundle isomorphism P C−1 of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='12) induces a bundle isomorphism (still denoted by P C−1) h∗(H ×ρ sl(n)) ≃ h∗(THC) → M φ �→ hφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Also, ˆ∇ = P C ◦ ∇ ◦ P C−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='17) 36 GEORGIOS DASKALOPOULOS AND CHIKAKO MESE In particular, since θ = −1 2h−1dh ∈ Ω1(M, End0(E)) ≃ Ω1(M, h∗(H ×ρ sl(n))), we have P C−1θ = hθ = −1 2dh ∈ Ω1(M, h∗(THC)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='18) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='1 follows from the the following implications: h is harmonic ⇒ 0 = −1 2d∗ ∇dh = d∗ ∇hθ = d∗ ∇P C−1θ by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='18) ⇒ 0 = P Cd∗ ∇P C−1θ = d∗ ˆ∇θ = d∗ Dθ by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' References [BH] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Bridson and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Haefliger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Metric Spaces of Non-Positive Curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Springer-Verlag, Berlin (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [Co] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Corlette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Flat G-bundles with canonical metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Differential Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 28 (1988) 361-382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [Do] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Donaldson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Twisted harmonic maps and the self-duality equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 55 (1987) 127-131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [ES] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Eells and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Sampson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Harmonic mappings of Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 86 (1964) 109-160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [GS] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Gromov and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Schoen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' IHES 76 (1992) 165-246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [J] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Jost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Nonlinear Methods in Riemannian and K¨ahlerian Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Birkh¨auser Verlag 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [KL] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Kleiner and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Lieb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Rigidity of quasi-isometries for symmetric spaces and Euclidean build- ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Publications mathematiques de I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='S, tome 86 (1997), 115-197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [KS1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Korevaar and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Schoen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Global existence theorems for harmonic maps to non-locally com- pact spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 5 (1997) 213-266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [KS2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Korevaar and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Schoen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Global existence theorem for harmonic maps to non-locally com- pact spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 5 (1997), 333-387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [Li] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' An Introduction to Higgs Bundles via Harmonic Maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' SIGMA 15 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [LSY] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Sun, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Yang and ST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Yau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='Curvatures of moduli space of curves and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Asian J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 21, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 5, (2017) 841-854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [LY] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Liu and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Hermitian harmonic maps and non-degenerate curvatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Mathematical Research Letters 21 (2014) 831-862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [M] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Mochizuki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Asymptotic behaviour of tame harmonic bundles and an application to pure twistor D-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Memoirs of the AMS 185 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [Sa] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Sampson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Harmonic maps in K¨ahler geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Harmonic mappings and minimal im- mersions, 193–205, Lecture Notes in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=', 1161, Springer, Berlin, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [S] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Schoen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Analytic Aspects of the Harmonic Map Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Seminar on Nonlinear Partial Differential Equations, 1984, MSRI Publications book series, Volume 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' [Siu] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Siu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' The complex analyticity of harmonic maps and the strong rigidity of compact K¨ahler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} +page_content=' 112 (1980) 73-111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE2T4oBgHgl3EQf5ggG/content/2301.04190v1.pdf'} diff --git a/2NE2T4oBgHgl3EQfjAcm/content/2301.03963v1.pdf b/2NE2T4oBgHgl3EQfjAcm/content/2301.03963v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d0552339c40ee54ccefd4efb7d9b74b14966af37 --- /dev/null +++ b/2NE2T4oBgHgl3EQfjAcm/content/2301.03963v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f1d197903cadc80534ac1f3ff52f9b37e9ba48afb79984657d534a0e5950c3ba +size 5207905 diff --git a/2dAzT4oBgHgl3EQfe_zq/content/tmp_files/2301.01447v1.pdf.txt b/2dAzT4oBgHgl3EQfe_zq/content/tmp_files/2301.01447v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9e63711181dc4618a4e4d8dea93eb24a4d1d8b23 --- /dev/null +++ b/2dAzT4oBgHgl3EQfe_zq/content/tmp_files/2301.01447v1.pdf.txt @@ -0,0 +1,3536 @@ +LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +YAO LI, MOLEI TAO, AND SHIROU WANG +Abstract. This paper proposes a probabilistic approach to investigate the shape of the +landscape of multi-dimensional potential functions. By using an appropriate coupling +scheme, two copies of the overdamped Langevin dynamics of the potential function are +coupled. For unimodal and multimodal potential functions, the tail distributions of the +coupling times are shown to have qualitatively different dependency on the noise mag- +nitude. More specifically, for a single-well potential function that is strongly convex, the +exponential tail of the coupling time distribution is independent of noise and uniformly +bounded away from zero by the convexity parameter; while for a multi-well potential +function, the exponential tail is exponentially small with respect to the noise magnitude +with the exponents determined by the essential barrier height, a quantity proposed in +this paper which, in a sense, characterizes the “non-convexity” of the potential function. +This provides a promising approach to detect the shape of a potential landscape through +the coupling time distributions which, in certain sense, shares the similar spirit with the +well-know problem “Can one hear the shape of a drum?” proposed by Kac in his fa- +mous paper [18]. Theoretical results are numerically verified and discussed for a variety +of examples in different contexts, including the Rosenbrock function, the potential of +interacting particle systems, as well as the loss functions of artificial neural networks. +1. Introduction +In 1966, a famous paper “Can one hear the shape of a drum?” by Mark Kac investigates +the possibility of inferring the shape of a domain from the spectral property of the Laplace +operator defined on this domain [18]. Although it is eventually proved that the shape of +a domain cannot be uniquely determined except a certain class of planar domain with +analytic boundary [36], some information about the domain can still be inferred from +the eigenvalue set of the Laplace operator; please refer to [35, 16] for further details. +Motivated by this spirit, the present paper investigates the approach of inferring the +property of a multi-dimensional landscape by “knocking” it and “listening” the sound, +where by “knocking” a multi-dimensional landscape, we mean to simulate a overdamped +Langevin dynamics along the potential landscape. More specifically, we run a large number +of samples of two coupled trajectories of the overdamped Langevin dynamics to infer the +landscape properties from the statistics of the coupling times. By the coupling lemma, the +tail distribution of the coupling times provides a lower bound of the spectral gap of the +Fokker-Planck operator of the overdamped Langevin dynamics. In this way, our approach, +similar to that in Kac’s paper, makes an attempt to establish a connection between the +characteristics of the potential landscape and the spectral property of the corresponding +operator. +We are interested in the multi-dimensional potential functions that have finitely many +local minima. Potential landscapes arise naturally in various areas [19, 21, 34] which can +2020 Mathematics Subject Classification. Primary 37H10; Secondary 60H10, 60J22, 60J60. +Key words and phrases. Potential landscape, coupling method, overdamped Langevin dynamics, essen- +tial barrier height, non-convexity. +1 +arXiv:2301.01447v1 [math.DS] 4 Jan 2023 + +2 +YAO LI, MOLEI TAO, AND SHIROU WANG +be of simple type that has only one equilibrium, or be of more complex types that support +multiple basins of attractions with the emergence of many local minima. To put in the +mathematical way, let U be a smooth function on a regular domain D ⊆ Rk(k ≥ 1) with +the local minima x1, ..., xL. Generically, under the deterministic negative gradient flow +(ϕt)t≥0 of U, xi’s are the stable equilibria of ϕt. For each i ∈ {1, ..., L}, call Bi = {x ∈ +D : ϕt(x) → xi as t → ∞} the basin (of attraction) of xi. A smooth function U is called a +single-well potential if it has only one local minimum x1 such that D = B1; U is called a +multi-well potential if 2 ≤ L < ∞ and D = +�� +1≤i≤L Bi +� � N, where N is a measure-zero +set. A multi-well potential is in particular called a double-well potential if L = 2. +This paper proposes a probabilistic approach to classify between the landscapes of +single- and multi-well potential functions. Our approach makes strong use of the coupling +idea in probability. Given two stochastic processes X = {Xt; t ≥ 0} and Y = {Yt; t ≥ 0}, +a coupling of X and Y is a stochastic process {(Xt, Yt); t ≥ 0} such that (i) for any t > 0, +Xt and Yt are respectively identically distributed with Xt and Yt; (ii) if Xs = Ys for certain +s > 0, then Xt = Yt for all t ≥ s. The first meeting time of Xt and Yt, called the coupling +time, is denoted by a random variable +τc = inft≥0{Xt = Yt}. +(1) +A coupling {(Xt, Yt); t ≥ 0} is said to be successful if τc < ∞ almost surely. +In our setting, the two stochastic processes to be coupled, i.e., X and Y as above, will +be the overdamped Langevin dynamics of U given by the following stochastic differential +equation(SDE) +dZt = −∇U(Zt)dt + εdBt, +(2) +where {Bt; t ≥ 0} is the Brownian motion, and ε > 0 scales the noise magnitude. Through- +out the paper, the following is always assumed1 for the potential function U. +(U1) The potential function U ∈ C3(D), where D is open, convex and connected, such +that limx→∂D U(x) = ∞, and if D is unbounded, it further holds that +limx→∂D |∇U| = ∞, limx→∂D |∇U(x)| − 2∆U(x) = ∞, +where | · | denotes the Euclidean norm. +To couple two stochastic processes, various coupling methods can be used in the different +contexts [24]. In this paper, to achieve numerical efficiency, a mixture use of the reflection +and maximal coupling methods is applied. More specifically, with certain threshold dis- +tance d > 0, the coupling (Xt, Yt) is switched between the reflection and maximal couplings +according to whether the distance |Xt − Yt| is greater than d or not. To be more precise, +let (Xt, Yt) evolve according to the reflection coupling if |Xt − Yt| > d and be switched to +the maximal coupling whenever |Xt − Yt| ≤ d; see Section 2 for more details. We call such +mixed use of the reflection and maximal coupling methods the reflection-maximal coupling +scheme. +How can the threshold d be chosen? In our numerical scheme based on the reflection- +maximal coupling, d will be closely related to the time step size h > 0. More precisely, +denote ˆ +Xh = { ˆXh +n; n ≥ 0} (resp. ˆY h = { ˆY h +n ; n ≥ 0}) the Euler–Maruyama scheme of X +(resp. Y ) with the time step size h, i.e., ˆXh +n = ˆXh +n−1 − ∇U( ˆXh +n−1)h + ε +√ +hNn (resp. ˆY h +n = +1In the single-well setting, (U1) ensures the existence of a global strong solution of (2); in the multi-well +setting, further assumptions on the finiteness and non-degeneracy on the saddle points and local minima, +as stated in (U2) or (U3)(iii), guarantees this [3]. + +LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +3 +ˆY h +n−1−∇U( ˆY h +n−1)h+ε +√ +hNn), where {Nn} are i.i.d standard normal random variables. The +threshold distance d is chosen to be proportional to the (directional) standard deviations +of the distribution of the random variable ε +√ +hNn, i.e., d = O(ε +√ +h). This guarantees a +sufficient overlap between the distributions of ˆXh +n and ˆY h +n if, assume at the previous step, +| ˆXh +n−1 − ˆY h +n−1| < d. Then under the reflection-maximal coupling scheme, ( ˆXh +n, ˆY h +n ) is a +maximal coupling and the probability P[ ˆXh +n = ˆY h +n ] is in order O(1); see Lemma 2.3 for +more details. Henceforth, by an h-reflection-maximal coupling we put particular emphasis +on the choice of the time step size h, and by a reflection-maximal coupling (without h) +we mean that the coupling is implemented under the reflection-maximal coupling scheme +with a generally small h and no further emphasis on its value. +We investigate how the exponential tail of the coupling time distributions of the over- +damped Langevin dynamics depends on the noise magnitudes. Our main message is, this +dependency will be both quantitatively and qualitatively different between potential func- +tions with only a single well and those with double or multiple wells. More specifically, we +prove that under the reflection-maximal coupling scheme, for a strongly convex single-well +potential (see (3) below), the exponential tail of P[τc > t] is uniformly bounded away from +zero independent of ε; see Theorem 1.1. For double- or multi-well potential functions, the +exponential tail, however, is exponentially small with respect to the noise magnitude; see +Theorem 1.2 and Theorem 1.3. +A single-well potential function U is said to be strongly convex with constant m0 > 0 if +⟨∇U(x) − ∇U(y), x − y⟩ ≥ m0|x − y|2, +∀x, y ∈ D, +(3) +where ⟨·, ·⟩ denotes the standard inner product in Rk. The supremum of all positive m0 +satisfying (3) is called the convexity parameter. Throughout this paper, m0 always denotes +the convexity parameter. +Theorem 1.1. Let U be a single-well potential satisfying (U1). Assume that U is strongly +convex with constant m0 > 0. Then, given any δ > 0 there exists h0 > 0 such that for any +h ∈ (0, h0), if {(Xt, Yt); t ≥ 0} is an h-reflection-maximal coupling of two solutions of (2) +satisfying E[|X0 − Y0|] < ∞, for all ε > 0, it holds that +lim sup +t→∞ +1 +t log P[τc > t] ≤ −m0 + δ. +In the situation that the potential function U is double-well with only two local minima, +a crucial quantity is the least barrier height to be passed by any continuous path connecting +the two local minima. Given any two subsets A, B ⊆ D, define the communication height +between A and B as +Φ(A, B) = +inf +φ(t)∈C([0,1],D), +φ(0)∈A, φ(1)∈B +supt∈[0,1] U(φ(t)), +(4) +where the infimum runs over all the continuous paths in D. For a double-well potential U +with two local minima x1, x2, define the essential barrier height +HU = min +� +Φ(x1, x2) − U(x1), Φ(x1, x2) − U(x2) +� +, +(5) +the lower height of the two barriers that has to be crossed from one local minimum to the +other. +In the double-well setting, the following potential functions, which are generic in certain +sense, are to be considered. + +4 +YAO LI, MOLEI TAO, AND SHIROU WANG +(U2) Let U : D → R be a double-well potential function satisfying (U1) with two local +minima x1, x2 such that +(i) The communication height between x1 and x2 is reached at a unique saddle point +z∗(x1, x2), i.e., +U(z∗(x1, x2)) = Φ(x1, x2); +(ii) U is non-degenerate (i.e., the Hessian of U has only non-zero eigenvalues) at the two +local minima x1, x2, and the saddle point z∗(x1, x2). +Besides assumptions for potential functions, in the double-well, and more generally, the +multi-well settings, the coupling scheme is also required to satisfy certain intuitive condi- +tions which, in particular for the reflection-maximal coupling scheme, can be numerically +verified. In the double-well setting, certain“local” coupling properties are assumed when +the two processes are lying in the same basin; see (H1)-(H2) in Section 4. +To include all possible initial conditions, in Theorem 1.2 and Theorem 1.3, the coupling +process is assumed to be initially related with all local minima. A probability measure µ +on D × D is said to be fully supported (w.r.t all the local minima) if for any δ > 0, +µ(Bδ(xi) × Bδ(xj)) > 0, +∀i, j ∈ {1, ..., L}, +where Bδ(x) denotes the ball centered at x with radius δ. Note that any probability +measure equivalent with the Lebesgue measure is fully supported. A coupling (X, Y ) is +said to be fully supported if the distribution of (X, Y ) is fully supported. Similarly in the +same way, a probability measure µ on D is said to be fully supported if for any δ > 0, +µ(Bδ(xi)) > 0, +∀i ∈ {1, ..., L}, +and a random variable X is said to be fully supported if the distribution of X is fully +supported. +Throughout this paper, by a ≲ b (resp. a ≳ b) we mean a ≤ const·b (resp. a ≥ const·b), +where the const is independent of any parameter of the process. By a ≃ b we mean both +a ≲ b and a ≳ b hold. +Theorem 1.2. Let U be a double-well potential satisfying (U2), and {(Xt, Yt); t ≥ 0} be +a coupling of two solutions of (2) such that (X0, Y0) is fully supported. Then, if (Xt, Yt) +is a reflection-maximal coupling satisfying (H1)-(H2), for any ε > 0 sufficiently small, it +holds that +lim +t→∞ +1 +t log P[τc > t] ≃ −C0e−2HU/ε2, +where HU is defined in (5), and the constant C0 > 0 is independent of ε and h. +The result in Theorem 1.2 can be generalized to potential functions that have any +finitely many local minima. +In this case, besides the uniqueness of saddle points and +the degeneracy conditions in (U2), the potential function U is also assumed to satisfy a +generic condition that U has different potential values and depths corresponding to the +different local minima. +(U3) Let U : D → R be a multi-well potential function satisfying (U1) with the local +minima x1, ..., xL such that +(i) U has different potential values at the different local minima. In particular, U admits +the unique global minimum; + +LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +5 +(ii) The different basins of potential U admit different depths. More precisely, there exists +some δ > 0 such that the L local minima of U, x1, ..., xL, can be labeled in such way that +Φ(xi, Ni−1) − U(xi) ≤ min1≤ℓ 0 sufficiently small, it +holds that +lim +t→∞ +1 +t log P[τc > t] ≃ −C0e−2HU/ε2, +where HU is defined in (7), and the constant C0 > 0 is independent of ε and h. + +6 +YAO LI, MOLEI TAO, AND SHIROU WANG +As an application of Theorem 1.2 and Theorem 1.3, the essential barrier height of a +potential function can be computed by empirically estimating the coupling time distribu- +tions. The essential barrier height captures the global nature of the non-convexity of a +potential function which, in an intuitive sense, would be of essential importance in the +non-convex optimization problems arising in the various areas. Based on the linear ex- +trapolation of the exponential tails, a numerical algorithm for the estimate of the essential +barrier height is developed. This algorithm is then validated in Section 5 for both 1D +double-well potential and multi-dimensional interacting particle systems, and the numeri- +cal results of the essential barrier heights are shown to be very close to the theoretical ones. +The algorithm is also applied to detect the loss landscapes of artificial neural networks. +The conclusion is that the loss landscapes of large artificial neural networks generally have +lower essential barrier heights than the small artificial neural networks, which is largely +consistent with the previously known results. +This paper is organized as follows. Section 2 prepares basic facts and results that will +be used in the subsequent sections including estimations related to the reflection and +maximal coupling methods, multi-well potentials and first hitting times, as well as the +probability generating functions. Section 3 studies the case of single-well potential and +proves Theorem 1.1. +Section 4 investigates the double- and multi-well situations and +proves Theorem 1.2 and Theorem 1.3. Section 5 explores various examples of single and +multi-well potentials for which the theoretical results and assumptions in the previous +sections are numerically verified. +2. Preliminary +This section prepares some instrumental facts that shall be used in the rest of the paper. +2.1. Reflection coupling and single-well potential. Let X, Y be two solutions of +dZt = g(Zt)dt + εdBt, +Zt ∈ Rk, +(8) +where g : Rk → Rk is Lipschitz continuous. +A reflection coupling of X and Y is a +stochastic process {(Xt, Yt); t ≥ 0} taking values in Rk × Rk such that +dXt = g(Xt)dt + εdBt, +dYt = g(Yt)dt + εPtdBt, +0 < t < τc; +Yt = Xt, +t ≥ τc, +(9) +where Pt = Ik − 2ete⊤ +t is the orthogonal matrix in which et = (Xt − Yt)/|Xt − Yt|, and τc +is the coupling time defined in (1). +The reflection coupling, as its name suggests, is to make the noise terms in Xt and Yt +the mirror reflection of each other [25]. It is particularly efficient in high-dimension by +only keeping the noise projections on the vertical direction of the hyperplane between Xt +and Yt while projections in other directions are cancelled so that the coupling effect in +making Xt and Yt towards each other are maximized. +Under the reflection coupling, the exponential tail of coupling time distributions of the +over-damped Langevin dynamics along a uniformly convex single-well potential is bounded +away from zero. +Proposition 2.1. Let U be a single-well potential satisfying (U1). Assume that U is +strongly convex with constant m0 > 0. Then there exists a constant c0 > 0 such that, if + +LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +7 +{(Xt, Yt); t ≥ 0} is a reflection coupling of two solutions of (2) with X0 = x0, Y0 = y0, for +any ε > 0 and any t > 0, it holds that +P[τc > t] ≤ c0|x0 − y0| +2ε +e−m0t. +Proof. Denote Rt = |Xt−Yt|/2ε. It is not hard to see that {Rt; t ≥ 0} is a one-dimensional +stochastic process satisfying +dRt = −R−1 +t ⟨∇U(Xt) − ∇U(Yt), Xt − Yt⟩dt + 2εd ¯Bt, 0 ≤ t < τc, +(10) +where { ¯Bt; t ≥ 0} is a one-dimensional Brownian motion. +By the strong convexity of U, the drift term in (10) is upper bounded by −m0Rt. Then +there exists a one-dimensional Ornstein-Uhlenbeck(O-U) process {St; t ≥ 0} +dSt = −m0Stdt + d ¯Bt, +S0 = |x0 − y0|/2ε, +(11) +such that Rt is always bounded by St for t ∈ [0, τc). Let +τ0 = inf{t ≥ 0 : St = 0}. +Then it is sufficient to estimate P[τ0 > t]. +Under the change of variables that t replaced by m0t and S replaced by √m0S, (11) +becomes a standard O-U process +dSt = −Stdt + d ¯Bt, +S0 = √m0|x0 − y0|/2ε. +(12) +By Proposition 1 in [22], the density function of τ0 of (12) has an analytic expression +p(t) = S0 +1 +� +2π sinh(t)3 e +t +2 − +e−tS2 +0 +2 sinh(t) , +t ≥ 0. +Thus, +P[τ0 ≥ t] += +� ∞ +m0t +p(s)ds +≤ +�m0 +2π |x0 − y0|(2ε)−1 +� ∞ +m0t +e +s +2 +sinh(s) +3 +2 +ds +≤ +c0|x0 − y0|(2ε)−1 +� ∞ +m0t +e−sds = c0|x0 − y0| +2ε +e−m0t +where the constant c0 > 0 is independent of x0, y0 and ε. +□ +2.2. Maximal coupling and estimations. Let µ1 and µ2 be two probability distribu- +tions on Rk. Call (X, Y ) a coupling of µ1 and µ2 if X ∼ µ1, Y ∼ µ2. By the well-known +coupling inequality (see, for instance, Lemma 3.6 in [2]), +TV(µ1, µ2) ≤ 2P[X ̸= Y ], +(13) +where TV(µ1, µ2) := 2 supA⊆Rk |µ1(A) − µ2(A)| denotes the total variation distance be- +tween probability measures on Rk. A coupling (X, Y ) is said to be a maximal coupling if +the equality in (13) is attained, i.e., the probability P[X = Y ] is maximized. +A particular way to obtain a maximal coupling is as follows. Denote the “minimum” +distribution of µ1 and µ2 by ν(·) = α−1 min{µ1(·), µ2(·)}, where α is the normalizer. With +probability (1 − α), let X and Y be independently sampled such that +X ∼ (1 − α)−1(µ1 − αν), +Y ∼ (1 − α)−1(µ2 − αν), +(14) + +8 +YAO LI, MOLEI TAO, AND SHIROU WANG +and with probability α, let X = Y ∼ ν. It is not hard to see that P[X ̸= Y ] = TV(µ1, µ2)/2. +Hence, (X, Y ) is a maximal coupling. +The following proposition is about the maximal coupling of normal distributions. +Proposition 2.2. Let X ∼ N(¯x, σ2Id), Y ∼ N(¯y, σ2Id) be normal random variables +taking values in Rk, where ¯x, ¯y ∈ Rk, σ > 0. Assume that (X, Y ) is a maximal coupling. +Then there exist a universal constant c0 > 0 and a constant ck > 0 only depending on k +such that +(i) P[|X − Y | > 0] ≤ c0σ−1|¯x − ¯y|; (ii) E[|X − Y |] ≤ ckσ + 2|¯x − ¯y| +Proof. (i) By the definition of maximal coupling, +P[|X − Y | > 0] = 1 +2TV(X, Y ) ≤ c0σ−1|¯x − ¯y|, +where by TV(X, Y ) we mean TV(µ, ν) if X ∼ µ, Y ∼ ν. The last inequality follows from +the standard calculation on Gaussian distributions [9]. +(ii) Let p¯x,σ and p¯y,σ be the probability density functions of X and Y , respectively. By +the definition of maximal coupling, +E[|X − Y |] += +� +Rk×Rk |x − y| · +� +p¯x,σ(x) − min{p¯x,σ(x), p¯y,σ(x)} +� +· +� +p¯y,σ(y) − min{p¯x,σ(y), p¯y,σ(y)} +� +dxdy +≤ +2 +� +x∈A1,y∈A2 +p¯x,σ(x)p¯y,σ(y)|x − y|dxdy, +(15) +where A1 = {z ∈ Rk : p¯x,σ(z) ≥ p¯y,σ(z)}, A2 = {z ∈ Rk : p¯x,σ(z) < p¯y,σ(z)}. +Further split the upper bound in (15) into three terms +(15) +≤ +2 +� +x∈A1,y∈A2 +p¯x,σ(x)p¯y,σ(y)|x − ¯x|dxdy + 2 +� +x∈A1,y∈A2 +p¯x,σ(x)p¯y,σ(y)|¯x − ¯y|dxdy ++ +2 +� +x∈A1,y∈A2 +p¯x,σ(x)p¯y,σ(y)|y − ¯y|dxdy := ψ1 + ψ2 + ψ3. +(16) +Plainly, ψ2 ≤ 2|¯x − ¯y|. For the estimate of ψ1, note that p¯x,σ(x) = +1 +(2π)k/2σk e− +1 +2σ2 |x−¯x|2. +Then +ψ1 ≤ 2 +� +Rk p¯x,σ(x)|x − ¯x|dx = 2(2π)k/2σ−k +� +Rk e− +1 +2σ2 |x−¯x|2|x − ¯x|dx, +which, under the spherical coordinate, yields +ψ1 ≤ 2Sk(2π)k/2σ−k +� ∞ +0 +e− rk +2σ2 r2dr := 1 +2ckσ, +where Sk denotes the unit sphere volume, and the constant ck only depends on k. +Similarly, ψ3 ≤ 1 +2ckσ. Hence +E[|X − Y |] ≤ ckσ + 2|¯x − ¯y|. +□ +In the context of stochastic process, the maximal coupling is defined in terms of the +conditional distributions of the associated discrete-time chains. Let Xh = {Xh +n}, Y h = +{Y h +n } be the time-h sample chains of two solutions of (8), and {(X h +n , Yh +n)} be a coupling +of Xh and Y h. Assume at step (n − 1), (X h +n−1, Yh +n−1) takes the value (x, y) ∈ Rk × Rk, + +LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +9 +and let µx (resp. µy) be the probability distribution of X h +n (resp. Yh +n) conditioning on +X h +n−1 = x (resp. Yh +n−1 = y). Then (X h, Yh) is a maximal coupling at step n if +TV(µx, µy) = 2P[X h +n ̸= Yh +n|X h +n−1 = x, Yh +n−1 = y]. +(17) +In the reflection-maximal coupling scheme, the maximal coupling is implemented if and +only if it is triggered at the previous step, i.e., (X h +n , Yh +n) is a maximal coupling if and +only if |X h +n−1 − Yh +n−1| ≤ d, where d = O(ε +√ +h) is the threshold distance. In the numerical +implementations, at each step, the distance between X h and Yh is checked to determine +whether the maximal coupling is triggered for the next step. If at certain step, the distance +between X h and Yh is greater than d, then the maximal coupling is not triggered and the +reflection coupling is implemented instead at the next step. +The trigger of maximal coupling is the key mechanism, especially under the numerical +scheme, to achieving a successful coupling. This is because under the maximal coupling, a +positive success rate, which is robust against small perturbations, is guaranteed. Without +the maximal coupling, numerical errors may cause the numerical trajectories of X h +n and +Yh +n “miss” each other even if theoretically, they should have been coupled successfully. In +addition, with an appropriate choice of the threshold distance d, the coupling probability +of X h +n and Yh +n has a lower bound independent of h; see Lemma 2.3 below. +Lemma 2.3. Let {(X h +n , Yh +n); n ≥ 0} be a coupling of the time-h sample chain of two +solutions of (8). +Assume for n ≥ 1, (X h +n , Yh +n) is a maximal coupling conditional on +X h +n−1 = x0, Yh +n−1 = y0, where x0, y0 ∈ Rk satisfying |x0 − y0| ≤ d := O(ε +√ +h). Then there +exist a universal constant γ ∈ (0, 1) and a constant Ck > 0 only depending on k such that +for any ε > 0, any h > 0 sufficiently small, the followings hold. +(i) P[|X h +n − Yh +n| > 0|X h +n−1 = x0, Yh +n−1 = y0] ≤ γ, +∀n ≥ 1; +(ii) E[|X h +n − Yh +n| +��X h +n−1 = x0, Yh +n−1 = y0] ≤ Ckε +√ +h, +∀n ≥ 1. +We note that Lemma 2.3 is a conditional version of Proposition 2.2 in the setting of +stochastic processes. For its proof, we need the following estimations (18) - (19). +Intuitively, the time-h sample chain {Xh +n; n ≥ 0} of solution of (8) should be “close +to” the time-h sample chain of its numerical integrator which, recall in the Introduction +section, is denote by { ˆXh +n; n ≥ 0}. In fact, by combining Girsanov Theorem and Pinsker’s +inequality, it is a standard result (see, for instance, Proposition 4.2 in [29]) that the total +variation distance between Xh +n and ˆXh +n conditional on Xh +n−1 = x0, ˆXh +n−1 = x0 is of order +O(h). If denote the probability density function of the distribution of {Xh +n; n ≥ 0} (resp. +{ ˆXh +n; n ≥ 0}) conditional on Xh +n−1 = x0 (resp. ˆXh +n−1 = x0) as ρh +x0 (resp. ˆρh +x0), then +� +Rk |ρh +x0(x) − ˆρh +x0(x)|dx → 0, +as h → 0. +(18) +Under the approximation in (18), the proof of Lemma 2.3 follows the same line of the +proof of Proposition 2.2 by the following approximations +lim +h→0 +� +Rk |x − x0|ρh +x0(x)dx = 0 +(resp. +lim +h→0 +� +Rk |x − x0|ˆρh +x0(x)dx = 0). +(19) +To see (19), let Zx0 +t +be the solution of SDE (8) with initial condition Z0 = x0. By the +Gronwall’s Lemma and standard computations, +E|Zx0 +t +− x0| ≤ (|g(x0)|t + εHk +√ +t)e|g|Lipt, +∀t > 0, + +10 +YAO LI, MOLEI TAO, AND SHIROU WANG +where |g|Lip denotes the Lipschitz constant of g, and Hk denotes the expectation of the +standard normal random variable in Rk. Thus +E|Zx0 +t +− x0| → 0, +as h → 0. +Then (19) is yielded by taking Zx0 +t +as Xh +n and ˆXh +n, respectively. +Now, we are prepared to prove Lemma 2.3. +Proof of Lemma 2.3. (i) Since ˆ +X h +n , ˆYh +n are normal random variables when conditioning on +ˆ +X h +n−1 = x0, ˆYh +n−1 = y0, respectively. Applying Proposition 2.2 (i) with σ = ε +√ +h, +P[| ˆ +X h +n − ˆYh +n| > 0| ˆ +X h +n−1 = x0, ˆYh +n−1 = y0] ≤ c0(ε +√ +h)−1|¯x − ¯y|, +where ¯x = x0 + g(x0)h, ¯y = y0 + g(y0)h. +Since g is Lipschitz continuous with the Lipschitz constant |g|Lip. Then for any h > 0 +sufficiently small, it holds that +|¯x − ¯y| ≤ |x0 − y0| + |g|Lip|x0 − y0|h ≤ O(ε +√ +h). +(20) +By making the coefficient in the term O(ε +√ +h) small enough, there exists a universal +constant γ ∈ (0, 1) such that +P[| ˆ +X h +n − ˆYh +n| > 0| ˆ +X h +n−1 = x0, ˆYh +n−1 = y0] < γ. +By the definition of maximal coupling, +P[|X h +n − Yh +n| > 0|X h +n−1 = x0, Yh +n−1 = y0] = TV(X h +n , Yh +n)/2 +(resp. +P[| ˆ +X h +n − ˆYh +n| > 0| ˆ +X h +n−1 = x0, ˆYh +n−1 = y0] = TV( ˆ +X h +n , ˆYh +n))/2 ). +Then combined with (18) that +TV(X h +n , ˆ +X h +n ) → 0, +TV(Yh +n, ˆYh +n) → 0, +as h → 0, +it holds that for any h > 0 sufficiently small, +P[|X h +n − Yh +n| > 0|X h +n−1 = x0, Yh +n−1 = y0] ≤ γ. +(ii) As in the proof of Proposition 2.2 (ii), by splitting E[|X h +n −Yh +n||X h +n−1 = x0, Yh +n−1 = y0] +into three terms, it similarly holds that +E[|X h +n − Yh +n||X h +n−1 = x0, Yh +n−1 = y0] ≤ ψ1 + ψ2 + ψ3, +where ψi’s are the same as in (16) with p¯x,σ (resp. p¯y,σ) being replaced by ρh +x0 (resp. ρh +y0). +Note that |¯x − x0| = |g(x0)|h → 0 (resp. |¯y − y0| = |g(y0)|h → 0) as h → 0. Then it +follows from (19) that for any δ > 0 and any h > 0 sufficiently small, +ψ1 ≤ ˆψ1 + 2 +� +Rk(ˆρh +¯x(x) − ρh +¯x(x))|x − ¯x|dx ≤ ˆψ1 + δ +(resp. ψ3 ≤ ˆψ3 + 2 +� +Rk(ˆρh +¯x(x) − ρh +¯x(x))|x − ¯x|dx ≤ ˆψ3 + δ) +where ˆψ1 (resp. ˆψ3) is as in Proposition 2.2 with p¯x,σ (resp. p¯y,σ) being ˆρh +x0 (resp. ˆρh +y0). +Applying Proposition 2.2(ii) with σ being ˆσ = ε +√ +h, we have +ˆψ1 ≤ ckε +√ +h, ˆψ3 ≤ ckε +√ +h, + +LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +11 +where ck > 0 is as in Proposition 2.2 only depending on k. Still, ˆψ2 ≤ 2|¯x − ¯y|, which, by +(20), yields ˆψ2 ≤ O(ε +√ +h). Thus, with certain constant Ck > 0 only depending on k, it +holds that +E[|X h +n − Yh +n||X h +n−1 = x0, Yh +n−1 = y0] ≤ Ckε +√ +h. +This completes the proof of Lemma 2.3. +□ +In concluding this subsection, we remark that as proposed for the numerical effi- +ciency, the maximal coupling is of discrete essentials. However, the theoretical results +of the reflection-maximal coupling stated in this paper is for the continuous-time setting. +To ensure a consistency between the discrete-time numerical scheme and its theoretical +continuous-time counterpart, it is assumed that all the values of the processes between +the discrete steps are ignored when the maximal coupling is implemented. In other words, +as long as (Xt, Yt) is a maximal coupling, for all the time that follows before Xt, Yt are +successfully coupled or (Xt, Yt) is switched to the reflection coupling, only the values of +(Xt, Yt) at t = nh matters and it does not make any difference which value (Xt, Yt) take +for all the t’s between the h-discrete steps. +2.3. Multi-well potentials and first hitting time. Let U : D → R be a multi-well +potential satisfying (U3). Henceforth and throughout this paper, all the L local minima +of U are labeled in such a way that +U(x1) < · · · < U(xL). +(21) +In particular, x1 always denotes the unique global minimum of U. Denote +Mi = {x1, ..., xi}, +1 ≤ i ≤ L. +(22) +The following proposition gives an equivalent characterization of the essential barrier +height HU. +Proposition 2.4. Let U be a multi-well potential on D with L local minima xi(1 ≤ i ≤ L). +Then +HU = max2≤i≤L +� +Φ(xi, Mi−1) − U(xi) +� +. +(23) +Proof. Since for each i ∈ {2, ..., L}, x1 ∈ Mi−1. Then +HU ≥ Φ(xi, x1) − U(xi) ≥ Φ(xi, Mi−1) − U(xi), +(24) +and hence +HU ≥ max2≤i≤L +� +Φ(xi, Mi−1) − U(xi) +� +. +(25) +It remains to show that “≥” in (25) can only be “=”. This can be shown by contradic- +tion. Suppose that “=” does not hold, i.e., for each i ∈ {2, ..., L}, +Φ(xi, Mi−1) − U(xi) < HU. +(26) +Then we claim that for each i0 ∈ {2, ..., L}, there exists a continuous path ψ : [0, 1] → D +such that +supt∈[0,1] U(ψ(t)) − U(xi0) < HU, +ψ(0) = xi0, ψ(1) = x1. +(27) +This would yield a contradiction that +HU = max2≤i0≤L{Φ(xi0, x1) − U(xi0)} < HU. +Hence, “=” in (25) must be attained. + +12 +YAO LI, MOLEI TAO, AND SHIROU WANG +To prove the claim, we construct such a continuous path ψ. +By (26), for each i ∈ +{2, ..., L}, there exists a continuous path φi : [0, 1] → D such that +supt∈[0,1] U(φi(t)) − U(xi) < HU, +φi(0) = xi, φi(1) ∈ Mi−1. +(28) +First take the path φi0 and denote xi1 = φi0(1) ∈ Mi0−1. By the definition of Mi, it must +be that 1 ≤ i1 < i0. If i1 = 1, then (27) is proved by letting ψ = φi0; for otherwise, turn +to the path φi1 and obtain a new index 1 ≤ i2 < i1 with φi1(1) = xi2. The procedure is +repeated until ik = 1 for some finite k. Then by gluing all the continuous paths φi0, ..., φik +one by one and up to a rescaling of t, we end up with a new continuous path ψ : [0, 1] → D +satisfying ψ(0) = xi0, ψ(1) = x1. +It remains to verify (27). Since U(xi0) ≥ U(xij), 0 ≤ j ≤ k. Then +supt∈[0,1] U(ψ(t)) − U(xi0) += +max0≤j≤k supt∈[0,1] U(φij(t)) − U(xi0) +≤ +max0≤j≤k supt∈[0,1] +� +U(φij(t)) − U(xij) +� +< HU, +where the last inequality follows from (28). +□ +Remark 2.5. In fact, (23) still holds if the maximum is taken on certain subset of +{2, ..., L}. Let I collect all the indexes at which the maximum in (7) is attained, i.e., +I = {2 ≤ i ≤ L : Φ(xi, x1) − U(x1) = HU}. +(29) +Then we have the following stronger version of Proposition 2.4 that +HU = maxi∈I{Φ(xi, Mi−1) − U(xi)}. +(30) +The proof of (30) follows the same line of the proof of Proposition 2.4. The “≥” directly +follows from (24) and we only need to show that “≥” must be an equality. Still, this +can be proved by contradiction. Suppose for each i ∈ I, Φ(xi, Mi−1) − U(xi) < HU. +Then corresponding to each i ∈ I, there exists a continuous path φi(t) connecting xi with +certain xji ∈ Mi−1 such that supt U(φi(t)) − U(xi) < HU. Following the constructing +procedure in the proof of Proposition 2.4, starting from any xi0 with i0 ∈ I, we can +construct a continuous path ψ by gluing certain φi’s piece by piece, passing a sequence of +local minima xi1, ..., xik−1 with ij ∈ I and ending with a local minimum xik with ik /∈ I. +If ik = 1, then ψ(t) is a continuous path connecting xi0 and x1 such that +supt U(ψ(t)) − U(xik) < HU. +(31) +This contradicts with i0 ∈ I. If ik ̸= 1, we can construct a new path ˜ψ satisfying (31) by +choosing a continuous path ˜φ(t) connecting xik with x1 satisfying supt U(˜φ(t)) − U(xi0) < +HU (such a path ˜φ exists because ik /∈ I) and then glue ψ and ˜φ together. Still, the +contradiction is yielded. +Let {Zt; t ≥ 0} be a solution of (2), and A ⊆ D be a subset. Denote the first hitting +time of Zt to the set A as +κ{Zt}(A) = inf{t > 0 : Zt ∈ A}. +(32) +The large deviation theory tells us that the first hitting time from a local minimum to an +appropriate subset A can be characterized by the related barrier height. + +LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +13 +Proposition 2.6. (Exponential law of first hitting time [4, 5]2) Let Z = {Zt; t ≥ 0} +be a solution of (2) with initial condition Z0 = xi, where 2 ≤ i ≤ L. Let A ⊆ D be a +closed subset such that ∪i−1 +ℓ=1Bε(xℓ) ⊂ A and dist(z∗(xi, Mi−1), A) > δ for certain δ > 0 +independent of ε. Then there exists ε0 > 0 such that for any ε ∈ (0, ε0) and any t > 0, +P[κ{Zt}(A) > t] ≃ exp +� +− C0e−2(Φ(xi,Mi−1)−U(xi))/ε2t +� +, +(33) +where Mi’s are defined in (22) and C0 > 0 is a constant independent of t and ε. +In the multi-well setting, the essential barrier height HU plays a similar role in the global +sense that it characterizes the first hitting time to the basin of the global minimum from +any of the local ones. +Lemma 2.7. Let {Zt; t ≥ 0} be a solution of (8). Then for any t > 0 and any ε > 0 +sufficiently small, +P[κ{Zt}(B1) > t] ≲ exp{−C0e−2HU/ε2t}, +(34) +where C0 is independent of ε and t. +Moreover, if Z0 is fully supported, then “≲” in (34) becomes “≃”. +Proof. Note that when Z0 ∈ B1, (34) automatically holds. +In the following, we only +consider the case Z0 ∈ Bi for 2 ≤ i ≤ L. For each 1 ≤ i ≤ L, let ˜Bi ⊆ Bi be an open +neighborhood of xi such that Bε(xi) ⊂ ˜Bi and dist( ˜Bi, ∂Bi) > δ holds for certain δ > 0 +and any i ∈ {1, ..., L}. Denote Di = �i +ℓ=1 ˜Bℓ, and define the stopping times +Ti = inf{t > 0 : Zt ∈ Di}, +1 ≤ i ≤ L, +(35) +and set TL+1 = 0, DL+1 = D. Note that each Ti is the infimum time at which Zt enters the +neighborhood of a new lower local minimum, where by “new lower” we mean that the local +minimum has a potential value that is lower than all the local minima the neighborhoods +of which have been passed by Zt. Plainly, 0 = TL+1 ≤ TL ≤ · · · ≤ T1 < ∞, and +κ{Zt}(B1) ≤ κ{Zt}( ˜B1) = T1. +Denote ∆Ti = Ti − Ti+1, 1 ≤ i ≤ L. Then +T1 = +L +� +i=1 +∆Ti. +We claim that for each i ∈ {1, ..., L} satisfying ∆Ti > 0, it must be that ZTi+1 ∈ ˜Bi+1, In +fact, suppose ∆Ti > 0 and ZTi+1 /∈ ˜Bi+1. Then ZTi+1 ∈ Di+1\ ˜Bi+1 = Di which, by (35), +yields Ti ≤ Ti+1. Since it generally holds that Ti+1 ≤ Ti. This yields a contradiction that +∆Ti = Ti − Ti+1 = 0. +By Proposition 2.6, for any 1 ≤ i ≤ L − 1, +supx∈ ˜Bi+1 Px[κ{Zt}(Di) > t] ≃ exp{−C0e−2(Φ(xi+1,Mi)−U(xi+1))/ε2t}, +which, by (23), yields +supx∈ ˜Bi+1 Px[κ{Zt}(Di) > t] ≲ exp{−C0e−2HU/ε2t}. +2It is well-known that the first hitting time is asymptotically exponentially distributed according to the +large deviation theory [8, 13]. It is relatively recent that the exponential tail is precisely estimated up to +a multiplicative error by techniques from the potential theory in [4, 5]. + +14 +YAO LI, MOLEI TAO, AND SHIROU WANG +Note that +κ{Zt}(Di) ◦ θTi+1 = ∆Ti, +1 ≤ i ≤ L − 1. +By the Markov property, +P[∆Ti > t|ZTi+1 ∈ ˜Bi+1] ≲ PZTi+1[κ{Zt}(Di) > t] ≲ exp{−C0e−2HU/ε2t}. +Hence +P[T1 > t] +≤ +� +1≤i≤L,∆Ti>0 P[∆Ti > t/L] += +� +1≤i≤L P[∆Ti > t/L|ZTi+1 ∈ ˜Bi+1] ≲ exp{−C0e−2HU/ε2t}, +where C0 denotes a constant independent of ε and t. +For the other side of (34), by Proposition 2.4, there exists i0 ∈ {2, ..., L} such that +HU = Φ(xi0, Mi0−1) − U(xi0). +Since Z0 is fully supported, we have for any δ > 0, P[Z0 ∈ Bδ(xi0)] > 0. Note that if +Z0 ∈ Bδ(xi0), 0 = TL = · · · = Ti0+1 < Ti0 ≤ T1. Hence, by Proposition 2.6, +P[T1 > t] ≥ P[∆Ti0 > t, Z0 ∈ Bδ(xi0)] ≃ exp{−C0e−2HU/ε2t}. +□ +The result in Lemma 2.7 can be strengthened to certain set containing B1. More specif- +ically, let +J = {1 ≤ j ≤ L : U(xj) < U(xi) ∀ i ∈ I}, +(36) +where the index set I is defined in (29). Note that J collects the indexes of all the local +minima whose potential values are smaller than U(xi) for any i ∈ I. Plainly, 1 ∈ J . We +claim that (34) holds for B1 = ∪j∈J Bj, i.e., for any ε > 0 sufficiently small and any t > 0, +P[κ{Zt}(B1) > t] ≃ exp{−C0e−2HU/ε2t}. +(37) +The estimation (37) can be proved following the same line of that of Lemma 2.7. The only +modification is in obtaining “≳”, we apply (30) (i.e., the stronger version of Proposition +2.4) instead of (23). +2.4. An upper bound of probability generating function. In this subsection, we +introduce a function that plays a key role in the estimation of exponential tails of the +coupling time distributions. +Given C0 > 0, λ0 > 1, define +g(λ; C0, λ0) = λ + C0 +∞ +� +n=1 +(λn+1 − λn)λ−n +0 , +λ ∈ R. +(38) +Then g(λ; C0, λ0) < ∞ holds for any λ ∈ (1, λ0). Plainly, +g(λ; C0, λ0) → 1, +as λ → 1. +(39) +In Section 3, a quantitative characterization of the limit in (39) will be needed. For +ρ > 1, λ0 > 1, C0 > 0, define +β(ρ; C0, λ0) = min +�√ρ − 1 +λ0 − 1 , +√ρ − 1 +C0 + √ρ − 1 +� +. +(40) +Obviously, β ∈ (0, 1). + +LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +15 +Proposition 2.8. Given ρ > 1, λ0 > 1 and C0 > 0. Then for any λ ∈ (1, 1 + β(λ0 − 1)) +where β = β(ρ; C0, λ0) is defined in (40), +g(λ; C0, λ0) < ρ. +Proof. Write λ = 1 + ˜β(λ0 − 1) with ˜β ∈ (0, 1), we have +g(λ; C0, λ0) += +λ + C0(λ − 1) +∞ +� +n=1 +(λ/λ0)n += +λ(1 + C0 ˜β/(1 − ˜β)) ≤ (1 + ˜β)(1 + C0 ˜β/(1 − ˜β)). +To obtain g(λ; C0, λ0) < ρ, we only need +(1 + ˜β) < √ρ, +1 + C0 +˜β +1 − ˜β +< √ρ. +This tells how (40) is defined. +□ +The function g in (38) is motivated by the probability generating function in probability. +Actually, if a random variable T admits an exponentially decaying tail, then E[λT ], i.e., +the generating function of T is non-negative integer valued, is upper bounded by g. This +elementary fact is often used throughout the paper for estimating the exponential tails of +coupling time distributions. +Proposition 2.9. Let T be a random variable taking values in R>0. Assume that for any +t > 0, P[T > t] ≤ C0λ−t +0 . Then for any λ ∈ (1, λ0), it holds that +E[λT ] ≤ g(λ; C0, λ0) < ∞. +Proof. Note that +E[λT ] = λP[T > 0] + +∞ +� +n=1 +(λn+1 − λn)P[T > n] ≤ λ + +∞ +� +n=1 +(λn+1 − λn)P[τ > n]. +□ +3. Single-well potential and proof of Theorem 1.1 +Throughout this section, U is assumed to be a strongly convex single-well potential +satisfying (U1). Let {(Xt, Yt); t ≥ 0} be an h-reflection-maximal coupling of two solutions +of (2). Recall that there is a threshold d = O(ε +√ +h) at which (Xt, Yt) is switched between +the reflection and maximal couplings. Throughout this paper, the threshold d is set as +2ε +√ +h. +Define +τ (1) +h += inf +� +t ≥ 0 : |Xt − Yt| ∈ (0, 2ε +√ +h], and for certain s ∈ (0, t), |Xs − Ys| > 2ε +√ +h +� +, +and let τ (1) +h += ∞ if the set is empty. We see that τ (1) +h +captures the infimum time when +|Xt − Yt| attains the threshold d (from far away) at which (Xt, Yt) is switched from the +reflection coupling to the maximal coupling. +It may happen that the distance |Xt − Yt| has always been not exceeding the threshold +d before being successfully coupled, and hence τ (1) +h += ∞. Let +τh = τ (1) +h +∧ τc. + +16 +YAO LI, MOLEI TAO, AND SHIROU WANG +Then τh < ∞ holds almost surely. As will be seen later, the coupling time τc is of a finite +iterations of τh. +3.1. Estimation of τh. In this subsection, τh is estimated under different initial condi- +tions of {(Xt, Yt)} in terms of |X0 − Y0| > 2ε +√ +h and |X0 − Y0| ≤ 2ε +√ +h, respectively. +Note that when |X0−Y0| > 2ε +√ +h, (Xt, Yt) has always been a reflection coupling until τh +at which (Xt, Yt) is switched to the maximal coupling. Then Proposition 2.1 immediately +yields the following. +Lemma 3.1. Assume |X0 − Y0| = r0 > 2ε +√ +h. Then τh = τ (1) +h +holds P-a.s. such that +P[τh > t] ≤ c0r0 +2ε e−m0t, +∀t > 0, +(41) +where c0 > 0 is as in Proposition 2.1. Consequently, by Proposition 2.9, for any λ ∈ +(1, em0), +E[λτh] ≤ g(λ; c0r0/2ε, em0). +(42) +Remark 3.2. Note that the estimation in (41) is for the continuous-time process instead +of its time-h sample chain which the numerical scheme truly approximates. Let τ 0 +h (resp. +τ h +h ) be the first passage time of the coupling process (Xt, Yt) (resp. its time-h sample chain +(X h +n , Yh +n)) to the set {(x, y) ∈ Rk ×Rk : |x−y| < 2ε +√ +h}. It is easy to see that τ h +h ≥ τ 0 +h, and +it is intuitive that their difference, which is usually difficult to be theoretically estimated, +should approach to zero as h vanishes, i.e., +limh→0(τ h +h − τ 0 +h) = 0, +P-a.s. +(43) +Throughout this section, (43) is always assumed and will be numerically verified in Section +5 for the example of symmetric quadratic potential functions. Hence, the estimation (41) +applies to the time-h sample chain (X h +n , Yh +n) (slightly enlarge c0 if necessary) whenever h +is sufficiently small. +It becomes more complicated when |X0−Y0| < 2ε +√ +h since the coupling method between +Xt and Yt may change during (0, τh). More precisely, there exists n > 0 such that (Xih, Yih) +is a maximal coupling for any integer 0 ≤ i < n, and for i = n, it holds either that +Xnh = Ynh, i.e., Xt, Yt are coupled successfully, or |Xnh −Ynh| > 2ε +√ +h. In the former case, +τc = τh, while for the latter one, (Xt, Yt) is a reflection coupling ever since until it again +holds that |Xt − Yt| < 2ε +√ +h. +The following proposition provides the estimation of τh under the initial condition |X0− +Y0| < 2ε +√ +h. +Lemma 3.3. Assume |X0 − Y0| ≤ 2ε +√ +h. Then there exists constants h0 > 0, C0 > 0 such +that for any h ∈ (0, h0), +P[τh > t] ≤ C0 +√ +hλ−n +h , +∀t > 0, +where n = ⌊t/h⌋. Consequently, by Proposition 2.9, for any λ ∈ (1, em0), +E[λτh] +≤ +g(λ; C0 +√ +h, em0). +(44) + +LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +17 +Proof. Recall in the proof of Proposition 2.1, we denote Rt = |Xt−Yt|/2ε. Then {Rt; t ≥ 0} +is a one-dimensional stochastic process on R≥0 induced by the coupling {(Xt, Yt); t ≥ 0}. +By the above analysis on the coupling behavior between Xt and Yt on (0, τh), +P[τh > t] +≤ +P[τh > nh] += +n−1 +� +j=1 +P[Rih ∈ (0, +√ +h], 0 ≤ i ≤ j − 1] +· +� +P[Rjh > +√ +h|R(j−1)h ∈ (0, +√ +h]] · P[τ (1) +h +◦ θjh > t − jh|Rjh > +√ +h] +� ++ +P[Rih ∈ (0, +√ +h], 0 ≤ i ≤ n − 1] · P[Rnh > 0|R(n−1)h ∈ (0, +√ +h]], +where θ is the usual shift operator. +Note that +P[Rih ∈ (0, +√ +h], 0 ≤ i ≤ j − 1] = +� +1 +j = 1 +�j−1 +i=1 P[Rih ∈ (0, +√ +h]|R(i−1)h ∈ (0, +√ +h]], +j > 1. +Since (Xih, Yih) is a maximal coupling whenever |X(i−1)h − Y(i−1)h| ≤ 2ε +√ +h. It follows +from Lemma 2.3 (i) that for certain γ ∈ (0, 1), P[Rih > 0|R(i−1)h ∈ (0, +√ +h]] < γ. Hence, +P[Rih ∈ (0, +√ +h], 0 ≤ i ≤ j − 1] ≤ γj−1, +and +P[Rih ∈ (0, +√ +h], 0 ≤ i ≤ n − 1] · P[Rnh > 0|R(n−1)h ∈ (0, +√ +h]] ≤ γn. +Therefore, +P[τh > nh] ≤ γn + +n−1 +� +j=1 +γj−1� +P[Rjh > +√ +h|R(j−1)h ∈ (0, +√ +h]] · P[τh ◦ θjh > t − jh|Rjh > +√ +h] +� +. +Now, it only remains to estimate +P[Rjh > +√ +h|R(j−1)h ∈ (0, +√ +h]] · P[τh ◦ θjh > t − jh|Rjh > +√ +h] +(45) +for 1 ≤ j ≤ n − 1. +By the Markov property, +(45) += +� ∞ +√ +h +P[τh ◦ θjh > t − jh|Rjh = s]P[Rjh = ds|R(j−1)h ∈ (0, +√ +h]] += +� ∞ +√ +h +P[τh > t − jh|R0 = s]P[Rjh = ds|R(j−1)h ∈ (0, +√ +h]]. +Then it follows from Lemma 3.1 that +(45) +≤ +c0e−m0(t−jh) +� ∞ +√ +h +rP[Rjh = dr|R(j−1)h ∈ (0, +√ +h]] +≤ +c0e−m0h(n−j)E[Rjh|R(j−1)h ∈ (0, +√ +h]]. +By Lemma 2.3 (ii), +E[Rjh|R(j−1)h ∈ (0, +√ +h]] ≤ 1 +2c2 +√ +h. +Thus, for certain constant C0 > 0 independent of h and ε, +(45) ≤ C0 +√ +he−m0h(n−j). + +18 +YAO LI, MOLEI TAO, AND SHIROU WANG +Therefore, +P[τh > t] ≤ +n−1 +� +j=1 +C0 +√ +hγj−1e−m0h(n−j) + γn = C0 +√ +he−m0h(n−1) +n−1 +� +j=1 +(γem0h)j−1 + γn +which, by letting h > 0 sufficiently small and enlarging C0 if necessary, yields +P[τh > t] ≤ C0 +√ +he−m0t. +□ +Combining Lemma 3.1 and Lemma 3.3, we have the following. +Lemma 3.4. Assume E[|X0 − Y0|] < ∞. Then for any h ∈ (0, h0) where h0 > 0 is as in +Lemma 3.3, for any λ ∈ (1, λh), it holds that +E[λτh] < ∞. +Proof. Recall the one-dimensional stochastic process Rt = |Xt − Yt|/(2ε), t ≥ 0. Then, if +denote µ as the initial distribution of {Rt; t ≥ 0}, we have +E[λτh] += +� √ +h +0 +Er[λτh]µ(dr) + +� ∞ +√ +h +Er[λτh]µ(dr) +≤ +g(λ; C0 +√ +h, λh) +√ +h + g(λ; c0 +� ∞ +√ +h +rµ(dr), λh) +≤ +g(λ; C0 +√ +h, λh) +√ +h + g(λ; c0E[R0], λh), +where Er[·] denotes the expectation with respect to the initial condition R0 = r. +Since E[R0] = E[|X0 − Y0|]/2ε < ∞, the lemma is proved. +□ +3.2. Iteration of τh and coupling times. The coupling time τc is in fact certain itera- +tion of τh. To see this, define +τ 0 +h = 0, +τ k +h = τ k−1 +h ++ τh ◦ θτ k−1 +h +, k ≥ 1, +and let +η = inf +� +k ≥ 1 : Xτ k +h = Yτ k +h +� +. +By the definition of τh, the following proposition immediately follows. +Proposition 3.5. Given any h > 0, any k ≥ 1. The followings hold. +(i) |Xτ k +h − Yτ k +h | = 2ε +√ +h or 0, where Xτ k +h = Yτ k +h if and only if k ≥ η. +(ii) If k > 1, then +P[|Xτ k +h − Yτ k +h | > 0|Fτ k−1 +h +] < γ. +where γ is as in Lemma 2.3. +By Proposition 3.5 (i), +τc = τ η +h, +P-a.s. +Thus, the estimation of τc is reduced to the estimation of τ η +h. +Theorem 3.6. Assume E[|X0 − Y0|] < ∞. Then for any δ > 0, there exists h0 > 0 such +that for any h ∈ (0, h0) and any λ ∈ (1, em0−δ), it holds that +E[λτ η +h ] < ∞. + +LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +19 +Proof. The idea of the proof follows Lemma 2.9 in [28]. Note that +E[λτ η +h ] +≤ +∞ +� +k=1 +E[λτ k +h Iη≥k] +(46) += +E[λτhIη≥1] + +∞ +� +k=2 +E[Iη≥kλτ k−1 +h +E[λτh◦θτk−1 +h +|Fτ k−1 +h +]], +where the last equality is by the observation that λτ k−1 +h +∈ Fτ k−1 +h +, {η ≥ k} ∈ Fτ k−1 +h +. +Still, we use the notation Rt = |Xt −Yt|/2ε, and Er denotes the expectation with initial +condition R0 = r. By Proposition 3.5 (i), Rτ k−1 +h += +√ +h whenever 2 ≤ k < η. By (44) and +strong Markov property, +E[λτh◦θτk−1 +h +|Fτ k−1 +h +] ≤ E√ +h[λτh] ≤ g(λ; C0 +√ +h, em0), +∀k ≥ 1, +where C0 > 0 is as in Lemma 3.3. Thus, +E[λτ η +h ] +≤ +E[λτhIη≥1] + g(λ; C0 +√ +h, em0) +∞ +� +k=2 +E[Iη≥kλτ k−1 +h +]. +(47) +Now we estimate E[Iη≥kλτ k−1 +h +] for k ≥ 2. First, by writing I{η≥k} = I{η≥k−1}I{Rτk−1 +h +>0}, +we obtain +E[I{η≥k}λτ k−1 +h +] += +E[I{η≥k−1}λτ k−2 +h +E[I{Rτk−1 +h +>0}λτh◦θτk−2 +h +|Fτ k−2 +h +]] += +E[I{η≥k−1}λτ k−2 +h +ERτk−2 +h +[I{Rτh>0}λτh]]. +Since Rτ k−2 +h +∈ [0, +√ +h] for k > 2, by the strong Markov property, +ERτk−2 +h +[I{Rτh>0}λτh] +≤ +� +E[λτh], +k = 2, +E√ +h[I{Rτh>0}λτh], +k > 2. +Note that by H¨older inequality, for any p ∈ (0, 1), +E[I{Rτh>0}λτh] ≤ (E[I{Rτh>0}])1−p · (E[λτh/p])p = (P[Rτh > 0])1−p · (E[λτh/p])p. +Then it follows from Proposition 3.5 (ii) and Lemma 3.3 that +E√ +h[I{Rτh>0}λτh/p] +≤ +γ1−pg(λ1/p; C0 +√ +h, em0)p. +Therefore, +E[I{η≥k}λτ k−1 +h +] +≤ +� +E[Iη≥1λτh], +k = 2 +γ1−pg(λ1/p; C0 +√ +h, em0)pE[I{η≥k−1}λτ k−2 +h +], +k > 2. +By induction, we have for k ≥ 2, +E[I{η≥k}λτ k−1 +h +] +≤ +γ(1−p)(k−2)g(λ1/p; C0 +√ +h, em0)p(k−2)E[λτh]. +Hence, (47) yields +E[λτ η +h ] +≤ +E[λτh] +� +1 + g(λ; C0 +√ +h, em0) +∞ +� +k=0 +� +γ1−pg(λ1/p; C0 +√ +h, em0)p�k� + +20 +YAO LI, MOLEI TAO, AND SHIROU WANG +By Lemma 3.4, E[λτh] < ∞. Thus, to guarantee E[λτ η +h ] < ∞, it only requires +g(λ1/p; C0 +√ +h, em0) < γ−(1−p)/p +(48) +By Proposition 2.8, (48) holds for any λ > 1 such that λ1/p ∈ (1, 1+βh(em0 −1)), where +by choosing h > 0 sufficiently small, +βh = min +�γ−(1−p)/p − 1 +em0 − 1 +, +γ−(1−p)/p − 1 +C0 +√ +h + γ−(1−p)/p − 1 +� += +γ−(1−p)/p − 1 +C0 +√ +h + γ−(1−p)/p − 1 +. +Note that βh → 1 as h → 0. Since p can be arbitrarily close to 1. Then by choosing h +sufficiently small, we have +E[λτ η +h ] < ∞ +holds for any λ ∈ (1, em0−δ) with δ > 0 being arbitrarily small. +□ +Now, the proof of Theorem 1.1 is straightforward. +Proof of Theorem 1.1: Note that for any λ ∈ (1, ∞) satisfying E[λτ η +h ] < ∞, we have for +any t > 0, +P[τ η +h > t] ≤ E[λτ η +h ]λ−t. +Hence, by Theorem 3.6, for any λ ∈ (1, em0−δ) and any δ > 0, +lim sup +t→∞ +1 +t log P[τ η +h > t] ≤ −m0 + m0δ. +Theorem 1.1 is proved by taking δ as m0δ. +4. Multi-well potentials and proof of Theorem 1.2 and Theorem 1.3 +In this section, potential functions with multiple wells are considered. +The case of +double-well potential is first studied in Section 4.1-4.3, and following the same idea, po- +tential functions with more wells are investigated in Section 4.4. +4.1. Key stopping times for double-well potentials. In this subsection, several key +stopping times for the estimate of τc are introduced for the double-well situation. +Let U be a double-well potential satisfying (U2) with two basins B1 and B2, and +{(Xt, Yt); t ≥ 0} be a coupling of two solutions of (2). Denote +τ (1) +ε += inf +� +t > h : (Xt, Yt) ∈ B1 × B1 or B2 × B2 +� +the infimum time when Xt and Yt lie in the same basin3 of U. To avoid the possible +repeated boundary-crossing in an initial infinitesimal time interval when Xt (or Yt) starts +from the basin boundaries, let τ (1) +ε +be measured after some small positive time which we +choose to be the step size h to make it compatible with the numerical simulations. +Plainly, τ (1) +ε +< ∞ for P-a.s. If initially Xt and Yt already belong to the same basin, then +τ (1) +ϵ += h with probability close to 1. Now, let Xt and Yt be initially lie in the different +basins, and assume, without loss of generality, that Yt starts from B1. Then +τ (1) +ε += κ{Xt}(B1) ∧ κ{Yt}(B2), +where recall that κ{Xt}(·) and κ{Yt}(·) are the first hitting times defined in (32). +3The subscript “ε” is used for the emphasis that in the multi-well setting, the coupling time is essentially +determined by the “basin-switching” behavior of the processes in which the noise magnitude ε plays the +fundamental role. + +LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +21 +Denote +λε = exp{C0e−2HU/ε2}, +(49) +where C0 > 0 is certain constant independent of ε. By Lemma 2.7, +P[τ (1) +ε +> t] ≤ P[κ{Xt}(B1) > t] ≲ λ−t +ε , +∀t > 0. +(50) +The reverse of (50) also holds under appropriate conditions. For a multi-well potential +U, recall the index sets I and J defined in (29) and (36), respectively. It is easy to see +that +Φ(xj, xi) − U(xj) > Φ(xi, xj) − U(xi) = HU, ∀ i ∈ I, j ∈ J . +(51) +Denote +B1 = +� +j∈J Bj, +B2 = +� +i∈I Bi, +where B1 has been defined at the end of Section 2.3. Plainly, B1 ⊆ B1. We note that B2 +collects all the “far-away” basins from B1, where by “far-away” we mean that any path +starting from such a basin has to overcome the largest barrier height (i.e., the essential +barrier height HU) to enter B1. +In this paper, for the multi-well situation (which, of course, includes the double-well +case), the following (H1) is always assumed and will be numerically verified in Section 5. +(H1) There exist δ0 > 0, γ0 > 0 such that, if {(Xt, Yt)} is a reflection-maximal coupling +of two solutions of (2) satisfying X0 ∈ ∪i∈IBδ0(xi), Y0 ∈ ∪j∈J Bδ0(xj), then for any t > 0 +and any ε > 0 sufficiently small, +P +� +Ys ∈ B1 for all s ∈ [0, t] +��κ{Xt}(B1) > t +� +> γ0. +(52) +(H1) states that when Xt and Yt start from the “bottom” of the basins from B2 and +B1 respectively, the probability of Yt not leaving B1 conditioning that Xt has not entered +B1 yet is uniformly positive and independent of ε and t. This is intuitive because by (51), +processes starting from the local minima with indexes from I have to overcome a higher +barrier to exist B1 than the vice versa. +In the double-well setting, Bi is simply Bi for i = 1, 2, and (52) is reduced to +P[κ{Yt}(B2) > t|κ{Xt}(B1) > t] > γ0. +(53) +Then under condition (H1), if X0 ∈ Bδ0(x1) and Y0 ∈ Bδ0(x2), the reverse of (50) holds +P[τ (1) +ε +> t] +≥ +P[κ{Xt}(B1) > t, κ{Yt}(B2) > t] += +P[κ{Yt}(B2) > t|κ{Xt}(B1) > t] · P[κ{Xt}(B1) > t] +≥ +γ0 · P[κ{Xt}(B1) > t] ≃ λ−t +ε , +(54) +where the last “≃” holds by letting δ0 > 0 be sufficiently small. +In a rather contrary sense to τ (1) +ε , define another stopping time +τ (2) +ε += +inf +� +t > h : (Xt, Yt) ∈ B1 × B2 or B2 × B1, and for certain s ∈ (0, t), +(Xs, Ys) ∈ B1 × B1 or B2 × B2 +� +, + +22 +YAO LI, MOLEI TAO, AND SHIROU WANG +and let τ (2) +ε += ∞ if the set is empty. Then τ (2) +ε +captures the infimum time when Xt, Yt +are separated (again) by the two different basins, where “again” applies when X0, Y0 are +already belong to the different basins. +Denote +τε = τ (2) +ε +∧ τc. +It is not hard to see that τε = τ (2) +ε +< τc when Xt, Yt are not coupled when staying within +the same basin, and τε = τc < τ (2) +ε += ∞ for otherwise. As will be seen in Section 4.3, the +coupling time τc is a finite iteration of τε for P-a.s. +4.2. Estimation of τε. In the multi-well situation, the following (H2) characterizes the +local coupling behaviors when both Xt and Yt are lying in the same basin. +(H2) Let {(Xt, Yt)} be a reflection-maximal coupling of two solutions of (2) such that +(X0, Y0) ∈ � +1≤i≤L Bi × Bi. The followings hold. +(i) There exists γ1 < 1 such that +P[Xτε ̸= Yτε] < γ1. +(ii) There exist r1, r2 > 0 such that for any t > 0 and any ε > 0 sufficiently small, +P[τε > t|Xτε = Yτε] ≲ e−r1t, +P[τε > t|Xτε ̸= Yτε] ≲ e−r2t. +We note that (H2) is intuitive that (H2)(i) suggests a positive probability for a suc- +cessful coupling when Xt, Yt belong to the same basin. The first inequality in (H2)(ii) is +a “conditional” version of Theorem 1.1 conditioning that Xt and Yt are being successfully +coupled within the same basin, and the second inequality states that for otherwise, the +exponential tail of τε remains largely unchanged (although the probability P[Xτε ̸= Yτε] +drops dramatically as ε decreases). +In this paper, for the multi-well situation, (H2) is always assumed and will be numeri- +cally verified in Section 5. We see that when X0 and Y0 belong to the same basin, the two +inequalities in (H2)(ii) together yield the following estimate of τε. +Proposition 4.1. Let {(Xt, Yt); t ≥ 0} be a reflection-maximal coupling of two solutions +of (2) such that (X0, Y0) ∈ B1 × B1 or B2 × B2. Then there exists r0 > 0 such that for +any t > 0 and any ε > 0 sufficiently small, it holds that +P[τε > t] ≲ e−r0t. +In the estimate of τε, it becomes more complicated when X0 and Y0 belong to the +different basins. Typically in this case, the coupling process {(Xt, Yt)} should experience +two stages during (0, τε). In Stage I, Xt, Yt are lying in the different basins until one of +them, Xt or Yt, jumps out of the basin it initially belongs to and enters the other basin +so that Xt, Yt are staying in the same basin; then it comes to the Stage II that Xt and Yt +are lying in the same basin for another period of time until they are either successfully +coupled, or not coupled and one of them, Xt or Yt, jumps out of the same basin again. +Hence, we write +τε = τε ◦ θτ (1) +ε ++ τ (1) +ε , +P- a.s., +(55) +where θ is the usual shift operator. +We note that Stage I and Stage II illustrate different time scales: Stage I has the +slow time-scale which typically lasts for an exponentially long period of time with the + +LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +23 +tail exponent being exponentially small; while Stage II has the fast time-scale for which, +according to Proposition 4.1, the tail exponent of the distribution is uniformly bounded +away from zero. +According to above analysis, we have the following estimate of τε when Xt, Yt initially +belong to the different basins. +Lemma 4.2. Let {(Xt, Yt); t ≥ 0} be a reflection-maximal coupling of two solutions of +(2) such that (X0, Y0) ∈ B1 × B2 or B2 × B1. Then there exists C1 > 0 such that for any +t > 0 and any ε > 0 sufficiently small, +P[τε > t] ≤ C1λ−t +ε . +Consequently, by Proposition 2.9, for any λ ∈ (1, λε), +E[λτε] ≤ g(λ; C1, λε) < ∞. +(56) +Proof. According to (55), +P[τε > t] = +� t +0 +P[τ (1) +ε += s]P[τε > t|τ (1) +ε += s]ds. +(57) +Note that for any given δ > 0, P[τ (1) +ε +< δ] → 0 as ε → 0. Hence, for any δ′, δ > 0 +sufficiently small, +(57) +≤ +� t +δ +P[τ (1) +ε +> s − δ]P[τε > t|τ (1) +ε += s]ds + δ′ +≤ +� t +δ +P[τ (1) +ε +> s − δ]P[τε ◦ θs > t − s|τ (1) +ε += s]ds + δ′ +By equation (50), there exists C2 > 0 such that +P[τ (1) +ε +> s − δ] ≤ C2λ−(s−δ) +ε +. +Since Xt, Yt belong to the same basin for t = τ (1) +ε . By Proposition 4.1, there exists C3 > 0 +such that +P[τε ◦ θs > t − s|τ (1) +ε += s] ≤ C3e−r0(t−s). +Therefore, +(57) ≤ C2C3λδ +ε +� t +0 +λ−s +ε e−r0(t−s)ds + δ′ ≤ 2C2C3λ−t +ε +� t +0 +(λεe−r0)t−sds + δ′, +where the last inequality is by the arbitrarily small of δ. +Since λεe−r0 < 1 for ε > 0 sufficiently small, we have, with certain constant C1 > 0 +independent of ε and t, +(57) ≤ C1λ−t +ε + δ′ +The lemma is proved since δ′ can be arbitrarily small. +□ +By noting that 1 < λε < er0 whenever ε sufficiently small, Proposition 4.1 and Lemma +4.2 directly yield the following. +Proposition 4.3. Let {(Xt, Yt)} be a reflection-maximal coupling of two solutions of (2) +such that (X0, Y0) is fully supported. Then for any ε > 0 sufficiently small, any λ ∈ (1, λε), +it holds that +E[λτε] < ∞. + +24 +YAO LI, MOLEI TAO, AND SHIROU WANG +4.3. Proof of Theorem 1.2. This subsection proves Theorem 1.2. Similar to the proof +of Theorem 1.1, a sequence of random times are inductively defined +τ 0 +ε = 0, τ k +ε = τ k−1 +ε ++ τε ◦ θτ k−1 +ε +, +k ≥ 1. +(58) +where θ is the usual shift operator. Note that by definition of τε, for each k ≥ 1, either +Xτ k +ε = Yτ k +ε , or Xτ k +ε and Yτ k +ε belong to the different basins. Let +η = inf{k ≥ 1 : Xτ k +ε = Yτ k +ε }. +(59) +The following Proposition 4.4 and Theorem 4.5 are respectively the analogues of Propo- +sition 3.5 and Theorem 3.6 in the double-well setting. +Proposition 4.4. For any ε > 0 sufficiently small, the followings hold. +(i) Xτ k +ε = Yτ k +ε if and only if k ≥ η; +(ii) For any ε > 0 sufficiently small, it holds that +P[Xτ k +ε ̸= Yτ k +ε |Fτ k−1 +ε +] < γ1, +∀k ≥ 1, +where γ1 < 1 is as in (H2)(i). +Plainly, Proposition 4.4 (i) holds and further yields +τc = τ η +ε , +P-a.s. +By the strong Markov property, Proposition 4.4 (ii) directly follows from (H2)(i). +Theorem 4.5. Let {(Xt, Yt); t ≥ 0} be a reflection-maximal coupling of two solutions of +(2) such that (X0, Y0) is fully supported. Then for any ε > 0 sufficiently small and any +λ ∈ (1, λε), it holds that +E[λτ η +ε ] < ∞. +Proof. The proof follows the same line of the proof of Theorem 3.6. With I{Rτkε >0} in the +proof of Theorem 3.6 replaced by I{Xτkε ̸=Yτkε }, we have +E[λτ η +ε ] ≤ E[λτε] +� +1 + g(λ; C1, λε) +∞ +� +k=0 +� +γ1−pg(λ1/p; C1, λε)p�k� +, +where C1 is as in Lemma 4.2, and p can be any number in (0, 1). Thus, E[λτ η +ε ] < ∞ if +g(λ1/p; C1, λε) < γ−(1−p)/p. +(60) +By Proposition 2.9, (60) holds for any λ1/p ∈ (1, (1 + β(λε − 1)) where +β = +γ−(1−p)/2p − 1 +C4 + γ−(1−p)/2p − 1 ∈ (0, 1). +Since ln λε ≃ C0e−2HU/ε2, it is not hard to see that ln λ is in the same order, i.e., there +exists constant ˜C0 > 0 independent of ε such that +lim +ε→0 ln λ · e2H/ε2 = ˜C0. +Then with λε = exp{ ˜C0e−2HU/ε2}, the lemma is proved by letting ε > 0 sufficiently +small. +□ + +LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +25 +Proof of Theorem 1.2: Note that for any λ > 1 satisfying E[λτ η +ε ] < ∞, it holds that +P[τ η +ε > t]λt ≤ E[λτ η +ε ]. +It then follows from Proposition 4.4 (ii) and Theorem 4.5 that for any t > 0, +P[τc > t] ≲ λ−t +ε . +For the other side of the inequality, since (X0, Y0) is fully supported, for any δ > 0 +sufficiently small, +P[τc > t] ≥ P[τc > t, (X0, Y0) ∈ Bδ(x1) × Bδ(x2) or Bδ(x2) × Bδ(x1)]. +Note that when X0 and Y0 belong to the different basins, τc ≥ τ (1) +ε . Thus, by (54) +P[τc > t, (X0, Y0) ∈ Bδ(x1) × Bδ(x2) or Bδ(x2) × Bδ(x1)] +≥ +P[τ (1) +ε +> t, (X0, Y0) ∈ Bδ(x1) × Bδ(x2) or Bδ(x2) × Bδ(x1)] ≳ λ−t +ε . +This completes the proof of Theorem 1.2. +□ +4.4. Multi-well potentials and proof of Theorem 1.3. In this subsection, we study +the general case of multi-well potentials. Let U be a potential function satisfying (U3) +with L(L ≥ 3) local minima x1, · · · , xL and the corresponding basins B1, · · · , BL. Let +{(Xt, Yt); t ≥ 0} be a coupling of two solutions of (2). +In the multi-well setting, the estimate of coupling time follows the same idea in the +double-well case and some key stopping times have to be defined. Without any confusion, +the notation τ (1) +ε +is continued to denote the infimum time when Xt, Yt lie in the same +basin, i.e., +τ (1) +ε += inf +� +t > h : (Xt, Yt) ∈ ∪1≤i≤LBi × Bi +� +. +Define +τ (3) +ε += +inf +� +t > h : (Xt, Yt) ∈ +� +1≤i,j≤L,i̸=j Bi × Bj, and for certain s ∈ (0, t), +(Xs, Ys) ∈ +� +1≤i≤L Bi × Bi +� +, +and let τ (3) +ε += ∞ if the set is empty. We see that τ (3) +ε +is a generalization of τ (2) +ε +in the +multi-well setting and coincides with τ (2) +ε +when L = 2. +In the multi-well setting, of particular interest is the time when both Xt and Yt are +lying around the (unique) global minimum x1. Denote ξ1 the infimum time when both Xt +and Yt lie in the basin B1. Recall that κ{Xt}(B1) (resp. κ{Yt}(B1)), as defined in (32), +denotes the infimum time when Xt (resp. Yt) lies in B1. Then it must be that +ξ1 ≥ max +� +κ{Xt}(B1), κ{Yt}(B1) +� +. +(61) +We note that both Xt and Yt may go into and out of the basin B1 many times before +ξ1. However, as long as ε is sufficiently small, the typical scenario is that one of the two +processes, say Xt, first enters B1 and “waits” the other process Yt to come. Although Xt +may leave B1 before Yt arrives, it should be very likely that Xt stays in the nearby basins +and goes back to B1 quickly before Yt jumps out of B1. +The following (H3) is assumed which states that ξ1 is no greater than κ{Xt}(B1) (or +κ{Yt}(B1)) by an infinitesimal the same order with κ{Xt}(B1) and κ{Yt}(B1). + +26 +YAO LI, MOLEI TAO, AND SHIROU WANG +(H3) Let {(Xt, Yt); t ≥ 0} be a reflection-maximal coupling of two solutions of (2) such +that (X0, Y0) ∈ � +1≤i,j≤L,i̸=j Bi × Bj. Then for any ε > 0 sufficiently small, +lim sup +t→∞ +1 +t log P +�� +ξ1 − max +� +κ{Xt}(B1), κ{Yt}(B1) +�� +> t +� +≲ e−2HU/ε2. +(62) +Note that unlike (H2) which is the local characterization of the coupling behavior +between Xt and Yt, (H3) is a global condition on the coupling behavior when Xt and Yt +run around the entire potential landscape. In Section 5.4, (H3) is numerically verified. +Still, as in (49), denote +λε = exp{C0e−2HU/ε2}, +where HU is defined in (7). +Under Assumption (H3), by Lemma 2.7, for the initial +condition (X0, Y0) ∈ � +1≤i,j≤L,i̸=j Bi × Bj, we have +P[ξ1 > t] ≲ λ−t +ε . +(63) +Since τ (1) +ε +≤ ξ1, (63) further yields +P[τ (1) +ε +> t] ≲ λ−t +ε . +(64) +Let +τε = τ (3) +ε +∧ τc. +The subsequent analysis follows the same line as in the double-well case: if Xt, Yt initially +belong to the same basin, then under condition (H2), Proposition 4.1 holds without any +change. If Xt, Yt initially belong to the different basins, then during the time interval +(0, τε), the coupling behavior between (Xt, Yt) is further “decomposed” into two stages. +Stage I: Xt and Yt are lying in the same basin before τ (1) +ε , and then Stage II: Xt, Yt are +either successfully coupled within the same basin, or not coupled before any one of them +exists the basin. +In the multi-well setting when more than two wells are present, (H1)-(H3) are always +assumed. The following Lemma 4.6 is a “multi-well version” of Lemma 4.2 for which the +proof is omitted. +Lemma 4.6. Let {(Xt, Yt); t ≥ 0} be a reflection-maximal coupling of two solutions of (2) +such that (X0, Y0) ∈ � +1≤i,j≤L,i̸=j Bi × Bj. Then for any t > 0, +P[τε > t] ≲ λ−t +ε . +Now, the proof of Theorem 1.3 follows the same line of Theorem 1.2. +Proof of Theorem 1.3: As to the single and double-well cases, the coupling time τc is a +finite iteration of τε with τc = τ η +ε , where η is as in (59). Applying the same arguments in +Theorem 3.6, for any λ ∈ (1, λε), +E[λτε] < ∞, +and hence +P[τc > t] ≲ λ−t +ε . +For the other side of the inequality, since (X0, Y0) is fully supported, we only consider +the initial condition that Xt starts from certain basin “far-away” from B1 (i.e., any basin + +LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +27 +with the index from I) and Yt starts from any of the basins with “low” values (i.e., any +basin with the index from J ). By (H1), under such initial condition, the event that Xt +enters B1 before Yt has ever existed B1 occurs with a positive probability and independent +of ε and t. Thus +P[τc > t] +≥ +P +� +Ys ∈ B1 for all s ∈ [0, t] +��κ{Xt} > t +� +· P[κ{Xt} > t] +≥ +γ0 · P[κ{Xt} > t] ≃ λ−t +ε +(65) +where “≃” comes from (37), the strengthened version of Lemma 2.7. +This completes the proof of Theorem 1.3. +5. Numerical Examples +In this section, various numerical examples are presented to verify the theoretical results +as well as assumptions that have been assumed in the previous sections. Please refer to +[23] for a detailed description on the coupling algorithm that will be used. +We first propose an algorithm for a precise numerical estimate of the exponential tail +of coupling times. +5.1. An algorithm for exponential tail estimation. Let τc be the coupling time. The +first task is to estimate the exponential tail of P[τc > t] with respect to t, or more precisely, +− lim +t→∞ +1 +t log P[τc > t] . +Since only finitely many coupling events are to be sampled, an efficient algorithm is needed +to both statistically confirm the existence of such exponential tail and to estimate it. +The main difficulty is that P[τc > t] usually does not behave like an exponential distri- +bution until t is sufficiently large. A suitable t∗ then needs to be determined such that on +one hand, 1{τc>t∗}(τc −t∗) truly behaves like an exponential distribution, and on the other +hand, such t∗ needs to be as small as possible so that enough samples with τc > t∗ are +collected. However, most exponentiality tests we have tried tend to provide a too small t∗ +for which in the log-linear plot, 1{τc>t∗}(τc −t∗) has not “stabilized” to a good exponential +distribution probably due to the sensitivity of log-linear plot in terms of small changes of +the tail distribution. +The purpose of our algorithm is to capture a suitable t∗ such that in the log-linear plot, +P[τc > t] v.s. t statistically forms a straight line. In other words, the confidence interval +of P[τc > t] should cover the straight line in the log-linear plot for all t > t∗. Choose +a sequence of times t0, t1, · · · tN, where tN is usually set to be the maximum of all the +coupling times obtained in the simulation. Denote the total sample size as M, and for +each i, let ni be the number of samples with τc > ti. Then the Agresti-Coull method [1] +provides a confidence interval +[˜p− +i , ˜p+ +i ] := [˜pi − z +� +˜pi +˜ +M +(1 − ˜pi) , ˜pi + z +� +˜pi +˜ +M +(1 − ˜pi)] , +where ˜ +M = M + z2, ˜pi = (ni + z2 +2 )/ ˜ +M, and z = Φ−1(1 − α/2) is the α-quantile of the +standard normal distribution. In the simulations, we usually choose z = 1.96, α = 0.05 +A weighted linear regression is used to fit the points (ti, log ˜pi) for i = N0, N0+1, · · · , N, +where the weight of (ti, log ˜pi) equals ni/M. The number N0 is chosen in the following + +28 +YAO LI, MOLEI TAO, AND SHIROU WANG +way. Suppose that the weighted linear regression provides a linear function y = at + b. A +number N0 is accepted if it satisfies +|{i | ati + b /∈ [˜p− +i , ˜p+ +i ]| < α(N − N0 + 1) . +In other words, statistically, 1{τc>tN0}(τc − tN0) is indistinguishable from an exponential +distribution. The smallest possible value of such N0 is the N0 we finally choose which can +be found through the binary search method. Then t∗ is given by tN0, and the exponential +tail is given by the index a in the weighted linear regression corresponding to N0. +5.2. Quadratic potential function. The first example is the quadratic potential func- +tion. The main purpose of this example is to numerically verify the theoretical result of +Theorem 1.1. Another purpose is to verify (43) which assumes that the first passage time +of the discrete-time trajectory approaches to that of the continuous-time trajectory as the +time step size vanishes. +Consider the quadratic potential function +U(x) = 1 +2xT Ax, +x ∈ Rk, +where A is a k × k Lehmer matrix, i.e., the entries Aij = min(i, j)/ max(i, j), which is +symmetric and positive definite [27]. The corresponding SDE is +dZt = −AZtdt + εdWt , +(66) +where Wt is a k-dimensional Wiener process, and ε > 0 is the strength of noise. +In the numerical simulations, the time step size h is always set to be 0.001 unless +otherwise specified. In Figure 1, the probability distribution of τc is demonstrated. The +four panels are P[τc > t] v.s. t in log-linear plot with respect to the 2 × 2, 4 × 4, 6 × 6, and +8 × 8 Lehmer matrices, respectively. In all the four cases, the strength of noise takes the +value 0.02, 0.1, 0.5, and 1.5. The slope of each curve of P[τc > t] v.s. t in the log-linear +plot is measured by the algorithm introduced in subsection 5.1. +In all the four cases, although as noise changes the probability distribution of τc is +very different, the slope of the exponential tail remains unchanged. +In addition, the +least eigenvalue of A, which can be explicitly computed, is very close to the slope of the +corresponding exponential tails with an error less than 0.01. This numerically verifies the +result of Theorem 1.1 that the slope of the exponential tail is only determined by the +convexity of the potential function and independent of the noise magnitudes. +Now, we use this example to verify (43) in Remark 3.2 that +limh→0 |τ 0 +h − τ h +h | = 0, +where τ 0 +h = inft>0{|Xt − Yt| = 2 +√ +h} is the continuous-time first passage time, and τ h +h = +h · infn>0{|Xnh − Ynh| = 2 +√ +h} is the first passage time of the time-h sample chain. +This can be done by the extrapolation argument. Let h1 = h/n for some integer n, and +define the first passage time of the time-h1 sample chain +τ h1 +h += h1 · inf +n>0{|Xnh1 − Ynh1| = 2 +√ +h} . +By the strong approximation of the Euler-Maruyama scheme of the SDE, i.e., +limh1→0 τ h1 +h += τ 0 +h, +P-a.s., +we only need to compare τ h +h and τ h1 +h , where the latter is an approximation of τ 0 +h, concerning +the same trajectory. + +LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +29 +Figure 1. P[τc > t] v.s. t in log-linear plots and their exponential tails. +Four panels are for Lehmer matrices with size 2, 4, 6, and 8. The least +eigenvalue of each matrix is demonstrated in the title of each subplot. +In Figure 2 Left, (τ h +h − τ h1 +h ) demonstrates a linear growth with respect to a decreasing +√h1, and an extrapolation at h1 = 0 provides an estimate of (τ h +h − τ 0 +h). In Figure 2 Right, +(τ h +h − τ 0 +h) is estimated for h = 0.0002, 0.0005, 0.001, 0.005, and 0.01, respectively. We see +that the error (τ h +h − τ 0 +h) decreases with respect to a decreasing h. A linear fit shows that +(τ h +h − τ 0 +h) is approximately proportional to +√ +h, which is consistent with the results in +[14, 15]. +Figure 2. Left: (τ h +h − τ h1 +h ) v.s. √h1 for five different values of h. Right: +(τ h +h − τ 0 +h) v.s. +√ +h and a linear fitting. + +2x2 Lehmer matrix with least Eigenvalue 0.5 +4x4 Lehmer matrix with least Eigenvalue 0.20778 +noise = 0.02 P[T。 > t] +01 +noise = 0.02 P[T。 > t] + noise = 0.02 tail slope = -0.49702 +-2 +noise = 0.02 tail slope = -0.20526 +noise = 0.1 P[T。 >t] +noise = 0.1 P[T。 > t] +noise = 0.1 tail slope = -0.49379 + noise = 0.1 tail slope = -0.20921 +-4 +-4 +noise = 0.5 P[T。 > t] +noise = 0.5 P[T。> t] +noise = 0.5 tail slope = -0.49844 + noise = 0.5 tail slope = -0.21201 +-6 +noise = 1.5 P[T。 > t] +-6 +noise = 1.5 P[T。 > t] + noise = 1.5 tail slope = -0.50858 + noise = 1.5 tail slope = -0.21128 +-8 +-8 +-10 +-10 +-12 +-12 +-145 +-14 5 +10 +35 +10 +0 +5 +15 +20 +25 +30 +0 +20 +30 +40 +50 +60 +70 +80 +6x6 Lehmer matrix with least Eigenvalue 0.12401 +8x8 Lehmer matrix with least Eigenvalue 0.087073 +0 +noise = 0.02 P[↑ > t] +noise = 0.02 P[T。 > t] + noise = 0.02 tail slope = -0.12335 +noise = 0.02 tail slope = -0.08785 +-2 +noise = 0.1 P[T。 > t] +noise = 0.1 P[T。 > t] + noise = 0.1 tail slope = -0.12536 +noise = 0.1 tail slope = -0.090324 +-4 +noise = 0.5 P[。 > t] +noise = 0.5 P[T。> t] + noise = 0.5 tail slope = -0.12573 +noise = 0.5 tail slope = -0.092452 +-6 +noise = 1.5 P[T。 > t] +noise = 1.5 P[T。 > t] +noise = 1.5 tail slope = -0.13157 +noise = 1.5 tail slope = -0.088732 +-8 +-8 +-10 +-10 +-12 +-12 +-14 +100 +*0 +20 +40 +60 +80 +120 +140 +50 +100 +150 +200 +2500.09 +h, = 0.01 +0.09 + Th from extrapolation + h, = 0.01 linear fit +linear fit +0.08 +0.08 +h, = 0.05 +h, = 0.05 linear fit +0.07 +0.07 +h, = 0.001 +h, = 0.001 linear fit +0.06 +0.06 +h, = 0.005 +-h, = 0.005 linear fit +0.05 +-h, = 0.002 +oh +-h, = 0.002 linear fit +0.04 ++0.04 +hh +0.03 +0.03 +0.02 +0.02 +0.01 +0.01 +0 +0 +-0.01 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +Vhi +Vh30 +YAO LI, MOLEI TAO, AND SHIROU WANG +5.3. 1D double-well potential. In this subsection we consider an asymmetric one- +dimensional double-well potential +U(x) = x4 − 2x2 + 0.2x, +x ∈ R. +It is easy to see that U has two local minima, 0.9740 and −1.0241, respectively. The +barrier heigh for any trajectory to overcome when moving between the two local minima +is 1.2074 if from left to right, and 0.8076 if vice-versa; see Figure 3 Top. +The first goal of this example is to verify the theoretical result of Theorem 1.2 that the +tail of the coupling time distribution is determined by the lower barrier height. The time +step size and the coupling method are the same as before. The coupling time distribution +of τc is estimated under the different noise magnitudes ε = 0.32, 0.36, 0.4, 0.45, 0.5, 0.6, +and 0.7. In Figure 3 Top, the exponential tail corresponding to each noise is estimated by +the weighted linear regression described in Section 5.1. We see that the exponential tail +r(ε) changes dramatically as ε decreases. In Figure 3 Bottom right,a linear relationship +of −ε2 log r(ε) v.s. ε2 is clearly presented. A linear extrapolation of ε2, as ε → 0, further +shows that r(0) = 1.617. This matches well with the theoretical value of r(0) = 2HU +which, in this example, equals 1.615. +Figure 3. Top: Coupling time distributions for different noise magni- +tudes. +Bottom left: Asymmetric double well potential. +Bottom right: +linear extrapolation for barrier height. +The second goal of this example is to numerically verify (H2) in Section 4.2. +Let +dZt = −∇U(Zt)dt be the deterministic gradient flow of U, and xu = 0.05129 be the +unstable equilibrium Zt for which the basins of the two local minima are B1 = (−∞, xu) +and B2 = (xu, ∞). + +Coupling time distribution for double well potential +Coupling time distribution for double well potential + noise = 0.45 + noise = 0.5 +-2 +-2 +slope = 0.013406 +noise=0.6 +slope = 0.098569 +noise = 0.7 +-4 +V +-4 +slope = 0.34001 +°ild +aild + noise = 0.32 + slope = 1.1495e-06 +9- +noise = 0.36 +-6 +log +slope = 3.266e-05 +noise = 0.4 +slope = 0.00033893 +-8 +-8 +-10 +-10 +0 +2 +4 +6 +8 +10 +0 +500 +1000 +1500 +2000 +2500 +3000 +t +t +×104 +Asymmetric double well potential +Linear extrapolation for double well potential +2 +2 +1.5 +1.5 +slope r(e) +log(r(e) +linear fit +1 +1 +theoretical value +0.5 +height = 1.2074 +height = 0.80757 +0.5 +-0.5 +0 +-1 +-0.5 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +?LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +31 +Let both Xt and Yt start from B2. Recall that, as defined in Section 4.1, τc, τ (2) +ε +and +τε denote the coupling time, first passage time to B1, and the minimum of the former +two, respectively. +To numerically verify (H2)(ii), we examine the distributions of τε +conditioning on τε = τc and τε = τ (2) +ε , respectively. If under different noise magnitudes ε, +the exponential tails of both two conditional distributions have a lower bound independent +of ε, then (H2)(ii) is verified. +In Figure 4, X0 and Y0 are independently and uniformly sampled from (0.1, 1.5). Let +the magnitude of noise change from 0.1 to 0.6, and under each noise magnitude, 1 × 108 +samples of the coupling processes are run until τε. Then the distribution of τε is further +conditioned on each of the two cases that (i) Xt, Yt are couple before exiting to B1, and +(ii) any one of them, X t or Yt, exits to B1 before they are being coupled, respectively. +As shown in Figure 4 Left and Middle, both the conditional exit time and the conditional +coupling time remain largely unchanged with respect to the decreasing ε. +Finally, we verify (H2)(i) that the coupling probability is uniformly away from zero at +τε, i.e., P[τc = τε] > γ for some γ > 0, regardless of the initial conditions. The result is +shown in Figure 4 Right. Let X0 be a fixed value X0 and Y0 be uniformly distributed in +(0.1, 1.5). Under three different values of X0, the probability P[τc = τε] is estimated with +ε ranged from 0.005 to 0.6. We see that P[τc = τε] is uniformly away from zero regardless +of X0 and ε. Note that while the first initial value (the blue plot) is at the boundary of +B1, the coupling probability P[τc = τε] is still uniformly positive. +Figure 4. Left: Conditional exit time distribution. Middle: Conditional +coupling time distribution. Right: Probability of couple at τε +5.4. Interacting particle system in the double-well potential. In this section, we +study a variance of the double well potential of the previous section. Denote +V (x) = x4 − 2x2 + 0.2x, +x ∈ R +be the double-well potential as in the previous section, and let three particles move along +V under the over-damped Langevin dynamics. Assume, besides the potential V , there is +also a pairwise interaction potential between the three particles. The energy potential for +the interacting particle system is +U(x1, x2, x3) = +�3 +i=1 V (xi) + σ +� +i,j=1,2,3, i̸=j(xi − xj)2 . +where σ > 0 is the interaction strength. +It is not hard to see that U has two trivial local minima x1 = x2 = x3 = 0.9740 and +x1 = x2 = x3 = −1.0241, respectively, corresponding to which all the three particles + +conditional exit time distribution +conditional coupling time distribution +100 +noise = 0.1 +noise = 0.1 +-2 +noise = 0.2 +noise = 0.2 +noise = 0.3 +noise = 0.3 +noise = 0.4 +noise = 0.4 +-4 +-5 +noise = 0.5 +noise = 0.5 +T +-6 +noise = 0.6 +noise = 0.6 +1 +-8 +II +-10 +7 +V +P[Te +P. +-15 +-14 +。= 0.050126 +X。= 0.06 +-16 +X = 0.08 +-18 +-20 +10-2 +2 +3 +4 +0 +0 +2 +3 +1 +4 +5 +10-2 +10-1 +100 +t +f +E32 +YAO LI, MOLEI TAO, AND SHIROU WANG +are lying at the same local minimum of V (x). When σ > 0 is sufficiently small, there are +another six local minima under which the three particles are staying in the different basins +of V (x); see Figure 5 Top for a sample trajectory of the solution of (2). +We see that there are two extreme cases in terms of the interactions of the interacting +particle system. One extreme case is when there are no interactions between the three +particles, i.e., σ = 0. In this case, the three particles are independent and passing between +the two wells one by one, resulting the same barrier heights with that of the double- +well potential V . +The other extreme case corresponds to σ = ∞, which means that +the interactions between the three particles are so strong that they always have to move +together. In this case, the energy potential U has the same local minima as that of V and +the essential barrier height is the triple of that of V . Hence, the essential barrier height +HU of the interacting particle system should lie between that of the two extreme cases, +i.e., 0.8076 and 3 × 0.8076 = 2.4228, respectively, and HU increases as σ increases. +The distribution of the coupling time τc is computed for ε = 0.4, 0.41, 0.42, 0.43, 0.45, +0.47, 0.5, 0.55, 0.6, and 0.7 when σ = 0.05, and ε = 0.41, 0.42, 0.43, 0.44, 0.45, 0.47, 0.5, +0.55, 0.6 and 0.7 when σ = 0.1, and the negative slopes r(ε) are estimated for both cases. +A similar relation of r(ε) v.s. ε is observed with the double-well potential, and a linear +extrapolation of −ε2 log r(ε) provides an estimation of the the essential barrier height. +In Figure 5 Middle right, the linear extrapolation yields r(0) = 1.7374 for σ = 0.05 and +r(0) = 1.9598 for σ = 0.1, both of which are approximately equal to the double of the +barrier height 2HU. As expected, the barrier height increases as the interaction between +particles becomes stronger. +The next task is to numerically verify (H1)-(H3) in Section 4 for the setting of multi- +well potentials. In the following, the interaction strength is always set as σ = 0.05. +Numerical verification of (H1). Let X0 = (1, 1, 1), and Y0 = (−1, −1, −1) so that Yt +starts near the global minimum. According to the definition of I and J , it is easy to see +from height of barriers in Figure 8 that B1 is in fact the complement of the basin of attrac- +tion that contains (1, 1, 1). Our simulation uses four noise magnitudes ε = 0.6, 0.65, 0.7, +and 0.75. After each step of the Euler-Maruyama scheme, it will be numerically checked +to determine whether Xt and Yt are in B1.4 Interestingly, we have never seen Yt leaves B1 +before Xt enters B1 after tens of millions of samples collected. The same thing happens for +the 1D double well potential: numerically Yt never leaves B1 before Xt enters B1. Hence, +(H1) is numerically concluded with a even stronger result that +P +� +Ys ∈ B1 for all s ∈ [0, t] +��κ{Xt}(B1) > t +� +≈ 1. +Numerical verification of (H2). Let Xt and Yt both start from B1. We simulate the +coupling process until either that Xt and Yt are coupled successfully before any one of +them exists B1, or one of them, Xt or Yt, exits B1 before they are coupled within B1, +respectively. In Figure 6 Left and Middle panel, the conditional exit time distribution +and conditional coupling time distribution are demonstrated, respectively. We see that +the slopes of both the two tail distributions are largely unchanged as the noise magnitude +decrease from 0.5 to 0.1. This is consistent with the result for the double-well potential. +In Figure 6 Right, the coupling probability is demonstrated in term of noise magnitude. +Again, similar to the case of double-well potential, when starting within the same basin, +the probability P[τε = τc] significantly increases with respect to a decreasing ε as the +4The criterion is as follows. If for i = 1, 2, 3, it holds that either xi > 0.11, or that 0 ≤ xi < 0.11 +whenever −∂U/∂xi > 0, then (x1, x2, x3) is not in B1. We note that this is sufficient for all the samples +in our simulation. + +LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +33 +Figure 5. Top: Sample trajectory of a 3-particle interacting particle sys- +tem in a double well potential. Middle: coupling time distribution and ε +v.s. −ε2 log r(ε) with σ = 0.05. Bottom: coupling time distribution and ε2 +v.s. −ε2 log r(ε) with σ = 0.1. Theoretical values of r(0) are given by the +minimum energy path. +Figure 6. Left: Conditional exit time distribution. Middle: Conditional +coupling time distribution. Right: Probability of coupling at τε +chance of Xt, Yt to escape from the basin becomes lower. However, for those successfully +leave B1 before being coupled, the exit time distribution does not change much with +respect to the noise magnitudes. +Numerical verification of (H3). The (H3) is numerically verified by computing the +overshoot time. The criterion of deciding whether a trajectory hits the basin B1 is the + +Trajectory of interacting particles with = 0.1 +2 +partcle 1 +particle 2 +position +particle 3 +-1 +-2 +0 +5 +10 +15 +20 +25 +time +Coupling time distribution for interacting particle system, = 0.1 +Linear extrapolation for interacting particle system, = 0.1 +2 +noise = 0.6 +t +noise = 0.5 +slope = 0.0007 +1.8 +noise = 0.45 +-5 +slope = 0.0001 +P +noise = 0.43 +1.6 +slope r(e) +slope = 0.0000 +log +2c +noise = 0.41 +linear fit +slope = 0.0000 +theoretical value +1.4 +-10 +1 +2 +3 +4 +5 +6 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +t +e? +×104 +Coupling time distribution for interacting particle system, = 0.05 +Linear extrapolation for interacting particle system, = 0.05 +2 +oise = 0.6 +slope = 0.0122 +noise = 0.5 +V +slope = 0.0017 +noise = 0.45 +-5 +slope = 0.0004 +P +slope r(e) +log 1 +slope = 0.0001 +2c +linear fit +1.4 +slope = 0.0000 +theoretical value +-10 +1.2 +2 +1 +4 +5 +6 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +t +2 +×104conditional exit time distribution +conditional coupling time distribution +0 +0.8 +noise = 0.1 +noise = 0.1 +noise = 0.2 +noise = 0.2 +0.7 +noise = 0.3 +noise = 0.3 +noise = 0.4 +noise = 0.4 +-5 +0.6 +T +noise = 0.5 +T +noise = 0.5 +0.5 +T +t +11. 0.4 +-10 +V +>-10 +log P[ +-15 +0.2 +0.1 +-20 +-20 +0 +0 +3 +0.1 +0.2 +0.3 +1 +2 +3 +0 +1 +2 +4 +0.4 +0.5 +t34 +YAO LI, MOLEI TAO, AND SHIROU WANG +Figure 7. Comparison of probability distributions of the “overshoot time” +ξ1 − max{κXt, κYt} and the coupling time. Left, middle, and right panels +are for ε = 0.6, 0.55, and 0.5 respectively. +same as above. The probability distribution of the overshoot time ξ1 −max{κXt, κYt}, i.e., +the infimum time when both Xt and Yt are staying in B1 after each of them has visited B1 +is computed. The magnitudes of noise are chosen as 0.5, 0.55, and 0.6, and for each case, +the probability distribution of ξ1 −max{κXt, κYt} is computed by running 1×107 samples. +As shown in Figure 7, the tail of ξ1 − max{κXt, κYt} has two phases. The second phase +has a slower decrease of the exponential tail but is still faster than that of the coupling +time. The second phase comes from the event that one of the two trajectories, Xt or Yt, +takes an excursion to other basins after visiting B1 and then returns back, while the other +trajectory has always been staying in B1. Note that the probability of such event is low +and a large number of samples are required to capture the exponential tail. In Figure 7, +the distributions of the overshoot time and coupling time are compared. We see that the +slope of tail distribution of the overshoot time decays quickly as noise vanishes, but still +remains steeper than that of the coupling time in the log-linear plot. Since it has been +numerically verified that the coupling time distribution is consistent with the theoretical +barrier height HU, (H3) is verified automatically. +Finally, the String method [12] is used to compute the heights of various barriers between +the local minima (0.9740, 0.9740, 0.9740) and (−1.0241, −1.0241, −1.0241) in the energy +landscape to validate the essential barrier height inferred from our coupling approach. +As shown in Figure 8, the essential barrier, which is the highest barrier that a trajectory +needs to overcome to enter the basin of the global minimum, is leftmost barrier in the lower +left panel of Figure 8 (A)(B). In this example, since the three particles are indistinct, the +energy potential is rather symmetric for which the eight local minima are of only two types +consists of four particular cases that all the three particles are lying in the same well (global +or local), or two of the three particles are lying in one well (global or local) with the other +particle lying in the other well. We see in Figure 8 that the minimal energy path(MEP) +connecting the two minima (0.9740, 0.9740, 0.9740) and (−1.0241, −1.0241, −1.0241) has +actually passed all the four cases. Hence, the essential barrier height HU can be attained +by such an MEP although in principle, it has to be taken over all the paths connecting any +of the local minima to the global one. In Figure 8, it shows that HU = 0.8961 for σ = 0.05 + +A comparison of coupling time distribution and "overshoot time" distribution +0 +0 +- overshoot time, noise = 0.55 +0 + overshoot time, noise = 0.6 +overshoot time, noise = 0.5 + slope = 0.0055186 + slope = 0.012271 +slope = 0.001725 +-2 + coupling time, noise = 0.55 + coupling time, noise = 0.6 +-2 +-2 +coupling time, noise = 0.5 +slope = 0.0024833 + slope = 0.0067741 + slope = 0.00068099 +-4 F +-4 +-4 +-6 +-6 +-6 +-8 +P(Te +ild +1og +-10 +-12 +-12 +-12 +-14 +-14 +-14 +-16 +-16 +-16 +-18 +-18 +-18 +0 +1000 +2000 +3000 +500 +1500 +1000 +2000 4000 6000 8000 10000 +t +t +tLANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +35 +and HU = 0.9916 for σ = 0.1, corresponding to the theoretical value r(0) = 1.7922 for +σ = 0.05 and r(0) = 1.9832 for σ = 0.1, respectively. +The result from the String method is further confirmed through the equivalent char- +acterization (23) by numerically solving all the 27 critical points (including all the local +minima and saddle points) of U. +The essential barrier height obtained in this way is +HU = 0.8962 for σ = 0.05 and HU = 0.9916 for σ = 0.1, which are almost the same +with the essential barrier heights given by the String method. As shown in Figure 5, both +values are very close to r(0)/2 in the linear extrapolation. +5.5. Rosenbrock function. In this example, we test on the famous non-convex landscape +of the Rosenbrock function in 2D and 4D, respectively. For N ∈ N+, the Rosenbrock +function is defined as +RN(x) = +N−1 +� +i=1 +[b(xi+1 − x2 +i )2 + (a − xi)2], +x ∈ RN, +where a, b are two constants. In this example, we choose a = 1, b = 20. When N = 2, RN +has a unique minimum at (1, 1), and when N = 4, RN admits a global minimum (1, 1, 1, 1) +and a local minimum (−1, 1, 1, 1). In Figure 9, the landscape of log R2(x) is demonstrated +in the Top Left panel and a slice of log R4(x) at x3 = x4 = 1 is demonstrated in the +Bottom Left panel. We note that the logarithm scale is used to visualize the detailed +landscape near each minimum. We see that near each minimum, the landscape looks like +a valley which is only convex on a very small area around the minimum. The landscape +of R4 cannot be completely visualized by the heat map at a slice. However, it is not hard +to check that for a = 1, b = 20, the region on which R4 being convex is very small. +In the test of the landscape of R2, the noise magnitude is set to be ε = 0.001, 0.01, 0.1, +1.0, 1.5, and 2.0, and ε = 0.001, 0.003, 0.01, 0.03, 0.1, 0.3, and 1.0 for R4. The coupling +time distributions are given in the two right panels of Figure 9. We see that when the noise +is sufficiently small, the tail distributions of coupling times are parallel in the log-linear +plot since the coupling time is almost determined by the convexity of the convex area near +the global minimum. This is consistent with the result of Theorem 1.1. However, as noise +magnitude increases, the probability of the coupling process to couple in the entire valley +instead of just the vicinity of the global minimum becomes larger, and this changes the +tail of the coupling time distributions. +Another interesting phenomenon is that unlike the case of double-well potential, for +the potential function R4, exponentially small tails of the coupling time distribution with +respect to the noise magnitude is not observed even when the noise is as small as 0.001. +Even if one of the coupled processes start at the local minimum (−1, 1, 1, 1), the tail of the +coupling time distribution has very little change; see the plot with legend “noise = 0.001 +fixed”. This is because the basin of the local minimum is so shallow with such a low +barrier that the stochastic trajectories can easily pass the barrier and enter the valley of +the global minimum. +5.6. Loss functions of artificial neural networks. In the last example, the perfor- +mance of the coupling method in high dimension is examined. We consider the training +process of an artificial neural network(ANN) including two hidden layers with N1, N2 + +36 +YAO LI, MOLEI TAO, AND SHIROU WANG +(a) σ = 0.05 +(b) σ = 0.1 +Figure 8. Minimum Energy Path (MEP) computed by String method [12] +at high numerical resolution. MEP is the mostly likely path for transition +from one metastable state to another in the zero-temperature limit of over- +damped Langevin, and it is known (e.g., [11]) to reveal barrier heights and +descent depths along the transition path, which are labeled by dV values in +the bottom left panel. Top panel visualizes the MEP in 3D (i.e., x1, x2, x3) +where legend lists the potential U’s value at each local minimum. Bottom +right panel is the integrand of the Freidlin-Wentzell action functional plot- +ted as a function of arc-length parameterization of the path, which serves +as a sanity check to ensure the MEP is computed correctly. + +Steps=31300 Change=1.0054e-08 +☆-3.6073 +☆ +-2.4077 +-2.8238 +-2.8238 +0 +-2.8238 +-2.4231 +-1 +-2.4231 +-2.4231 +0.5 +0 +0.5 +-0.5 +-0.5 +0 +-1 +-1.5 +-1.5 +-1 +u +action integrand +-1.5 +25 +-2 +dU=0.9115 +dU-0.896|14 +20 +dU=0.71175 +dU=1.1125 +-2.5 +15 +dU=0.54397 +-3 +=1.3275 +10 +-3.5 +5 +-4 +0 +0 +100 +200 +300 +400 +500 +0 +100 +200 +300 +400 +500Steps=34400 Change=1.0205e-08 +☆-3.6073 +-2.4077 +-2.4721 +-2.4721 +0 +-2.4721 +-2.0682 +-1 +-2.0682 +-2.0682 +0.5 +0 +0.5 +-0.5 +-0.5 +0 +-1 +-1.5 +-1.5 +-1 +U +action integrand +-1 +25 +-1.5 +20 +dU=0.62307dU=0.65195 +-2 +U=1.027 +dU=0.99157 +15 +-2.5 +10 +dU=1.458 +-3 +-3.5 +5 +-4 +0 +0 +100 +200 +300 +400 +500 +0 +100 +200 +300 +400 +500LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +37 +Figure 9. Top Left: landscape of R2(x). +Top Right: Coupling time +distribution for R2(x). Bottom Left: landscape of R4(x). Bottom Right: +Coupling time distribution for R4(x). +neurons, respectively, which has the following form +h1 = ReLU(W1x + b1) +(67) +h2 = ReLU(W2h1 + b2) +(68) +y = W3h2 + b3 , +(69) +where x ∈ R2, y ∈ R, b1 ∈ RN1, b2 ∈ RN2, b3 ∈ R, and W1, W2, W3 are N1 × 2, N2 × N1, +1×N2 matrices, respectively. Let θ collect the entries of W1, W2, W3, b1, b2, and b3 which +are the unknown parameters to be determined in the training. Note that the dimension +of θ is (N1N2 + 3N1 + 2N2 + 1). For simplicity, (67) - (69) are collectively written as +y = NN(θ, x). +For a given training set {x1, · · · , xM; y1, · · · yM}, let the loss function +L(θ) = +�M +i=1 (yi − NN(θ, xi))2 , +where the size of the training set M = 100. +The collocation points x1, · · · , x100 are +uniformed sampled from [−1 , 1]2, yi = |xi|2; see the first column of Figure 11 for the +distribution of collocation points and the target function y = |x|2. What we are interested +in is the landscape of the loss surface L(θ). +Our coupling algorithm is tested on three ANNs for which the size of hidden layers are +N1 = 4, N2 = 3, N1 = N2 = 10, and N1 = N2 = 20, which are called the “small network”, +“medium network”, and “big network”, respectively. Note that in this example, the small +network is under-parameterized and the big network is over-parameterized. It is believed +that over-parametrization lowers the barrier heights of ANNs (e.g., [17, 33, 6, 26, 30, 31, + +Heat map of 2D Rosenbrock function +Coupling time distribution for 2D Rosenbrock function +1.5 +noise = 0.001 +noise = 0.01 +1 +noise = 0.1 +2 +noise = 1 +0.5 +-5 +noise = 1.5 +0 +V +noise = 2 +0 +-2 +-0.5 +-4 +-10 +-6 +-1 +-8 +X +-1.5 +-15 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +0 +10 +20 +30 +40 +50 +60 +70 +t +Heatmap of 4D Rosenbrock function at slice z = w = 1 +Coupling time distribution for 4D Rosenbrock function +1.5 +noise = 0.001 +-2 +noise = 0.003 +1 +noise = 0.01 +noise = 0.03 +2 +-4 +0.5 +noise = 0.1 +V +-6 +noise = 0.3 +0 +log P[Te +0 +noise = 1 +noise = 0.001 (fixed) +-8 +-2 +-0.5 +-10 +4 +-1 +-12 +-6 +-1.5 +-14 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +10 +20 +30 +40 +50 +0 +60 +70 +t38 +YAO LI, MOLEI TAO, AND SHIROU WANG +10]) which, however, is not easy to justify because the loss landscape is usually in high- +dimension and very complicated. Our coupling approach provides a feasible way to explore +this by computing the essential barrier height of the entire landscape. +In Figure 10, the coupling time distribution of each ANN is computed under 10 different +magnitudes of noise (while only 5 of them are demonstrated in the figure to avoid a too +crowded legend though). Similar to the previous examples, all slopes are estimated through +the weighted linear fitting. The six smallest values of ε2 are used in the linear extrapolation +of −ε2 log r(ε) v.s. ε2 which are demonstrated in the three lower panels. We see that the +larger neural network has a lower essential barrier height. This is consistent with the study +in [10] in which MEPs are computed between the local minima of loss functions of ANNs. +The small neural network in this example is under-parametrized because the number +of unknown parameters to be determined is 31 while the size of the training set is 100. +As shown in Figure 10, the loss function of the small neural network has a much larger +essential barrier height than that of both the medium and big neural networks. In the +training process, when start from random initial conditions, the small neural network may +converge to a “bad” local minimum which does not fit the target function very well (see +the middle panels of Figure 11). +In contrast, for all the initial values that have been +tested, both the medium and big neural networks converge to a “good” local minimum +that approximates the target function reasonably well (see the right panels of Figure 11). +This is consistent with some known results on the loss surface of artificial neural networks +in the literature [7, 20, 32]. +Figure +10. Coupling time distribution and linear extrapolation of +−ε2 log r(ε) v.s. ε2. Left: small network. Middle: medium network, Right: +large network. + +Small NN, coupling time distribution +Medium NN, coupling time distribution +Large NN, coupling time distribution +noise = 0.35 P[T。 > t] +noise = 0.25 P[T。 > t] +noise = 0.2 P[T。 > t] +2 +noise = 0.35 tail slope = -0.104 +2 +noise = 0.25 tail slope = -0.229 +2 +noise = 0.2 tail slope = -0.455 +noise = 0.2 P[T。 > t] +noise = 0.17 P[r。 > t] +noise = 0.12 P[。 > t] +noise = 0.2 tail slope = -0.029 +noise = 0.17 tail slope = -0.102 +noise = 0.12 tail slope = -0.155 +noise = 0.12 P[T。 > t] +0 +noise = 0.1 P[。 > t] +noise = 0.07 P[7。 > t] +0 +0.12 tail slop +e = -0.006 +noise = 0.1 tail slope = -0.022 +noise = 0.07 tail slop +noise = 0.095 P[T。 > t] +noise = 0.07 P[T。 > t] +noise = 0.043 P[↑。> t] +-2 +-0.001 +州 +noise = 0.07 tail slope = -0.004 +-2 +0.09 P[T,> t] +noise = 0.065 P[T。 > t] +2 +V +V +V +noise = 0.038 P[T, > t] + 0.09 tail slc +e = -0.001 +log P[re +oise = 0.038 tail slope = -0.003 +P +-4 +-4 +P +-4 +-6 +-6 +-6 +-8 +-8 +-8 +-10 +-10 5 +-10 +0 +2000 +4000 +6000 +8000 +10000 +0 +1000 +2000 +3000 +1000 +2000 +3000 +4000 +0 +t +t +t +Small NN, linear extrapolation +Medium NN, linear extrapolation +Large NN, linear extrapolation +Estimated barrier height = 0.017355 +Estimated barrier height = 0.009744 +Estimated barrier height = 0.0038681 +0.1 +0.05 +0.2 +0.08 +0.04 +2e +0.1 +0.04 +0.05 +0.02 +0.01 +0 +0 +0.02 +0.04 +0.06 +0 +0.01 +0.02 +0.03 +0.04 +0.005 +0.01 +0.0150.02 +0.025 +0 +noise magnitude e +noise magnitude e +noise magnitude eLANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +39 +Figure 11. Left column: training set and target function. Middle column: +y = NN(θ, x) for θ at bad and good local minima of the small network. +Right column: y = NN(θ, x) from medium and big neural network. +6. Conclusion and further discussions +This paper investigates the relations between the landscape of a multi-dimensional po- +tential function and the coupling time distributions of the overdamped Langevin system +with respect to the given potential. This is motivated by the fact that the exponential tail +of the coupling time distribution gives a lower bound of the spectral gap of the Fokker- +Planck operator of the overdamped Langevin dynamics. It has been known for a long time +that some global information can be inferred from the spectrum of a differential operator, +such as detailed by the famous problem “Can one hear the shape of a drum?” in [18]. As +expected, certain connections between the shape of the landscape and the coupling time +distributions are established. +This paper shows that as the magnitude of the noise vanishes, the variation of the +exponential tails of the distribution of the overdamped Langevin dynamics of a single-well +potential differs from that of a multi-well potential in a qualitative way. More specifically, +for single-well potential functions that are strongly convex, the exponential tails is bounded +from above by a constant depending on the convexity of the potential function; for a multi- +well potential function, however, the exponential tail decreases exponentially fast as the +noise magnitude goes to zero. A linear extrapolation can then be used to infer the slope +of the exponential tail in the vanishing limit of the noise, which is defined and called the +essential barrier height, characterizing the barrier height of the potential landscape in the +global way. All these claims are justified both theoretically and numerically. In particular, +the numerical results for the loss surface of artificial neural networks are corroborated by +other studies in certain different ways. +The coupling scheme used in this paper is a combination of two coupling methods, the +reflection coupling and maximal coupling, for the purpose of coupling efficiency. We find +that the bound provided by the reflection-maximal coupling scheme is reasonably close to + +Training set +Small NN, bad local min +medium NN, local min +1r +-1 +0 +8o +8 +0.5 +00 +-0.5 +1.5 +-0.5 +1.5 +0 +0 +oo +0 +0 +0 +0 +1 +0 +00 +000 +0 +@o +0 +0o0 +-0.5 +0 +oo +0.5 +0.5 +0.5 +0.5 +0 +0 +00 +oo +oo +00 +1 +1 +-7 +-1 +-0.5 +0 +0.5 +1 +-1 +-0.5 +0 +0.5 +1 +-1 +-0.5 +0 +0.5 +> +Ground truth +Small NN, good local min +big NN, local min +-1 +1 +-1 +-0.5 +1.5 +-0.5 +1.5 +-0.5 +1.5 +0 +0 +1 +1 +0 +1 +0.5 +0.5 +0.5 +0.5 +0.5 +0.5 +0 +1 +1 +-0.5 +0 +0.5 +-1 +-0.5 +0 +0.5 +-1 +-0.5 +0.5 +-1 +1 +1 +040 +YAO LI, MOLEI TAO, AND SHIROU WANG +the optimal one (i.e., the one that makes the coupling inequality becomes an equality). +Although in this paper only the coupling time is examined, more information are expected +to be inferred from the coupling result in future work, e.g., the coupling location may pro- +vide us additional information about the landscape. In addition, this paper only considers +the tail of the coupling time distribution that is related to the principal eigenvalue of +the Fokker-Planck operator. The entire coupling time distribution, however, may provide +additional information on the non-principal eigenvalue of the Fokker-Planck operator. For +example, for a multi-well potential, the conditional coupling time distribution conditioning +on not being coupled in the deepest “well” may be related to the heights of lower barriers. +These could be potentially interesting future directions. +References +[1] Alan Agresti and Brent A. Coull, Approximate is better than ’exact’ for interval estimation of binomial +proportions, The American Statistician. 52 (1998), no. 2, 119–126. +[2] David Aldous, Random walks on finite groups and rapidly mixing Markov chains, S´eminaire de Prob- +abilit´es XVII 1981/82, Springer, 1983, pp. 243–297. +[3] Erich Baur, Metastabilit¨at von reversiblen diffusionsprozessen., Diploma thesis, Bonn University +(2011). +[4] Anton Bovier, Michael Eckhoff, V`eronique Gayrard, and Markus Klein, Metastability in reversible +diffusion processes. i. sharp asymptotics for capacities and exit times, J. Eur. Math. Soc. 6 (2004), +no. 4, 399—-424. +[5] Anton Bovier, V`eronique Gayrard, and Markus Klein, Metastability in reversible diffusion processes. +ii. precise asymptotics for small eigenvalues, J. Eur. Math. Soc. 7 (2005), no. 1, 69—-99. +[6] Lenaic Chizat, Edouard Oyallon, and Francis Bach, On lazy training in differentiable programming, +Advances in Neural Information Processing Systems 32 (2019). +[7] Anna Choromanska, Mikael Henaff, Michael Mathieu, G´erard Ben Arous, and Yann LeCun, The loss +surfaces of multilayer networks, Artificial intelligence and statistics, PMLR, 2015, pp. 192–204. +[8] Martin Day, On the exponential exit law in the small parameter exit problem, Stochastics 8 (1983), +no. 4, 297––323. +[9] Luc Devroye, Mehrabian Abbas, and Reddad Tommy, The total variation distance between high- +dimensional gaussians, arXiv:1810.08693 (2018). +[10] Felix Draxler, Kambis Veschgini, Manfred Salmhofer, and Fred Hamprecht, Essentially no barri- +ers in neural network energy landscape, International conference on machine learning, PMLR, 2018, +pp. 1309–1318. +[11] Weinan E, Weiqing Ren, and Eric Vanden-Eijnden, String method for the study of rare events, Physical +Review B 66 (2002), no. 5, 052301. +[12] +, Simplified and improved string method for computing the minimum energy paths in barrier- +crossing events, Journal of Chemical Physics 126 (2007), no. 16, 164103. +[13] Mark Freidlin and Alexander Wentzell, Random Perturbations of Dynamical Systems, Springer, 1998. +[14] Emmanuel Gobet and St´ephane Menozzi, Exact approximation rate of killed hypoelliptic diffusions +using the discrete euler scheme, Stochastic Processes and their Applications 112 (2004), no. 2, 201– +223. +[15] +, Stopped diffusion processes: boundary corrections and overshoot, Stochastic Processes and +their Applications 120 (2010), no. 2, 130–162. +[16] Victor Yakovlevich Ivrii, On the second term of the spectral asymptotics for the laplace–beltrami op- +erator on manifolds with boundary and for elliptic operators acting in fiberings, Doklady Akademii +Nauk, vol. 250, Russian Academy of Sciences, 1980, pp. 1300–1302. +[17] Arthur Jacot, Franck Gabriel, and Cl´ement Hongler, Neural tangent kernel: Convergence and gener- +alization in neural networks, Advances in neural information processing systems 31 (2018). +[18] Mark Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (2020), no. 4 Part II, 1–23. +[19] Stuart Kauffman and Simon Levin, Towards a general theory of adaptive walks on rugged landscapes, +J. Theor. Biol. 128 (1987), no. 1, 11–45. + +LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD +41 +[20] Kenji Kawaguchi, Jiaoyang Huang, and Leslie Pack Kaelbling, Every local minimum value is the global +minimum value of induced model in nonconvex machine learning, Neural Computation 31 (2019), +no. 12, 2293–2323. +[21] Paul Krugman, Complex landscapes in economic geography, Am. Econ. Rev. 84 (1994), no. 2, 412–416. +[22] Boris Leblanc, Renault Olivier, and Olivier Scaillet, A correction note on the first passage time of an +ornstein-uhlenbeck process to a boundary, Finance Stoch. 4 (2000), no. 1, 109–111. +[23] Yao Li and Shirou Wang, Numerical computations of geometric ergodicity for stochastic dynamics, +Nonlinearity 33 (2020), no. 12, 6935–6970. +[24] Torgny Lindvall, Lectures on the coupling method, Courier Corporation, 2002. +[25] Torgny Lindvall and L. C. G Rogers, Coupling of multidimensional diffusions by reflection, Ann. +Probab. 14 (1986), no. 3, 860—-872. +[26] Song Mei, Theodor Misiakiewicz, and Andrea Montanari, Mean-field theory of two-layers neural +networks: dimension-free bounds and kernel limit, Conference on Learning Theory, PMLR, 2019, +pp. 2388–2464. +[27] Morris Newman and John Todd, The evaluation of matrix inversion programs, J. Soc. Indust. Appl. +Math. 6 (1958), 466—-476. +[28] Esa Nummelin and Pekka Tuominen, Geometric ergodicity of harris recurrent markov chains with +applications to renewal theory, Stochastic Process. Appl. 12 (1982), no. 2, 187—-202. +[29] Gilles Pag`es and Fabien Panloup, Unajusted langevin algorithm with multiplicative noise: Total vari- +ation and wasserstein bounds, arXiv:2012.14310 (2020). +[30] Grant M Rotskoff and Eric Vanden-Eijnden, Trainability and accuracy of neural networks: An inter- +acting particle system approach, arXiv preprint arXiv:1805.00915 (2018). +[31] Justin Sirignano and Konstantinos Spiliopoulos, Mean field analysis of neural networks: A central +limit theorem, Stochastic Processes and their Applications 130 (2020), no. 3, 1820–1852. +[32] Mahdi Soltanolkotabi, Adel Javanmard, and Jason D Lee, Theoretical insights into the optimization +landscape of over-parameterized shallow neural networks, IEEE Transactions on Information Theory +65 (2018), no. 2, 742–769. +[33] Mei Song, Andrea Montanari, and P Nguyen, A mean field view of the landscape of two-layers neural +networks, Proceedings of the National Academy of Sciences 115 (2018), no. 33, E7665–E7671. +[34] David Wales, Energy landscapes: Applications to clusters, biomolecules and glasses, Cambridge Uni- +versity Press, 2003. +[35] Hermann Weyl, ¨Uber die asymptotische verteilung der eigenwerte, Nachrichten von der Gesellschaft +der Wissenschaften zu G¨ottingen, Mathematisch-Physikalische Klasse 1911 (1911), 110–117. +[36] Steve Zelditch, Spectral determination of analytic bi-axisymmetric plane domains, Geometric & Func- +tional Analysis GAFA 10 (2000), no. 3, 628–677. +Yao Li: Department of Mathematics and Statistics, University of Massachusetts Amherst, +Amherst, MA, 01002, USA +Email address: yaoli@math.umass.edu +Molei Tao: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332 +USA +Email address: mtao@gatech.edu +Shirou Wang: School of Mathematics, Jilin University, Changchun 130012, PRC, and De- +partment of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB +T6G 2G1, Canada +Email address: shirou@jlu.edu.cn + diff --git a/2dAzT4oBgHgl3EQfe_zq/content/tmp_files/load_file.txt b/2dAzT4oBgHgl3EQfe_zq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f9ce6e239311254affdbf4215d8d0fa0c7173878 --- /dev/null +++ b/2dAzT4oBgHgl3EQfe_zq/content/tmp_files/load_file.txt @@ -0,0 +1,1859 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf,len=1858 +page_content='LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD YAO LI, MOLEI TAO, AND SHIROU WANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' This paper proposes a probabilistic approach to investigate the shape of the landscape of multi-dimensional potential functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' By using an appropriate coupling scheme, two copies of the overdamped Langevin dynamics of the potential function are coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' For unimodal and multimodal potential functions, the tail distributions of the coupling times are shown to have qualitatively different dependency on the noise mag- nitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' More specifically, for a single-well potential function that is strongly convex, the exponential tail of the coupling time distribution is independent of noise and uniformly bounded away from zero by the convexity parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' while for a multi-well potential function, the exponential tail is exponentially small with respect to the noise magnitude with the exponents determined by the essential barrier height, a quantity proposed in this paper which, in a sense, characterizes the “non-convexity” of the potential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' This provides a promising approach to detect the shape of a potential landscape through the coupling time distributions which, in certain sense, shares the similar spirit with the well-know problem “Can one hear the shape of a drum?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' proposed by Kac in his fa- mous paper [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Theoretical results are numerically verified and discussed for a variety of examples in different contexts, including the Rosenbrock function, the potential of interacting particle systems, as well as the loss functions of artificial neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Introduction In 1966, a famous paper “Can one hear the shape of a drum?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' by Mark Kac investigates the possibility of inferring the shape of a domain from the spectral property of the Laplace operator defined on this domain [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Although it is eventually proved that the shape of a domain cannot be uniquely determined except a certain class of planar domain with analytic boundary [36], some information about the domain can still be inferred from the eigenvalue set of the Laplace operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' please refer to [35, 16] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Motivated by this spirit, the present paper investigates the approach of inferring the property of a multi-dimensional landscape by “knocking” it and “listening” the sound, where by “knocking” a multi-dimensional landscape, we mean to simulate a overdamped Langevin dynamics along the potential landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' More specifically, we run a large number of samples of two coupled trajectories of the overdamped Langevin dynamics to infer the landscape properties from the statistics of the coupling times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' By the coupling lemma, the tail distribution of the coupling times provides a lower bound of the spectral gap of the Fokker-Planck operator of the overdamped Langevin dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' In this way, our approach, similar to that in Kac’s paper, makes an attempt to establish a connection between the characteristics of the potential landscape and the spectral property of the corresponding operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' We are interested in the multi-dimensional potential functions that have finitely many local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Potential landscapes arise naturally in various areas [19, 21, 34] which can 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Primary 37H10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Secondary 60H10, 60J22, 60J60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Potential landscape, coupling method, overdamped Langevin dynamics, essen- tial barrier height, non-convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='01447v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='DS] 4 Jan 2023 2 YAO LI, MOLEI TAO, AND SHIROU WANG be of simple type that has only one equilibrium, or be of more complex types that support multiple basins of attractions with the emergence of many local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' To put in the mathematical way, let U be a smooth function on a regular domain D ⊆ Rk(k ≥ 1) with the local minima x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=', xL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Generically, under the deterministic negative gradient flow (ϕt)t≥0 of U, xi’s are the stable equilibria of ϕt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' For each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=', L}, call Bi = {x ∈ D : ϕt(x) → xi as t → ∞} the basin (of attraction) of xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' A smooth function U is called a single-well potential if it has only one local minimum x1 such that D = B1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' U is called a multi-well potential if 2 ≤ L < ∞ and D = �� 1≤i≤L Bi � � N, where N is a measure-zero set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' A multi-well potential is in particular called a double-well potential if L = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' This paper proposes a probabilistic approach to classify between the landscapes of single- and multi-well potential functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Our approach makes strong use of the coupling idea in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Given two stochastic processes X = {Xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' t ≥ 0} and Y = {Yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' t ≥ 0}, a coupling of X and Y is a stochastic process {(Xt, Yt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' t ≥ 0} such that (i) for any t > 0, Xt and Yt are respectively identically distributed with Xt and Yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' (ii) if Xs = Ys for certain s > 0, then Xt = Yt for all t ≥ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' The first meeting time of Xt and Yt, called the coupling time, is denoted by a random variable τc = inft≥0{Xt = Yt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' (1) A coupling {(Xt, Yt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' t ≥ 0} is said to be successful if τc < ∞ almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' In our setting, the two stochastic processes to be coupled, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=', X and Y as above, will be the overdamped Langevin dynamics of U given by the following stochastic differential equation(SDE) dZt = −∇U(Zt)dt + εdBt, (2) where {Bt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' t ≥ 0} is the Brownian motion, and ε > 0 scales the noise magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Through- out the paper, the following is always assumed1 for the potential function U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' (U1) The potential function U ∈ C3(D), where D is open, convex and connected, such that limx→∂D U(x) = ∞, and if D is unbounded, it further holds that limx→∂D |∇U| = ∞, limx→∂D |∇U(x)| − 2∆U(x) = ∞, where | · | denotes the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' To couple two stochastic processes, various coupling methods can be used in the different contexts [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' In this paper, to achieve numerical efficiency, a mixture use of the reflection and maximal coupling methods is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' More specifically, with certain threshold dis- tance d > 0, the coupling (Xt, Yt) is switched between the reflection and maximal couplings according to whether the distance |Xt − Yt| is greater than d or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' To be more precise, let (Xt, Yt) evolve according to the reflection coupling if |Xt − Yt| > d and be switched to the maximal coupling whenever |Xt − Yt| ≤ d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' see Section 2 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' We call such mixed use of the reflection and maximal coupling methods the reflection-maximal coupling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' How can the threshold d be chosen?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' In our numerical scheme based on the reflection- maximal coupling, d will be closely related to the time step size h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' More precisely, denote ˆ Xh = { ˆXh n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' n ≥ 0} (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' ˆY h = { ˆY h n ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' n ≥ 0}) the Euler–Maruyama scheme of X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Y ) with the time step size h, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=', ˆXh n = ˆXh n−1 − ∇U( ˆXh n−1)h + ε √ hNn (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' ˆY h n = 1In the single-well setting, (U1) ensures the existence of a global strong solution of (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' in the multi-well setting, further assumptions on the finiteness and non-degeneracy on the saddle points and local minima, as stated in (U2) or (U3)(iii), guarantees this [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD 3 ˆY h n−1−∇U( ˆY h n−1)h+ε √ hNn), where {Nn} are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='d standard normal random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' The threshold distance d is chosen to be proportional to the (directional) standard deviations of the distribution of the random variable ε √ hNn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=', d = O(ε √ h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' This guarantees a sufficient overlap between the distributions of ˆXh n and ˆY h n if, assume at the previous step, | ˆXh n−1 − ˆY h n−1| < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Then under the reflection-maximal coupling scheme, ( ˆXh n, ˆY h n ) is a maximal coupling and the probability P[ ˆXh n = ˆY h n ] is in order O(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='3 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Henceforth, by an h-reflection-maximal coupling we put particular emphasis on the choice of the time step size h, and by a reflection-maximal coupling (without h) we mean that the coupling is implemented under the reflection-maximal coupling scheme with a generally small h and no further emphasis on its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' We investigate how the exponential tail of the coupling time distributions of the over- damped Langevin dynamics depends on the noise magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Our main message is, this dependency will be both quantitatively and qualitatively different between potential func- tions with only a single well and those with double or multiple wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' More specifically, we prove that under the reflection-maximal coupling scheme, for a strongly convex single-well potential (see (3) below), the exponential tail of P[τc > t] is uniformly bounded away from zero independent of ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' For double- or multi-well potential functions, the exponential tail, however, is exponentially small with respect to the noise magnitude;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' A single-well potential function U is said to be strongly convex with constant m0 > 0 if ⟨∇U(x) − ∇U(y), x − y⟩ ≥ m0|x − y|2, ∀x, y ∈ D, (3) where ⟨·, ·⟩ denotes the standard inner product in Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' The supremum of all positive m0 satisfying (3) is called the convexity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Throughout this paper, m0 always denotes the convexity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Let U be a single-well potential satisfying (U1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Assume that U is strongly convex with constant m0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Then, given any δ > 0 there exists h0 > 0 such that for any h ∈ (0, h0), if {(Xt, Yt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' t ≥ 0} is an h-reflection-maximal coupling of two solutions of (2) satisfying E[|X0 − Y0|] < ∞, for all ε > 0, it holds that lim sup t→∞ 1 t log P[τc > t] ≤ −m0 + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' In the situation that the potential function U is double-well with only two local minima, a crucial quantity is the least barrier height to be passed by any continuous path connecting the two local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Given any two subsets A, B ⊆ D, define the communication height between A and B as Φ(A, B) = inf φ(t)∈C([0,1],D), φ(0)∈A, φ(1)∈B supt∈[0,1] U(φ(t)), (4) where the infimum runs over all the continuous paths in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' For a double-well potential U with two local minima x1, x2, define the essential barrier height HU = min � Φ(x1, x2) − U(x1), Φ(x1, x2) − U(x2) � , (5) the lower height of the two barriers that has to be crossed from one local minimum to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' In the double-well setting, the following potential functions, which are generic in certain sense, are to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' 4 YAO LI, MOLEI TAO, AND SHIROU WANG (U2) Let U : D → R be a double-well potential function satisfying (U1) with two local minima x1, x2 such that (i) The communication height between x1 and x2 is reached at a unique saddle point z∗(x1, x2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=', U(z∗(x1, x2)) = Φ(x1, x2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' (ii) U is non-degenerate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=', the Hessian of U has only non-zero eigenvalues) at the two local minima x1, x2, and the saddle point z∗(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Besides assumptions for potential functions, in the double-well, and more generally, the multi-well settings, the coupling scheme is also required to satisfy certain intuitive condi- tions which, in particular for the reflection-maximal coupling scheme, can be numerically verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' In the double-well setting, certain“local” coupling properties are assumed when the two processes are lying in the same basin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' see (H1)-(H2) in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' To include all possible initial conditions, in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='3, the coupling process is assumed to be initially related with all local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' A probability measure µ on D × D is said to be fully supported (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='t all the local minima) if for any δ > 0, µ(Bδ(xi) × Bδ(xj)) > 0, ∀i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=', L}, where Bδ(x) denotes the ball centered at x with radius δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Note that any probability measure equivalent with the Lebesgue measure is fully supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' A coupling (X, Y ) is said to be fully supported if the distribution of (X, Y ) is fully supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Similarly in the same way, a probability measure µ on D is said to be fully supported if for any δ > 0, µ(Bδ(xi)) > 0, ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=', L}, and a random variable X is said to be fully supported if the distribution of X is fully supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Throughout this paper, by a ≲ b (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' a ≳ b) we mean a ≤ const·b (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' a ≥ const·b), where the const is independent of any parameter of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' By a ≃ b we mean both a ≲ b and a ≳ b hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Let U be a double-well potential satisfying (U2), and {(Xt, Yt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' t ≥ 0} be a coupling of two solutions of (2) such that (X0, Y0) is fully supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' Then, if (Xt, Yt) is a reflection-maximal coupling satisfying (H1)-(H2), for any ε > 0 sufficiently small, it holds that lim t→∞ 1 t log P[τc > t] ≃ −C0e−2HU/ε2, where HU is defined in (5), and the constant C0 > 0 is independent of ε and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' The result in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='2 can be generalized to potential functions that have any finitely many local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' In this case, besides the uniqueness of saddle points and the degeneracy conditions in (U2), the potential function U is also assumed to satisfy a generic condition that U has different potential values and depths corresponding to the different local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' (U3) Let U : D → R be a multi-well potential function satisfying (U1) with the local minima x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=', xL such that (i) U has different potential values at the different local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' In particular, U admits the unique global minimum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' LANDSCAPE CLASSIFICATION THROUGH COUPLING METHOD 5 (ii) The different basins of potential U admit different depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=' More precisely, there exists some δ > 0 such that the L local minima of U, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfe_zq/content/2301.01447v1.pdf'} +page_content=', xL, can be labeled in such way that Φ(xi, Ni−1) − U(xi) ≤ min1≤ℓ 180◦—the cooler morning limb—are as- +signed temperature Twest. Pressures in the atmosphere +run as low as 1 µbar, as one of the benefits of HRCCS +is that it can probe low pressures such as these (e.g., +Kempton et al. 2014; Gandhi et al. 2020; Hood et al. +2020). The bottom of the atmosphere is set at 0.5 bar; +our previous 3D forward models run in Savel et al. (2022) +across the optical and near-infrared indicate that for our +test case of WASP-76b (West et al. 2016), this region +is the deepest that can be probed given the expected +continuum opacity. The parameterized modeled atmo- +spheres in this study have no set wind fields, as in our +models (motivated by and assuming chemical equilib- +rium), winds do not control AWE—only the chemical +abundance of a given cell does. +We calculate AWE to assess the relative strength of the +scale height and equilibrium chemistry effects. To infer +the strength of the scale height effect, we construct pairs +of model atmospheres. In each pair, one atmosphere is +constructed self-consistently: pressure falls off per hy- +drostatic equilibrium, with the scale height set by the +temperature on either limb. For the models here, we +hold composition constant across both limbs, thereby +holding µ constant at 2.36 (appropriate for a solar- +composition gas dominated by molecular H2; Kempton +& Rauscher 2012). See Section 2.4 for a discussion of this +caveat. The other atmosphere in the pair is constructed +on the same pressure grid as the western limb at all lon- +gitudes. That is, the eastern limb is not simulated as +inflated compared to the western limb—removing the +scale height effect from the projected model atmosphere +in transmission. +2.3. Equilibrium chemistry +To calculate the number densities of our species in +each modeled atmospheric cell (nα), we construct a grid +in temperature–pressure space using the FastChem equi- +librium chemistry code (Stock et al. 2018) and interpo- +late the grid based on local atmospheric cell temper- +ature and pressure. +We initialize the code with solar +abundances from Lodders (2003). Our chemistry code +does not explicitly include any condensation or cloud- +formation processes. +Even disregarding questions of species detectability in +HRCCS data, it is worth considering that not all species +with FastChem thermochemical data have freely avail- +able opacity data. With this constraint in mind, we re- +strict our AWE molecule calculations to molecules with +opacity data available on ExoMol,2 a popular opacity +database for exoplanet atmosphere modeling. +2.4. Asymmetry metric: application +We calculate AWE for our grid of parameterized at- +mospheres. Disregarding the scale height effect, we find +that positive ions tend to form preferentially on the +hotter limb of our models at an equilibrium tempera- +ture of 2200 K (Figure 1). This is expected, as ther- +mal ionization should increase the abundance of positive +ions at higher temperatures. Furthermore, larger east– +west temperature asymmetries lead to larger abundance +asymmetries. +Including the scale height effect increases the asymme- +try for neutral atoms and molecules, as can be seen by +comparing the right-hand sides of Figures 1–2. Further- +more, there is more homogeneity across the AWE values +2 https://www.exomol.com/ + +6 +Savel et al. +(a) +(b) +Figure 1. Asymmetry (as defined in Equation 1) of all chemical species considered in this study in our parameterized at- +mospheres at an equilibrium temperature of 2200 K. These models do not self-consistently inflate the hotter limb of the +parameterized model (i.e., they do not observe the “scale height effect”). The shading of each species represents the normalized +temperature difference, ˜∆T, across the two limbs of our parameterized atmospheres; the lightest boxes have ˜∆T = 0.1, whereas +the darkest have ˜∆T = 0.6. For illustrative purposes, we color in green tick marks for species with detections noted in Guillot +et al. (2022) (and including the recent CO2 detection; Ahrer et al. 2022). We also draw a vertical line denoting 0 asymmetry. +Without taking the scale height effect into account, positive ions form much more predominantly on the warmer limb (i.e., have +negative asymmetry) than other species and reach the greatest asymmetry values. +across positive ions, negative ions, neutral atoms, and +neutral molecules (Figure 2). In particular, while higher +˜∆T still implies higher absolute asymmetry in neutral +species, the scale height effect makes it such that the +warmer limb almost always has higher projected asym- +metry. +It is therefore clear that the scale height effect strongly +tamps down genuine variation in species abundance due +to equilibrium chemistry. However, the fact that inter- +species variation in asymmetry remains implies that this +variation in abundance is not completely washed out by +the scale height effect; if the scale height effect truly and +fully dominated, all species would have the same AWE +value. +When considering individual species more closely, we +find that certain species are particularly differentially +affected by the scale height effect. +For example, Fig- +ure 3 shows that there is a stark difference in whether +the scale height effect is included for Fe. However, this +is not as much the case for, e.g., Sr II. The meaning be- +hind this result is evident in the equilibrium abundance +calculations of Fe and Sr II: Fe is less sensitive to tem- + += 2200K, scaled=False +JS +Rb +Ge +Ga +Sc - +Ca +Li - +Zn +V- +Ti +si - +s +p- +0 +-IN +Na +N- +K +H- +Fe +Cu +Cr : +Co +CI - +C +Al - +Zr IIII +YIII +Sr III +Rb IlI +Ge llI +Ga IlI +Zn III +Cu llI +Ni III +Co IIII +Fe IIII +Mn III +Cr IIII +VII +Ti II +Sc III +S +Ca II +e +KII +Ar IIII +Positive-ions +CI II +ads +S III +Negative ions +P III +Si III +Al II +Mg IIII +Na III +Nelll +F III +O I=I +N III +C III +Li I +Rb II +Ge ll +Ga lI +Li II +Zn II +Si lI +s II +PII +Ni lI +Ne ll +Na II +NII +Mg II +KII +He lI +H3O lI +H2 II +HO II +HII +F II +Cu II +Co II +CI II +C II +Ar II +AIlII +ScllI +YII +Sr II +Zr II +VII +CrlI +Mn II +Call +Ti lI +Fel +-102 +-101 +-100 +100 +101 +AsymmetryTeg = 2200K, scaled=False +sis +PS +SO3 +SiO2 +SO2 +02 +sio +PO +N20 +N2 +NS +PN +NO +SiH2 +H202 +TiH +HSi +SH +HP +NiH +NaH +HNO3 +HN +MgH +HKO +FeH +PF3 +NaF +FMg +HF +CrH +CINa +CIK +CIH +Cao +CaH +CaF +C2H4 +C2 +CS +CP +COS +CN +CH3 +S +CH20 +e +CH +AIO +AlH +AIF +AICI +OH +ov +S +O!! +PH3 +NH3 +H2S +H20 +CO2 +CO +CH4 +C2 H2 +H2 +Zn +s +P +Ne +N +Ge +F +Cl +Ar +Co +Al +Z +Sc +Y +Sr +Rb +Ga +Cu +IS +0 +Li +Mn +Ni +Cr +c +IL +Ca +Neutral atoms +Mg +Fe +Na +Neutral molecules +He +H +-102 +-101 +-100 +100 +101 +0 +AsymmetryDiagnosing limb asymmetries in transmission +7 +(a) +(b) +Figure 2. Similar to Figure 1, but now including the scale height effect (inflating the hotter limb in our parameterized models). +Now, all species have asymmetries that favor the hotter limb (negative asymmetry)—simply because the hotter limb subtends +more solid angle on the sky. However, there still exists inter-species variability in asymmetry, implying that the scale height +effect does not entirely swamp genuine differences in equilibrium chemistry across limbs. Furthermore, negative ions still have +larger asymmetries than positive ions or neutral species. +perature variations than Sr II. This result is expected, +as the onset of Sr II is determined by the temperature +at which Sr I can be effectively ionized. This is gen- +erally the case for positive ions—the temperature effect +on chemical abundance wins out over the scale height ef- +fect, as seen by the left-hand sides of Figures 1–2. Physi- +cally, this behavior is because the Saha equation is more +strongly dependent on temperature than most chemical +equilibrium reaction rates. +The results of this experiment indicate that the most +temperature-sensitive species are strongly influenced by +both abundance changes and scale height differences. +Conversely, to isolate the scale height effect, it would be +therefore useful to consider a species with very weakly +temperature-dependent abundance; in this case, if a +strong asymmetry were detected, it could be attributed +to a scale height effect (or other non-equilibrium chem- +istry or physics). We explore this idea further in Sec- +tion 3. +Note that this approach, aside from its simplified +temperature–pressure structure, does not account for a +variety of physics. Namely, it does not include the ef- +fects of hydrogen dissociation and recombination that +occurs in the ultra-hot Jupiter regime (Tan & Komacek +2019). Inclusion of this physics would serve to decrease +the mean molecular weight in the atmosphere, increasing +the scale height for the hotter, eastern limb, thereby am- +plifying the observed asymmetry. Additionally, at the + += 2200K, scaled=True +Zr - +Sr +Rb +Ge +Ga +Sc - +Ca +Li - +Zn +V- +Ti +Si - +s +p- +0 +-IN +Na +N- +K +H- +Fe +F . +no +Cr: +Co +CI - +c +Al - +Zr IIII +Y III +Sr III! +Rb II +Ge llI +Ga IlI +Zn III +Cu llI +Ni III +Co IIII +Fe IIII +Mn IIII +Cr III +VII +Ti II +Sc III +S +Ca llI +e +KIII +Ar IIII +Positive-ions +CI IIII +S III +PIII +Negative ions +Si III +Al III +Mg IIII +Na II +Ne III +FII +o II +NIII +C II +Li lII +Rb II +Ge ll +Ga lI +LilI +Zn II +Si lI +sII +PII +Ni lI +Ne lI +Na llI +NII +Mg II +KII +He lI +H3O II +H2 II +HO II +H II +F II +Cu lI +Co II +CI II +C II +Ar II +AIlII +ScllI +YII +SrlI +ZrlI +viII +CrlI +Mn II +Ca lI +Ti II +Fe ll +-102 +-100 +100 +101 +-101 +0 +Asymmetry= 2200K, scaled=True +sis +PS +SO3 +SiO2 +Neutralatoms +SO2 +02 +Neutral-molecules +sio + PO +N20 +N2 +NS +PN +NO +SiH2 +H202 +TiH +HSi +SH +HP +NiH +NaH +HNO3 +NH +H6W +HKO +FeH +PF3 +NaF +FMg +HF +CrH +CINa +CIK +CIH +Cao +CaH +CaF +C2H4 +C2 +cs +CP +COS +CN +CH3 +S +CH20 +CH +AIO +AlH +Speo +AIF +AICI +HO +ov +S +O!! +PH3 +NH3 +H2S +H20 +CO2 +co +CH4 +C2 H2 +H2 1 +Zn +s +P +Ne +N +Ge +F +C1 +Co +Al +Z +Sc +Y +Sr +Rb +Ga +Cu +Si +0 +Li +Mn +Ni +Cr +c +V +IL +Ca +Mg +Na +K +He +H +-102 +100 +-101 +101 +-100 +0 +Asymmetry8 +Savel et al. +Figure 3. Asymmetry (per Equation 1) for Sr II, Fe, H2O, and CO in our parameterized atmospheres. Our grid sweeps over +equilibrium equilibrium temperature and normalized temperature difference across limbs, and includes models that observe the +scale height effect (circles) and do not (squares). We find that species with strong temperature-dependent abundances (e.g., +Sr II) are less dominated by the scale height effect than species with weaker temperature-dependent abundances. +lower-temperature end, we did not include the effects of +certain species being sequestered into clouds (e.g., sili- +cate clouds). We will model the Doppler shift impact +of optically thick clouds in Section 3.1.2. Finally, our +approach does not include disequilibrium effects (e.g., +vertical / horizontal mixing) that may alter asymme- +tries. Therefore, the results shown here motivate asym- +metries due to equilibrium chemistry alone, which we +expect to be a first-order driver of asymmetry; disequi- +librium chemistry is not expected to be significant in the +ultrahot Jupiter regime (e.g., Tsai et al. 2021). +We further did not include the effect of temperature- +and pressure-dependent opacities. +At the spectrum +level, a temperature asymmetry would be exaggerated +by the fact that, e.g,. Fe absorbs more on the hotter +limb than the colder limb because its opacity increases +with temperature. This would mean that the detected +net Doppler shift is even more strongly weighted to the +hotter limb. +Despite these limitations in our modeling, the trends +listed above should hold to first order and provide intu- +ition about the relative strengths of two potential drivers +of asymmetry in exoplanet atmospheres. +Broadly, it +holds that the scale height effect appears to dominate in +general, but relative differences in abundances of species +as a function of temperature still matter. Given the lim- + +Al +CO +0.6 +101. + No scale height effect +101. +· Scale height effect +least +口 +口 +0.5 +口 +口 +0 +0 +口 +口 +口 +口 +C +口 +口 +0.4 +O +0 +-101. +-101. +1000 +2000 +3000 +1000 +2000 +3000 +T +H20 +SrlII +101 +101. +0.3 +口 +口 +least +口 +口 +口 +口 +口 +-0 +0 +Iwest +口 +0.2 +口 +口 +口 +口 +口 +口 +口 +08 +口O +0000 +: +DO +00 +000 +-101. +-101. +0.1 +1000 +2000 +3000 +1000 +2000 +3000 + (K) +Teq (K) +2Diagnosing limb asymmetries in transmission +9 +itations of simple models, we will move on to more self- +consistent atmospheric modeling in the following sec- +tions. +3. SELECTED DIAGNOSTICS +3.1. Diagnostics for specific mechanisms +Per Section 2, even differentiating between two drivers +of asymmetry in exoplanet atmospheres is nontrivial. +Drivers can compete to varying degrees to produce a +similar result: an asymmetric trend in net Doppler shifts +in HRCCS. +However, +by exploiting nuances in the HRCCS +Doppler shift signal and by independent means, it may +be possible to disentangle even drivers that produce sim- +ilar effects. Table 1 lists example drivers of asymmetries +in HRCCS and how they might be diagnosed. The asso- +ciated works listed in the table may not directly propose +these diagnostics, but at minimum they provide founda- +tional material for them. +Of course, exhibiting a single diagnostic does not not +mean that a given physical mechanism is in play. Other +mechanisms could surely be present, and uniquely con- +straining a single mechanism as dominant would require +ruling out the others, as well. For instance, both day– +night winds and morning limb condensation could result +in a net blueshifted CCF. But if, for example, a night- +side temperature were derived from a phase curve that +was far too hot for any known condensate to form, then +day–night winds would be much preferred to conden- +sation as a physical solution. Together, collections of +diagnostics are hence able to test the dominance of in- +dividual mechanisms. +In the following sections, we explore a few tests for +specific physical mechanisms of asymmetry: using CO +as a baseline molecule to identify the scale height effect +and tracking the blueshifts of multiple species to identify +the presence of clouds. We furthermore evaluate the ef- +fectiveness of diagnostics that may be used to evaluate +a number of different mechanisms: averaging HRCCS +data into two phase bins and using finely phase-resolved +HRCCS data. We additionally show how these diagnos- +tics can further motivate or rule out “toy models” that +at first may appear convincing. +3.1.1. CO as a baseline molecule +We have demonstrated (Section 2.4) that species +with strongly temperature-dependent abundances are +the least susceptible to the scale height effect. +Con- +versely, observing a species with very weak temperature- +dependent abundance could indicate whether the scale +height effect is in play. +Figure 4. Volume mixing ratio of CO as a function of pres- +sure and temperature as calculated by FastChem. Overplot- +ted are the onset of ultra-hot Jupiters (as defined by their +dayside temperature; Parmentier et al. 2018), the CO/CH4 +equivalency curve from Visscher (2012) as a function of pres- +sure, the Fe condensation curve from Mbarek & Kemp- +ton (2016), and 1D temperature–pressure profiles for a hot +Jupiter (WASP-39b) and an ultra-hot Jupiter (WASP-18b) +as computed with HELIOS (Malik et al. 2017). Both the con- +densation curve and the equivalency curve are computed at +solar metallicity. Considering the regime of ultra-hot Jupiter +atmospheres, CO is a relatively stable chemical species. +Consider CO. In Figure 3, its AWE values are clustered +around 0 without the scale height effect, with relatively +weak dependence on ˜∆T. However, CO’s AWE values +are strongly negative when the scale height effect is in- +cluded. We propose using CO as a tracer of the scale +height (and other chemistry-unrelated) effects. +As shown in Figure 4, the abundance of CO is +relatively stable between 1000 K and 3500 K. Beltz +et al. (2022) note that this stability holds over the +temperature–pressure range of the observable atmo- +sphere of the ultra-hot Jupiter WASP-76 b. +Indeed, +this feature remains true over the general temperature– +pressure range of ultra-hot Jupiters. For illustrative pur- +poses, we calculate the 1D temperature–pressure pro- +files of a hot Jupiter (WASP-39b; Faedi et al. 2011) and +an ultra-hot Jupiter (WASP-18b; Hellier et al. 2009). +These profiles, calculated with the HELIOS 1D radiative- +convective model (with full heat redistribution), indi- +cate that the observable atmosphere for these planets is +largely within a region of near-constant CO mixing ratio. +The stability of CO is due to three factors: its strong +chemical bonding, its lack of participation in gas-phase +chemical reactions, and its lack of condensation. + +10-6 +-3 +Ultra-hot Jupiter +dayside onset +-4 +CO/CH4 equivalency +10-5 +Fe condensation +-5 +WASP-18 b 1D profile +Pressure (bars) +WASP-39 b 1D profile +-6 +10-4 +(xp) 0x +-7 +10-31 +-8 +-9 +10-21 +-10 +10-1 +-11 +-12 +100 +1000 +2000 +3000 +4000 +5000 +Temperature +(K)10 +Savel et al. +Since the strong triple bond of CO makes it diffi- +cult to thermally dissociate, CO remains stable at tem- +peratures that would dissociate molecules with weaker +bonds, such as H2O (Parmentier et al. 2018), which +has two single bonds. +Beyond roughly 3500 K, even +the triple bond becomes susceptible to thermal dissocia- +tion; hence, the few exoplanets with significant portions +of their atmosphere hotter than this temperature (e.g., +KELT-9b, with Teq ≈ 4050 K; Gaudi et al. 2017) would +likely exhibit spatial variation in CO abundance. Most +ultra-hot Jupiters, though, should fall shy of this regime. +Furthermore, the high photoionization threshold of CO +(relative to, e.g., H2O; Heays et al. 2017) means that +it is not commonly photodissociated (Van Dishoeck & +Black 1988). +Even when it is photodissociated, recy- +clying pathways exist in hot Jupiters that can replenish +CO abundance, keeping it near equilibrium abundance +even inclusive of photochemistry (Moses et al. 2011). +Hence, the assumption of non-dissociation of CO is rea- +sonably justified across much of the ultra-hot Jupiter +population. +Additionally, CO does not commonly participate in +thermochemical reactions and is the dominant car- +bon carrier in our temperature–pressure range of inter- +est. While at lower temperatures the dominant carbon +carrier becomes CH4, the ultra-hot Jupiter regime is +squarely beyond the CO/CH4 equivalency curve (Fig- +ure 4; Visscher 2012). Therefore, even aside from ther- +mal dissociation, CO should not participate in gas-phase +thermochemistry that would alter its abundance. +Finally, CO does not form any high-temperature con- +densates expected in ultra-hot Jupiter atmospheres. +The condensation temperature of CO (≈80 K at 1 bar; +Lide 2006; Fray & Schmitt 2009) is far below the +temperature–pressure range of ultra-hot Jupiters. This +quality makes CO a less complicated tracer of, e.g., at- +mospheric dynamics than species that do condense in +this region of parameter space, such as Fe, Mg, or Mn +(Mbarek & Kempton 2016). Therefore, while the calcu- +lations of Figure 4 do not include gas-phase condensa- +tion, the resultant spatial constancy of CO should still +be robust even when condensation is considered. CO +is thus a more straightforward molecule to model than +other, condensing species, as it does not participate in +the complex microphysics of condensation and cloud for- +mation (see, e.g., Gao et al. 2021). +Beyond its spatial uniformity, there are further obser- +vational reasons that CO is an appealing species to tar- +get. Namely, CO has very strong spectroscopic bands +placed across the infrared wavelength range (e.g., Li +et al. 2015) that do not overlap with other strong ab- +sorbers and are relatively well understood (Li et al. +2015). +Additionally, the high cosmic abundance of C +and O (Lodders 2003) means that, unlike many of the +species in the previous section, CO is readily detectable +(and has been become a standard detection in HRCCS; +Snellen et al. 2010; de Kok et al. 2013; Rodler et al. +2013; Brogi et al. 2014, 2016; Flowers et al. 2019; Gia- +cobbe et al. 2021; Line et al. 2021; Pelletier et al. 2021; +Zhang et al. 2021; Guilluy et al. 2022; van Sluijs et al. +2022). +Given its stability and observational advantage, we +propose that CO can be used as a faithful tracer of the +atmosphere—whether it is inflated in some regions, what +its wind profile is, whether regions are blocked by clouds, +etc. In turn, CO may then be leveraged to better mo- +tivate sources of asymmetry that affect other species. +While other species with low AWE in Figure 1 (e.g., He, +Fe, MgH, Rb II) would also appear to be good candi- +dates for baseline species, these species are either largely +spectroscopically inactive, have variable abundance over +broader temperature–pressure ranges, or can condense. +A caveat to the above is that while CO is a faithful longi- +tudinal tracer, it is not an unbiased radial tracer (as seen +in Figure 4). As with all chemical species, CO has its +own balance between deep and strong lines that depends +on the waveband considered (see, e.g., Section 3.3.1). +Therefore, the net CO Doppler signal does not uniformly +weight the wind profile across all altitudes. Again, this +is a bias inherent to all chemical species. +3.1.2. A decreasing blueshift test for clouds +As noted in Table 1, clouds may introduce strong +asymmetry into HRCCS data. +Savel et al. (2022) +demonstrated that gray, optically thick clouds produce +stronger blueshifts in the Doppler shift signal of WASP- +76b than the blueshifts in clear models, also changing +the trend of Doppler shift with phase. +But, again as +shown in Table 1, these changes at the Doppler shift +level are not sufficient to uniquely identify clouds as the +driver of an observed asymmetry. Combinations of ob- +servable quantities that would uniquely identify clouds +as the source of observed HRCCS asymmetry are there- +fore necessary. +To devise such a test, we investigate in this work a +limiting-case cloudy model. As in Savel et al. (2022), +we construct gray, optically thick, post-processed clouds +in our 3D ray-striking code. We here make another as- +sumption, though: that the clouds are confined to the +cooler, morning limb, as opposed to having a distribu- +tion dictated by a specific species’ condensation curve. +This distribution is based on planetary longitude (be- +tween longitudes of 180◦ and 360◦). This approach is +motivated by the results of Roman et al. (2021), who + +Diagnosing limb asymmetries in transmission +11 +found that a subset of cloudy GCMs exhibited a cloud +distribution strongly favoring the western limb.3 +Our +approach benefits from providing limiting-case intuition +for how cloudiness affects Doppler shift signals while +avoiding the complex questions of how clouds form and +which species contribute the most opacity (Gao et al. +2021; Gao & Powell 2021). +Briefly, our modeling methodology is as follows: +1. Double-gray, two-stream GCM. GCMs such as this +one solve the primitive equations of meteorology, +which are a reduced form of the Navier-Stokes +equations solved on a spherical, rotating sphere +with a set of simplifying assumptions.4 The out- +put of these models is temperature, pressure, and +wind velocity as a function of latitude, longitude, +and altitude. We use the GCM that was shown +to best fit the Ehrenreich et al. (2020) WASP-76b +data in Savel et al. (2022). +2. Equilibrium chemistry with FastChem. As in Sec- +tion 2.3, we interpolate a model grid of chem- +istry to determine local abundances of a number +of chemical species as determined by temperature +and pressure conditions of the GCM output. +3. Ray-striking radiative transfer. +Using a code +modified from Kempton & Rauscher (2012) (as +detailed in Savel et al. 2022), we compute the +high-resolution absorption spectrum of our plan- +etary atmosphere by calculating the net absorp- +tion of stellar light along lines of sight through +our GCM output. This absorption is calculated +inclusive of net motions along the lines of sight +from atmospheric winds and rotation, inducing +Doppler shifts relative to that of a static atmo- +sphere’s spectrum. Limb-darkening is calculated +with a quadratic limb-darkening law in the observ- +able planetary atmosphere and with the batman +code (Kreidberg 2015) for the portion of the star +blocked by the optically thick planetary interior. +Given its increasing utility as a benchmark planet for +HRCCS studies (e.g., Ehrenreich et al. 2020; Kesseli & +3 These GCMs produced clouds on a temperature–pressure basis, +and did not model clouds as tracers. +Therefore, they do not +capture potential disequilibrium cloud transport (e.g., as done in +Komacek et al. 2022), which may alter the degree of patchiness +within the cloud deck. +4 These assumptions are 1) local hydrostatic equilibrium, such that +vertical motions are caused purely by the convergence and di- +vergence of horizontal flow, 2) the “traditional approximation,” +which removes the vertical coordinate from the Coriolis effect, +and 3) a thin atmosphere. +Figure 5. Atmospheric Doppler shifts, which should remain +in the HRCCS signal after the orbital motion is subtracted, +as a function of orbital phase for our forward models. Shown +are representative species that span Doppler shifts and are +noted as potentially observable by Kesseli et al. (2022): Fe, +Sr II, and Sc. Cloud-free models are represented with solid +lines, whereas models with fully cloudy morning limbs are +represented with dashed lines. The first half of transit (RV1) +and second half of transit (RV2) Doppler shifts for Fe from +Kesseli et al. (2022) are overplotted as horizontal lines, with +width determined by observational errors. Our cloudy mod- +els are much more strongly blueshifted than their cloud-free +counterparts, become less blueshifted over transit, and do +not have significant CCF peaks at early phases. +Snellen 2021; Landman et al. 2021; Seidel et al. 2021; +Wardenier et al. 2021; Kesseli et al. 2022; S´anchez-L´opez +et al. 2022), we model the ultra-hot Jupiter WASP-76b +(West et al. 2016). We calculate 25 spectra inclusive of +Doppler effects equally spaced in phase from the begin- +ning to end of transit. For our cross-correlation tem- +plate, T , we use a model that does not include Doppler +effects. +We then cross-correlate our template against our cal- +culated spectrum, y: +c(v) = +N +� +i=0 +yi(λ)Ti(v, λ), +(4) +where the mask or template is Doppler-shifted by ve- +locity v and interpolated onto the wavelength grid, λ, +of y for summing. Our CCF is computed on a grid of +velocities from −250 km s−1 and 250 km s−1 with a step +of 1 km s−1. The final net planet-frame Doppler shift is +calculated by fitting a Gaussian to the peak of the CCF. +The results of our experiment are shown in Figure 5. +When we allow clouds to extend over the entire morn- +ing limb, note that all species become less blueshifted +over time. Because the limb that is rotating away from +the observer (the “receding limb”) is entirely blocked off +by clouds, there is no wavelength-dependent absorption + +2 +Kesseli+22 Fe RVi +Sc +Kesseli+22 Fe RV2 +Sr plus +Planet-frame RV (km/s) +0 +Fe +-2 +6 +8 +一10 +12 +-15 +-10 +-5 +0 +5 +10 +15 +Phase (degrees)12 +Savel et al. +for that limb. Therefore, the contribution of redshift- +ing from solid-body rotation on the receding limb is not +present—the only Doppler shift contributions are from +evening limb rotation and evening limb winds, which are +generally in the same direction as the rotation. Hence, +there are much stronger blueshifts at earlier phases than +in the clear models. +However, at later phases, the non-cloudy regions of the +atmosphere rotate into the receding limb, thereby con- +tributing some rotational redshift to the net Doppler +shift signal.5 +At the earliest phases, the cloudy mod- +els do not have enough wavelength-dependent absorp- +tion to produce a significant CCF peak. +Notably, all +species follow this trend, as the blocking of clouds as +modeled here is wavelength-independent and altitude- +independent. +This behavior is shown in Figure 5 for +Fe, Sc, and Sr II—all species identified in Kesseli et al. +(2022) has having high potential observability for ultra- +hot Jupiters. +From Figure 5, it is also apparent that the cloud- +driven trend of decreasing blueshift in phase is not +matched by the observations of Kesseli et al. (2022). +As found in Savel et al. (2022) in comparison to the +Ehrenreich et al. (2020) data, while the absolute magni- +tude of the cloudy model’s Doppler shift better match +the data than the clear model’s, the cloudy model trend +over Doppler shift is not matched by the data. In sum, +this limiting-case model of opaque, morning limb clouds +does not appear to be a first-order effect driving ex- +isting observational trends. +This does not necessarily +mean that clouds are not the driving factor behind limb +asymmetries; it may simply be that a more physically +motivated model for partial cloud coverage of the limb +could fit the available data better. +Also of note in Figure 5 is that the egress signatures +of the clear and cloudy models are quite distinct. Near a +phase of roughly 14 degrees, the clear model produces a +sharp change in Doppler shift for all species as the lead- +ing (rotationally redshifted) limb begins to leave the stel- +lar disk. This sharply blueshifting behavior continues to +the end of egress, until the last sliver of the trailing (rota- +tionally blueshifted) limb has left the stellar disk as well. +In the cloudy case, however, the leading limb leaving the +stellar disk has no effect, as it is fully cloudy. While this +effect is evident in these models, it may be less evident +5 The degree of rotation during transit varies as a function of +semimajor axis and host star radius, and hence from planet to +planet. While we only model WASP-76b, other planets also have +large (e.g., compared to the angles probed by transmission spec- +troscopy) rotations during transit (Wardenier et al. 2022). +in observations, which naturally cannot finely sample +ingress and egress phases. +3.2. Phase bins +We have thus far examined drivers of asymmetry and +potential diagnostics of specific mechanisms. Next, we +will evaluate a few HRCCS data types to determine how +robust they are and their potential ability to constrain +a number of different physical mechanisms that give rise +to HRCCS asymmetry. +The first of these data types is HRCCS Doppler shifts +that are binned in phase. +A substantial fraction of +HRCCS studies present detections and Doppler shifts +integrated over the entirety of transit (e.g., Giacobbe +et al. 2021). This approach maximizes detection SNR, +which may be necessary for a given set of observations +(e.g., because of a low-resolution spectrograph, small +telescope aperture, faint star, low species abundance, +low number of absorption lines, or weak intrinsic ab- +sorption line strengths). While it is possible to reveal +aspects of limb asymmetry with this approach, espe- +cially when comparing detections of multiple species to +one another, phase-resolving the transit (and observing +isolated ingresses and egresses when possible) will cer- +tainly give a more direct probe of east–west asymme- +tries. Binning HRCCS data in phase across transit may +provide a desirable balance between revealing asymme- +try and maintaining high SNR. +We seek to address this question by phase-binning +modeled Doppler shifts to examine its biases with re- +spect to the underlying model. We follow this experi- +ment with a comparison to the phase-binned observa- +tions of Kesseli et al. (2022). +3.2.1. Theoretical phase binning +We average our phase-resolved calculations into two +bins: the first and second half of transit. +Once our +CCFs are calculated we average them in phase to ef- +fectively reduce our data to two single bins: the first +half of transit and the second half of transit. We make +versions of these two half-transit bins that include or +exclude the ingress and egress phases (when the planet +is only partially occulting the star). +Motivated by recent detections in the near-infrared +(Landman et al. 2021; S´anchez-L´opez et al. 2022), we +search for absorption from various molecules6 in our +models, focusing on the CARMENES (Quirrenbach +et al. 2014) wavelength range and resolution for direct +6 We use the MoLLIST (Brooke et al. 2016), POKAZATEL +(Polyansky et al. 2018), and Li2015 (Li et al. 2015) linelists for +OH, H2O, and CO, respectively. + +Diagnosing limb asymmetries in transmission +13 +Figure 6. Single-species (OH, CO, H2O, HCN) 3D forward- +modeled spectra of WASP-76b. These spectra are simulated +over the CARMENES waveband and resolution. +Doppler +effects are not included in these spectra, which are modeled +at center of transit. H2O is the dominant absorber in this +bandpass, followed by OH. HCN exhibits no spectral features +above the continuum for WASP-76b in this bandpass. +comparison against observational results using that in- +strument. Of these molecules, we find that OH, H2O, +and CO produce significant absorption over the mod- +eled wavelength range, with OH and H2O producing the +strongest features (Figure 6). We find that HCN does +not produce any noticeable absorption under the as- +sumption of chemical equilibrium and solar composition, +implying either more exotic chemistry for WASP-76b’s +atmosphere (i.e., photochemistry or non-solar abun- +dances; Moses et al. 2012), or that the detection of +HCN in this atmosphere (S´anchez-L´opez et al. 2022) +was spurious (perhaps due to the nature of the HCN +opacity function; Zhang et al. 2020). We furthermore +find a moderate (≈ 4 km s−1) increase in blueshift for +our modeled H2O. +While this increase in blueshift is +commensurate with the increase in blueshift described +for H2O in S´anchez-L´opez et al. (2022), we are once +again unable to match the high reported velocities (here +-14.3 km s−1) with our self-consistent forward models. +Figure 7 shows the results of this experiment. As the +error for each phase bin, we take the average error of +phase bins from Kesseli et al. (2022) (1.55 km s−1). We +define the two phase bins as inconsistent if the peak of +their respective CCFs are inconsistent at 2σ. +We find that excluding ingress and egress phases can +strongly reduce the difference in derived Doppler shift +between phase bins. Furthermore, we find that, as ex- +pected from Kempton & Rauscher (2012), differences +between bins are maximized when just considering the +ingress and egress phases.7 +While higher-order drivers of asymmetry are clearly +not detectable with phase bins (e.g,. at what longitude +condensation may begin to play a role; Wardenier et al. +2021), certain drivers of asymmetry are accessible with +this method. For example, ignoring for now the exact +details of error budgets, all species in Figure 7 clearly +blueshift over the course of transit. This provides poten- +tial evidence for, among other things, a spatially vary- +ing wind field, condensation, optically thick clouds, or a +scale height effect. Furthermore, per the results of Sec- +tion 3.1.1 the detection of CO’s blueshifting indicates +that something besides equilibrium chemistry is driv- +ing at least some of the asymmetry in the atmosphere. +These underlying models are cloud-free, so these results +imply sensitivity to, e.g., the scale height effect. +3.2.2. Comparison to Kesseli et al. (2022) +With our models calculated, we can now explore the +ability of phase-resolved spectra to confront toy mod- +els by comparing the models to observations. A prime +observational work that made use of phase binning is +Kesseli et al. (2022); there, the authors search for asym- +metries in two phase bins for a wide variety of species, +motivated by the strength of those species’ opacity func- +tions in the data’s wavelength range. +To consider a toy model: based on previous studies +(Ehrenreich et al. 2020; Tabernero et al. 2021; Savel +et al. 2022), it appears that Ca II does not follow the +Fe-like Doppler shift trend first observed by Ehrenreich +et al. (2020). Rather, it appears that Ca II, with its +strong opacity and resultant deep lines, may be probing +a non-hydrostatic region of the atmosphere (Casasayas- +Barris et al. 2021; Deibert et al. 2021; Tabernero et al. +2021). This region of the atmosphere cannot be cap- +tured by the models of this work and Savel et al. (2022). +Without a model of atmospheric escape, it seems dif- +ficult to elevate the above picture beyond “toy model” +status. However, by phase-resolving multiple species, a +clearer picture can emerge. +For our comparison with Kesseli et al. (2022), we use +the same line lists as in that study: the National In- +stitute of Standards and Technology (NIST; Kramida +et al. 2019) line lists. It is crucial to use the same line +7 We would expect that binning with fewer spectra (just includ- +ing the ingress phases) would increase the associated error on +Doppler shift at each bin. However, the point of this exercise is to +illustrate the magnitude of ingress/egress Doppler shift discrep- +ancy; observational strategies such as stacking multiple transits +could reduce errors in practice and make these differences dis- +cernible. + +H20 +CO +1.375 +HO +HCN +1.350 +Transit depth (%) +1.325 +1.300 +1.275 +1.250 +1.225 +1.200 +1.0 +1.1 +1.2 +1.3 +1.4 +1.5 +1.6 +1.7 +Wavelength (microns)14 +Savel et al. +Figure 7. CCFs of individual species averaged over two phase bins. Each column corresponds to different species (OH, CO, +H2O), and each row corresponds to different bin selection: without including ingress and egress, including the full transit, and +only including ingress and egress. Central bars between the CCFs are colored blue if the difference between the CCFs is greater +than optimal Doppler shift errors (1.55 km/s, in black; Kesseli et al. 2022); otherwise, they are colored red. In our models, CO +only displays detectable CCF differences when only including ingress and egress. The SNR in each plot refers to the difference +between the two phase bins’ CCF peaks relative to the optimal Doppler shift errors. +lists for comparisons of HRCCS studies—different line +list databases can contain vastly discrepant numbers of +line transition, which greatly affects the resultant opac- +ity function (see, for instance, Figure 11 of Grimm et al. +2021). +The results of our comparison with the species de- +tected in Kesseli et al. (2022) are shown in Figure 8. As +in Savel et al. (2022), these baseline models—no clouds, +no condensation, no orbital eccentricity—cannot fully +explain the Doppler shifts of Fe observed in WASP- +76b. +However, the comparison across multiple differ- +ent species provides further constraints. Figure 8 shows +that Fe, V, Cr, Ca II, and Sr II are strongly discrepant +from our models for at least one half of transit, whereas +Na, Mg, Mn, and Ni are reasonably well described by +our models for both the first and second half of transit. +Furthermore, Fe, V, and Cr all have stronger blueshifts +in the second phase bin than in our models. The similar + +CO, no ingress / egress +OH, no ingress / egress +H2O, no ingress / egress +1.4 - +2nd half peak: i +1st half peak: + 2nd half peak: i +1st half peak: +2nd half peak:i! +1st half peak: +-3.52 km s-1 +-1.3 km s-1 +-5.41 km s-1 +-3.15 km s-1 +-5.93 km s-1 +-3.73 km s-1 +1.2 +SNR: 1.4 +SNR: 1.5 +SNR: 1.4 +1.0 +0.8 +3333333333 +8 0.6 +0.4 - +0.2 +CO, full transit +H20, full transit +OH, full transit +2nd half peak: +1st half peak: +2nd half peak:! +1st half peak: +2nd half peak: +1.4 - +1st half peak: +-3.73 km s- +-0.38 km s-1 +-6.08 km s- +-2.27 km s-1 +-6.23 km s-1 +-2.81 km s-1 +1.2 - +0 +SNR: 2.2 +SNR: 2.5 +SNR: 2.2 +1.0 - +0.8 +CF +0 0.6 +0.4 - +0.2 +CO, only ingress / egress +OH, only ingress / egress +H2O, only ingress / egress +2nd half peak:i +1.4 +1st half peak: +2nd half peakl +1st half peak: +2nd half peakl +1st half peak: +-6.28 km s-1 +4.27 km s-1 +-8.04 km s-1T +2.24 km s-1 +-7.98 km s-1T +1.65 km s-1 +1.2 +■ +SNR: 6.8 +SNR: 6.6 +SNR: 6.2 +0.8 +CF +8 0.6 - +0.4 +0.2 +-40 +20 +0 +20 +40 +-40 +20 +0 +20 +40 +-40 +20 +0 +20 +40 +Velocity (km/s) +Velocity (km/s) +Velocity (km/s)Diagnosing limb asymmetries in transmission +15 +Figure 8. The net Doppler shifts of Kesseli et al. (2022) (error bars) as compared to this work’s models (crosses). The first +phase bin is drawn thinner than the second phase bin; observed phase bins are connected by a dotted line for visibility’s sake. +The species are ordered and colored by total observed detection SNR. Rows without crosses correspond to species that we could +not recover via cross-correlation in our models. Our models are able to explain some species (e.g., Na), fail to explain others +(e.g., Cr) and fail to detect yet others (e.g,. K). +level of disagreement between Fe, V, and Cr implies that +they share a common driver of asymmetry. This result +in turn implies that whatever driver affects them affects +the regions in which these species form similarly — be +it clouds, condensation, etc. +To bridge the toy models presented in Section 2.3 to +our Kesseli et al. (2022) comparison, we compute a set +of high-resolution spectra exactly as above, but with +the same altitude grid at all latitudes and longitudes +in an effort to effectively turn off the scale height ef- +fect while maintaining chemical limb inhomogeneities. +Post-processing this (self-inconsistent) model yields less +than half the Doppler shift asymmetry as compared to +our self-consistent models. +This experiment confirms +the intuition that the scale height effect is a first-order +asymmetry effect. +Finally, we consider the Ca II toy model previously de- +scribed. Certain lightweight and/or ionized species may +be entrained in an outflow, as indicated by some pre- +vious observations (e.g., Tabernero et al. 2021) of very +deep absorption lines in transmission that must extend +very high up in altitude. The differential behavior of the +Ca II and Sr II Doppler shifts lends more credence to +this hypothesis. +In sum, by taking advantage of phase-binned spectra, +it is possible to better identify drivers of HRCCS asym- +metry. Additionally, our predictions in Figure 8 indi- +cate that most species should have roughly the same +Doppler shift patterns. In stark contrast, observations +reveal much larger variations in velocity across different +species. While some interpretation may be due to spuri- +ous detections, physics that is not included in our model +(e.g., outflows, condensation) may be playing a driving +role. +3.3. Full phase-resolved spectra +Currently, the most information-rich diagnostic avail- +able to probe asymmetry in HRCCS is phase-resolved +cross-correlation functions (e.g., Ehrenreich et al. 2020; +Borsa et al. 2021)—that is, net Doppler shifts associated +with the absorption spectrum evaluated over multiple +points in transit. With these data, one should be able +to directly constrain longitudinally dependent drivers of +asymmetry, providing the best chance of disentangling +the physical mechanisms outlined in Section 2. But how +far can we push these data? +3.3.1. Example: probing physics in the NIR +To explore this question, we take as an example a +three-species (OH, H2O, and CO) near-infrared (NIR) +dataset over a CARMENES-like waveband as in Sec- +tion 3.2. +Figure 9 shows the Doppler shifts of these + +-14 +Sr+ +Ni +Kesseli+22 SNR of species detection +Co +Model, first bin +12 +Fe +Model, second bin +Kesseli+22, +Mn +first bin +Kesseli+22, +10 +cies +Cr +second bin +V +d +S'Ca+ +8 +K +Mg +Na +6 +Li +H +-15 +-10 +-5 +0 +5 +10 +Planet-frame Doppler shift (km/s)16 +Savel et al. +Figure 9. Modeled phase-resolved Doppler shifts for select NIR-absorbing species, with representative error bars (Ehrenreich +et al. 2020) drawn on. We find that OH and H2O have distinct Doppler signatures from CO; however, OH and H2O have Doppler +shifts that are indistinguishable from one another with current best-case error bars (e.g., Ehrenreich et al. 2020). Considering +CO as a “baseline species” here allows one to better understand how H2O and OH may change through the atmosphere. +species as a function of phase, produced for single species +at a time as in Section 3.2, but without any averaging. +Without considering any data, a compelling toy model +would be as follows: H2O is thermally dissociated on +the hotter, approaching limb, so it preferentially exists +on the receding limb. OH is a product of H2O photodis- +sociation, so it forms preferentially on the approaching +limb. CO is constant everywhere; therefore, CO should +not experience much of a trend in Doppler shift, OH +should be more blueshifted than CO, and H2O should +be more redshifted than CO. +We shall see, however, that additional, complicating +physics is revealed by fully phase-resolved spectra. For +our models, the relevant underlying physics is as follows: +1. Altitude-dependent winds: H2O lines are more +strongly blueshifted than CO lines at all phases +because the H2O line cores over the wavelength +range of the CARMENES bandpass more predom- +inantly form at higher altitudes. At high altitudes, +the atmospheric flow switches from dominantly ro- +tational (via an eastward equatorial jet) to dom- +inantly divergent (via day–night winds) (Ham- +mond & Lewis 2021). This result is the opposite of +what would be expected from the above-described +toy model, revealing the shortcomings of simple +models and how they can sometimes mislead us. +2. Equilibrium chemistry: H2O and CO are less +blueshifted than OH because OH preferentially +forms on the approaching, blueshifted limb of the +planet. OH being more blueshifted than the other +molecules is in agreement with the predictions of +the toy model. +3. Equilibrium chemistry: The relationship be- +tween H2O and OH changes as a function of phase +because the ratio OH/H2O increases a function of +temperature, and hotter regions of the planet ro- +tate into view over transit. +This finding is also +qualitatively in agreement with the toy model. +However, per Figure 9, this effect is unfortunately +not likely to be observable given the error bars in +current data sets. +Now the question remains: Can we observe in real +data the trends matching these model explanations? As +a simple experiment, we can apply error bars representa- +tive of the best observing nights on the best instrument +with the most observable chemical species (roughly 2 +km/s, as drawn as vertical error bars in Figure 8; Ehren- +reich et al. 2020) and determine whether these trends are +still detectable. With our errorbars now applied to our +simulated data, only the first explanation—that H2O +forms at higher altitudes than CO—can fully be ad- +dressed, assuming that Doppler shifts for both species + +CO +OH +- H20 +5.0 +Planet-frame RV (km/s) +2.5 +0.0 +-2.5 +-5.0 +7.5 +-10.0 +12.5 +-15 +-10 +5 +5 +10 +15 +Phase (degrees)Diagnosing limb asymmetries in transmission +17 +(a) +(b) +Figure 10. Results of an investigation into anomalous Ca II blueshift between different model runs. In panel (a), it can be +seen that forward models that include absorption due to Sc opacity yield a larger Ca II blueshift than models that lack Sc (Fe +Doppler shift is included for comparison). Panel (b) illustrates the cause of this anomalous blueshift: a Sc line overlapping one +line in the optical Ca II doublet. These results imply that overlapping line profiles can subtly contaminate calculated Doppler +shifts. +can be obtained. The second explanation can only be +partially addressed—we can still determine that CO is +less blueshifted than OH. +3.3.2. Warning: blending of Doppler shifts +The disentangling of physics in Section 3.3.1 rests on +a fundamental assumption: that each cross-correlation +template directly tracks only a single species. Indeed, +one of the promises of HRCCS is the ability to uniquely +constrain individual species’ abundance; with individual +line profiles resolved, different species should be readily +identifiable from one another in cross-correlation space +(e.g., Brogi & Line 2019). Furthermore, our noiseless +models should be even less susceptible to degeneracies +between different species’ spectral manifestations. +Panel (a) of Figure 10 seems to contradict the notion +of complete line profile independence across species. For +models run in Savel et al. (2022), Sc was excluded. Mo- +tivated by the search for atoms in Kesseli et al. (2022), +however, we included Sc in this work’s models. +Sur- +prisingly, we found a subsequent significant difference in +the Doppler shifts recovered from our cross-correlation +analysis in our Sc-inclusive models. +Panel (b) of Figure 10 reveals the source of the dis- +crepancy. In the optical, Ca II opacity is dominated by +a doublet; one of the lines in this doublet partially over- +laps with a strong, narrow Sc line. When both species +combined in a forward model, the Sc line produces ab- +sorption just blueward of this Ca II line’s core; hence, +the cross-correlation of the Ca II template yields a spuri- +ous blueshift. There did exist other modeling differences +between the two spectra (e.g., the Savel et al. (2022) +models included TiO and VO), but none of these differ- +ences strongly impacted the Doppler shift of Ca II. +Because Ca II in the optical has only two strong lines, +it is particularly susceptible to this type of error. All it +takes is one slight overlap with another species near a +Ca II doublet core, and the Ca II Doppler signal can be +significantly biased. Species with forests of lines (e.g., +Fe in the optical) should hence be more robust to chance +overlaps with other species’ lines. +To guard against this error for species with few lines, +we recommend cross-correlating templates against one +another to get a first-order sense for the extent of species +overlap in Doppler space. +Furthermore, we recom- +mend performing these analyses on HRCCS with com- +bined species models, as opposed to single-species mod- +els. This approach could involve a retrieval framework +(Brogi & Line 2019; Gandhi et al. 2019; Gibson et al. +2020), which couples a statistical sampler to an atmo- +spheric forward model to determine the exoplanet spec- +trum that best fits the data, inclusive of multiple chem- +ical species at once. +4. CONCLUSION +The past few years have yielded asymmetric Doppler +signals from exoplanet atmospheres as a function of +phase. +Compelling “toy models” notwithstanding, a +number of physical processes can drive these asymme- +tries, and it can be difficult to uniquely constrain the +cause of an asymmetry. +In this study, we determine that if an asymmetry is +observed: +1. It may be due to a scale height difference across +the atmosphere, not a chemistry difference across + +2 +Ca+ with Sc +Fe +Ca+ +Planet-frame RV (km/s) +0 +-2 +-6 +-8 +-1015 +-10 +-5 +0 +5 +10 +15 +Phase (degrees)Sc +1.40 +Ca+ template +(%) +ransit depth ( +1.35 +1.30 +1.25 +1.20 +0.3931 +0.3932 0.3933 0.3934 0.3935 0.3936 0.3937 0.3938 +Wavelength (microns)18 +Savel et al. +the atmosphere. Comparing a signal of a species in +HRCCS to a baseline species that is guaranteed to +be chemically stable over the atmosphere can bet- +ter motivate whether the asymmetry could be due +to chemistry. CO is an excellent baseline species +for ultra-hot Jupiters, as it is stable over these +planets’ expected temperature–pressure space, has +many spectral lines in the near-infrared accessible +to ground-based spectrographs, and has been de- +tected in numerous studies. +2. The asymmetry can be highly informative even if +it is binned in phase, especially if multiple species +are considered. For instance, much larger Doppler +shifts (both blue and red) of certain species rela- +tive to the predictions of hydrostatic GCMs can +be used as evidence for outflowing material. +3. The asymmetry may be boosted by including +(and perhaps only considering) ingress and egress +phases. Ingress and egress spectra are the the gold +standard for asymmetric signals so long as the sig- +nal to noise is high enough. +4. The asymmetry may be influenced by line con- +fusion between species, even at high resolution. +Species with very few lines (e.g., a single doublet) +in the observed waveband are especially suscep- +tible to contamination by other species in cross- +correlation analysis, and they should be carefully +checked against theoretical models for possible +contaminating opacity sources. +5. If all species exhibit a similar asymmetry— +especially if they all become less blueshifted over +the course of transit—the asymmetry may be due +to a large-scale effect, such as clouds blanketing +the cooler limb. +6. Per our comparison of near-infrared absorbers in +the CARMENES waveband, the toy model pre- +dictions of the H2O Doppler shift relative to CO +was inaccurate, as it did not include information +about the vertical coordinate. With H2O lines on +average probing higher in the atmosphere than CO +in this waveband, they probed a different part of +the flow, departing from expectations of the toy +model. +By aiming to systematically understand even just a +few drivers of asymmetry, this work has made it clear +that HRCCS—already arguably abstract given its gen- +eral inability to produce visible planetary spectra—has +yet more nuance to uncover. As data quality continues +to increase, it will become increasingly necessary to un- +derstand the relationships between higher-order physical +effects. +ACKNOWLEDGMENTS +A.B.S., E.M.-R.K., and E.R. acknowledge funding +from the Heising-Simons Foundation. +We thank Michael Zhang for a thoughtful conversa- +tion on the cross-correlation signature of HCN. We also +thank Anusha Pai Asnodkar for a robust discussion of +degeneracies in HRCCS tests. Finally, we thank Serena +Cronin for providing useful insight into applications of +CO detections in extragalactic astronomy. +The authors acknowledge the University of Maryland +supercomputing resources (http://hpcc.umd.edu) made +available for conducting the research reported in this +paper. +This research has made use of NASA’s Astrophysics +Data System Bibliographic Services. +Software: +astropy (Price-Whelan et al. 2018), +batman (Kreidberg 2015), FastChem (Stock et al. 2018), +IPython (P´erez & Granger 2007), HELIOS-K (Grimm +et al. 2021), HELIOS (Malik et al. 2017), Matplotlib +(Hunter 2007), NumPy (Harris et al. 2020), Numba (Lam +et al. 2015), pandas (McKinney 2010), SciPy (Virtanen +et al. 2020), tqdm (da Costa-Luis 2019) +REFERENCES +Ahrer, E.-M., Alderson, L., Batalha, N. M., et al. 2022, +arXiv preprint arXiv:2208.11692 +Akinsanmi, B., Barros, S., Santos, N., et al. 2019, +Astronomy & Astrophysics, 621, A117 +Arcangeli, J., D´esert, J.-M., Line, M. R., et al. 2018, The +Astrophysical Journal Letters, 855, L30 +Beichman, C., Benneke, B., Knutson, H., et al. 2014, +Publications of the Astronomical Society of the Pacific, +126, 1134 +Beltz, H., Rauscher, E., Brogi, M., & Kempton, E. M.-R. +2020, The Astronomical Journal, 161, 1 +Beltz, H., Rauscher, E., Kempton, E. M. R., et al. 2022, +AJ, 164, 140, doi: 10.3847/1538-3881/ac897b + +Diagnosing limb asymmetries in transmission +19 +Beltz, H., Rauscher, E., Roman, M. T., & Guilliat, A. 2021, +The Astronomical Journal, 163, 35 +Benneke, B., Wong, I., Piaulet, C., et al. 2019, The +Astrophysical Journal Letters, 887, L14 +Birkby, J. L. 2018, arXiv preprint arXiv:1806.04617 +Blecic, J., Dobbs-Dixon, I., & Greene, T. 2017, The +Astrophysical Journal, 848, 127 +Borsa, F., Allart, R., Casasayas-Barris, N., et al. 2021, +Astronomy & Astrophysics, 645, A24 +Bourrier, V., Ehrenreich, D., Lendl, M., et al. 2020, +Astronomy & Astrophysics, 635, A205 +Brogi, M., De Kok, R., Albrecht, S., et al. 2016, The +Astrophysical Journal, 817, 106 +Brogi, M., De Kok, R., Birkby, J., Schwarz, H., & Snellen, +I. 2014, Astronomy & Astrophysics, 565, A124 +Brogi, M., & Line, M. R. 2019, The Astronomical Journal, +157, 114 +Brooke, J. S., Bernath, P. F., Western, C. M., et al. 2016, +Journal of Quantitative Spectroscopy and Radiative +Transfer, 168, 142 +Caldas, A., Leconte, J., Selsis, F., et al. 2019, Astronomy & +Astrophysics, 623, A161 +Casasayas-Barris, N., Orell-Miquel, J., Stangret, M., et al. +2021, A&A, 654, A163, +doi: 10.1051/0004-6361/202141669 +Charbonneau, D., Brown, T. M., Noyes, R. W., & Gilliland, +R. L. 2002, The Astrophysical Journal, 568, 377 +Cho, J. Y., Menou, K., Hansen, B. M., & Seager, S. 2008, +The Astrophysical Journal, 675, 817 +Cooper, C. S., & Showman, A. P. 2005, The Astrophysical +Journal, 629, L45 +Crossfield, I. J., & Kreidberg, L. 2017, The Astronomical +Journal, 154, 261 +da Costa-Luis, C. 2019, Journal of Open Source Software, +4, 1277 +de Kok, R. J., Brogi, M., Snellen, I. A., et al. 2013, +Astronomy & Astrophysics, 554, A82 +Deibert, E. K., de Mooij, E. J., Jayawardhana, R., et al. +2021, The Astrophysical Journal Letters, 919, L15 +Demory, B.-O., Gillon, M., De Wit, J., et al. 2016, Nature, +532, 207 +Ehrenreich, D., Lovis, C., Allart, R., et al. 2020, Nature, +580, 597 +Espinoza, N., & Jones, K. 2021, The Astronomical Journal, +162, 165 +Faedi, F., Barros, S. C., Anderson, D. R., et al. 2011, +Astronomy & Astrophysics, 531, A40 +Feng, Y. K., Line, M. R., Fortney, J. J., et al. 2016, The +Astrophysical Journal, 829, 52 +Flowers, E., Brogi, M., Rauscher, E., Kempton, E. M.-R., & +Chiavassa, A. 2019, The Astronomical Journal, 157, 209 +Fortney, J. J., Dawson, R. I., & Komacek, T. D. 2021, +Journal of Geophysical Research: Planets, 126, +e2020JE006629 +Fossati, L., Ayres, T., Haswell, C., et al. 2013, The +Astrophysical Journal Letters, 766, L20 +Fray, N., & Schmitt, B. 2009, Planetary and Space Science, +57, 2053 +Fromang, S., Leconte, J., & Heng, K. 2016, Astronomy & +Astrophysics, 591, A144 +Gandhi, S., Brogi, M., & Webb, R. K. 2020, Monthly +Notices of the Royal Astronomical Society, 498, 194 +Gandhi, S., Kesseli, A., Snellen, I., et al. 2022, Monthly +Notices of the Royal Astronomical Society +Gandhi, S., Madhusudhan, N., Hawker, G., & Piette, A. +2019, The Astronomical Journal, 158, 228 +Gao, P., & Powell, D. 2021, The Astrophysical Journal +Letters, 918, L7 +Gao, P., Wakeford, H. R., Moran, S. E., & Parmentier, V. +2021, Aerosols in exoplanet atmospheres, Wiley Online +Library +Gaudi, B. S., Stassun, K. G., Collins, K. A., et al. 2017, +Nature, 546, 514 +Giacobbe, P., Brogi, M., Gandhi, S., et al. 2021, Nature, +592, 205 +Gibson, N. P., Merritt, S., Nugroho, S. K., et al. 2020, +Monthly Notices of the Royal Astronomical Society, 493, +2215 +Grimm, S. L., Malik, M., Kitzmann, D., et al. 2021, The +Astrophysical Journal Supplement Series, 253, 30 +Guillot, T., Fletcher, L. N., Helled, R., et al. 2022, arXiv +preprint arXiv:2205.04100 +Guilluy, G., Giacobbe, P., Carleo, I., et al. 2022, Astronomy +& Astrophysics, 665, A104 +Hammond, M., & Lewis, N. T. 2021, Proceedings of the +National Academy of Sciences, 118, e2022705118 +Harrington, J., Hansen, B. M., Luszcz, S. H., et al. 2006, +Science, 314, 623 +Harris, C. R., Millman, K. J., van der Walt, S. J., et al. +2020, Nature, 585, 357–362, +doi: 10.1038/s41586-020-2649-2 +Heays, A., Bosman, AD, v., & Van Dishoeck, E. 2017, +Astronomy & Astrophysics, 602, A105 +Hellier, C., Anderson, D., Cameron, A. C., et al. 2009, +Nature, 460, 1098 +Herman, M. K., de Mooij, E. J., Nugroho, S. K., Gibson, +N. P., & Jayawardhana, R. 2022, The Astronomical +Journal, 163, 248 + +20 +Savel et al. +Hindle, A., Bushby, P., & Rogers, T. 2021, The +Astrophysical Journal, 922, 176 +Hoeijmakers, H. J., Seidel, J. V., Pino, L., et al. 2020, +Astronomy & Astrophysics, 641, A123 +Hood, C. E., Fortney, J. J., Line, M. R., et al. 2020, The +Astronomical Journal, 160, 198 +Hunter, J. D. 2007, Computing in science & engineering, 9, +90 +Kataria, T., Sing, D. K., Lewis, N. K., et al. 2016, The +Astrophysical Journal, 821, 9 +Kempton, E. M.-R., Perna, R., & Heng, K. 2014, The +Astrophysical Journal, 795, 24 +Kempton, E. M.-R., & Rauscher, E. 2012, The +Astrophysical Journal, 751, 117 +Kesseli, A. Y., & Snellen, I. 2021, The Astrophysical +Journal Letters, 908, L17 +Kesseli, A. Y., Snellen, I., Casasayas-Barris, N., Molli`ere, +P., & S´anchez-L´opez, A. 2022, The Astronomical +Journal, 163, 107 +Knutson, H. A., Charbonneau, D., Allen, L. E., et al. 2007, +Nature, 447, 183 +Komacek, T. D., Showman, A. P., & Parmentier, V. 2019, +The Astrophysical Journal, 881, 152 +Komacek, T. D., Tan, X., Gao, P., & Lee, E. K. H. 2022, +ApJ, 934, 79, doi: 10.3847/1538-4357/ac7723 +Kramida, A., Olsen, K., & Ralchenko, Y. 2019, National +Institute of Standards and Technology, US Department +of Commerce +Kreidberg, L. 2015, Publications of the Astronomical +Society of the Pacific, 127, 1161 +Kreidberg, L., Koll, D. D., Morley, C., et al. 2019, Nature, +573, 87 +Lacy, B. I., & Burrows, A. 2020, The Astrophysical +Journal, 905, 131 +Lam, S. K., Pitrou, A., & Seibert, S. 2015, in Proceedings +of the Second Workshop on the LLVM Compiler +Infrastructure in HPC, 1–6 +Landman, R., S´anchez-L´opez, A., Molli`ere, P., et al. 2021, +A&A, 656, A119, doi: 10.1051/0004-6361/202141696 +Li, G., Gordon, I. E., Rothman, L. S., et al. 2015, The +Astrophysical Journal Supplement Series, 216, 15 +Lide, D. R. 2006, CRC Handbook of Chemistry and +Physics, ; Lide, DR, Ed, CRC Press: Boca Raton, Fla +Line, M. R., Brogi, M., Bean, J. L., et al. 2021, Nature, +598, 580 +Lodders, K. 2003, The Astrophysical Journal, 591, 1220 +Louden, T., & Wheatley, P. J. 2015, The Astrophysical +Journal Letters, 814, L24 +MacDonald, R. J., Goyal, J. M., & Lewis, N. K. 2020, The +Astrophysical Journal Letters, 893, L43 +MacDonald, R. J., & Lewis, N. K. 2022, The Astrophysical +Journal, 929, 20 +Madhusudhan, N., & Seager, S. 2009, The Astrophysical +Journal, 707, 24 +Malik, M., Grosheintz, L., Mendon¸ca, J. M., et al. 2017, +The astronomical journal, 153, 56 +Mansfield, M., Bean, J. L., Stevenson, K. B., et al. 2020, +The Astrophysical Journal Letters, 888, L15 +May, E. M., Komacek, T. D., Stevenson, K. B., et al. 2021, +The Astronomical Journal, 162, 158 +Mayne, N. J., Baraffe, I., Acreman, D. M., et al. 2014, +Astronomy & Astrophysics, 561, A1 +Mbarek, R., & Kempton, E. M.-R. 2016, The Astrophysical +Journal, 827, 121 +McKinney, W. 2010, in Proceedings of the 9th Python in +Science Conference, ed. St´efan van der Walt & Jarrod +Millman, 56 – 61, doi: 10.25080/Majora-92bf1922-00a +Menou, K., & Rauscher, E. 2009, The Astrophysical +Journal, 700, 887 +Miller-Ricci, E., Seager, S., & Sasselov, D. 2008, The +Astrophysical Journal, 690, 1056 +Montalto, M., Santos, N., Boisse, I., et al. 2011, Astronomy +& Astrophysics, 528, L17 +Moses, J., Madhusudhan, N., Visscher, C., & Freedman, R. +2012, The Astrophysical Journal, 763, 25 +Moses, J. I., Visscher, C., Fortney, J. J., et al. 2011, The +Astrophysical Journal, 737, 15 +Oklopˇci´c, A., & Hirata, C. M. 2018, The Astrophysical +Journal Letters, 855, L11 +Owen, J. E., Murray-Clay, R. A., Schreyer, E., et al. 2023, +Monthly Notices of the Royal Astronomical Society, 518, +4357 +Parmentier, V., & Crossfield, I. J. 2018, Handbook of +exoplanets, 116 +Parmentier, V., Showman, A. P., & Fortney, J. J. 2021, +Monthly Notices of the Royal Astronomical Society, 501, +78 +Parmentier, V., Line, M. R., Bean, J. L., et al. 2018, +Astronomy & Astrophysics, 617, A110 +Pelletier, S., Benneke, B., Darveau-Bernier, A., et al. 2021, +The Astronomical Journal, 162, 73 +P´erez, F., & Granger, B. E. 2007, Computing in Science & +Engineering, 9, 21 +Perna, R., Heng, K., & Pont, F. 2012, The Astrophysical +Journal, 751, 59 +Pino, L., Ehrenreich, D., Allart, R., et al. 2018, Astronomy +& Astrophysics, 619, A3 +Pluriel, W., Leconte, J., Parmentier, V., et al. 2022, A&A, +658, A42, doi: 10.1051/0004-6361/202141943 + +Diagnosing limb asymmetries in transmission +21 +Pluriel, W., Zingales, T., Leconte, J., & Parmentier, V. +2020, Astronomy & Astrophysics, 636, A66 +Polyansky, O. L., Kyuberis, A. A., Zobov, N. F., et al. +2018, Monthly Notices of the Royal Astronomical +Society, 480, 2597 +Price-Whelan, A. M., Sip˝ocz, B. M., G¨unther, H. M., et al. +2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f +Quirrenbach, A., Amado, P., Caballero, J., et al. 2014, in +Ground-based and airborne instrumentation for +astronomy V, Vol. 9147, SPIE, 531–542 +Rauscher, E., & Menou, K. 2010, The Astrophysical +Journal, 714, 1334 +Rodler, F., K¨urster, M., & Barnes, J. R. 2013, Monthly +Notices of the Royal Astronomical Society, 432, 1980 +Roman, M. T., Kempton, E. M.-R., Rauscher, E., et al. +2021, The Astrophysical Journal, 908, 101 +S´anchez-L´opez, A., Landman, R., Molli`ere, P., et al. 2022, +Astronomy & Astrophysics, 661, A78 +Savel, A. B., Kempton, E. M.-R., Malik, M., et al. 2022, +The Astrophysical Journal, 926, 85 +Seidel, J., Ehrenreich, D., Allart, R., et al. 2021, Astronomy +& Astrophysics, 653, A73 +Showman, A., & Guillot, T. 2002, Astronomy & +Astrophysics, 385, 166 +Showman, A. P., Fortney, J. J., Lewis, N. K., & Shabram, +M. 2013, The Astrophysical Journal, 762, 24 +Showman, A. P., Fortney, J. J., Lian, Y., et al. 2009, The +Astrophysical Journal, 699, 564 +Snellen, I. A., De Kok, R. J., De Mooij, E. J., & Albrecht, +S. 2010, Nature, 465, 1049 +Spake, J. J., Sing, D. K., Evans, T. M., et al. 2018, Nature, +557, 68 +Stock, J. W., Kitzmann, D., Patzer, A. B. C., & Sedlmayr, +E. 2018, Monthly Notices of the Royal Astronomical +Society, 479, 865 +Tabernero, H., Osorio, M. Z., Allart, R., et al. 2021, +Astronomy & Astrophysics, 646, A158 +Tan, X., & Komacek, T. D. 2019, The Astrophysical +Journal, 886, 26 +Tsai, S.-M., Lyons, J. R., Grosheintz, L., et al. 2017, The +Astrophysical Journal Supplement Series, 228, 20 +Tsai, S.-M., Malik, M., Kitzmann, D., et al. 2021, The +Astrophysical Journal, 923, 264 +Van Dishoeck, E. F., & Black, J. H. 1988, The +Astrophysical Journal, 334, 771 +van Sluijs, L., Birkby, J. L., Lothringer, J., et al. 2022, +arXiv preprint arXiv:2203.13234 +Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, +Nature methods, 17, 261 +Visscher, C. 2012, The Astrophysical Journal, 757, 5 +Wardenier, J. P., Parmentier, V., & Lee, E. K. 2022, +Monthly Notices of the Royal Astronomical Society, 510, +620 +Wardenier, J. P., Parmentier, V., Lee, E. K., Line, M. R., & +Gharib-Nezhad, E. 2021, Monthly Notices of the Royal +Astronomical Society, 506, 1258 +Welbanks, L., & Madhusudhan, N. 2022, The Astrophysical +Journal, 933, 79 +West, R., Hellier, C., Almenara, J.-M., et al. 2016, +Astronomy & Astrophysics, 585, A126 +Wyttenbach, A., Molli`ere, P., Ehrenreich, D., et al. 2020, +Astronomy & Astrophysics, 638, A87 +Yan, F., Casasayas-Barris, N., Molaverdikhani, K., et al. +2019, Astronomy & Astrophysics, 632, A69 +Zhang, M., Chachan, Y., Kempton, E. M.-R., Knutson, +H. A., et al. 2020, The Astrophysical Journal, 899, 27 +Zhang, X., & Showman, A. P. 2018, The Astrophysical +Journal, 866, 2 +Zhang, Y., Snellen, I. A. G., & Molli`ere, P. 2021, A&A, +656, A76, doi: 10.1051/0004-6361/202141502 + diff --git a/39AzT4oBgHgl3EQfuv0b/content/tmp_files/load_file.txt b/39AzT4oBgHgl3EQfuv0b/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f37258f8d9857512278ebbf1f89fbcc657450feb --- /dev/null +++ b/39AzT4oBgHgl3EQfuv0b/content/tmp_files/load_file.txt @@ -0,0 +1,1817 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf,len=1816 +page_content='Draft version January 5, 2023 Typeset using LATEX twocolumn style in AASTeX63 Diagnosing limb asymmetries in hot and ultra-hot Jupiters with high-resolution transmission spectroscopy Arjun B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Savel ,1, 2 Eliza M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Kempton ,2 Emily Rauscher ,3 Thaddeus D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Komacek ,2 Jacob L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Bean ,4 Matej Malik ,2 and Isaac Malsky 3 1Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA 2Astronomy Department, University of Maryland, College Park, 4296 Stadium Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', College Park, MD 207842 USA 3Department of Astronomy, University of Michigan, 1085 South University Avenue, Ann Arbor, MI 48109, USA 4Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA Submitted to ApJ Abstract Due to their likely tidally synchronized nature, (ultra)hot Jupiter atmospheres should experience strongly spatially heterogeneous instellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The large irradiation contrast and resulting atmospheric circulation induce temperature and chemical gradients that can produce asymmetries across the eastern and western limbs of these atmospheres during transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' By observing an (ultra)hot Jupiter’s transmis- sion spectrum at high spectral resolution, these asymmetries can be recovered—namely through net Doppler shifts originating from the exoplanet’s atmosphere yielded by cross-correlation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Given the range of mechanisms at play, identifying the underlying cause of observed asymmetry is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' In this work, we explore sources and diagnostics of asymmetries in high-resolution cross-correlation spectroscopy of hot and ultra-hot Jupiters using both parameterized and self-consistent atmospheric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' If an asymmetry is observed, we find that it can be difficult to attribute it to equilibrium chemistry gradients because many other processes can produce asymmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Identifying a molecule that is chemically stable over the temperature range of a planetary atmosphere can help establish a “baseline” to disentangle the various potential causes of limb asymmetries observed in other species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We identify CO as an ideal molecule, given its stability over nearly the entirety of the ultra-hot Jupiter temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Furthermore, we find that if limb asymmetry is due to morning terminator clouds, blueshifts for a number of species should decrease during transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Finally, by comparing our forward models to Kesseli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022), we demonstrate that binning high-resolution spectra into two phase bins provides a desirable trade-off between maintaining signal to noise and resolving asymmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Keywords: Exoplanet atmospheric composition (2021) — Radiative transfer simulations (1967) — High resolution spectroscopy (2096) — Hot Jupiters (753) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' INTRODUCTION Exoplanet atmospheres vary spatially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This is es- pecially the case for tidally locked exoplanets, which feature permanent daysides and permanent nightsides;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' such strong gradients in instellation in turn drive strong latitudinal and longitudinal variations in atmospheric temperature, dynamics, and chemistry (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Showman & Guillot 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Cooper & Showman 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Harrington Corresponding author: Arjun B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Savel asavel@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='edu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Menou & Rauscher 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Showman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Rauscher & Menou 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Perna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Mayne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Demory et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Fro- mang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Kataria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Parmentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Zhang & Showman 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Kreidberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Komacek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Tan & Komacek 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Parmentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Roman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The spatial variations in exoplanet atmospheres have increasingly observable ramifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Even with the in- sight gained from modeling exoplanet atmospheres as one-dimensional objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Madhusudhan & Seager 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Crossfield & Kreidberg 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='01694v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='EP] 4 Jan 2023 ID2 Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Benneke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019), a substantial and growing liter- ature demonstrates that upcoming JWST (Beichman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2014) data will require consideration of 3D pro- cesses for accurate interpretation of exoplanet atmo- spheric data (Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Blecic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Caldas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Lacy & Burrows 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' MacDonald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Pluriel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Espinoza & Jones 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Pluriel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' MacDonald & Lewis 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Welbanks & Madhusudhan 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Perhaps more urgently, ground- based high-resolution (R ≥ 15, 000) cross-correlation spectroscopy (HRCCS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' for a review, see Birkby 2018) datasets already show signs of significant multidimen- sionality (Flowers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Beltz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Gandhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Herman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' van Sluijs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' In transit geometry, HRCCS is similar to the more traditional transmission spectroscopy technique (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Charbonneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Both methods leverage the idea that, as an exoplanet passes between its host star and an observer, stellar light is attenuated on a wavelength-dependent basis as it passes through the up- per layers of the planet’s atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' But with HRCCS, the planetary absorption spectrum is buried in the stel- lar and telluric noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Therefore, models of planetary ab- sorption often cannot be directly compared to HRCCS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1 However, by leveraging cross-correlation tech- niques, researchers can combine the signal from the many planetary absorption lines resolved at high resolu- tion to yield a combined, statistically significant signal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Snellen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The resolving of individual spectral lines allows for more than just binary detection/non-detection of plane- tary absorption: crucially, the Doppler shifts of plan- etary absorption lines are recoverable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The Doppler shifting of planetary lines due to the planet’s orbital motion is in fact central for extracting the planetary signal with cross-correlation techniques, as the stellar and telluric lines are comparatively static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Specifically, a template spectrum is chosen to model the planetary absorption signal, and it is cross-correlated against the combined planet, star, and telluric signal by Doppler shifting the template at varying velocities and multiply- ing the shifted template against the combined observed signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The resulting cross-correlation function (CCF, a function of Doppler-shifted velocity) is maximized at the Doppler shift where the template best matches the com- bined observed signal—that is, at the Doppler shift of the planet signal in the observed combined data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Again, 1 There are a few notable exceptions in the high-resolution spec- troscopy literature in which planetary absorption is strong enough (and data quality high enough) that individual planetary absorption lines can be analyzed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Tabernero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' this method requires that the planet’s spectral lines move across a spectrograph’s pixels during observations, with the stellar and telluric lines largely remaining on the same pixel (or being easily detrended in time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' With current instruments, this assumption is certainly justi- fied for tidally locked ultra-hot Jupiters, which tend to have high orbital velocities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Fortney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' With the planetary signal identified, further Doppler shifting and line broadening that is not associated with planetary orbital motion, telluric lines, or stellar lines is attributable to the 3D manifestations of planetary ro- tation and winds (Kempton & Rauscher 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Show- man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Kempton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Brogi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Ehrenreich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Thus, the multidimensionality of exoplanetary atmospheres is imprinted on HRCCS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Recent years have seen the intrinsic 3-dimensionality of these objects be uniquely constrained with transit HRCCS results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Observational studies such as Louden & Wheatley (2015), Ehrenreich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2020), and Kesseli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022) have isolated signals from the morning and evening limbs of planetary atmospheres, unveiling Doppler shifts of multiple chemical species at multiple points in transit—and hence over multiple longitudi- nal slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Such studies have revealed asymmetries in the probed Doppler velocity field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', changes in the Doppler shift of the CCF maximum as a function of orbital phase), which are often attributed to physical asymmetries in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' However, as reviewed in Section 2, an asymmetric sig- nal in HRCCS can arise from a combination of different classes of mechanisms: 1) chemistry, 2) clouds, 3) dy- namics, 4) orbital properties, and 5) thermal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Disentangling these effects is not a straightforward pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This may especially be the case if transmission spectra must be stacked together to achieve a higher signal-to-noise ratio (SNR), thereby smearing phase in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' In this work, we aim to explore the general question of asymmetry in exoplanet atmospheres, with particu- lar focus on its manifestations in high-resolution trans- mission spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Section 2 examines what drives asymmetry in exoplanet atmospheres;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' we here define a metric that quantifies limb-to-limb asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' In Sec- tion 3, we elaborate on diagnostics of specific mecha- nisms that may drive such asymmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This section additionally emphasizes how these diagnostics may be used to support or falsify compelling “toy models” mo- tivated by the drivers described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Finally, we summarize our results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' SELECTED DRIVERS Diagnosing limb asymmetries in transmission 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Example drivers of phase asymmetries of ultra-hot Jupiters Mechanism Proposed diagnostic Selected work(s) Escaping atmosphere H Lyman-α transit duration Owen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2023) Very deep, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Ca II lines Fossati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2013) Strong vertical winds Seidel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2021) Very broadened, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Na I lines Hoeijmakers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2020) Large diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' between (Na) doublet lines Hoeijmakers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2020) Strong H-α absorption Wyttenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2020) Blueshifting CCF w/ phaseP Bourrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2020) Excess He absorption (10830˚A) Oklopˇci´c & Hirata (2018), Spake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2018) Compare Doppler shifts of ions w/ different masses This work Scale height difference Blueshifting CCF w/ phase Kempton & Rauscher (2012) Strongly varying CO Doppler shift w/ phase Wardenier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2021), this work H2 dissoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='/recomb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Small phase curve amplitude Mansfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2020) High continuum/muted spectrum from H− Arcangeli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2018) Weak drag state Blueshifted CCF Wardenier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2021), Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022) Large phase curve offset May & Komacek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2021) Cold interior Blueshifted CCF Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022) Large phase curve amplitude May & Komacek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2021) Cold nightside May & Komacek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2021) Superrotating jet CCF FWHM exceeding solid-body rotation Brogi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2016) Phase curve offset Knutson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2007) Day-night winds Blueshifted CCF Snellen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2010) Equilibrium chemistry Limb-to-limb abundance discrepancy This work Compare chemical, dynamical timescales Showman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2013) Photochemistry Disequilibrium abundance of species Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2017) Increase in product, decrease in parent w/ phaseP Future work Condensation Model with GCM Wardenier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2021), Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022) Strongly blueshifting CCF w/ phase Ehrenreich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2020) Eccentricity All lines similarly Doppler shifted Montalto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2011), Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022) Independent orbital constraints Montalto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2011) Clouds All species become less blueshifted w/ phase This work Blueshifted CCF Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022) Lines absent in low resolution present at high resolution Kempton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2014), Hood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2020) Comparing CCFs of water bands Pino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2018) Lorentz forces Reduced hotspot offset Beltz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2021) Increased phase curve amplitude Beltz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2021) Westward hotspot offset Hindle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2021) Spatially varying winds Variable Doppler shift over phase Kempton & Rauscher (2012) Compare ingress / egress Doppler shifts Kempton & Rauscher (2012) Compare Doppler shifts of different-strength lines Kempton & Rauscher (2012) Tidal deformation/lag Blueshifting CCF over phaseP Future work Light curve fitting Akinsanmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2019) T-dependent opacity Blueshifting CCF over phase Wardenier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2021) Note—Tests with a “P” superscript have been proposed but not explicitly modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 4 Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' There exist a number of potential drivers of asymme- try in high-resolution transmission spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' But what are the relative strengths of these drivers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Previous works have considered the effects of conden- sation, longitude-dependent winds, and orbital eccen- tricity in producing such asymmetries (Wardenier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Table 1 includes these and a number of other potential drivers of asymmetry (along with potential diagnostics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' While many drivers are listed in Table 1, we consider in this work the relative strengths of two potentially first-order effects: the “scale height effect” and differences in equilibrium chemistry abundance across the limbs of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Be- ing both temperature-dependent effects, the distinction between the two is particularly subtle from an observa- tional perspective, and hence interesting from a theoret- ical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The scale height effect is due to the larger scale height in hotter regions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Miller-Ricci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2008), such that they are “puffed up” and cover more solid angle on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' These hotter regions therefore contribute more to the observed net Doppler signal in HRCCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The scale height effect is seen in Kempton & Rauscher (2012) as a slight, increasing blueshift over transit and as slight ingress/egress differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The effect there is not as dra- matic as in planets with larger east–west limb asymme- try, such as WASP-76b (West et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Wardenier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' With respect to equilibrium chemistry: because of the strong day–night contrasts in (ultra)hot Jupiter atmo- spheres, there exist strong spatial variations in temper- ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The day–night contrasts result in east–west con- trasts because the equatorial jet advects hot gas ahead of the substellar point to the evening limb and relatively cold gas from the antistellar point to the morning limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Furthermore, as the planet rotates on its spin axis dur- ing transit, the hotter side of the planet progressively ro- tates into view, exacerbating these differences at egress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Ignoring all disequilibrium processes and scale height differences, there should therefore exist strong spatial variations in gas-phase atmospheric composition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' at a given bulk composition, equilibrium chemistry implies variations in chemistry solely as a function of temper- ature and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' It is expected that asymmetries in transmission could hence vary as a function of temper- ature due to differences in chemistry alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Chemical gradients are invoked to explain a number of observational datasets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Ehrenreich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Kesseli & Snellen 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' However, other temperature- dependent effects, such as the scale height effect, may instead be driving observed asymmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' With this dis- tinction in mind, it is prudent to consider the difference in strength between these two effects and whether one considerably outweighs the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Asymmetry metric To quantify the asymmetry of chemical abundance in a planetary atmosphere, we construct a west–east asym- metry metric, AWE: AW E = 1 C � west log10 �� nα(T, P) dl � dΩ − 1 C � east log10 � � nα(T, P) dl � dΩ, (1) where, for species α, n is the number density in an atmo- spheric cell, dΩ is the solid angle subtended by a given sky-projected radius–latitude cell, and there are C total cells per limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' By equilibrium chemistry, n is a func- tion solely of temperature T and pressure P within a given cell in the modeled 3D atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' For each 2D sky-projected radius–latitude cell, dl is integrated along the line of sight through the planet’s 3D modeled atmo- sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This metric takes into account regions of the planet outside the terminator (which impacts transmis- sion spectra even at low resolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Caldas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019, Wardenier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022) by ray-striking through a 3D atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' AWE essentially reduces to the difference in mean (log) abundance between the two limbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The sign of this quantity encodes information about the asymmetry, as well: positive AWE implies that the western limb is more abundant in a species, whereas negative AWE implies that the eastern limb is more abundant in a species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Model atmospheres As of yet, we have remained agnostic to the model that generates the temperature–pressure structure and defines the grid cells for an AWE calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Some of the most complex and physics-rich descriptions of 3D exoplanet temperature–pressure structures are given by general circulation models (GCMs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Showman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' In this study, however, we seek to gain intuition for the basic scaling of asymmetry with planetary tem- perature (which drives the scale height and equilibrium chemistry gradients), and the added physical complexity of GCMs could add “noise” to this “signal”—it would be difficult to isolate the effect of increasing planetary temperature alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Furthermore, we here consider un- physical situations in order to determine the magnitude of the resulting difference with the correct physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Fi- nally, GCMs are very computationally expensive to run and have a number of free parameters to tune, and we here aim to explore a nontrivial grid of models over a representative range of parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Diagnosing limb asymmetries in transmission 5 We opt for a simple, parameterized approach instead of pursuing a full GCM description of our atmospheres for this specific experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Our model atmospheres have two parameters: a normalized east–west contrast ˜∆T = (Teast − Twest)/Teast and an equilibrium temper- ature Teq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' A normalized east–west contrast is a natural choice over an absolute east–west contrast for this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' namely, it prevents negative temperatures at low Teq, and it has physical meaning motivated by dynamical theory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Tan & Komacek 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' In these models, the choice of ˜∆T also uniquely enforces the east-west tem- perature differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The limb-to-limb difference cannot exceed the day–night difference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' based on the GCMs of Tan & Komacek (2019) and a set of phase curve ob- servations (Parmentier & Crossfield 2018), we do not expect a day–night contrast to exceed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='6, so we hold our east–west contrast below this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Hence, we here sweep our parameterized atmospheric models in ˜∆T from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='6, in addition to sweeping in Teq from 1000 K – 4000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Each atmosphere is charac- terized by two isothermal temperature–pressure profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Defining Teast = Teq + ∆T/2 Twest = Teq − ∆T/2 (2) and noting that ∆T = Teast−Twest, it therefore follows that Teast = Teq 1 − ˜∆T/2 Twest = Teq 1 + ˜∆T/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (3) With the substellar longitude at 0◦, all cells with a longitude φ < 180◦—the warmer evening limb—are as- signed temperature Teast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Conversely, all cells with a longitude φ > 180◦—the cooler morning limb—are as- signed temperature Twest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Pressures in the atmosphere run as low as 1 µbar, as one of the benefits of HRCCS is that it can probe low pressures such as these (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Kempton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Gandhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Hood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The bottom of the atmosphere is set at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='5 bar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' our previous 3D forward models run in Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022) across the optical and near-infrared indicate that for our test case of WASP-76b (West et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2016), this region is the deepest that can be probed given the expected continuum opacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The parameterized modeled atmo- spheres in this study have no set wind fields, as in our models (motivated by and assuming chemical equilib- rium), winds do not control AWE—only the chemical abundance of a given cell does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We calculate AWE to assess the relative strength of the scale height and equilibrium chemistry effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' To infer the strength of the scale height effect, we construct pairs of model atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' In each pair, one atmosphere is constructed self-consistently: pressure falls off per hy- drostatic equilibrium, with the scale height set by the temperature on either limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' For the models here, we hold composition constant across both limbs, thereby holding µ constant at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='36 (appropriate for a solar- composition gas dominated by molecular H2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Kempton & Rauscher 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' See Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='4 for a discussion of this caveat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The other atmosphere in the pair is constructed on the same pressure grid as the western limb at all lon- gitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' That is, the eastern limb is not simulated as inflated compared to the western limb—removing the scale height effect from the projected model atmosphere in transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Equilibrium chemistry To calculate the number densities of our species in each modeled atmospheric cell (nα), we construct a grid in temperature–pressure space using the FastChem equi- librium chemistry code (Stock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2018) and interpo- late the grid based on local atmospheric cell temper- ature and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We initialize the code with solar abundances from Lodders (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Our chemistry code does not explicitly include any condensation or cloud- formation processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Even disregarding questions of species detectability in HRCCS data, it is worth considering that not all species with FastChem thermochemical data have freely avail- able opacity data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' With this constraint in mind, we re- strict our AWE molecule calculations to molecules with opacity data available on ExoMol,2 a popular opacity database for exoplanet atmosphere modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Asymmetry metric: application We calculate AWE for our grid of parameterized at- mospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Disregarding the scale height effect, we find that positive ions tend to form preferentially on the hotter limb of our models at an equilibrium tempera- ture of 2200 K (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This is expected, as ther- mal ionization should increase the abundance of positive ions at higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Furthermore, larger east– west temperature asymmetries lead to larger abundance asymmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Including the scale height effect increases the asymme- try for neutral atoms and molecules, as can be seen by comparing the right-hand sides of Figures 1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Further- more, there is more homogeneity across the AWE values 2 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='exomol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='com/ 6 Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (a) (b) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Asymmetry (as defined in Equation 1) of all chemical species considered in this study in our parameterized at- mospheres at an equilibrium temperature of 2200 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' These models do not self-consistently inflate the hotter limb of the parameterized model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', they do not observe the “scale height effect”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The shading of each species represents the normalized temperature difference, ˜∆T, across the two limbs of our parameterized atmospheres;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' the lightest boxes have ˜∆T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1, whereas the darkest have ˜∆T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' For illustrative purposes, we color in green tick marks for species with detections noted in Guillot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022) (and including the recent CO2 detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Ahrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We also draw a vertical line denoting 0 asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Without taking the scale height effect into account, positive ions form much more predominantly on the warmer limb (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', have negative asymmetry) than other species and reach the greatest asymmetry values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' across positive ions, negative ions, neutral atoms, and neutral molecules (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' In particular, while higher ˜∆T still implies higher absolute asymmetry in neutral species, the scale height effect makes it such that the warmer limb almost always has higher projected asym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' It is therefore clear that the scale height effect strongly tamps down genuine variation in species abundance due to equilibrium chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' However, the fact that inter- species variation in asymmetry remains implies that this variation in abundance is not completely washed out by the scale height effect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' if the scale height effect truly and fully dominated, all species would have the same AWE value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' When considering individual species more closely, we find that certain species are particularly differentially affected by the scale height effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' For example, Fig- ure 3 shows that there is a stark difference in whether the scale height effect is included for Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' However, this is not as much the case for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Sr II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The meaning be- hind this result is evident in the equilibrium abundance calculations of Fe and Sr II: Fe is less sensitive to tem- = 2200K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' scaled=False ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='JS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Rb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ga ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Sc - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ca ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Li - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Zn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='V- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ti ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='si - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='p- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='IN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Na ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='N- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='H- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Fe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Cu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Cr : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Co ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='CI - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Al - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Zr IIII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='YIII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Sr III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Rb IlI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ge llI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ga IlI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Zn III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Cu llI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ni III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Co IIII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Fe IIII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Mn III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Cr IIII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='VII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ti II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Sc III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ca II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='KII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ar IIII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Positive-ions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='CI II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='ads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='S III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Negative ions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='P III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Si III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Al II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Mg IIII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Na III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Nelll ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='F III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='O I=I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='N III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='C III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Li I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Rb II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ge ll ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ga lI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Li II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Zn II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Si lI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='s II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='PII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ni lI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ne ll ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Na II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='NII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Mg II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='KII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='He lI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='H3O lI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='H2 II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='HO II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='HII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='F II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Cu II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Co II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='CI II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='C II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ar II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='AIlII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='ScllI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='YII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Sr II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Zr II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='VII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='CrlI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Mn II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Call ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ti lI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Fel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='AsymmetryTeg = 2200K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' scaled=False sis PS SO3 SiO2 SO2 02 sio PO N20 N2 NS PN NO SiH2 H202 TiH HSi SH HP NiH NaH HNO3 HN MgH HKO FeH PF3 NaF FMg HF CrH CINa CIK CIH Cao CaH CaF C2H4 C2 CS CP COS CN CH3 S CH20 e CH AIO AlH AIF AICI OH ov S O!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' PH3 NH3 H2S H20 CO2 CO CH4 C2 H2 H2 Zn s P Ne N Ge F Cl Ar Co Al Z Sc Y Sr Rb Ga Cu IS 0 Li Mn Ni Cr c IL Ca Neutral atoms Mg Fe Na Neutral molecules He H 102 101 100 100 101 0 AsymmetryDiagnosing limb asymmetries in transmission 7 (a) (b) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Similar to Figure 1, but now including the scale height effect (inflating the hotter limb in our parameterized models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Now, all species have asymmetries that favor the hotter limb (negative asymmetry)—simply because the hotter limb subtends more solid angle on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' However, there still exists inter-species variability in asymmetry, implying that the scale height effect does not entirely swamp genuine differences in equilibrium chemistry across limbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Furthermore, negative ions still have larger asymmetries than positive ions or neutral species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' perature variations than Sr II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This result is expected, as the onset of Sr II is determined by the temperature at which Sr I can be effectively ionized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This is gen- erally the case for positive ions—the temperature effect on chemical abundance wins out over the scale height ef- fect, as seen by the left-hand sides of Figures 1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Physi- cally, this behavior is because the Saha equation is more strongly dependent on temperature than most chemical equilibrium reaction rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The results of this experiment indicate that the most temperature-sensitive species are strongly influenced by both abundance changes and scale height differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Conversely, to isolate the scale height effect, it would be therefore useful to consider a species with very weakly temperature-dependent abundance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' in this case, if a strong asymmetry were detected, it could be attributed to a scale height effect (or other non-equilibrium chem- istry or physics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We explore this idea further in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Note that this approach, aside from its simplified temperature–pressure structure, does not account for a variety of physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Namely, it does not include the ef- fects of hydrogen dissociation and recombination that occurs in the ultra-hot Jupiter regime (Tan & Komacek 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Inclusion of this physics would serve to decrease the mean molecular weight in the atmosphere, increasing the scale height for the hotter, eastern limb, thereby am- plifying the observed asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Additionally, at the = 2200K, scaled=True Zr - Sr Rb Ge Ga Sc - Ca Li - Zn V- Ti Si - s p- 0 IN Na N- K H- Fe F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' no Cr: Co CI - c Al - Zr IIII Y III Sr III!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Rb II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ge llI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ga IlI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Zn III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Cu llI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ni III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Co IIII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Fe IIII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Mn IIII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Cr III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='VII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ti II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Sc III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ca llI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='KIII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ar IIII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Positive-ions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='CI IIII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='S III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='PIII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Negative ions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Si III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Al III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Mg IIII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Na II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ne III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='FII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='o II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='NIII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='C II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Li lII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Rb II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ge ll ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ga lI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='LilI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Zn II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Si lI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='sII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='PII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ni lI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ne lI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Na llI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='NII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Mg II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='KII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='He lI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='H3O II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='H2 II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='HO II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='H II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='F II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Cu lI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Co II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='CI II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='C II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ar II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='AIlII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='ScllI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='YII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='SrlI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='ZrlI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='viII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='CrlI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Mn II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ca lI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Ti II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Fe ll ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='Asymmetry= 2200K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' scaled=True sis PS SO3 SiO2 Neutralatoms SO2 02 Neutral-molecules sio PO N20 N2 NS PN NO SiH2 H202 TiH HSi SH HP NiH NaH HNO3 NH H6W HKO FeH PF3 NaF FMg HF CrH CINa CIK CIH Cao CaH CaF C2H4 C2 cs CP COS CN CH3 S CH20 CH AIO AlH Speo AIF AICI HO ov S O!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' PH3 NH3 H2S H20 CO2 co CH4 C2 H2 H2 1 Zn s P Ne N Ge F C1 Co Al Z Sc Y Sr Rb Ga Cu Si 0 Li Mn Ni Cr c V IL Ca Mg Na K He H 102 100 101 101 100 0 Asymmetry8 Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Asymmetry (per Equation 1) for Sr II, Fe, H2O, and CO in our parameterized atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Our grid sweeps over equilibrium equilibrium temperature and normalized temperature difference across limbs, and includes models that observe the scale height effect (circles) and do not (squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We find that species with strong temperature-dependent abundances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Sr II) are less dominated by the scale height effect than species with weaker temperature-dependent abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' lower-temperature end, we did not include the effects of certain species being sequestered into clouds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', sili- cate clouds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We will model the Doppler shift impact of optically thick clouds in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Finally, our approach does not include disequilibrium effects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', vertical / horizontal mixing) that may alter asymme- tries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Therefore, the results shown here motivate asym- metries due to equilibrium chemistry alone, which we expect to be a first-order driver of asymmetry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' disequi- librium chemistry is not expected to be significant in the ultrahot Jupiter regime (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We further did not include the effect of temperature- and pressure-dependent opacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' At the spectrum level, a temperature asymmetry would be exaggerated by the fact that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Fe absorbs more on the hotter limb than the colder limb because its opacity increases with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This would mean that the detected net Doppler shift is even more strongly weighted to the hotter limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Despite these limitations in our modeling, the trends listed above should hold to first order and provide intu- ition about the relative strengths of two potential drivers of asymmetry in exoplanet atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Broadly, it holds that the scale height effect appears to dominate in general, but relative differences in abundances of species as a function of temperature still matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Given the lim- Al CO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='6 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' No scale height effect 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Scale height effect least 口 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='5 口 口 0 0 口 口 口 口 C 口 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='4 O 0 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 1000 2000 3000 1000 2000 3000 T H20 SrlII 101 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3 口 口 least 口 口 口 口 口 0 0 Iwest 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2 口 口 口 口 口 口 口 08 口O 0000 : DO 00 000 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1 1000 2000 3000 1000 2000 3000 (K) Teq (K) 2Diagnosing limb asymmetries in transmission 9 itations of simple models, we will move on to more self- consistent atmospheric modeling in the following sec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' SELECTED DIAGNOSTICS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Diagnostics for specific mechanisms Per Section 2, even differentiating between two drivers of asymmetry in exoplanet atmospheres is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Drivers can compete to varying degrees to produce a similar result: an asymmetric trend in net Doppler shifts in HRCCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' However, by exploiting nuances in the HRCCS Doppler shift signal and by independent means, it may be possible to disentangle even drivers that produce sim- ilar effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Table 1 lists example drivers of asymmetries in HRCCS and how they might be diagnosed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The asso- ciated works listed in the table may not directly propose these diagnostics, but at minimum they provide founda- tional material for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Of course, exhibiting a single diagnostic does not not mean that a given physical mechanism is in play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Other mechanisms could surely be present, and uniquely con- straining a single mechanism as dominant would require ruling out the others, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' For instance, both day– night winds and morning limb condensation could result in a net blueshifted CCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' But if, for example, a night- side temperature were derived from a phase curve that was far too hot for any known condensate to form, then day–night winds would be much preferred to conden- sation as a physical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Together, collections of diagnostics are hence able to test the dominance of in- dividual mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' In the following sections, we explore a few tests for specific physical mechanisms of asymmetry: using CO as a baseline molecule to identify the scale height effect and tracking the blueshifts of multiple species to identify the presence of clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We furthermore evaluate the ef- fectiveness of diagnostics that may be used to evaluate a number of different mechanisms: averaging HRCCS data into two phase bins and using finely phase-resolved HRCCS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We additionally show how these diagnos- tics can further motivate or rule out “toy models” that at first may appear convincing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' CO as a baseline molecule We have demonstrated (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='4) that species with strongly temperature-dependent abundances are the least susceptible to the scale height effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Con- versely, observing a species with very weak temperature- dependent abundance could indicate whether the scale height effect is in play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Volume mixing ratio of CO as a function of pres- sure and temperature as calculated by FastChem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Overplot- ted are the onset of ultra-hot Jupiters (as defined by their dayside temperature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Parmentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2018), the CO/CH4 equivalency curve from Visscher (2012) as a function of pres- sure, the Fe condensation curve from Mbarek & Kemp- ton (2016), and 1D temperature–pressure profiles for a hot Jupiter (WASP-39b) and an ultra-hot Jupiter (WASP-18b) as computed with HELIOS (Malik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Both the con- densation curve and the equivalency curve are computed at solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Considering the regime of ultra-hot Jupiter atmospheres, CO is a relatively stable chemical species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Consider CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' In Figure 3, its AWE values are clustered around 0 without the scale height effect, with relatively weak dependence on ˜∆T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' However, CO’s AWE values are strongly negative when the scale height effect is in- cluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We propose using CO as a tracer of the scale height (and other chemistry-unrelated) effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' As shown in Figure 4, the abundance of CO is relatively stable between 1000 K and 3500 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Beltz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022) note that this stability holds over the temperature–pressure range of the observable atmo- sphere of the ultra-hot Jupiter WASP-76 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Indeed, this feature remains true over the general temperature– pressure range of ultra-hot Jupiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' For illustrative pur- poses, we calculate the 1D temperature–pressure pro- files of a hot Jupiter (WASP-39b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Faedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2011) and an ultra-hot Jupiter (WASP-18b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Hellier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' These profiles, calculated with the HELIOS 1D radiative- convective model (with full heat redistribution), indi- cate that the observable atmosphere for these planets is largely within a region of near-constant CO mixing ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The stability of CO is due to three factors: its strong chemical bonding, its lack of participation in gas-phase chemical reactions, and its lack of condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 10-6 3 Ultra-hot Jupiter dayside onset 4 CO/CH4 equivalency 10-5 Fe condensation 5 WASP-18 b 1D profile Pressure (bars) WASP-39 b 1D profile 6 10-4 (xp) 0x 7 10-31 8 9 10-21 10 10-1 11 12 100 1000 2000 3000 4000 5000 Temperature (K)10 Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Since the strong triple bond of CO makes it diffi- cult to thermally dissociate, CO remains stable at tem- peratures that would dissociate molecules with weaker bonds, such as H2O (Parmentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2018), which has two single bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Beyond roughly 3500 K, even the triple bond becomes susceptible to thermal dissocia- tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' hence, the few exoplanets with significant portions of their atmosphere hotter than this temperature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', KELT-9b, with Teq ≈ 4050 K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Gaudi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2017) would likely exhibit spatial variation in CO abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Most ultra-hot Jupiters, though, should fall shy of this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Furthermore, the high photoionization threshold of CO (relative to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', H2O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Heays et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2017) means that it is not commonly photodissociated (Van Dishoeck & Black 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Even when it is photodissociated, recy- clying pathways exist in hot Jupiters that can replenish CO abundance, keeping it near equilibrium abundance even inclusive of photochemistry (Moses et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Hence, the assumption of non-dissociation of CO is rea- sonably justified across much of the ultra-hot Jupiter population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Additionally, CO does not commonly participate in thermochemical reactions and is the dominant car- bon carrier in our temperature–pressure range of inter- est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' While at lower temperatures the dominant carbon carrier becomes CH4, the ultra-hot Jupiter regime is squarely beyond the CO/CH4 equivalency curve (Fig- ure 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Visscher 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Therefore, even aside from ther- mal dissociation, CO should not participate in gas-phase thermochemistry that would alter its abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Finally, CO does not form any high-temperature con- densates expected in ultra-hot Jupiter atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The condensation temperature of CO (≈80 K at 1 bar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Lide 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Fray & Schmitt 2009) is far below the temperature–pressure range of ultra-hot Jupiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This quality makes CO a less complicated tracer of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', at- mospheric dynamics than species that do condense in this region of parameter space, such as Fe, Mg, or Mn (Mbarek & Kempton 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Therefore, while the calcu- lations of Figure 4 do not include gas-phase condensa- tion, the resultant spatial constancy of CO should still be robust even when condensation is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' CO is thus a more straightforward molecule to model than other, condensing species, as it does not participate in the complex microphysics of condensation and cloud for- mation (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Beyond its spatial uniformity, there are further obser- vational reasons that CO is an appealing species to tar- get.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Namely, CO has very strong spectroscopic bands placed across the infrared wavelength range (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2015) that do not overlap with other strong ab- sorbers and are relatively well understood (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Additionally, the high cosmic abundance of C and O (Lodders 2003) means that, unlike many of the species in the previous section, CO is readily detectable (and has been become a standard detection in HRCCS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Snellen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' de Kok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Rodler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Brogi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2014, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Flowers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Gia- cobbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Line et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Pelletier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Guilluy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' van Sluijs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Given its stability and observational advantage, we propose that CO can be used as a faithful tracer of the atmosphere—whether it is inflated in some regions, what its wind profile is, whether regions are blocked by clouds, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' In turn, CO may then be leveraged to better mo- tivate sources of asymmetry that affect other species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' While other species with low AWE in Figure 1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', He, Fe, MgH, Rb II) would also appear to be good candi- dates for baseline species, these species are either largely spectroscopically inactive, have variable abundance over broader temperature–pressure ranges, or can condense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' A caveat to the above is that while CO is a faithful longi- tudinal tracer, it is not an unbiased radial tracer (as seen in Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' As with all chemical species, CO has its own balance between deep and strong lines that depends on the waveband considered (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Therefore, the net CO Doppler signal does not uniformly weight the wind profile across all altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Again, this is a bias inherent to all chemical species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' A decreasing blueshift test for clouds As noted in Table 1, clouds may introduce strong asymmetry into HRCCS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022) demonstrated that gray, optically thick clouds produce stronger blueshifts in the Doppler shift signal of WASP- 76b than the blueshifts in clear models, also changing the trend of Doppler shift with phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' But, again as shown in Table 1, these changes at the Doppler shift level are not sufficient to uniquely identify clouds as the driver of an observed asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Combinations of ob- servable quantities that would uniquely identify clouds as the source of observed HRCCS asymmetry are there- fore necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' To devise such a test, we investigate in this work a limiting-case cloudy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' As in Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022), we construct gray, optically thick, post-processed clouds in our 3D ray-striking code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We here make another as- sumption, though: that the clouds are confined to the cooler, morning limb, as opposed to having a distribu- tion dictated by a specific species’ condensation curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This distribution is based on planetary longitude (be- tween longitudes of 180◦ and 360◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This approach is motivated by the results of Roman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2021), who Diagnosing limb asymmetries in transmission 11 found that a subset of cloudy GCMs exhibited a cloud distribution strongly favoring the western limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3 Our approach benefits from providing limiting-case intuition for how cloudiness affects Doppler shift signals while avoiding the complex questions of how clouds form and which species contribute the most opacity (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Gao & Powell 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Briefly, our modeling methodology is as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Double-gray, two-stream GCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' GCMs such as this one solve the primitive equations of meteorology, which are a reduced form of the Navier-Stokes equations solved on a spherical, rotating sphere with a set of simplifying assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='4 The out- put of these models is temperature, pressure, and wind velocity as a function of latitude, longitude, and altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We use the GCM that was shown to best fit the Ehrenreich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2020) WASP-76b data in Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Equilibrium chemistry with FastChem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' As in Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3, we interpolate a model grid of chem- istry to determine local abundances of a number of chemical species as determined by temperature and pressure conditions of the GCM output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Ray-striking radiative transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Using a code modified from Kempton & Rauscher (2012) (as detailed in Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022), we compute the high-resolution absorption spectrum of our plan- etary atmosphere by calculating the net absorp- tion of stellar light along lines of sight through our GCM output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This absorption is calculated inclusive of net motions along the lines of sight from atmospheric winds and rotation, inducing Doppler shifts relative to that of a static atmo- sphere’s spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Limb-darkening is calculated with a quadratic limb-darkening law in the observ- able planetary atmosphere and with the batman code (Kreidberg 2015) for the portion of the star blocked by the optically thick planetary interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Given its increasing utility as a benchmark planet for HRCCS studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Ehrenreich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Kesseli & 3 These GCMs produced clouds on a temperature–pressure basis, and did not model clouds as tracers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Therefore, they do not capture potential disequilibrium cloud transport (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', as done in Komacek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022), which may alter the degree of patchiness within the cloud deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 4 These assumptions are 1) local hydrostatic equilibrium, such that vertical motions are caused purely by the convergence and di- vergence of horizontal flow, 2) the “traditional approximation,” which removes the vertical coordinate from the Coriolis effect, and 3) a thin atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Atmospheric Doppler shifts, which should remain in the HRCCS signal after the orbital motion is subtracted, as a function of orbital phase for our forward models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Shown are representative species that span Doppler shifts and are noted as potentially observable by Kesseli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022): Fe, Sr II, and Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Cloud-free models are represented with solid lines, whereas models with fully cloudy morning limbs are represented with dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The first half of transit (RV1) and second half of transit (RV2) Doppler shifts for Fe from Kesseli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022) are overplotted as horizontal lines, with width determined by observational errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Our cloudy mod- els are much more strongly blueshifted than their cloud-free counterparts, become less blueshifted over transit, and do not have significant CCF peaks at early phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Snellen 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Landman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Seidel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Wardenier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Kesseli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' S´anchez-L´opez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022), we model the ultra-hot Jupiter WASP-76b (West et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We calculate 25 spectra inclusive of Doppler effects equally spaced in phase from the begin- ning to end of transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' For our cross-correlation tem- plate, T , we use a model that does not include Doppler effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We then cross-correlate our template against our cal- culated spectrum, y: c(v) = N � i=0 yi(λ)Ti(v, λ), (4) where the mask or template is Doppler-shifted by ve- locity v and interpolated onto the wavelength grid, λ, of y for summing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Our CCF is computed on a grid of velocities from −250 km s−1 and 250 km s−1 with a step of 1 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The final net planet-frame Doppler shift is calculated by fitting a Gaussian to the peak of the CCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The results of our experiment are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' When we allow clouds to extend over the entire morn- ing limb, note that all species become less blueshifted over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Because the limb that is rotating away from the observer (the “receding limb”) is entirely blocked off by clouds, there is no wavelength-dependent absorption 2 Kesseli+22 Fe RVi Sc Kesseli+22 Fe RV2 Sr plus Planet-frame RV (km/s) 0 Fe 2 6 8 一10 12 15 10 5 0 5 10 15 Phase (degrees)12 Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' for that limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Therefore, the contribution of redshift- ing from solid-body rotation on the receding limb is not present—the only Doppler shift contributions are from evening limb rotation and evening limb winds, which are generally in the same direction as the rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Hence, there are much stronger blueshifts at earlier phases than in the clear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' However, at later phases, the non-cloudy regions of the atmosphere rotate into the receding limb, thereby con- tributing some rotational redshift to the net Doppler shift signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='5 At the earliest phases, the cloudy mod- els do not have enough wavelength-dependent absorp- tion to produce a significant CCF peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Notably, all species follow this trend, as the blocking of clouds as modeled here is wavelength-independent and altitude- independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This behavior is shown in Figure 5 for Fe, Sc, and Sr II—all species identified in Kesseli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022) has having high potential observability for ultra- hot Jupiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' From Figure 5, it is also apparent that the cloud- driven trend of decreasing blueshift in phase is not matched by the observations of Kesseli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' As found in Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022) in comparison to the Ehrenreich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2020) data, while the absolute magni- tude of the cloudy model’s Doppler shift better match the data than the clear model’s, the cloudy model trend over Doppler shift is not matched by the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' In sum, this limiting-case model of opaque, morning limb clouds does not appear to be a first-order effect driving ex- isting observational trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This does not necessarily mean that clouds are not the driving factor behind limb asymmetries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' it may simply be that a more physically motivated model for partial cloud coverage of the limb could fit the available data better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Also of note in Figure 5 is that the egress signatures of the clear and cloudy models are quite distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Near a phase of roughly 14 degrees, the clear model produces a sharp change in Doppler shift for all species as the lead- ing (rotationally redshifted) limb begins to leave the stel- lar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This sharply blueshifting behavior continues to the end of egress, until the last sliver of the trailing (rota- tionally blueshifted) limb has left the stellar disk as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' In the cloudy case, however, the leading limb leaving the stellar disk has no effect, as it is fully cloudy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' While this effect is evident in these models, it may be less evident 5 The degree of rotation during transit varies as a function of semimajor axis and host star radius, and hence from planet to planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' While we only model WASP-76b, other planets also have large (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', compared to the angles probed by transmission spec- troscopy) rotations during transit (Wardenier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' in observations, which naturally cannot finely sample ingress and egress phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Phase bins We have thus far examined drivers of asymmetry and potential diagnostics of specific mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Next, we will evaluate a few HRCCS data types to determine how robust they are and their potential ability to constrain a number of different physical mechanisms that give rise to HRCCS asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The first of these data types is HRCCS Doppler shifts that are binned in phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' A substantial fraction of HRCCS studies present detections and Doppler shifts integrated over the entirety of transit (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Giacobbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This approach maximizes detection SNR, which may be necessary for a given set of observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', because of a low-resolution spectrograph, small telescope aperture, faint star, low species abundance, low number of absorption lines, or weak intrinsic ab- sorption line strengths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' While it is possible to reveal aspects of limb asymmetry with this approach, espe- cially when comparing detections of multiple species to one another, phase-resolving the transit (and observing isolated ingresses and egresses when possible) will cer- tainly give a more direct probe of east–west asymme- tries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Binning HRCCS data in phase across transit may provide a desirable balance between revealing asymme- try and maintaining high SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We seek to address this question by phase-binning modeled Doppler shifts to examine its biases with re- spect to the underlying model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We follow this experi- ment with a comparison to the phase-binned observa- tions of Kesseli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Theoretical phase binning We average our phase-resolved calculations into two bins: the first and second half of transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Once our CCFs are calculated we average them in phase to ef- fectively reduce our data to two single bins: the first half of transit and the second half of transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We make versions of these two half-transit bins that include or exclude the ingress and egress phases (when the planet is only partially occulting the star).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Motivated by recent detections in the near-infrared (Landman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' S´anchez-L´opez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022), we search for absorption from various molecules6 in our models, focusing on the CARMENES (Quirrenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2014) wavelength range and resolution for direct 6 We use the MoLLIST (Brooke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2016), POKAZATEL (Polyansky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2018), and Li2015 (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2015) linelists for OH, H2O, and CO, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Diagnosing limb asymmetries in transmission 13 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Single-species (OH, CO, H2O, HCN) 3D forward- modeled spectra of WASP-76b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' These spectra are simulated over the CARMENES waveband and resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Doppler effects are not included in these spectra, which are modeled at center of transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' H2O is the dominant absorber in this bandpass, followed by OH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' HCN exhibits no spectral features above the continuum for WASP-76b in this bandpass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' comparison against observational results using that in- strument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Of these molecules, we find that OH, H2O, and CO produce significant absorption over the mod- eled wavelength range, with OH and H2O producing the strongest features (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We find that HCN does not produce any noticeable absorption under the as- sumption of chemical equilibrium and solar composition, implying either more exotic chemistry for WASP-76b’s atmosphere (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', photochemistry or non-solar abun- dances;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Moses et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2012), or that the detection of HCN in this atmosphere (S´anchez-L´opez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022) was spurious (perhaps due to the nature of the HCN opacity function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We furthermore find a moderate (≈ 4 km s−1) increase in blueshift for our modeled H2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' While this increase in blueshift is commensurate with the increase in blueshift described for H2O in S´anchez-L´opez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022), we are once again unable to match the high reported velocities (here 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3 km s−1) with our self-consistent forward models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Figure 7 shows the results of this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' As the error for each phase bin, we take the average error of phase bins from Kesseli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='55 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We define the two phase bins as inconsistent if the peak of their respective CCFs are inconsistent at 2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We find that excluding ingress and egress phases can strongly reduce the difference in derived Doppler shift between phase bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Furthermore, we find that, as ex- pected from Kempton & Rauscher (2012), differences between bins are maximized when just considering the ingress and egress phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='7 While higher-order drivers of asymmetry are clearly not detectable with phase bins (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' at what longitude condensation may begin to play a role;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Wardenier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021), certain drivers of asymmetry are accessible with this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' For example, ignoring for now the exact details of error budgets, all species in Figure 7 clearly blueshift over the course of transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This provides poten- tial evidence for, among other things, a spatially vary- ing wind field, condensation, optically thick clouds, or a scale height effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Furthermore, per the results of Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1 the detection of CO’s blueshifting indicates that something besides equilibrium chemistry is driv- ing at least some of the asymmetry in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' These underlying models are cloud-free, so these results imply sensitivity to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', the scale height effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Comparison to Kesseli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022) With our models calculated, we can now explore the ability of phase-resolved spectra to confront toy mod- els by comparing the models to observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' A prime observational work that made use of phase binning is Kesseli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' there, the authors search for asym- metries in two phase bins for a wide variety of species, motivated by the strength of those species’ opacity func- tions in the data’s wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' To consider a toy model: based on previous studies (Ehrenreich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Tabernero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022), it appears that Ca II does not follow the Fe-like Doppler shift trend first observed by Ehrenreich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Rather, it appears that Ca II, with its strong opacity and resultant deep lines, may be probing a non-hydrostatic region of the atmosphere (Casasayas- Barris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Deibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Tabernero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This region of the atmosphere cannot be cap- tured by the models of this work and Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Without a model of atmospheric escape, it seems dif- ficult to elevate the above picture beyond “toy model” status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' However, by phase-resolving multiple species, a clearer picture can emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' For our comparison with Kesseli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022), we use the same line lists as in that study: the National In- stitute of Standards and Technology (NIST;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Kramida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019) line lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' It is crucial to use the same line 7 We would expect that binning with fewer spectra (just includ- ing the ingress phases) would increase the associated error on Doppler shift at each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' However, the point of this exercise is to illustrate the magnitude of ingress/egress Doppler shift discrep- ancy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' observational strategies such as stacking multiple transits could reduce errors in practice and make these differences dis- cernible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' H20 CO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='375 HO HCN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='350 Transit depth (%) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='325 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='300 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='275 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='250 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='225 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='7 Wavelength (microns)14 Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' CCFs of individual species averaged over two phase bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Each column corresponds to different species (OH, CO, H2O), and each row corresponds to different bin selection: without including ingress and egress, including the full transit, and only including ingress and egress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Central bars between the CCFs are colored blue if the difference between the CCFs is greater than optimal Doppler shift errors (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='55 km/s, in black;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Kesseli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' otherwise, they are colored red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' In our models, CO only displays detectable CCF differences when only including ingress and egress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The SNR in each plot refers to the difference between the two phase bins’ CCF peaks relative to the optimal Doppler shift errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' lists for comparisons of HRCCS studies—different line list databases can contain vastly discrepant numbers of line transition, which greatly affects the resultant opac- ity function (see, for instance, Figure 11 of Grimm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The results of our comparison with the species de- tected in Kesseli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022) are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' As in Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022), these baseline models—no clouds, no condensation, no orbital eccentricity—cannot fully explain the Doppler shifts of Fe observed in WASP- 76b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' However, the comparison across multiple differ- ent species provides further constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Figure 8 shows that Fe, V, Cr, Ca II, and Sr II are strongly discrepant from our models for at least one half of transit, whereas Na, Mg, Mn, and Ni are reasonably well described by our models for both the first and second half of transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Furthermore, Fe, V, and Cr all have stronger blueshifts in the second phase bin than in our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The similar CO, no ingress / egress OH, no ingress / egress H2O, no ingress / egress 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='4 - 2nd half peak: i 1st half peak: 2nd half peak: i 1st half peak: 2nd half peak:i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 1st half peak: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='52 km s-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3 km s-1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='41 km s-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='15 km s-1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='93 km s-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='73 km s-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2 SNR: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='4 SNR: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='5 SNR: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='8 3333333333 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2 CO, full transit H20, full transit OH, full transit 2nd half peak: 1st half peak: 2nd half peak:!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 1st half peak: 2nd half peak: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='4 - 1st half peak: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='73 km s- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='38 km s-1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='08 km s- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='27 km s-1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='23 km s-1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='81 km s-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2 - 0 SNR: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2 SNR: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='5 SNR: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='8 CF 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2 CO, only ingress / egress OH, only ingress / egress H2O, only ingress / egress 2nd half peak:i 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='4 1st half peak: 2nd half peakl 1st half peak: 2nd half peakl 1st half peak: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='28 km s-1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='27 km s-1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='04 km s-1T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='24 km s-1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='98 km s-1T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='65 km s-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2 ■ SNR: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='8 SNR: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='6 SNR: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='8 CF 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='6 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2 40 20 0 20 40 40 20 0 20 40 40 20 0 20 40 Velocity (km/s) Velocity (km/s) Velocity (km/s)Diagnosing limb asymmetries in transmission 15 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The net Doppler shifts of Kesseli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022) (error bars) as compared to this work’s models (crosses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The first phase bin is drawn thinner than the second phase bin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' observed phase bins are connected by a dotted line for visibility’s sake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The species are ordered and colored by total observed detection SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Rows without crosses correspond to species that we could not recover via cross-correlation in our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Our models are able to explain some species (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Na), fail to explain others (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Cr) and fail to detect yet others (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' level of disagreement between Fe, V, and Cr implies that they share a common driver of asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This result in turn implies that whatever driver affects them affects the regions in which these species form similarly — be it clouds, condensation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' To bridge the toy models presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3 to our Kesseli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022) comparison, we compute a set of high-resolution spectra exactly as above, but with the same altitude grid at all latitudes and longitudes in an effort to effectively turn off the scale height ef- fect while maintaining chemical limb inhomogeneities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Post-processing this (self-inconsistent) model yields less than half the Doppler shift asymmetry as compared to our self-consistent models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This experiment confirms the intuition that the scale height effect is a first-order asymmetry effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Finally, we consider the Ca II toy model previously de- scribed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Certain lightweight and/or ionized species may be entrained in an outflow, as indicated by some pre- vious observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Tabernero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021) of very deep absorption lines in transmission that must extend very high up in altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The differential behavior of the Ca II and Sr II Doppler shifts lends more credence to this hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' In sum, by taking advantage of phase-binned spectra, it is possible to better identify drivers of HRCCS asym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Additionally, our predictions in Figure 8 indi- cate that most species should have roughly the same Doppler shift patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' In stark contrast, observations reveal much larger variations in velocity across different species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' While some interpretation may be due to spuri- ous detections, physics that is not included in our model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', outflows, condensation) may be playing a driving role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Full phase-resolved spectra Currently, the most information-rich diagnostic avail- able to probe asymmetry in HRCCS is phase-resolved cross-correlation functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Ehrenreich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Borsa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021)—that is, net Doppler shifts associated with the absorption spectrum evaluated over multiple points in transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' With these data, one should be able to directly constrain longitudinally dependent drivers of asymmetry, providing the best chance of disentangling the physical mechanisms outlined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' But how far can we push these data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Example: probing physics in the NIR To explore this question, we take as an example a three-species (OH, H2O, and CO) near-infrared (NIR) dataset over a CARMENES-like waveband as in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=" Figure 9 shows the Doppler shifts of these 14 Sr+ Ni Kesseli+22 SNR of species detection Co Model, first bin 12 Fe Model, second bin Kesseli+22, Mn first bin Kesseli+22, 10 cies Cr second bin V d S'Ca+ 8 K Mg Na 6 Li H 15 10 5 0 5 10 Planet-frame Doppler shift (km/s)16 Savel et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Modeled phase-resolved Doppler shifts for select NIR-absorbing species, with representative error bars (Ehrenreich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020) drawn on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We find that OH and H2O have distinct Doppler signatures from CO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' however, OH and H2O have Doppler shifts that are indistinguishable from one another with current best-case error bars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Ehrenreich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Considering CO as a “baseline species” here allows one to better understand how H2O and OH may change through the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' species as a function of phase, produced for single species at a time as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2, but without any averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Without considering any data, a compelling toy model would be as follows: H2O is thermally dissociated on the hotter, approaching limb, so it preferentially exists on the receding limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' OH is a product of H2O photodis- sociation, so it forms preferentially on the approaching limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' CO is constant everywhere;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' therefore, CO should not experience much of a trend in Doppler shift, OH should be more blueshifted than CO, and H2O should be more redshifted than CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We shall see, however, that additional, complicating physics is revealed by fully phase-resolved spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' For our models, the relevant underlying physics is as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Altitude-dependent winds: H2O lines are more strongly blueshifted than CO lines at all phases because the H2O line cores over the wavelength range of the CARMENES bandpass more predom- inantly form at higher altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' At high altitudes, the atmospheric flow switches from dominantly ro- tational (via an eastward equatorial jet) to dom- inantly divergent (via day–night winds) (Ham- mond & Lewis 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This result is the opposite of what would be expected from the above-described toy model, revealing the shortcomings of simple models and how they can sometimes mislead us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Equilibrium chemistry: H2O and CO are less blueshifted than OH because OH preferentially forms on the approaching, blueshifted limb of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' OH being more blueshifted than the other molecules is in agreement with the predictions of the toy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Equilibrium chemistry: The relationship be- tween H2O and OH changes as a function of phase because the ratio OH/H2O increases a function of temperature, and hotter regions of the planet ro- tate into view over transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This finding is also qualitatively in agreement with the toy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' However, per Figure 9, this effect is unfortunately not likely to be observable given the error bars in current data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Now the question remains: Can we observe in real data the trends matching these model explanations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' As a simple experiment, we can apply error bars representa- tive of the best observing nights on the best instrument with the most observable chemical species (roughly 2 km/s, as drawn as vertical error bars in Figure 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Ehren- reich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020) and determine whether these trends are still detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' With our errorbars now applied to our simulated data, only the first explanation—that H2O forms at higher altitudes than CO—can fully be ad- dressed, assuming that Doppler shifts for both species CO OH H20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='0 Planet-frame RV (km/s) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='5 15 10 5 5 10 15 Phase (degrees)Diagnosing limb asymmetries in transmission 17 (a) (b) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Results of an investigation into anomalous Ca II blueshift between different model runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' In panel (a), it can be seen that forward models that include absorption due to Sc opacity yield a larger Ca II blueshift than models that lack Sc (Fe Doppler shift is included for comparison).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Panel (b) illustrates the cause of this anomalous blueshift: a Sc line overlapping one line in the optical Ca II doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' These results imply that overlapping line profiles can subtly contaminate calculated Doppler shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The second explanation can only be partially addressed—we can still determine that CO is less blueshifted than OH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Warning: blending of Doppler shifts The disentangling of physics in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1 rests on a fundamental assumption: that each cross-correlation template directly tracks only a single species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Indeed, one of the promises of HRCCS is the ability to uniquely constrain individual species’ abundance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' with individual line profiles resolved, different species should be readily identifiable from one another in cross-correlation space (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Brogi & Line 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Furthermore, our noiseless models should be even less susceptible to degeneracies between different species’ spectral manifestations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Panel (a) of Figure 10 seems to contradict the notion of complete line profile independence across species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' For models run in Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022), Sc was excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Mo- tivated by the search for atoms in Kesseli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022), however, we included Sc in this work’s models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Sur- prisingly, we found a subsequent significant difference in the Doppler shifts recovered from our cross-correlation analysis in our Sc-inclusive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Panel (b) of Figure 10 reveals the source of the dis- crepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' In the optical, Ca II opacity is dominated by a doublet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' one of the lines in this doublet partially over- laps with a strong, narrow Sc line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' When both species combined in a forward model, the Sc line produces ab- sorption just blueward of this Ca II line’s core;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' hence, the cross-correlation of the Ca II template yields a spuri- ous blueshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' There did exist other modeling differences between the two spectra (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', the Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' (2022) models included TiO and VO), but none of these differ- ences strongly impacted the Doppler shift of Ca II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Because Ca II in the optical has only two strong lines, it is particularly susceptible to this type of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' All it takes is one slight overlap with another species near a Ca II doublet core, and the Ca II Doppler signal can be significantly biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Species with forests of lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Fe in the optical) should hence be more robust to chance overlaps with other species’ lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' To guard against this error for species with few lines, we recommend cross-correlating templates against one another to get a first-order sense for the extent of species overlap in Doppler space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Furthermore, we recom- mend performing these analyses on HRCCS with com- bined species models, as opposed to single-species mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This approach could involve a retrieval framework (Brogi & Line 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Gandhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Gibson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020), which couples a statistical sampler to an atmo- spheric forward model to determine the exoplanet spec- trum that best fits the data, inclusive of multiple chem- ical species at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' CONCLUSION The past few years have yielded asymmetric Doppler signals from exoplanet atmospheres as a function of phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Compelling “toy models” notwithstanding, a number of physical processes can drive these asymme- tries, and it can be difficult to uniquely constrain the cause of an asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' In this study, we determine that if an asymmetry is observed: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' It may be due to a scale height difference across the atmosphere, not a chemistry difference across 2 Ca+ with Sc Fe Ca+ Planet-frame RV (km/s) 0 2 6 8 1015 10 5 0 5 10 15 Phase (degrees)Sc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='40 Ca+ template (%) ransit depth ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3931 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3932 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3933 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3934 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3936 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3937 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3938 Wavelength (microns)18 Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Comparing a signal of a species in HRCCS to a baseline species that is guaranteed to be chemically stable over the atmosphere can bet- ter motivate whether the asymmetry could be due to chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' CO is an excellent baseline species for ultra-hot Jupiters, as it is stable over these planets’ expected temperature–pressure space, has many spectral lines in the near-infrared accessible to ground-based spectrographs, and has been de- tected in numerous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The asymmetry can be highly informative even if it is binned in phase, especially if multiple species are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' For instance, much larger Doppler shifts (both blue and red) of certain species rela- tive to the predictions of hydrostatic GCMs can be used as evidence for outflowing material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The asymmetry may be boosted by including (and perhaps only considering) ingress and egress phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Ingress and egress spectra are the the gold standard for asymmetric signals so long as the sig- nal to noise is high enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The asymmetry may be influenced by line con- fusion between species, even at high resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Species with very few lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', a single doublet) in the observed waveband are especially suscep- tible to contamination by other species in cross- correlation analysis, and they should be carefully checked against theoretical models for possible contaminating opacity sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' If all species exhibit a similar asymmetry— especially if they all become less blueshifted over the course of transit—the asymmetry may be due to a large-scale effect, such as clouds blanketing the cooler limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Per our comparison of near-infrared absorbers in the CARMENES waveband, the toy model pre- dictions of the H2O Doppler shift relative to CO was inaccurate, as it did not include information about the vertical coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' With H2O lines on average probing higher in the atmosphere than CO in this waveband, they probed a different part of the flow, departing from expectations of the toy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' By aiming to systematically understand even just a few drivers of asymmetry, this work has made it clear that HRCCS—already arguably abstract given its gen- eral inability to produce visible planetary spectra—has yet more nuance to uncover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' As data quality continues to increase, it will become increasingly necessary to un- derstand the relationships between higher-order physical effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' ACKNOWLEDGMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' acknowledge funding from the Heising-Simons Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We thank Michael Zhang for a thoughtful conversa- tion on the cross-correlation signature of HCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' We also thank Anusha Pai Asnodkar for a robust discussion of degeneracies in HRCCS tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Finally, we thank Serena Cronin for providing useful insight into applications of CO detections in extragalactic astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' The authors acknowledge the University of Maryland supercomputing resources (http://hpcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='edu) made available for conducting the research reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' This research has made use of NASA’s Astrophysics Data System Bibliographic Services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Software: astropy (Price-Whelan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2018), batman (Kreidberg 2015), FastChem (Stock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2018), IPython (P´erez & Granger 2007), HELIOS-K (Grimm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021), HELIOS (Malik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2017), Matplotlib (Hunter 2007), NumPy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020), Numba (Lam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2015), pandas (McKinney 2010), SciPy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020), tqdm (da Costa-Luis 2019) REFERENCES Ahrer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Alderson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Batalha, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022, arXiv preprint arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='11692 Akinsanmi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Barros, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Santos, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019, Astronomy & Astrophysics, 621, A117 Arcangeli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', D´esert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Line, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2018, The Astrophysical Journal Letters, 855, L30 Beichman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Benneke, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Knutson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2014, Publications of the Astronomical Society of the Pacific, 126, 1134 Beltz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Rauscher, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Brogi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Kempton, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020, The Astronomical Journal, 161, 1 Beltz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Rauscher, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Kempton, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022, AJ, 164, 140, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3847/1538-3881/ac897b Diagnosing limb asymmetries in transmission 19 Beltz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Rauscher, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Roman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Guilliat, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, The Astronomical Journal, 163, 35 Benneke, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Wong, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Piaulet, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019, The Astrophysical Journal Letters, 887, L14 Birkby, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2018, arXiv preprint arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='04617 Blecic, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Dobbs-Dixon, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Greene, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2017, The Astrophysical Journal, 848, 127 Borsa, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Allart, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Casasayas-Barris, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, Astronomy & Astrophysics, 645, A24 Bourrier, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Ehrenreich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Lendl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020, Astronomy & Astrophysics, 635, A205 Brogi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', De Kok, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Albrecht, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2016, The Astrophysical Journal, 817, 106 Brogi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', De Kok, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Birkby, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Schwarz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Snellen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2014, Astronomy & Astrophysics, 565, A124 Brogi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Line, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019, The Astronomical Journal, 157, 114 Brooke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Bernath, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Western, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2016, Journal of Quantitative Spectroscopy and Radiative Transfer, 168, 142 Caldas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Leconte, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Selsis, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019, Astronomy & Astrophysics, 623, A161 Casasayas-Barris, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Orell-Miquel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Stangret, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, A&A, 654, A163, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1051/0004-6361/202141669 Charbonneau, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Brown, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Noyes, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Gilliland, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2002, The Astrophysical Journal, 568, 377 Cho, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Menou, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Hansen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Seager, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2008, The Astrophysical Journal, 675, 817 Cooper, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Showman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2005, The Astrophysical Journal, 629, L45 Crossfield, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Kreidberg, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2017, The Astronomical Journal, 154, 261 da Costa-Luis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019, Journal of Open Source Software, 4, 1277 de Kok, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Brogi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Snellen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2013, Astronomy & Astrophysics, 554, A82 Deibert, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', de Mooij, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Jayawardhana, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, The Astrophysical Journal Letters, 919, L15 Demory, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='-O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Gillon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', De Wit, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2016, Nature, 532, 207 Ehrenreich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Lovis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Allart, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020, Nature, 580, 597 Espinoza, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Jones, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, The Astronomical Journal, 162, 165 Faedi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Barros, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Anderson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2011, Astronomy & Astrophysics, 531, A40 Feng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Line, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Fortney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2016, The Astrophysical Journal, 829, 52 Flowers, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Brogi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Rauscher, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Kempton, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Chiavassa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019, The Astronomical Journal, 157, 209 Fortney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Dawson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Komacek, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, Journal of Geophysical Research: Planets, 126, e2020JE006629 Fossati, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Ayres, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Haswell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2013, The Astrophysical Journal Letters, 766, L20 Fray, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Schmitt, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2009, Planetary and Space Science, 57, 2053 Fromang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Leconte, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Heng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2016, Astronomy & Astrophysics, 591, A144 Gandhi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Brogi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Webb, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020, Monthly Notices of the Royal Astronomical Society, 498, 194 Gandhi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Kesseli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Snellen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022, Monthly Notices of the Royal Astronomical Society Gandhi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Madhusudhan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Hawker, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Piette, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019, The Astronomical Journal, 158, 228 Gao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Powell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, The Astrophysical Journal Letters, 918, L7 Gao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Wakeford, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Moran, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Parmentier, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, Aerosols in exoplanet atmospheres, Wiley Online Library Gaudi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Stassun, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Collins, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2017, Nature, 546, 514 Giacobbe, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Brogi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Gandhi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, Nature, 592, 205 Gibson, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Merritt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Nugroho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020, Monthly Notices of the Royal Astronomical Society, 493, 2215 Grimm, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Malik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Kitzmann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, The Astrophysical Journal Supplement Series, 253, 30 Guillot, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Fletcher, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Helled, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022, arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='04100 Guilluy, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Giacobbe, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Carleo, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022, Astronomy & Astrophysics, 665, A104 Hammond, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Lewis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, Proceedings of the National Academy of Sciences, 118, e2022705118 Harrington, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Hansen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Luszcz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2006, Science, 314, 623 Harris, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Millman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', van der Walt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020, Nature, 585, 357–362, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1038/s41586-020-2649-2 Heays, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Bosman, AD, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Van Dishoeck, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2017, Astronomy & Astrophysics, 602, A105 Hellier, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Anderson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Cameron, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2009, Nature, 460, 1098 Herman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', de Mooij, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Nugroho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Gibson, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Jayawardhana, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022, The Astronomical Journal, 163, 248 20 Savel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Hindle, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Bushby, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Rogers, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, The Astrophysical Journal, 922, 176 Hoeijmakers, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Seidel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Pino, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020, Astronomy & Astrophysics, 641, A123 Hood, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Fortney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Line, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020, The Astronomical Journal, 160, 198 Hunter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2007, Computing in science & engineering, 9, 90 Kataria, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Sing, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Lewis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2016, The Astrophysical Journal, 821, 9 Kempton, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Perna, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Heng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2014, The Astrophysical Journal, 795, 24 Kempton, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Rauscher, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2012, The Astrophysical Journal, 751, 117 Kesseli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Snellen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, The Astrophysical Journal Letters, 908, L17 Kesseli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Snellen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Casasayas-Barris, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Molli`ere, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & S´anchez-L´opez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022, The Astronomical Journal, 163, 107 Knutson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Charbonneau, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Allen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2007, Nature, 447, 183 Komacek, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Showman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Parmentier, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019, The Astrophysical Journal, 881, 152 Komacek, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Tan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Gao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Lee, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022, ApJ, 934, 79, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3847/1538-4357/ac7723 Kramida, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Olsen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Ralchenko, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019, National Institute of Standards and Technology, US Department of Commerce Kreidberg, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2015, Publications of the Astronomical Society of the Pacific, 127, 1161 Kreidberg, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Koll, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Morley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019, Nature, 573, 87 Lacy, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Burrows, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020, The Astrophysical Journal, 905, 131 Lam, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Pitrou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Seibert, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2015, in Proceedings of the Second Workshop on the LLVM Compiler Infrastructure in HPC, 1–6 Landman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', S´anchez-L´opez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Molli`ere, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, A&A, 656, A119, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1051/0004-6361/202141696 Li, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Gordon, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Rothman, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2015, The Astrophysical Journal Supplement Series, 216, 15 Lide, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2006, CRC Handbook of Chemistry and Physics, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Lide, DR, Ed, CRC Press: Boca Raton, Fla Line, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Brogi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Bean, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, Nature, 598, 580 Lodders, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2003, The Astrophysical Journal, 591, 1220 Louden, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Wheatley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2015, The Astrophysical Journal Letters, 814, L24 MacDonald, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Goyal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Lewis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020, The Astrophysical Journal Letters, 893, L43 MacDonald, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Lewis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022, The Astrophysical Journal, 929, 20 Madhusudhan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Seager, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2009, The Astrophysical Journal, 707, 24 Malik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Grosheintz, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Mendon¸ca, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2017, The astronomical journal, 153, 56 Mansfield, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Bean, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Stevenson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020, The Astrophysical Journal Letters, 888, L15 May, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Komacek, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Stevenson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, The Astronomical Journal, 162, 158 Mayne, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Baraffe, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Acreman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2014, Astronomy & Astrophysics, 561, A1 Mbarek, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Kempton, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2016, The Astrophysical Journal, 827, 121 McKinney, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2010, in Proceedings of the 9th Python in Science Conference, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' St´efan van der Walt & Jarrod Millman, 56 – 61, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='25080/Majora-92bf1922-00a Menou, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Rauscher, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2009, The Astrophysical Journal, 700, 887 Miller-Ricci, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Seager, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Sasselov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2008, The Astrophysical Journal, 690, 1056 Montalto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Santos, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Boisse, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2011, Astronomy & Astrophysics, 528, L17 Moses, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Madhusudhan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Visscher, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Freedman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2012, The Astrophysical Journal, 763, 25 Moses, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Visscher, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Fortney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2011, The Astrophysical Journal, 737, 15 Oklopˇci´c, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Hirata, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2018, The Astrophysical Journal Letters, 855, L11 Owen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Murray-Clay, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Schreyer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2023, Monthly Notices of the Royal Astronomical Society, 518, 4357 Parmentier, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Crossfield, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2018, Handbook of exoplanets, 116 Parmentier, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Showman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Fortney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, Monthly Notices of the Royal Astronomical Society, 501, 78 Parmentier, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Line, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Bean, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2018, Astronomy & Astrophysics, 617, A110 Pelletier, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Benneke, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Darveau-Bernier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, The Astronomical Journal, 162, 73 P´erez, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Granger, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2007, Computing in Science & Engineering, 9, 21 Perna, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Heng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Pont, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2012, The Astrophysical Journal, 751, 59 Pino, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Ehrenreich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Allart, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2018, Astronomy & Astrophysics, 619, A3 Pluriel, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Leconte, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Parmentier, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022, A&A, 658, A42, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1051/0004-6361/202141943 Diagnosing limb asymmetries in transmission 21 Pluriel, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Zingales, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Leconte, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Parmentier, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020, Astronomy & Astrophysics, 636, A66 Polyansky, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Kyuberis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Zobov, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2018, Monthly Notices of the Royal Astronomical Society, 480, 2597 Price-Whelan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Sip˝ocz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', G¨unther, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2018, AJ, 156, 123, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='3847/1538-3881/aabc4f Quirrenbach, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Amado, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Caballero, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2014, in Ground-based and airborne instrumentation for astronomy V, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 9147, SPIE, 531–542 Rauscher, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Menou, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2010, The Astrophysical Journal, 714, 1334 Rodler, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', K¨urster, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Barnes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2013, Monthly Notices of the Royal Astronomical Society, 432, 1980 Roman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Kempton, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Rauscher, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, The Astrophysical Journal, 908, 101 S´anchez-L´opez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Landman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Molli`ere, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022, Astronomy & Astrophysics, 661, A78 Savel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Kempton, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Malik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022, The Astrophysical Journal, 926, 85 Seidel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Ehrenreich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Allart, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, Astronomy & Astrophysics, 653, A73 Showman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Guillot, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2002, Astronomy & Astrophysics, 385, 166 Showman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Fortney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Lewis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Shabram, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2013, The Astrophysical Journal, 762, 24 Showman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Fortney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Lian, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2009, The Astrophysical Journal, 699, 564 Snellen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', De Kok, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', De Mooij, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Albrecht, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2010, Nature, 465, 1049 Spake, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Sing, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Evans, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2018, Nature, 557, 68 Stock, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Kitzmann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Patzer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Sedlmayr, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2018, Monthly Notices of the Royal Astronomical Society, 479, 865 Tabernero, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Osorio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Allart, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, Astronomy & Astrophysics, 646, A158 Tan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Komacek, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019, The Astrophysical Journal, 886, 26 Tsai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Lyons, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Grosheintz, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2017, The Astrophysical Journal Supplement Series, 228, 20 Tsai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Malik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Kitzmann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, The Astrophysical Journal, 923, 264 Van Dishoeck, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Black, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 1988, The Astrophysical Journal, 334, 771 van Sluijs, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Birkby, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Lothringer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022, arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='13234 Virtanen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Gommers, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Oliphant, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020, Nature methods, 17, 261 Visscher, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2012, The Astrophysical Journal, 757, 5 Wardenier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Parmentier, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Lee, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022, Monthly Notices of the Royal Astronomical Society, 510, 620 Wardenier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Parmentier, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Lee, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Line, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Gharib-Nezhad, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, Monthly Notices of the Royal Astronomical Society, 506, 1258 Welbanks, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Madhusudhan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2022, The Astrophysical Journal, 933, 79 West, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Hellier, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Almenara, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2016, Astronomy & Astrophysics, 585, A126 Wyttenbach, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Molli`ere, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Ehrenreich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020, Astronomy & Astrophysics, 638, A87 Yan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Casasayas-Barris, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Molaverdikhani, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2019, Astronomy & Astrophysics, 632, A69 Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Chachan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Kempton, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Knutson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2020, The Astrophysical Journal, 899, 27 Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Showman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2018, The Astrophysical Journal, 866, 2 Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', Snellen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=', & Molli`ere, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content=' 2021, A&A, 656, A76, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} +page_content='1051/0004-6361/202141502' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfuv0b/content/2301.01694v1.pdf'} diff --git a/5dAzT4oBgHgl3EQffvz_/content/tmp_files/2301.01459v1.pdf.txt b/5dAzT4oBgHgl3EQffvz_/content/tmp_files/2301.01459v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8357c93b0c5409117cc9c114cd1f6c98e089a099 --- /dev/null +++ b/5dAzT4oBgHgl3EQffvz_/content/tmp_files/2301.01459v1.pdf.txt @@ -0,0 +1,1963 @@ +Modeling the Central Supermassive Black Holes Mass of Quasars via LSTM Approach +Seyed Sajad Tabasi,1, 2, ∗ Reyhaneh Vojoudi Salmani,3, 2, † Pouriya Khaliliyan,3, 2, ‡ and Javad T. Firouzjaee3, 2, 4, § +1Department of Physics, Sharif University of Technology, P. O. Box 11155-9161, Tehran, Iran +2PDAT Laboratory, Department of Physics, K. N. Toosi University of Technology, P.O. Box 15875-4416, Tehran, Iran +3Department of Physics, K. N. Toosi University of Technology, P. O. Box 15875-4416, Tehran, Iran +4 School of Physics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran +One of the fundamental questions about quasars is related to their central supermassive black +holes. The reason for the existence of these black holes with such a huge mass is still unclear and +various models have been proposed to explain them. However, there is still no comprehensive ex- +planation that is accepted by the community. The only thing we are sure of is that these black +holes were not created by the collapse of giant stars, nor by the accretion of matter around them. +Moreover, another important question is the mass distribution of these black holes over time. Ob- +servations have shown that if we go back through redshift, we see black holes with more masses, +and after passing the peak of star formation redshift, this procedure decreases. Nevertheless, the +exact redshift of this peak is still controversial. In this paper, with the help of deep learning and the +LSTM algorithm, we tried to find a suitable model for the mass of central black holes of quasars over +time by considering QuasarNET data. Our model was built with these data reported from redshift +3 to 7 and for two redshift intervals 0 to 3 and 7 to 10, it predicted the mass of the quasar’s central +supermassive black holes. We have also tested our model for the specified intervals with observed +data from central black holes and discussed the results. +Keywords: +Quasars, Supermassive Black Holes, Sloan Digital Sky Survey, QuasarNET Data Set, Deep +Learning, and LSTM Model +I. +INTRODUCTION +In recent years, the study of the high-redshift(z > 6) +quasars was a direct probe to explore the Universe at +the age less than 1 Gyr after the Big Bang. These early +forming quasars are essential to studying the early growth +of supermassive black holes (SMBHs) [1]. +By detecting the reverberation between the variations +of broad emission lines and the continuum we can deter- +mine SMBHs mass in quasars [2]. Until now, the time lag +of Hβ emission lines has been confirmed and measured +only in ∼100 quasars [3]. +The continuum and line emission from luminous +quasars which are one of the most luminous objects, over +a large wavelength range can be characterized by sev- +eral leading parts. +The broad emission line region [4] +the optical-to-ultraviolet continuum emission, which is +explained by a standard accretion disk extending down +to the innermost stable circular orbit [5], X-ray emission +with a power-law spectrum produced by inverse Compton +scattering of photons from the accretion disk of relativis- +tic electrons in the hot corona [6], and a soft X-ray ex- +cess [7]. Spectroscopic observations from optical to near- +infrared of these quasars suggest that such SMBHs are +already established when the universe is only 700Myr +old [8]. +To explain the existence of these SMBHs, many theo- +∗Electronic address: sstabasi98@gmail.com +†Electronic address: r.s.vojoudi@gmail.com +‡Electronic address: pouriya@email.kntu.ac.ir +§Electronic address: firouzjaee@kntu.ac.ir +retical models have been proposed like using primordial +density seeds [9–11] and appealing a super-Eddington ac- +cretion process [12]. +To utilize the spectroscopic observational data in phys- +ical studies, we need an exact classification and redshift +determination of astrophysical objects. Along the way, +the Sloan Digital Sky Survey Catalogue 16th Data Re- +lease Quasar Only(SDSS-DR16Q) [13], consists of two +files, being the quasar-only main catalog of 750414 en- +tries which contains sooner visually confirmed quasars +SDSS-I/II/III, and a 1440615-row “superset” of SDSS- +IV/eBOSS quasar object classifications. +The DR16Q catalogs present multiple redshifts per ob- +ject that are available, including the neural automated +QuasarNET [14] redshift which is claimed > 99% ef- +ficiency and > 99% accuracy, that rests on garnering +deeper insights into this triumvirate connection by co- +locating and analyzing observational data and simulated +data. Meanwhile, the enormous increase in computing +power over the last decades has allowed the application +of acquired statistical methods in the analysis of big and +complex data sets. +Using previously-fed data has brought huge opportuni- +ties for astronomers to develop intelligent tools and inter- +faces, utilizing pipeline classifiers, machine learning(ML), +and deep learning(DL) methods, to deal with data sets +and extract novel information with possible predictions +and estimate the relevant confidence which the behavior +new data will have. +In astronomy and astrophysics, ML [15, 16] and DL +[17, 18] have been used in a broad range of subjects(e.g. +quasars and other types of sources), such as redshift de- +termination [19, 20], morphological classification and ref- +erences therein [21, 22], source selection and classifica- +arXiv:2301.01459v1 [astro-ph.GA] 4 Jan 2023 + +2 +tion [23–25], image and spectral reconstruction [26], and +more. +ML methods for obtaining redshift estimation for +quasars are becoming progressively crucial in the epoch of +rich data astronomy. Redshift measurements of quasars +are important as they can enable quasar population +studies, and provide insight into the star formation +rate(SFR), the luminosity function(LF), and the density +rate evolution [27]. +In this work, we have used DL to model the mass of +quasars’ central SMBH as a function of their redshift. +Firstly, Sec. +II is dedicated to the available observa- +tional data and evidence on quasars. The estimation of +a quasar’s central SMBH mass is discussed in detail in +Sec. III. Furthermore, in Sec. IV, the mass evolution +of these black holes(BHs) is investigated. Sec. V is the +comparison between two newborn research platforms, +QuasarNET and FNET, and the reasons behind using +QuasarNET for our model are explained. Additionally, +we use two correction methods which are explained in +Sec. +VI. A detailed explanation of our DL model can +be found in Sec. +VII to X. In Sec. +VII we introduce +Long short-term memory(LSTM) which is the recurrent +neural network(RNN) that we build our model based +on. +We explain the chosen optimization function and +its validation loss in Sec. +VIII which is shown in +multiple figures. Sec. IX presents the topology design +of our model and finally, the comparison of the model +predictions with other data sets is discussed in Sec. X. +II. +OBSERVATIONAL EVIDENCE AND DATA +The most comprehensive observed quasi-stellar ob- +jects(QSOs) spectra to date are cataloged in the SDSS- +IV. SDSS has been operative since 2000 and catalogs of +quasars have been produced and made available since +2002. In addition to producing images, it performs spec- +troscopic surveys across a large area of the sky. We can +get about one million galaxies and 10,000 quasars spectra +from the survey images of the sky, which are obtained +by a 2.5m telescope equipped with a large format mo- +saic Charge-coupled device(CCD) camera, and two dig- +ital spectrographs. As part of its calibration, the SDSS +uses observations of the US Naval Observatory’s 1m tele- +scope to calibrate its photometry, and an array of astro- +metric CCDs control its astrometry [28]. +The SDSS provides data necessary to study the large- +scale structure of the universe. +As far as the obser- +vatory’s limit allows, the imaging survey should detect +∼ 5 × 107 galaxies, ∼ 106 quasars, and ∼ 8 × 107 stars. +By using photometric redshifts and angular correlation +functions, these photometric data allow studies of large- +scale structures that go beyond spectroscopic analysis. +Quasars can provide information on the structure at even +larger scales [28]. +The SDSS-DR16Q contains 750,414 quasars, with the +automated redshift range 1 ≤ z ≤ 7.1. The number of +sources reaches its maximum around z ≈ 2.5 and at ear- +lier epochs i.e. higher redshifts, they are comparatively +rare [29]. However, there is a problem with the SDSS- +DR16Q catalog. It contains non quasar sources due to +pipeline classification errors and incorrect redshift esti- +mations [13]. +For example, in a search for undeclared +quasars, the SDSS-DR16Q main quasars are found to +contain 81 entries that are not quasars. It must there- +fore be noted that the pipeline catalog is not an adequate +training samples for quasars because many objects with +z ≥ 6 as well as significant fractions of these objects at +z ≥ 4, may not be quasars or not quasars at the given +redshifts due to incorrect pipeline classifications [29]. +III. +MASS ESTIMATION OF QUASARS’ +CENTRAL SMBH +In terms of fundamental parameters of quasars, one +can mention the central SMBH mass and structure, along +with the ratio of the accretion rate to the Eddington +accretion rate [30]. +The central SMBH mass can be measured via the gas +or stellar dynamics [30] from optical or ultraviolet(UV) +spectroscopy using empirical relations [31]. The broad +emission line region(BLR) probably provides the best +probe of these characteristics [32]. +The size of BLRs +can be determined by reverberation mapping(RM) [33], +which is a measuring technique in astrophysics. RM pro- +vides invaluable information about the kinematic and +ionization distribution of the gas using the time lag be- +tween emission line and continuum variations [32]. +Assuming that gravity dominates the dynamics of the +BLR and the virial relationship between time lag and line +width exists, the BH mass can be estimated as [34] +MBH = fcτv2 +G +, +(1) +where τ is the mean time delay for the region of inter- +est, v is the velocity of the gas in that region, c is the +speed of light, G is the gravitational constant, and f is a +scaling factor of order unity that depends on the detailed +geometry and kinematics of the line-emitting region. +The worth mentioning point is that the virial relation- +ship claims a virialized system with individual clouds +moving in their Keplerian orbits. This leads to the pro- +portionality of mean cloud velocity and emissivity radius +[35] +v ∝ rBLR +1 +2 , +(2) +where rBLR is the emissivity radius. +In the absence of RM, the quasar continuum luminosity +is sufficient to estimate the BLR. With RM estimations, +the best-fitting RBLR − λLλ relations were derived for +quasars at monochromatic luminosity in both 3000 and +5100 ˚A rest-frames as follows [37] + +3 +RBLR = (18.5 ± 6.6)[λL3000/1037W] +(0.32±0.14), +(3) +RBLR = (26.4 ± 4.4)[λL5100/1037W] +(0.61±0.10). +(4) +Here, L is the luminosity measured at a wavelength λ. +In Eq. 1, an intrinsic Keplerian velocity of a broad-line +gas is related to the full width at half maximum (FWHM) +of a chosen broad emission line by the geometric factor +f as +VBLR = f × FWHM, +(5) +In other words, it is the width of a spectrum curve +measured between those points on the y-axis which are +half of the maximum amplitude. +As the geometry of the BLR in radio-quiet quasars is +currently unknown, it is generally agreed that f = +� +3/2, +which is appropriate for randomly oriented orbits of the +BLR gas. +However, FWHM measurements for broad +emission lines in radio-loud quasars indicate a disc-like +geometry [38]. +Given the similarity between the opti- +cal emission-line spectra of radio-loud and radio-quiet +quasars, it is not unreasonable to consider the possibil- +ity that BLRs of radio-quiet quasars that dominate the +SDSS data can follow the same equation as well [36] +VBLR = FWHM +(2 sin i) . +(6) +Here, i represents the angle between the line of sight +and the axis of the disc. +Our virial BH mass estimators are derived by substi- +tuting the calibrations of the RBLR–λLλ relations into +Eq. 1 and determining VBLR using MgII or Hβ [37]. +Based on the L5100 which is the monochromatic lu- +minosity at rest-frame 5100 ˚A and the Hβ line, a more +specific expression to calculate the mass of a BH can be +written as [39] +MBH(Hβ) = 1.05 × 108( +L5100 +1046ergs−1 )0.65 +(7) +× [FWHM(Hβ) +103kms−1 +]2M⊙, +where MBH(Hβ) represents BH mass by considering +Hβ line, FWHM(Hβ) is the full width at half maximum +of Hβ line, and M⊙ is the solar mass. +Large spectroscopic surveys like the SDSS observe +both broad Hβ and MgII lines. +Therefore, one can +be calibrated against the other and based on L3000 and +MgIIλ2798 line width, a similar expression can be de- +rived as [35] +MBH(MgIIλ2798) = 8.9 × 107( +L3000 +1046ergs−1 )0.58 +(8) +× [FWHM(MgIIλ2798) +103kms−1 +]2M⊙, +where MBH(MgIIλ2798) represents BH mass by con- +sidering Hβ line, and FWHM(MgIIλ2798) is the full +width at half maximum of MgII line. +Based on empirical estimation of f ≃ 1.1 for the Hβ +line, we can now write more specific expressions to calcu- +late MBH for several emission lines like MgII as follows +[39] +MBH +M⊙ += 4.7(λL5100 +1037W ) +0.61 +[FWHM(Hβ) +kms−1 +] +2 +, +(9) +MBH +M⊙ += 3.2(λL3200 +1037W ) +0.62 +[FWHM(MgII) +kms−1 +] +2 +. (10) +Besides, it is well-known that the relationship between +stellar velocity dispersion and BH mass can be written +as [39] +log(MBH +M⊙ +) = 4.38 × log( +σ∗ +200kms−1 ) + 8.49, +(11) +where σ∗ is the stellar velocity dispersion. +Furthermore, to estimate the mass of a BH, observa- +tions in the local universe reveal the existence of a corre- +lation between the central SMBH mass and the bulge of +the host galaxies [40]. +log(MBH +M⊙ +) = α + βlog(MBulge,∗ +1011M⊙ +), +(12) +where MBulge,∗ is the bulge stellar mass and the best- +fit of α and β should be +α = 7.93 ± 0.061; β = 1.15 ± 0.075. +(13) +IV. +MASS EVOLUTION OF QUASARS’ +CENTRAL SMBH +As studying the cosmic history of compact cosmolog- +ical objects is so crucial to track the history line of the +universe in a much bigger structure, we are so curious +about the evolution of SMBHs mass. In the presence of +a SMBH, there are obvious links between the physical +properties and those of its host. Due to high redshifts +that many quasars have, they are ideal to be studied to +recognize BH evolution through time back to the early +universe [41]. +According to the modelling of spectra from the SDSS +first data release, the virial mass of BHs for 12698 quasars + +4 +in the redshift interval 0.1 ≤ z ≤ 2.1 is estimated. +There is entirely consistent evidence to suggest that the +BH mass of SDSS quasars lies in 107M⊙ ≤ MBH ≤ +3 × 109M⊙. The local BH mass function for early-type +galaxies using the MBH − σ and MBH − Lbulge correla- +tions(Eq. 11 and Eq. 12) are also estimated. In addition, +by comparing the number density of active BHs at z ≈ 2 +with the local mass density of inactive ones, a lower limit +is set on the lifetime of quasars, which confirms that the +bulk of BHs with mass ≥ 108.5M⊙ are situated in place +by z ≈ 2 [36]. +There are several different ideas on the central SMBH +mass evolution through time in literature. Based on the +effective flux limit along with the role of the quasar con- +tinuum luminosity, most studies agree that the SMBH +mass increases as a function of redshift, namely most low +mass SMBHs can be found in the late universe(e.g. step- +ping down from ≈ 109M⊙ at z ≈ 2.0 to ≈ 108M⊙ at +z ≈ 0.2). Considering Eq. 9 and Eq. 10, redshift does +not alter the mean FWHM and it can be roughly consid- +ered to be constant. Therefore, the mean virial mass of +the SMBH should be increased as [Lλ]0.6 [36]. +Quasars undergo important cosmic evolution accord- +ing to optical, X-ray, and bolometric LFs. Interestingly, +based on predictions of [42] using an extended version of +the galaxy formation model, GALFORM code, quasars +evolution will be influenced by different physical pro- +cesses such as the accretion mode and the obscuration +prescription. Observational data have also reported sim- +ilar trends [43]. +Furthermore, SMBHs grow exponentially during a pe- +riod in which accretion governs their mass evolution. +When z ≳ 5, the growth of a SMBH in a quasar is as +follows [44] +MBH(t) = MBH(t0)etτ, +(14) +τ ≃ 0.4Gyr +η +1 − η +1 +µ, +(15) +µ ≡ +L +LEdd +× factive, +(16) +where MBH(t0) is the initial mass of BH i.e. the seed’s +mass, η is the radiative efficiency(see [45] for reported +values of η for several objects), L is the luminosity of the +quasar, LEdd is the luminosity at Eddington limit, factive +is the duty cycle, and µ is a constant which is determined +as a combination of L/LEdd and factive. Therefore, it is +possible to calculate the growth of the BH easily as +log MBH(z) = log MBH(z0) +(17) ++ log[exp (R(1 − η +η +)zd), +η ≡ +Lbol +˙Mc2 , +(18) +zd ≡ (1 + z)−3/2 − (1 + z0)−3/2. +(19) +In above equations, MBH(z0) is the mass of BHs’ seed +and R is a constant that is defined as follows +R ≡ 0.4Gyr +µ +, +(20) +R = +� +� +� +� +� +3.79322, +µ = 0.1 +18.9661, +µ = 0.5 +37.9322, +µ = 1.0. +(21) +V. +QUASARNET AND FNET +To investigate the mass evolution even more precisely, +QuasarNET and FNET are the two available research +platforms. Using ML, QuasarNET makes deployment of +data-driven modelling techniques possible by combining +and co-locating large observational data sets of quasars, +the high-redshift luminous population of accreting BHs, +at z ≥ 3 alongside simulated data spanning the same +cosmic epochs. The main quasar population data source +of QuasarNET is NASA Extra-galactic Database(NED) +which contains quasars retrieved from several indepen- +dent optical surveys, principally the magnitude-limited +SDSS. There is no comparison between quasars from +SDSS and those from other surveys when it comes to +spectra and photometry [46]. +NED contains all quasars in principle, but some are +missing because their photometric redshifts were incor- +rectly assigned. Photometric redshift estimation meth- +ods suffer from degeneracy, a well-known limitation of +current photometric redshift determination methods [47]. +QuasarNET fills in the missing sources by analyzing the +published catalogues from all surveys. It expands to in- +clude additional parameters used to derive BHs mass, in- +stead of archiving only the reported masses. It contains +136 quasars’ features, such as the position, redshift, lu- +minosity, mass, line width, and Eddington ratio. +Two observationally determined functions are used as +constraints in theoretical models to describe the assembly +history of the BHs population across time: the BH mass +function and the Quasar Luminosity Function(QLF). As +a statistical measurement of the combined distribution +of BHs mass through redshifts, the BH mass function +encodes the mass growth history. Similar to the QLF, +which reflects their accretion history, the BH mass func- +tion is a statistical measurement of the distribution of +quasars’ luminosities through redshift [46]. +On the other hand, by using DL, to study quasars in +the SDSS-DR16Q of eBOSS on a wide range of signal- +to-noise(SNR) ratios, there is a 1-dimensional convolu- +tional neural network(CNN) with a residual neural net- +work(ResNet) structure, named FNet. With its 24 con- +volutional layers and ResNet structure, which has dif- +ferent kernel sizes of 500, 200, and 15, FNET can use +a self-learning process to identify ”local” and ”global” +patterns in the entire sample of spectra [29]. + +5 +Although FNET seems to be similar to the recently +adopted CNN-based redshift estimator and classifier, i.e. +QuasarNET [14], their hidden layer implementations are +distinct. +The redshift estimation in FNET is done based on re- +lating the hidden pattern which lies in flux to a spe- +cific redshift, not using any information about emis- +sion/absorption lines, while QuasarNET follows the tra- +ditional redshift estimation procedure using the identified +emission lines in spectra. This makes FNET to outper- +form QuasarNET for some complex spectra(insufficient +lines, high noise, etc.) by recognizing the global pattern. +Moreover, FNET provides similar accuracy to Quasar- +NET, but it is applicable for a wider range of SDSS spec- +tra, especially for those missing the clear emission lines +exploited by QuasarNET. In more detail, from a statis- +tical point of view, FNET is capable to infer accurate +redshifts even for low SNRs or incomplete spectra. +It +predicts the redshift of 5,190 quasars with 91.6 % accu- +racy, while QuasarNET fails to estimate [29]. +It is important to know that the FNET vs. Quasar- +NET comes out on top in redshift prediction, but its +lack of quasars’ central SMBH mass information makes +QuasarNET the preferred option for some studies like +this work. However, if in the future SMBHs mass will be +estimated by using redshifts from FNET approach, our +study can be done again to achieve more accurate results. +VI. +FLUX AND VOLUME-LIMITED SAMPLES +Observations are affected by flux as we move to higher +redshifts and more distant objects. +This is why some +objects are not included in data sets. We suppose that +they are not even present because their low flux makes +them very difficult or in some cases impossible to observe. +This will influence the results of any model that is built +on a set of objects. To remove this bias, we must first +correct the data set. +Two correction methods can be put into use to build +a corrected data set and check if the result is solid or if +the correction can end up with a huge deviation from the +first result. +Using the friends-of-friends algorithm, quasars can be +linked into systems with a specific neighbourhood radius, +called linking length(LL). The size of the group can be +determined based on the choice of LL or more generally +on its scaling law. LL is parameterized upon a scaling +law as [48] +LL +LL0 += 1 + a arctan( z +z∗ +), +(22) +where a = 1.00, z∗ = 0.050 and LL0 is the value of LL +at initial redshift. +Setting a limit for absolute magnitude is needed for +creating volume-limited samples and all less luminous +FIG. 1: The total number of objects available in the +QuasarNET data set is 37648. As a result of data correction +methods, 34403 objects were removed (red dots). The accepted +data are the final flux and volume-limited samples, made of 3245 +Objects(blue dots). +quasars have to be excluded from the data set. +Flux- +limited samples, on the other hand, are formed from +dozens of cylinders containing quasars. +Flux-limited +samples can be made with both constant and varying +LL. The constant LL0 is set as [48] +LL0 = 250[kms−1], +(23) +LL0 = 0.25[h−1Mpc]. +(24) +Following the extraction of the necessary columns and +rejecting duplicate quasars from the data set, there is +only one step left, which is verifying if the quasars are +within the volume of cylinders generated by the LLs. +To do so first we generate a cylinder, then by using the +distance between quasars and comparing this distance +with the volume of the cylinder, we consider a quasar +to be an accepted object if it is located in the cylinder. +The distance can be easily obtained from the redshift +difference between them in the data set. This algorithm +should be repeated as a loop for each quasar. +As a result of applying the correction methods that are +described, we end up with 3246 objects to work with, in- +stead of 37648 objects that are available in QuasarNET. +In FIG. 1 accepted and rejected quasars’ central SMBH +of SDSS-DR16Q in terms of their redshift are illustrated. +VII. +LONG SHORT-TERM MEMORY +LSTM is one of the most powerful RNN that is used in +DL and artificial intelligence [49]. The RNN is a dynamic +system in which there is an internal state at each step of +the classification process [50, 51]. The circular connec- +tions between neurons at the higher and lower layers, as +well as the possibility of self-feedback, are responsible for +this. These feedback connections enable RNNs to propa- +gate data from earlier events to current processing steps. +Thus, RNNs build a memory of time series events. +A standard RNN is not capable of bridging more than +5 to 10 time steps. It is because back-propagated error +signals either grow or shrink with every time step [49]. As + +Removed Data +11.0 +Accepted Data +10.5 +10.0 +log(MBH / M。) +9.5 +9.0 +8.5 +8.0 +7.5 +3 +5 +6 +76 +a result, the error typically blows up or disappears over a +long period of time [52, 53]. When error signals are blown +up, the result is oscillating weights, while vanishing er- +rors mean learning takes too long or does not work at all. +It is possible to solve the vanishing error problem by us- +ing a gradient-based approach known as LSTM [53–56]. +More than 1,000 discrete time steps can be bridged us- +ing LSTM. LSTM uses constant error carousels(CECs), +which enforce a constant error flow within special cells. +Cell accessibility is handled by multiplicative gate +units, which learn when to grant access to cells [49]. Us- +ing a multiplicative input gate unit, memory contents +stored in j are protected from irrelevant inputs. We also +introduce a multiplicative output gate unit that protects +other units from being perturbed by currently irrelevant +memory contents stored in j [57]. Considering distinct +time steps t= 1, 2, etc., an individual step includes for- +ward and backward passes which are the update of all +units and calculation of error signals for all weights, re- +spectively. The Input yin and output yout gate activation +are computed as [54] +netoutj(t) = +� +m +ωoutjmym(t − 1), youtj(t) +(25) += foutj(netoutj(t)), +netinj(t) = +� +m +ωinjmym(t − 1), yinj(t) +(26) += finj(netinj(t)). +Here, netinj and netout are the input and output gate +activation, j indices are memory blocks, ωlm is the weight +on the connection from unit m to l. Index m ranges over +all source units, as specified by the network topology. For +gates, f is a logistic sigmoid in the range of [0, 1]. +Furthermore, there are adaptive gates, which learn to +reset memory blocks once their contents are out of date +and therefore, useless. Like the activation of the other +gates(Eq. 25 and Eq. 26), the forget gate activation yφ +is calculated as +netφj(t) = +� +m +ωφjmym(t − 1), yφj(t) +(27) += fφj(netφj(t)), +where netφj is the input from the network to the forget +gate. +The logistic sigmoid with range [0, 1] is used as +squashing function fφj and weighted by the hyperbolic +tangent function which has the overall task of memory +correction [54]. The forget gate stores all the 1 outputs +while forgetting all the 0 outputs. Finally, LSTM can be +written as [58] +it = σ(Wxixt + Whiht−1 + Wcict−1), +(28) +ft = σ(Wxfxt + Whfht−1 + Wcfct−1), +(29) +ot = σ(Wxoxt + Whoht−1 + Wcoct−1), +(30) +ht = ot ⊙ tanh(ct). +(31) +Here, it , ft, and ot are input gate, forget gate and +output gate of LSTM, ht represents LSTM output, σ is +LSTM logistic function, ⊙ denotes element-wise product, +W is the weight metric components, x is the input data +in time t, and c is LSTM memory cells. +In our application of LSTM, the forget gate and in- +put gate share the same parameters, but are computed +as ft = 1 − it. Note that bias terms are omitted in the +above equations, but they are applied by default. A lin- +ear dependence between LSTM memory cells(ct) and its +past(ct−1) are introduced as +ct = ft ⊙ ct−1 + it ⊙ tanh(Wxcxt + Whcxt−1). +(32) +VIII. +HYPERPARAMETER SELECTION +Hyperparameter selection in neural networks is repre- +sented by optimization functions. Therefore, specifying +hyperparameters such as the type of optimization func- +tion, learning rate, number of neurons in each layer, num- +ber of epochs, and validation are very important. Adam, +Stochastic gradient descent(SGD), RMSProp, AdaDelta, +and Ftrl are used as optimization functions. +We have considered about 20% of the learning data as +validation data. To determine the quality of the model, +we determine the loss. The cost function that we have +considered for the network is mean squared error(MSE). +The number of epochs for the network learning process +is equal to 50 and the batch size is equal to 25. Results +of the cost function values for each learning process with +different optimization functions and a learning rate of +0.0005 are shown in FIG. 2. +The results related to the loss value for learning and +testing data with different optimization functions are +reported in the TABLE I. +IX. +DATA AND NETWORK TOPOLOGY +Using QuasarNET data we predict the SMBHs mass +with the help of their redshift. +We use 3245 data for +modelling, 2596 data for the network learning process, +and 649 data for testing the network result. Data have a +redshift range of 3 to 7. In the first step, data are sorted +in ascending order of their redshifts. The reason is that + +7 +(a) +(b) +(c) +(d) +(e) +FIG. 2: (a) shows model evaluation for SGD optimization +function. Optimization function loss is illustrated by the blue line +and orange lines represent validation loss. (b) is the model +evaluation using RMSProp whose optimization function loss and +validation loss are shown in blue and orange. (c) illustrates the +Adam model evaluation by comparing the Optimization function +loss(blue line) and validation loss(orange line). The model +evaluation for Ftel is shown in (d). loss of the optimization +function is represented by the blue line and the validation +function loss is shown by the orange line. (e) shows model +evaluation for the AdaDelta optimization function. Optimization +function loss is illustrated by the blue line and orange lines +represent validation loss. +Optimization functions +Train data MSE +Test data MSE +SGD +0.38 +0.39 +RMSProp +0.37 +0.38 +Adam +0.23 +0.23 +AdaDelta +0.22 +0.23 +Ftrl +0.26 +0.27 +TABLE I: This table shows the result of algorithm evaluation +by SGD, RMSProp, Adam, Ftrl and AdaDelta optimization +functions. +redshift is a time series and LSTM has a recurrent archi- +tecture which creates memory through time. Then, the +learning and testing data are separated in chronological +order. +The network topology can be described by an LSTM +layer as the dynamic layer of the network, a drop-out +layer to prevent over-fitting, 3 dense layers as static lay- +ers, and the output of the network which is printed by the +last dense layer. We use the hyperbolic tangent which is +an active function for the LSTM layer and the first dense. +Because the hyperbolic tangent is a non-linear function +with a symmetric range. It is a suitable option to control +sudden changes when they are in chronological order. For +the second dense, we use the rectified linear unit(ReLU), +to transfer the magnitude of the positive value to the +next layer. For the third dense, which outputs the net- +work as a continuous number, we use a linear function. +TABLE II shows the network structure based on the hy- +perparameters of the network. +Layers +Neurons +Computational Parameters +Inputs +- +- +LSTM +(None,256) +264192 +Dropout +(None,256) +0 +Dense +(None,512) +131584 +Dense +(None,256) +131328 +Dense +(None,1) +257 +Total Computational Parameters +527361 +Trainable Computational Parameters +527361 +Non-Trainable Computational Parameter +0 +TABLE II: This table illustrates the network topology which +includes each layer along with neurons and computational +parameters. +One of the main challenges that always exists in ML +and DL is the issue of transparency. +Transparency is +a dynamic issue and solving this problem is different for +each task. There is no specific method to solve this prob- +lem. Many factors such as the design of an interpretable +learning experience, the fundamental determination of + +SGD LoSS +1.4 +SGD Validation Loss +1.2 +1.0 +SS0 +0.8 +0.6 +0.4 +0.2 +0 +10 +20 +GE +40 +50 +EpochsRMSProp Loss +1.0 +RMSProp ValidationLoss +0.8: +SS0 +0.6 +0.4 +0.2 +0 +10 +20 +GE +40 +50 +Epochs0.27 +Adam Loss +Adam Validation Loss +0.26 +0.25 +SSO +0.24 +0.23 +0.22 +0 +10 +20 +30 +40 +50 +EpochsFtel Loss +70 +Ftrl Validation Loss + 09 +50 +40 +30 +20 : +10 +0 +10 +20 +E +40 +50 +Epochs0.30 +AdaDelta Loss +AdaDelta Validation Loss +0.28 +0.26 +0.24 +0.22 +0.20 +10 +20 +CE +40 +50 +Epochs8 +FIG. 3: Model built using flux and volume-limited samples. +Corrected QuasarNET data are plotted with blue dots. The black +line represents our LSTM model best-fit. In addition, red dotted +lines represent our models that include 95 percent of all data. +hyperparameters by the task, the observance of the prin- +ciples of feature selection, and the determination of the +appropriate number of data based on characteristics can +allow us to have a transparent model. +Transparency in the structure of algorithms is also +noteworthy. +In this paper, we investigate the trans- +parency of the model built by the designed network. +Trained data are also based on redshifts from 3 to 7. +With the help of the built model, SMBHs mass at 0 < +z < 3 and 7 < z < 10 are then predicted. +We can see the predicted changes of SMBHs mass +through redshift in FIG. 3 based on our built model with +its 95 percent confidence level. FIG. 4 compares the lin- +ear best-fit with our LSTM model best-fit both before +and after applying correction methods. It can clearly be +seen that stated correcting methods change our model +significantly. +X. +COMPARING WITH OTHER DATA +Using corrected flux and volume-limited samples of +QuasarNET data, we build a DL model for quasars’ cen- +tral SMBH mass. By applying correction methods, only +≃ 8.62% of QuasarNET data is accepted to use for mod- +elling. FIG. 1 shows the accepted data along with the +removed data. +Moreover, FIG. 3 illustrated our model whose best- +fit contains 95 percent of corrected data samples. The +model shows that SMBHs mass increases in 0 < z < 4.72 +and reaches its peak at z ≃ 4.72. The mass then falls +exponentially with increasing redshift at z > 4.72. +It +should be noted that our model yields a different result +than what is shown in other recent works like [44], where +the peak is z < 4. Nevertheless, in some studies which +attempt to show quasars’ central SMBHs mass evolution, +Eq. 14 is used that does not include any peaks(e.g. see +[59]). +The model is then evaluated by using different data +sets which are available in multiple tables. We use the +results of the long-term spectroscopic monitoring of 15 +PG quasars that have relatively strong Fe II emission to +generate TABLE III [60]. Moreover, TABLE IV shows +(a) +(b) +FIG. 4: (a) compares our model with the linear best-fit. Blue +dots indicate train data, the LSTM model prediction is showed +the colour red, and the orange line is the linear best-fit. (b) +illustrates our model and compares it with linear best-fit based on +flux and volume-limited samples. Train data is shown as blue +dots, LSTM model prediction as red, and linear best-fit as an +orange line. +(a) +(b) +FIG. 5: (a) shows the examination of our model using multiple +data sets in the redshift range of 0 < z < 7. An overview of the +utilized data can be found in TABLE III and IV. (b) is also the +model examination at 3 < z < 10 whose data is available in +TABLE V and VI. + +11 +LSTM Model Best-fit +95% CI +QuasarNET Data +10 ++ +C. Hu et al. (2021) +M. Vestegaard et al. (2005) +9 +log(MBH / Mo) +7 +6 +0 +1 +2 +3 +4 +5 +611 +LSTM Model Best-fit +95% CI +QuasarNETData +Y.Aggarwal (2022) +10 +J.Yang et al. (2021) +9 +g(MBH / M。) +60 +8 +7 +6 +3 +4 +5 +6 +8 +9 +1011 +LSTM Model Best-fit +95% CI +QuasarNET Data +10- +log( +8 - +7 - +6 +0 +1 +2 +3 +5 +6 +711.0: +TrainData +LSTM Model Prediction +Linear Best-Fit +10.5 +10.0 +log(M_bh) +9.5 +0'6 +B.5 +7.5 +3.0 +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +6.5 +7.0Trained Data +LSTM Model Best-fit +10.5 +Linear Best-fit +. +10.0 +log(MBH / Mo) +9.5 +9.0 +8.5 +8.0 +3.0 +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +6.59 +relatively nearby quasars with redshifts obtained from +the NED and central SMBHs mass determined through +multi-epoch spectrophotometry and RM [61]. A list of +69 high-redshift quasars is also available in TABLE V +and TABLE VI. For each quasar, the most accurate es- +timation of its central SMBH mass using Mg II emis- +sion lines along with its uncertainty is shown [62, 63]. +While the model matches observational data quite well at +3 < z < 10, there is a minor deviation at lower redshifts, +i.e. 0 < z < 3. The comparison between our model pre- +dictions and the observational data for both low-redshift +and high-redshift quasars can be seen in FIG. 5. +In addition to gas being sucked into SMBHs, there is +an alternative process that turns them into stars. There +has been a comparison of SMBH accretion rate and SFR +on a galactic scale in several observational studies [64– +67]. +In our next work, we will address the SFR and its +effects on the model. +Thus, it is possible to fix the +minor deviation between the model and observations. +Further, there are more data available for lower redshift +quasars, compared to higher ones, whose reasons should +be studied and may have an impact on the final results +of our model. +XI. +CONCLUSIONS +The question of how the SMBHs that have been ob- +served in the universe came into being is one of the +biggest questions in cosmology. In recent years, it has +been established that stellar BHs cannot accrete mass, +resulting in such BHs. If we want to consider these BHs +as stellar BHs that have reached such incredible mass due +to accretion, the age of the universe should have been +much longer than it is. On the other hand, it is impossi- +ble for a star to form a SMBH as a result of its collapse. +In addition, there is another idea that states that these +BHs are actually primordial BHs. Although this idea is +very controversial, it has not been rejected yet. There +are even hopes to prove such a thing. +One of the most interesting surveys available for +quasars is the SDSS. In this paper, we have used SDSS- +DR16Q. In particular, we have taken advantage of the +QuasarNET research platform. QuasarNET specifically +has focused on the study of SMBHs. +Although 37648 +data in redshifts between 3 and 7 have been reported in +it, these data need accurate corrections to be used. These +corrections are flux and volume-limited, which makes the +right conditions to work on SMBHs over time for training +the machine. After applying these corrections, 3246 data +remained and 34403 data were removed. In FIG. 1 we +have plotted accepted and removed data after correcting +them. +Considering the remaining 3246 data of the mass of +BHs in the center of quasars at redshifts between 3 and +7, we have modeled them over time with the help of the +LSTM RNN. We have elaborated details of our used DL +approach in several sections. +The model we have pre- +sented with the help of QuasarNET data tries to predict +the mass of the central massive BHs of quasars at red- +shifts between 0 and 10. +Firstly, in FIG. 4, we have compared our prediction +with the linear best-fit of QuasarNET data before and +after correcting data. Then, we illustrated the best-fit +and a band that 95 percent of the QuasarNET data is +within 2 standard deviations of the mean for our model +in redshifts 0 to 10. +Eventually, we should have compared our model with +other observational data at redshifts between 0 and 3 +and also 7 and 10. This will enable us to see whether our +model works or not. We have used four data sets for this +comparison. Two of them are related to redshifts 0 to 3 +and the other two are related to redshifts 7 to 10. FIG. 5 +demonstrates two redshift ranges, 0 to 7 and 3 to 10. As +it is evident, at redshifts higher than 7, our model has a +very good description of the data and can make a reliable +prediction, but at redshifts below 3, it seems that there +is a slight deviation. +This deviation can be due to not considering other pa- +rameters describing quasars. We have only used the esti- +mation of the mass of the central SMBHs of quasars and +their redshift in QuasarNET data. However, data such as +the Eddington ratio and bolometric luminosity are also +available and can be used for subsequent modeling. +Another thing that can improve the model is to con- +sider star formation with the help of other observational +data sets. Accurately obtaining the time of star forma- +tion causes the redshift of the peak of the model we ob- +tained to change to lower redshifts. This issue makes our +model predict more massive central SMBHs at redshifts +below 3, and as a result, it fits better with other data. +Finally, we must state that this effort to model SMBHs +at high redshifts will help us to find out when and how +they have been formed and their role in the formation +of the structures. +Furthermore, if the process of their +growth through the accretion and merger of primordial +BHs is also studied in future works, it will probably yield +interesting results. Because by going back through time, +the initial masses of these central SMBHs can be exam- +ined. +Acknowledgement +Authors thank Shant Baghram for the great discus- +sions that helped us to model and correct the Quasar- +NET data and Rahim Moradi for helpful discussion. +Data availability +The catalogue underlying this paper is available in +the Sloan Digital Sky Survey Quasar catalogue: 16th + +10 +data release (DR16Q) at https://www.sdss.org/dr16/ +algorithms/qsocatalog/ [13]. +The data that support the findings of this study +are +openly +available +at +https://www.kaggle.com/ +datasets/quasarnet/quasarnet, +reference +number +[68]. +[1] Inayoshi, Kohei, Eli Visbal, and Zolt´an Haiman. ”The +assembly of the first massive black holes.” arXiv preprint +arXiv:1911.05791 (2019). +[2] Blandford, R. D., and C. F. McKee. ”Reverberation map- +ping of the emission line regions of Seyfert galaxies and +quasars.” The Astrophysical Journal 255 (1982): 419- +439. +[3] Du, Pu, and Jian-Min Wang. ”The radius–luminosity re- +lationship depends on optical spectra in active galactic +nuclei.” The Astrophysical Journal 886.1 (2019): 42. +[4] Antonucci, Robert. ”Unified models for active galactic +nuclei and quasars.” Annual review of astronomy and +astrophysics 31 (1993): 473-521. +[5] Shields, G. A. ”Thermal continuum from accretion disks +in quasars.” Nature 272.5655 (1978): 706-708. +[6] Svensson, Roland, and Andrzej A. Zdziarski. ”Black hole +accretion disks with coronae.” The Astrophysical Journal +436 (1994): 599-606. +[7] Arnaud, K. A., et al. ”EXOSAT observations of a strong +soft X-ray excess in MKN 841.” Monthly Notices of the +Royal Astronomical Society 217.1 (1985): 105-113. +[8] Yang, Jinyi, et al. ”P¯oniu¯a ‘ena: A Luminous z= 7.5 +Quasar Hosting a 1.5 Billion Solar Mass Black Hole.” +The Astrophysical Journal Letters 897.1 (2020): L14. +[9] Wise, John H., et al. ”Formation of massive black holes in +rapidly growing pre-galactic gas clouds.” Nature 566.7742 +(2019): 85-88. +[10] Kroupa, Pavel, et al. ”Very high redshift quasars and the +rapid emergence of supermassive black holes.” Monthly +Notices of the Royal Astronomical Society 498.4 (2020): +5652-5683. +[11] Bernal, Jos´e Luis, et al. ”Signatures of primordial black +holes as seeds of supermassive black holes.” Journal of +Cosmology and Astroparticle Physics 2018.05 (2018): +017. +[12] Volonteri, Marta, Joseph Silk, and Guillaume Dubus. +”The case for supercritical accretion onto massive black +holes at high redshift.” The Astrophysical Journal 804.2 +(2015): 148. +[13] Lyke, Brad W., et al. ”The Sloan Digital Sky Survey +Quasar Catalog: Sixteenth Data Release.” The Astro- +physical Journal Supplement Series 250.1 (2020): 8. +[14] Busca, Nicolas, and Christophe Balland. ”QuasarNET: +Human-level spectral classification and redshifting with +Deep Neural Networks.” arXiv preprint arXiv:1808.09955 +(2018). +[15] Ball, Nicholas M., and Robert J. Brunner. ”Data min- +ing and machine learning in astronomy.” International +Journal of Modern Physics D 19.07 (2010): 1049-1106. +[16] Baron, Dalya. ”Machine learning in astronomy: A prac- +tical overview.” arXiv preprint arXiv:1904.07248 (2019). +[17] Allen, +Gabrielle, +et +al. +”Deep +learning +for +multi- +messenger astrophysics: A gateway for discovery in the +big data era.” arXiv preprint arXiv:1902.00522 (2019). +[18] Meher, Saroj K., and Ganapati Panda. ”Deep learning in +astronomy: a tutorial perspective.” The European Phys- +ical Journal Special Topics 230.10 (2021): 2285-2317. +[19] Nakoneczny, S. J., et al. ”Photometric selection and red- +shifts for quasars in the Kilo-Degree Survey Data Release +4.” Astronomy and Astrophysics 649 (2021): A81. +[20] Wenzl, Lukas, et al. ”Random forests as a viable method +to select and discover high-redshift quasars.” The Astro- +nomical Journal 162.2 (2021): 72. +[21] Burhanudin, U. F., et al. ”Light-curve classification with +recurrent neural networks for GOTO: dealing with imbal- +anced data.” Monthly Notices of the Royal Astronomical +Society 505.3 (2021): 4345-4361. +[22] Vardoulaki, E., et al. ”FR-type radio sources at 3 +GHz VLA-COSMOS: Relation to physical properties and +large-scale environment.” Astronomy and Astrophysics +648 (2021): A102. +[23] Wang, Cunshi, et al. ”J-PLUS: Support vector machine +applied to STAR-GALAXY-QSO classification.” Astron- +omy and Astrophysics 659 (2022): A144. +[24] Xiao, H. B., et al. ”Efficient Fermi source identification +with machine learning methods.” Astronomy and Com- +puting 32 (2020): 100387. +[25] Parkinson, PM Saz, et al. ”Classification and ranking of +Fermi LAT gamma-ray sources from the 3FGL catalog +using machine learning techniques.” The Astrophysical +Journal 820.1 (2016): 8. +[26] Li, Yin, et al. ”AI-assisted superresolution cosmological +simulations.” Proceedings of the National Academy of +Sciences 118.19 (2021): e2022038118. +[27] Narendra, Aditya, et al. ”Predicting the redshift of +gamma-ray loud quasars using Supervised Machine +Learning: +Part 2.” arXiv preprint arXiv:2201.05374 +(2022). +[28] York, Donald G., et al. ”The sloan digital sky survey: +Technical summary.” The Astronomical Journal 120.3 +(2000): 1579. +[29] Rastegarnia, F., et al. ”Deep learning in searching the +spectroscopic redshift of quasars.” Monthly Notices of +the Royal Astronomical Society 511.3 (2022): 4490-4499. +[30] Xie, G.-Z., Chen, L.-E., Xie, Z.-H., Ma, L., and Zhou, +S.-B. (2005). Agn black hole masses and methods to esti- +mate the mass. Publications of the Astronomical Society +of Japan, 57(1):183–186. +[31] Vestergaard, Marianne. ”Determining central black hole +masses in distant active galaxies.” The Astrophysical +Journal 571.2 (2002): 733. +[32] Wandel, A., B. M. Peterson, and M. A. Malkan. ”Cen- +tral masses and broad-line region sizes of active galac- +tic nuclei. I. Comparing the photoionization and rever- +beration techniques.” The Astrophysical Journal 526.2 +(1999): 579. +[33] Rodriguez-Pascual, P. M., et al. ”Steps toward determi- +nation of the size and structure of the broad-Line region +in active galactic nuclei. IX. Ultraviolet observations of +fairall 9.” The Astrophysical Journal Supplement Series + +11 +110.1 (1997): 9. +[34] Bentz, Misty C., et al. ”The Lick AGN monitoring +project: Broad-line region radii and black hole masses +from reverberation mapping of Hβ.” The Astrophysical +Journal 705.1 (2009): 199. +[35] Netzer, Hagai. The physics and evolution of active galac- +tic nuclei. Cambridge university press, 2013. +[36] McLure, Ross J., and James S. Dunlop. ”The cosmo- +logical evolution of quasar black hole masses.” Monthly +Notices of the Royal Astronomical Society 352.4 (2004): +1390-1404. +[37] McLure, Ross J., and Matt J. Jarvis. ”Measuring the +black hole masses of high-redshift quasars.” Monthly No- +tices of the Royal Astronomical Society 337.1 (2002): +109-116. +[38] Wills, B. J, and I. W. A. Browne. ”Relativistic beaming +and quasar emission lines.” The Astrophysical Journal +302 (1986): 56-63. +[39] Koss, Michael, et al. ”BAT AGN spectroscopic survey. +I. Spectral measurements, derived quantities, and AGN +demographics.” The Astrophysical Journal 850.1 (2017): +74. +[40] Schutte, Zachary, Amy E. Reines, and Jenny E. Greene. +”The black hole–bulge mass relation including dwarf +galaxies hosting active galactic nuclei.” The Astrophysi- +cal Journal 887.2 (2019): 245. +[41] Willott, Chris J., et al. ”Eddington-limited accretion and +the black hole mass function at redshift 6.” The Astro- +nomical Journal 140.2 (2010): 546. +[42] Fanidakis, N., et al. ”The evolution of active galactic +nuclei across cosmic time: what is downsizing?.” Monthly +Notices of the Royal Astronomical Society 419.4 (2012): +2797-2820. +[43] Hopkins, Philip F., Gordon T. Richards, and Lars Hern- +quist. ”An observational determination of the bolometric +quasar luminosity function.” The Astrophysical Journal +654.2 (2007): 731. +[44] Trakhtenbrot, Benny. ”What do observations tell us +about the highest-redshift supermassive black holes?.” +Proceedings of the International Astronomical Union +15.S356 (2019): 261-275. +[45] Trakhtenbrot, Benny, Marta Volonteri, and Priyamvada +Natarajan. ”On the accretion rates and radiative efficien- +cies of the highest-redshift quasars.” The Astrophysical +Journal Letters 836.1 (2017): L1. +[46] Natarajan, Priyamvada, et al. ”QuasarNet: A new re- +search platform for the data-driven investigation of black +holes.” arXiv preprint arXiv:2103.13932 (2021). +[47] Salvato, Mara, Olivier Ilbert, and Ben Hoyle. ”The many +flavours of photometric redshifts.” Nature Astronomy 3.3 +(2019): 212-222. +[48] Tago, E., et al. ”Groups of galaxies in the SDSS Data +Release 7-Flux-and volume-limited samples.” Astronomy +and Astrophysics 514 (2010): A102. +lags.” The Astrophysical Journal 856.1 (2018): 6. +[49] Staudemeyer, +Ralf C., +and Eric Rothstein Morris. +”Understanding LSTM–a tutorial into long short-term +memory recurrent neural networks.” arXiv preprint +arXiv:1909.09586 (2019). +[50] Williams, Ronald J., and David Zipser. Gradient-based +learning algorithms for recurrent connectionist networks. +Boston, MA: College of Computer Science, Northeastern +University, 1990. +[51] Werbos, Paul J. ”Backpropagation through time: what +it does and how to do it.” Proceedings of the IEEE 78.10 +(1990): 1550-1560. +[52] Bengio, Yoshua, Patrice Simard, and Paolo Frasconi. +”Learning long-term dependencies with gradient descent +is difficult.” IEEE transactions on neural networks 5.2 +(1994): 157-166. +[53] Hochreiter, Sepp, and J¨urgen Schmidhuber. ”LSTM can +solve hard long time lag problems.” Advances in neural +information processing systems 9 (1996). +[54] Gers, Felix A., J¨urgen Schmidhuber, and Fred Cummins. +”Learning to forget: Continual prediction with LSTM.” +Neural computation 12.10 (2000): 2451-2471. +[55] Gers, +Felix A., +Nicol N. Schraudolph, +and J¨urgen +Schmidhuber. ”Learning precise timing with LSTM re- +current networks.” Journal of machine learning research +3.Aug (2002): 115-143. +[56] Hochreiter, Sepp, and J¨urgen Schmidhuber. ”Long short- +term memory.” Neural computation 9.8 (1997): 1735- +1780. +[57] Graves, Alex. ”Long short-term memory.” Supervised se- +quence labelling with recurrent neural networks (2012): +37-45. +[58] Yao, +Kaisheng, +et al. ”Depth-gated LSTM.” arXiv +preprint arXiv:1508.03790 (2015). +[59] Banados, Eduardo, et al. ”An 800 million solar mass +black hole in a significantly neutral universe at redshift +7.5.” arXiv preprint arXiv:1712.01860 (2017). +[60] Hu, Chen, et al. ”Supermassive Black Holes with High +Accretion Rates in Active Galactic Nuclei. XII. Reverber- +ation Mapping Results for 15 PG Quasars from a Long- +duration High-cadence Campaign.” The Astrophysical +Journal Supplement Series 253.1 (2021): 20. +[61] Vestergaard, P., L. Rejnmark, and L. Mosekilde. ”Rela- +tive fracture risk in patients with diabetes mellitus, and +the impact of insulin and oral antidiabetic medication on +relative fracture risk.” Diabetologia 48.7 (2005): 1292- +1299. +[62] Aggarwal, Yash. ”New insights into the origins and +growth of seeds of supermassive black holes.” +[63] Yang, Jinyi, et al. ”Probing Early Supermassive Black +Hole Growth and Quasar Evolution with Near-infrared +Spectroscopy of 37 Reionization-era Quasars at 6.3 z +7.64.” The Astrophysical Journal 923.2 (2021): 262. +[64] Netzer, Hagai, et al. ”Spitzer quasar and ULIRG evo- +lution study (QUEST). II. The spectral energy distribu- +tions of palomar-green quasars.” The Astrophysical Jour- +nal 666.2 (2007): 806. +[65] Wild, Vivienne, et al. ”Bursty stellar populations and ob- +scured active galactic nuclei in galaxy bulges.” Monthly +Notices of the Royal Astronomical Society 381.2 (2007): +543-572. +[66] Wild, Vivienne, Timothy Heckman, and St´ephane Char- +lot. ”Timing the starburst–AGN connection.” Monthly +Notices of the Royal Astronomical Society 405.2 (2010): +933-947. +[67] Rosario, D. J., et al. ”The mean star formation rate +of X-ray selected active galaxies and its evolution from +z 2.5: results from PEP-Herschel.” Astronomy and As- +trophysics 545 (2012): A45. +[68] QuasarNet +(2022). +QuasarNet +[Dataset]. +https://www.kaggle.com/datasets/quasarnet/quasarnet +[69] Wang, Feige, et al. ”A luminous quasar at redshift 7.642.” +The Astrophysical Journal Letters 907.1 (2021): L1. +a redshift of 7.5, Nature 553 473 (2017) + +12 +[70] Mortlock, Daniel J., et al. ”A luminous quasar at a red- +shift of z= 7.085.” Nature 474.7353 (2011): 616-619. +[71] Matsuoka, Yoshiki, et al. ”Discovery of the First Low- +luminosity Quasar at z¿ 7.” The Astrophysical Journal +Letters 872.1 (2019): L2. +[72] Wang, Feige, et al. ”The discovery of a luminous broad +absorption line quasar at a redshift of 7.02.” The Astro- +physical Journal Letters 869.1 (2018): L9. +[73] Wang, Feige, et al. ”A Significantly Neutral Intergalactic +Medium Around the Luminous z 7 Quasar J0252–0503.” +The Astrophysical Journal 896.1 (2020): 23. +[74] B.P. Venemans et al., Discovery of Three z 6.5 Quasars +in the VISTA Kilo-Degree Infrared Galaxy (VIKING) +Survey, ApJ 779 24 (2013) +[75] Reed, Sophie L., et al. ”Three new VHS–DES quasars at +6.7¡ z¡ 6.9 and emission line properties at z¿ 6.5.” Monthly +Notices of the Royal Astronomical Society 487.2 (2019): +1874-1885. +[76] Matsuoka, Yoshiki, et al. ”SUBARU HIGH-z EXPLO- +RATION OF LOW-LUMINOSITY QUASARS (SHEL- +LQs). I. DISCOVERY OF 15 QUASARS AND BRIGHT +GALAXIES.” The Astrophysical Journal 828.1 (2016): +26. +[77] Mazzucchelli, C., et al. ”Physical properties of 15 quasars +.” The Astrophysical Journal 849.2 (2017): 91. +[78] Eilers, Anna-Christina, et al. ”Detecting and character- +izing young quasars. I. Systemic redshifts and proximity +zone measurements.” The Astrophysical Journal 900.1 +(2020): 37. +[79] Onoue, Masafusa, et al. ”No Redshift Evolution in the +Broad-line-region Metallicity up to z= 7.54: Deep Near- +infrared Spectroscopy of ULAS J1342+ 0928.” The As- +trophysical Journal 898.2 (2020): 105. +[80] Mortlock, D. J., et al. ”Discovery of a redshift 6.13 quasar +in the UKIRT infrared deep sky survey.” Astronomy and +Astrophysics 505.1 (2009): 97-104. + +13 +Object +Redshift +MBH(×107M⊙) +PG 0003+199 +0.0259 +0.50+0.18 +−0.18 +PG 0804+761 +0.1005 +4.14+0.91 +−0.98 +PG 0838+770 +0.1316 +2.89+1.01 +−1.13 +PG 1115+407 +0.1542 +7.76+2.23 +−1.95 +PG 1322+659 +0.1678 +3.35+1.92 +−0.71 +PG 1402+261 +0.1643 +3.41+1.28 +−1.51 +PG 1404+226 +0.0972 +0.68+0.14 +−0.23 +PG 1415+451 +0.1132 +1.75+0.36 +−0.32 +PG 1440+356 +0.0770 +1.49+0.49 +−0.55 +PG 1448+273 +0.0646 +1.01+0.38 +−0.23 +PG 1519+226 +0.1351 +4.87+0.49 +−0.86 +PG 1535+547 +0.0385 +1.55+0.84 +−0.82 +PG 1552+085 +0.1187 +1.30+0.68 +−0.65 +PG 1617+175 +0.1144 +4.79+2.94 +−2.83 +PG 1626+554 +0.1316 +19.17+2.98 +−2.73 +TABLE III: This table contains 15 low redshift quasars at z < 1 with their central SMBH mass reported in [60]. +Object +Redshift +log(M/M⊙) (Hβ, rms) +Mrk 335 +0.02578 +7.152+0.101 +−0.131 +PG 0026+129 +0.14200 +8.594+0.095 +−0.122 +PG 0052+251 +0.15500 +8.567+0.081 +−0.100 +Fairall 9 +0.04702 +8.407+0.086 +−0.108 +Mrk 590 +0.02638 +7.677+0.063 +−0.074 +3C 120 +0.03301 +7.744+0.195 +−0.226 +Ark 120 +0.03230 +8.176+0.052 +−0.059 +PG 0804+761 +0.10000 +8.841+0.049 +−0.055 +PG 0844+349 +0.06400 +7.966+0.150 +−0.231 +Mrk 110 +0.03529 +7.400+0.094 +−0.121 +PG 0953+414 +0.23410 +8.441+0.084 +−0.104 +NGC 3783 +0.00973 +7.474+0.072 +−0.087 +NGC 4151 +0.00332 +7.124+0.129 +−0.184 +PG 1226+023 +0.15830 +8.947+0.083 +−0.103 +PG 1229+204 +0.06301 +7.865+0.171 +−0.285 +PG 1307+085 +0.15500 +8.643+0.107 +−0.142 +Mrk 279 +0.03045 +7.543+0.102 +−0.133 +PG 1411+442 +0.08960 +8.646+0.124 +−0.174 +NGC 5548 +0.01717 +7.827+0.017 +−0.017 +PG 1426+015 +0.08647 +9.113+0.113 +−0.153 +Mrk 817 +0.03145 +7.694+0.063 +−0.074 +PG 1613+658 +0.12900 +8.446+0.165 +−0.270 +PG 1617+175 +0.11240 +8.774+0.019 +−0.115 +PG 1700+518 +0.29200 +8.893+0.091 +−0.103 +3C 390.3 +0.05610 +8.458+0.087 +−0.110 +Mrk 509 +0.03440 +8.115+0.035 +−0.038 +PG 2130+099 +0.06298 +8.660+0.049 +−0.056 +NGC 7469 +0.01632 +7.086+0.047 +−0.053 +TABLE IV: This table contains 28 low redshift quasars at z < 1 with their central SMBH mass from [61]. + +14 +Object +Redshift +MBH(×109M⊙) +Refence +J0313-1806 +7.64 +0.16+0.4 +−0.4 +[69] +ULAS J1342+0928 +7.541 +0.91+0.13 +−0.14 +[59] +J100758.264+211529.207 +7.52 +1.5+0.2 +−0.2 +[8] +ULAS J1120+0641 +7.085 +2.0+1.5 +−0.7 +[70] +J124353.93+010038.5 +7.07 +0.33+0.2 +−0.2 +[71] +J0038-1527 +7.021 +1.33+0.25 +−0.25 +[72] +DES J025216.64–050331.8 +7 +1.39+0.16 +−0.16 +[73] +ULAS J2348-3054 +6.886 +2.1+0.5 +−0.5 +[74] +VDES J0020-3653 +6.834 +1.67+0.32 +−0.32 +[75] +PSO J172.3556+18.7734 +6.823 +3.7+1.3 +−1.0 +[74] +ULAS J0109-3047 +6.745 +1.0+0.1 +−0.1 +[74] +HSC J1205-0000 +6.73 +1.15+0.39 +−0.39 +[75] +VDES J0244-5008 +6.724 +3.7+1.3 +−1.0 +[74] +PSO J338.2298 +6.658 +3.7+1.3 +−1.0 +[74] +ULAS J0305-3150 +6.604 +1.0+0.1 +−0.1 +[74] +PSO J323.1382 +6.592 +1.39+0.32 +−0.51 +[77] +PSO J231.6575 +6.587 +3.05+0.44 +−2.24 +[77] +PSO J036.5078 +6.527 +3+0.92 +−0.77 +[77] +V DESJ0224 − 4711 +6.526 +2.12+0.42 +−0.42 +[75] +PSOJ167.6415 +6.508 +0.3+0.008 +−0.012 +[74] +PSOJ261 + 19 +6.483 +0.67+0.21 +−0.21 +[78] +PSOJ247.2970 +6.476 +5.2+0.22 +−0.25 +[77] +PSOJ011 + 09 +6.458 +1.2+0.51 +−0.51 +[78] +CFHQSJ0210 − 0456 +6.438 +0.08+0.055 +−0.04 +[41] +CFHQSJ2329 − 0301 +6.417 +2.5+0.4 +−0.4 +[41] +HSCJ0859 + 0022 +6.388 +0.038+0.001 +−0.0018 +[76] +HSCJ2239 + 0207 +6.245 +1.1+3 +−2 +[77] +V DESJ0330–4025 +6.239 +5.87+0.89 +−0.89 +[78] +V DESJ0323–4701 +6.238 +0.55+0.126 +−0.126 +[78] +PSOJ359–06 +6.164 +1.66+0.21 +−0.21 +[78] +CFHQSJ0221 − 0802 +6.161 +0.7+0.75 +−0.47 +[41] +HSCJ1208 − 0200 +6.144 +0.71+0.24 +−0.52 +[79] +ULASJ1319 + 0950 +6.13 +2.7+0.6 +−0.6 +[80] +CFHQSJ1509 − 1749 +6.121 +3.0+0.3 +−0.3 +[41] +PSOJ239–07 +6.114 +3.63+0.2 +−0.2 +[78] +HSCJ2216 − 0016 +6.109 +0.7+0.14 +−0.23 +[79] +CFHQSJ2100 − 1715 +6.087 +3.37+0.64 +−0.64 +[41] +PSOJ158–14 +6.057 +2.15+0.25 +−0.25 +[78] +CFHQSJ1641 + 3755 +6.047 +0.24+0.1 +−0.8 +[41] +CFHQSJ0055 + 0146 +5.983 +0.24+0.9 +−0.7 +[41] +PSOJ056–16 +5.975 +0.75+0.007 +−0.007 +[78] +TABLE V: This table contains 41 high-redshift quasars at z > 5 with their central SMBH mass from different +references which are identified in the fourth column. + +15 +Object +Redshift +MBH(×109M⊙) +J002429.77+391319.0 +6.620 ± 0.004 +0.27 ± 0.02 +J003836.10-152723.6 +6.999 ± 0.001 +1.36 ± 0.05 +J004533.57+090156.9 +6.441 ± 0.004 +0.63 ± 0.02 +J021847.04+000715.2 +6.766 ± 0.004 +0.61 ± 0.07 +J024655.90-521949.9 +6.86 ± 0.02 +1.05 ± 0.37 +J025216.64-050331.8 +6.99 ± 0.02 +1.28 ± 0.09 +J031343.84-180636.4 +7.611 ± 0.004 +1.61 ± 0.40 +J031941.66-100846.0 +6.816 ± 0.004 +0.40 ± 0.03 +J041128.63-090749.8 +6.827 ± 0.006 +0.95 ± 0.09 +J043947.08+163415.7 +6.519 ± 0.003 +0.63 ± 0.02 +J052559.68-240623.0 +6.543 ± 002 +0.002 0.29± +J070626.39+292105.5 +6.5925 ± 0.0004 +2.11 ± 0.04 +J082931.97+411740.4 +6.384 ± 0.004 +1.40 ± 0.16 +J083737.84+492900.4 +6.773 ± 0.007 +0.71 ± 0.18 +J083946.88+390011.5 +6.702 ± 0.001 +0.81 ± 0.02 +J091054.53-041406.8 +6.9046 ± 0.0003 +0.671 ± 0.003 +J092120.56+000722.9 +6.610 ± 0.003 +0.41 ± 0.03 +J092347.12+040254.4 +6.719 ± 0.005 +0.26 ± 0.01 +J092359.00+075349.1 +6.5654 ± 0.0002 +1.77 ± 0.02 +J100758.26+211529.2 +6.682 ± 0.002 +0.49 ± 0.15 +J105807.72+293041.7 +7.48 ± 0.01 +1.43 ± 0.22 +J110421.59+213428.8 +6.585 ± 0.005 +0.54 ± 0.03 +J112001.48+064124.3 +6.766 ± 0.005 +1.69 ± 0.15 +J112925.34+184624.2 +7.070 ± 0.003 +1.35 ± 0.04 +J113508.93+501133.0 +6.824 ± 0.001 +0.29 ± 0.02 +J121627.58+451910.7 +6.579 ± 0.001 +1.49 ± 0.05 +J131608.14+102832.8 +6.648 ± 0.003 +0.61 ± 0.20 +J134208.10+092838.6 +7.51 ± 0.01 +0.81 ± 0.18 +J153532.87+194320.1 +6.370 ± 0.001 +3.53 ± 0.33 +J172408.74+190143.0 +6.480 ± 0.001 +0.67 ± 0.08 +J200241.59-301321.7 +6.673 ± 0.001 +1.62 ± 0.27 +J210219.22-145854.0 +6.652 ± 0.003 +0.74 ± 0.11 +J221100.60-632055.8 +6.83 ± 0.01 +0.55 ± 0.24 +J223255.15+293032.0 +6.655 ± 0.003 +3.06 ± 0.36 +J233807.03+214358.2 +6.565 ± 0.009 +0.56 ± 0.03 +TABLE VI: This table contains 35 high-redshift quasars at z > 6 with their central SMBH mass from [63]. + diff --git a/5dAzT4oBgHgl3EQffvz_/content/tmp_files/load_file.txt b/5dAzT4oBgHgl3EQffvz_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a444d7d601ae7456c10c44ea073c39c3e5e65ef6 --- /dev/null +++ b/5dAzT4oBgHgl3EQffvz_/content/tmp_files/load_file.txt @@ -0,0 +1,1390 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf,len=1389 +page_content='Modeling the Central Supermassive Black Holes Mass of Quasars via LSTM Approach Seyed Sajad Tabasi,1, 2, ∗ Reyhaneh Vojoudi Salmani,3, 2, † Pouriya Khaliliyan,3, 2, ‡ and Javad T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Firouzjaee3, 2, 4, § 1Department of Physics, Sharif University of Technology, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Box 11155-9161, Tehran, Iran 2PDAT Laboratory, Department of Physics, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Toosi University of Technology, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Box 15875-4416, Tehran, Iran 3Department of Physics, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Toosi University of Technology, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Box 15875-4416, Tehran, Iran 4 School of Physics, Institute for Research in Fundamental Sciences (IPM), P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Box 19395-5531, Tehran, Iran One of the fundamental questions about quasars is related to their central supermassive black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The reason for the existence of these black holes with such a huge mass is still unclear and various models have been proposed to explain them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' However, there is still no comprehensive ex- planation that is accepted by the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The only thing we are sure of is that these black holes were not created by the collapse of giant stars, nor by the accretion of matter around them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Moreover, another important question is the mass distribution of these black holes over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Ob- servations have shown that if we go back through redshift, we see black holes with more masses, and after passing the peak of star formation redshift, this procedure decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Nevertheless, the exact redshift of this peak is still controversial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In this paper, with the help of deep learning and the LSTM algorithm, we tried to find a suitable model for the mass of central black holes of quasars over time by considering QuasarNET data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Our model was built with these data reported from redshift 3 to 7 and for two redshift intervals 0 to 3 and 7 to 10, it predicted the mass of the quasar’s central supermassive black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' We have also tested our model for the specified intervals with observed data from central black holes and discussed the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Keywords: Quasars, Supermassive Black Holes, Sloan Digital Sky Survey, QuasarNET Data Set, Deep Learning, and LSTM Model I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' INTRODUCTION In recent years, the study of the high-redshift(z > 6) quasars was a direct probe to explore the Universe at the age less than 1 Gyr after the Big Bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' These early forming quasars are essential to studying the early growth of supermassive black holes (SMBHs) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' By detecting the reverberation between the variations of broad emission lines and the continuum we can deter- mine SMBHs mass in quasars [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Until now, the time lag of Hβ emission lines has been confirmed and measured only in ∼100 quasars [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The continuum and line emission from luminous quasars which are one of the most luminous objects, over a large wavelength range can be characterized by sev- eral leading parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The broad emission line region [4] the optical-to-ultraviolet continuum emission, which is explained by a standard accretion disk extending down to the innermost stable circular orbit [5], X-ray emission with a power-law spectrum produced by inverse Compton scattering of photons from the accretion disk of relativis- tic electrons in the hot corona [6], and a soft X-ray ex- cess [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Spectroscopic observations from optical to near- infrared of these quasars suggest that such SMBHs are already established when the universe is only 700Myr old [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' To explain the existence of these SMBHs, many theo- ∗Electronic address: sstabasi98@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='com †Electronic address: r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='vojoudi@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='com ‡Electronic address: pouriya@email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='kntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='ir §Electronic address: firouzjaee@kntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='ir retical models have been proposed like using primordial density seeds [9–11] and appealing a super-Eddington ac- cretion process [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' To utilize the spectroscopic observational data in phys- ical studies, we need an exact classification and redshift determination of astrophysical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Along the way, the Sloan Digital Sky Survey Catalogue 16th Data Re- lease Quasar Only(SDSS-DR16Q) [13], consists of two files, being the quasar-only main catalog of 750414 en- tries which contains sooner visually confirmed quasars SDSS-I/II/III, and a 1440615-row “superset” of SDSS- IV/eBOSS quasar object classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The DR16Q catalogs present multiple redshifts per ob- ject that are available, including the neural automated QuasarNET [14] redshift which is claimed > 99% ef- ficiency and > 99% accuracy, that rests on garnering deeper insights into this triumvirate connection by co- locating and analyzing observational data and simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Meanwhile, the enormous increase in computing power over the last decades has allowed the application of acquired statistical methods in the analysis of big and complex data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Using previously-fed data has brought huge opportuni- ties for astronomers to develop intelligent tools and inter- faces, utilizing pipeline classifiers, machine learning(ML), and deep learning(DL) methods, to deal with data sets and extract novel information with possible predictions and estimate the relevant confidence which the behavior new data will have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In astronomy and astrophysics, ML [15, 16] and DL [17, 18] have been used in a broad range of subjects(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' quasars and other types of sources), such as redshift de- termination [19, 20], morphological classification and ref- erences therein [21, 22], source selection and classifica- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='01459v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='GA] 4 Jan 2023 2 tion [23–25], image and spectral reconstruction [26], and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ML methods for obtaining redshift estimation for quasars are becoming progressively crucial in the epoch of rich data astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Redshift measurements of quasars are important as they can enable quasar population studies, and provide insight into the star formation rate(SFR), the luminosity function(LF), and the density rate evolution [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In this work, we have used DL to model the mass of quasars’ central SMBH as a function of their redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Firstly, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' II is dedicated to the available observa- tional data and evidence on quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The estimation of a quasar’s central SMBH mass is discussed in detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Furthermore, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' IV, the mass evolution of these black holes(BHs) is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' V is the comparison between two newborn research platforms, QuasarNET and FNET, and the reasons behind using QuasarNET for our model are explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Additionally, we use two correction methods which are explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' A detailed explanation of our DL model can be found in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' VII to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' VII we introduce Long short-term memory(LSTM) which is the recurrent neural network(RNN) that we build our model based on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' We explain the chosen optimization function and its validation loss in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' VIII which is shown in multiple figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' IX presents the topology design of our model and finally, the comparison of the model predictions with other data sets is discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' OBSERVATIONAL EVIDENCE AND DATA The most comprehensive observed quasi-stellar ob- jects(QSOs) spectra to date are cataloged in the SDSS- IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' SDSS has been operative since 2000 and catalogs of quasars have been produced and made available since 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In addition to producing images, it performs spec- troscopic surveys across a large area of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' We can get about one million galaxies and 10,000 quasars spectra from the survey images of the sky, which are obtained by a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5m telescope equipped with a large format mo- saic Charge-coupled device(CCD) camera, and two dig- ital spectrographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' As part of its calibration, the SDSS uses observations of the US Naval Observatory’s 1m tele- scope to calibrate its photometry, and an array of astro- metric CCDs control its astrometry [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The SDSS provides data necessary to study the large- scale structure of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' As far as the obser- vatory’s limit allows, the imaging survey should detect ∼ 5 × 107 galaxies, ∼ 106 quasars, and ∼ 8 × 107 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' By using photometric redshifts and angular correlation functions, these photometric data allow studies of large- scale structures that go beyond spectroscopic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Quasars can provide information on the structure at even larger scales [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The SDSS-DR16Q contains 750,414 quasars, with the automated redshift range 1 ≤ z ≤ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The number of sources reaches its maximum around z ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 and at ear- lier epochs i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' higher redshifts, they are comparatively rare [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' However, there is a problem with the SDSS- DR16Q catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' It contains non quasar sources due to pipeline classification errors and incorrect redshift esti- mations [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' For example, in a search for undeclared quasars, the SDSS-DR16Q main quasars are found to contain 81 entries that are not quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' It must there- fore be noted that the pipeline catalog is not an adequate training samples for quasars because many objects with z ≥ 6 as well as significant fractions of these objects at z ≥ 4, may not be quasars or not quasars at the given redshifts due to incorrect pipeline classifications [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' MASS ESTIMATION OF QUASARS’ CENTRAL SMBH In terms of fundamental parameters of quasars, one can mention the central SMBH mass and structure, along with the ratio of the accretion rate to the Eddington accretion rate [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The central SMBH mass can be measured via the gas or stellar dynamics [30] from optical or ultraviolet(UV) spectroscopy using empirical relations [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The broad emission line region(BLR) probably provides the best probe of these characteristics [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The size of BLRs can be determined by reverberation mapping(RM) [33], which is a measuring technique in astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' RM pro- vides invaluable information about the kinematic and ionization distribution of the gas using the time lag be- tween emission line and continuum variations [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Assuming that gravity dominates the dynamics of the BLR and the virial relationship between time lag and line width exists, the BH mass can be estimated as [34] MBH = fcτv2 G , (1) where τ is the mean time delay for the region of inter- est, v is the velocity of the gas in that region, c is the speed of light, G is the gravitational constant, and f is a scaling factor of order unity that depends on the detailed geometry and kinematics of the line-emitting region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The worth mentioning point is that the virial relation- ship claims a virialized system with individual clouds moving in their Keplerian orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' This leads to the pro- portionality of mean cloud velocity and emissivity radius [35] v ∝ rBLR 1 2 , (2) where rBLR is the emissivity radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In the absence of RM, the quasar continuum luminosity is sufficient to estimate the BLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' With RM estimations, the best-fitting RBLR − λLλ relations were derived for quasars at monochromatic luminosity in both 3000 and 5100 ˚A rest-frames as follows [37] 3 RBLR = (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='6)[λL3000/1037W] (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='32±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='14), (3) RBLR = (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='4 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='4)[λL5100/1037W] (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='61±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' (4) Here, L is the luminosity measured at a wavelength λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 1, an intrinsic Keplerian velocity of a broad-line gas is related to the full width at half maximum (FWHM) of a chosen broad emission line by the geometric factor f as VBLR = f × FWHM, (5) In other words, it is the width of a spectrum curve measured between those points on the y-axis which are half of the maximum amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' As the geometry of the BLR in radio-quiet quasars is currently unknown, it is generally agreed that f = � 3/2, which is appropriate for randomly oriented orbits of the BLR gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' However, FWHM measurements for broad emission lines in radio-loud quasars indicate a disc-like geometry [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Given the similarity between the opti- cal emission-line spectra of radio-loud and radio-quiet quasars, it is not unreasonable to consider the possibil- ity that BLRs of radio-quiet quasars that dominate the SDSS data can follow the same equation as well [36] VBLR = FWHM (2 sin i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' (6) Here, i represents the angle between the line of sight and the axis of the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Our virial BH mass estimators are derived by substi- tuting the calibrations of the RBLR–λLλ relations into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 1 and determining VBLR using MgII or Hβ [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Based on the L5100 which is the monochromatic lu- minosity at rest-frame 5100 ˚A and the Hβ line, a more specific expression to calculate the mass of a BH can be written as [39] MBH(Hβ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='05 × 108( L5100 1046ergs−1 )0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='65 (7) × [FWHM(Hβ) 103kms−1 ]2M⊙, where MBH(Hβ) represents BH mass by considering Hβ line, FWHM(Hβ) is the full width at half maximum of Hβ line, and M⊙ is the solar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Large spectroscopic surveys like the SDSS observe both broad Hβ and MgII lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Therefore, one can be calibrated against the other and based on L3000 and MgIIλ2798 line width, a similar expression can be de- rived as [35] MBH(MgIIλ2798) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='9 × 107( L3000 1046ergs−1 )0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='58 (8) × [FWHM(MgIIλ2798) 103kms−1 ]2M⊙, where MBH(MgIIλ2798) represents BH mass by con- sidering Hβ line, and FWHM(MgIIλ2798) is the full width at half maximum of MgII line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Based on empirical estimation of f ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 for the Hβ line, we can now write more specific expressions to calcu- late MBH for several emission lines like MgII as follows [39] MBH M⊙ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='7(λL5100 1037W ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='61 [FWHM(Hβ) kms−1 ] 2 , (9) MBH M⊙ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2(λL3200 1037W ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='62 [FWHM(MgII) kms−1 ] 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' (10) Besides, it is well-known that the relationship between stellar velocity dispersion and BH mass can be written as [39] log(MBH M⊙ ) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='38 × log( σ∗ 200kms−1 ) + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='49, (11) where σ∗ is the stellar velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Furthermore, to estimate the mass of a BH, observa- tions in the local universe reveal the existence of a corre- lation between the central SMBH mass and the bulge of the host galaxies [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' log(MBH M⊙ ) = α + βlog(MBulge,∗ 1011M⊙ ), (12) where MBulge,∗ is the bulge stellar mass and the best- fit of α and β should be α = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='061;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='075.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' (13) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' MASS EVOLUTION OF QUASARS’ CENTRAL SMBH As studying the cosmic history of compact cosmolog- ical objects is so crucial to track the history line of the universe in a much bigger structure, we are so curious about the evolution of SMBHs mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In the presence of a SMBH, there are obvious links between the physical properties and those of its host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Due to high redshifts that many quasars have, they are ideal to be studied to recognize BH evolution through time back to the early universe [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' According to the modelling of spectra from the SDSS first data release, the virial mass of BHs for 12698 quasars 4 in the redshift interval 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 ≤ z ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 is estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' There is entirely consistent evidence to suggest that the BH mass of SDSS quasars lies in 107M⊙ ≤ MBH ≤ 3 × 109M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The local BH mass function for early-type galaxies using the MBH − σ and MBH − Lbulge correla- tions(Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 11 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 12) are also estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In addition, by comparing the number density of active BHs at z ≈ 2 with the local mass density of inactive ones, a lower limit is set on the lifetime of quasars, which confirms that the bulk of BHs with mass ≥ 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5M⊙ are situated in place by z ≈ 2 [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' There are several different ideas on the central SMBH mass evolution through time in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Based on the effective flux limit along with the role of the quasar con- tinuum luminosity, most studies agree that the SMBH mass increases as a function of redshift, namely most low mass SMBHs can be found in the late universe(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' step- ping down from ≈ 109M⊙ at z ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 to ≈ 108M⊙ at z ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Considering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 9 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 10, redshift does not alter the mean FWHM and it can be roughly consid- ered to be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Therefore, the mean virial mass of the SMBH should be increased as [Lλ]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='6 [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Quasars undergo important cosmic evolution accord- ing to optical, X-ray, and bolometric LFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Interestingly, based on predictions of [42] using an extended version of the galaxy formation model, GALFORM code, quasars evolution will be influenced by different physical pro- cesses such as the accretion mode and the obscuration prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Observational data have also reported sim- ilar trends [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Furthermore, SMBHs grow exponentially during a pe- riod in which accretion governs their mass evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' When z ≳ 5, the growth of a SMBH in a quasar is as follows [44] MBH(t) = MBH(t0)etτ, (14) τ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='4Gyr η 1 − η 1 µ, (15) µ ≡ L LEdd × factive, (16) where MBH(t0) is the initial mass of BH i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' the seed’s mass, η is the radiative efficiency(see [45] for reported values of η for several objects), L is the luminosity of the quasar, LEdd is the luminosity at Eddington limit, factive is the duty cycle, and µ is a constant which is determined as a combination of L/LEdd and factive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Therefore, it is possible to calculate the growth of the BH easily as log MBH(z) = log MBH(z0) (17) + log[exp (R(1 − η η )zd), η ≡ Lbol ˙Mc2 , (18) zd ≡ (1 + z)−3/2 − (1 + z0)−3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' (19) In above equations, MBH(z0) is the mass of BHs’ seed and R is a constant that is defined as follows R ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='4Gyr µ , (20) R = � � � � � 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='79322, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='9661, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='9322, µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' (21) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' QUASARNET AND FNET To investigate the mass evolution even more precisely, QuasarNET and FNET are the two available research platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Using ML, QuasarNET makes deployment of data-driven modelling techniques possible by combining and co-locating large observational data sets of quasars, the high-redshift luminous population of accreting BHs, at z ≥ 3 alongside simulated data spanning the same cosmic epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The main quasar population data source of QuasarNET is NASA Extra-galactic Database(NED) which contains quasars retrieved from several indepen- dent optical surveys, principally the magnitude-limited SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' There is no comparison between quasars from SDSS and those from other surveys when it comes to spectra and photometry [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' NED contains all quasars in principle, but some are missing because their photometric redshifts were incor- rectly assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Photometric redshift estimation meth- ods suffer from degeneracy, a well-known limitation of current photometric redshift determination methods [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' QuasarNET fills in the missing sources by analyzing the published catalogues from all surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' It expands to in- clude additional parameters used to derive BHs mass, in- stead of archiving only the reported masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' It contains 136 quasars’ features, such as the position, redshift, lu- minosity, mass, line width, and Eddington ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Two observationally determined functions are used as constraints in theoretical models to describe the assembly history of the BHs population across time: the BH mass function and the Quasar Luminosity Function(QLF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' As a statistical measurement of the combined distribution of BHs mass through redshifts, the BH mass function encodes the mass growth history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Similar to the QLF, which reflects their accretion history, the BH mass func- tion is a statistical measurement of the distribution of quasars’ luminosities through redshift [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' On the other hand, by using DL, to study quasars in the SDSS-DR16Q of eBOSS on a wide range of signal- to-noise(SNR) ratios, there is a 1-dimensional convolu- tional neural network(CNN) with a residual neural net- work(ResNet) structure, named FNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' With its 24 con- volutional layers and ResNet structure, which has dif- ferent kernel sizes of 500, 200, and 15, FNET can use a self-learning process to identify ”local” and ”global” patterns in the entire sample of spectra [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 5 Although FNET seems to be similar to the recently adopted CNN-based redshift estimator and classifier, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' QuasarNET [14], their hidden layer implementations are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The redshift estimation in FNET is done based on re- lating the hidden pattern which lies in flux to a spe- cific redshift, not using any information about emis- sion/absorption lines, while QuasarNET follows the tra- ditional redshift estimation procedure using the identified emission lines in spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' This makes FNET to outper- form QuasarNET for some complex spectra(insufficient lines, high noise, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=') by recognizing the global pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Moreover, FNET provides similar accuracy to Quasar- NET, but it is applicable for a wider range of SDSS spec- tra, especially for those missing the clear emission lines exploited by QuasarNET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In more detail, from a statis- tical point of view, FNET is capable to infer accurate redshifts even for low SNRs or incomplete spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' It predicts the redshift of 5,190 quasars with 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='6 % accu- racy, while QuasarNET fails to estimate [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' It is important to know that the FNET vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Quasar- NET comes out on top in redshift prediction, but its lack of quasars’ central SMBH mass information makes QuasarNET the preferred option for some studies like this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' However, if in the future SMBHs mass will be estimated by using redshifts from FNET approach, our study can be done again to achieve more accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' FLUX AND VOLUME-LIMITED SAMPLES Observations are affected by flux as we move to higher redshifts and more distant objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' This is why some objects are not included in data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' We suppose that they are not even present because their low flux makes them very difficult or in some cases impossible to observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' This will influence the results of any model that is built on a set of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' To remove this bias, we must first correct the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Two correction methods can be put into use to build a corrected data set and check if the result is solid or if the correction can end up with a huge deviation from the first result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Using the friends-of-friends algorithm, quasars can be linked into systems with a specific neighbourhood radius, called linking length(LL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The size of the group can be determined based on the choice of LL or more generally on its scaling law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' LL is parameterized upon a scaling law as [48] LL LL0 = 1 + a arctan( z z∗ ), (22) where a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='00, z∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='050 and LL0 is the value of LL at initial redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Setting a limit for absolute magnitude is needed for creating volume-limited samples and all less luminous FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 1: The total number of objects available in the QuasarNET data set is 37648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' As a result of data correction methods, 34403 objects were removed (red dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The accepted data are the final flux and volume-limited samples, made of 3245 Objects(blue dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' quasars have to be excluded from the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Flux- limited samples, on the other hand, are formed from dozens of cylinders containing quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Flux-limited samples can be made with both constant and varying LL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The constant LL0 is set as [48] LL0 = 250[kms−1], (23) LL0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='25[h−1Mpc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' (24) Following the extraction of the necessary columns and rejecting duplicate quasars from the data set, there is only one step left, which is verifying if the quasars are within the volume of cylinders generated by the LLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' To do so first we generate a cylinder, then by using the distance between quasars and comparing this distance with the volume of the cylinder, we consider a quasar to be an accepted object if it is located in the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The distance can be easily obtained from the redshift difference between them in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' This algorithm should be repeated as a loop for each quasar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' As a result of applying the correction methods that are described, we end up with 3246 objects to work with, in- stead of 37648 objects that are available in QuasarNET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 1 accepted and rejected quasars’ central SMBH of SDSS-DR16Q in terms of their redshift are illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' LONG SHORT-TERM MEMORY LSTM is one of the most powerful RNN that is used in DL and artificial intelligence [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The RNN is a dynamic system in which there is an internal state at each step of the classification process [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The circular connec- tions between neurons at the higher and lower layers, as well as the possibility of self-feedback, are responsible for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' These feedback connections enable RNNs to propa- gate data from earlier events to current processing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Thus, RNNs build a memory of time series events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' A standard RNN is not capable of bridging more than 5 to 10 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' It is because back-propagated error signals either grow or shrink with every time step [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' As Removed Data 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 Accepted Data 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 log(MBH / M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=') 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 3 5 6 76 a result, the error typically blows up or disappears over a long period of time [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' When error signals are blown up, the result is oscillating weights, while vanishing er- rors mean learning takes too long or does not work at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' It is possible to solve the vanishing error problem by us- ing a gradient-based approach known as LSTM [53–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' More than 1,000 discrete time steps can be bridged us- ing LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' LSTM uses constant error carousels(CECs), which enforce a constant error flow within special cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Cell accessibility is handled by multiplicative gate units, which learn when to grant access to cells [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Us- ing a multiplicative input gate unit, memory contents stored in j are protected from irrelevant inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' We also introduce a multiplicative output gate unit that protects other units from being perturbed by currently irrelevant memory contents stored in j [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Considering distinct time steps t= 1, 2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', an individual step includes for- ward and backward passes which are the update of all units and calculation of error signals for all weights, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The Input yin and output yout gate activation are computed as [54] netoutj(t) = � m ωoutjmym(t − 1), youtj(t) (25) = foutj(netoutj(t)), netinj(t) = � m ωinjmym(t − 1), yinj(t) (26) = finj(netinj(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Here, netinj and netout are the input and output gate activation, j indices are memory blocks, ωlm is the weight on the connection from unit m to l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Index m ranges over all source units, as specified by the network topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' For gates, f is a logistic sigmoid in the range of [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Furthermore, there are adaptive gates, which learn to reset memory blocks once their contents are out of date and therefore, useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Like the activation of the other gates(Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 25 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 26), the forget gate activation yφ is calculated as netφj(t) = � m ωφjmym(t − 1), yφj(t) (27) = fφj(netφj(t)), where netφj is the input from the network to the forget gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The logistic sigmoid with range [0, 1] is used as squashing function fφj and weighted by the hyperbolic tangent function which has the overall task of memory correction [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The forget gate stores all the 1 outputs while forgetting all the 0 outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Finally, LSTM can be written as [58] it = σ(Wxixt + Whiht−1 + Wcict−1), (28) ft = σ(Wxfxt + Whfht−1 + Wcfct−1), (29) ot = σ(Wxoxt + Whoht−1 + Wcoct−1), (30) ht = ot ⊙ tanh(ct).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' (31) Here, it , ft, and ot are input gate, forget gate and output gate of LSTM, ht represents LSTM output, σ is LSTM logistic function, ⊙ denotes element-wise product, W is the weight metric components, x is the input data in time t, and c is LSTM memory cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In our application of LSTM, the forget gate and in- put gate share the same parameters, but are computed as ft = 1 − it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Note that bias terms are omitted in the above equations, but they are applied by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' A lin- ear dependence between LSTM memory cells(ct) and its past(ct−1) are introduced as ct = ft ⊙ ct−1 + it ⊙ tanh(Wxcxt + Whcxt−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' (32) VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' HYPERPARAMETER SELECTION Hyperparameter selection in neural networks is repre- sented by optimization functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Therefore, specifying hyperparameters such as the type of optimization func- tion, learning rate, number of neurons in each layer, num- ber of epochs, and validation are very important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Adam, Stochastic gradient descent(SGD), RMSProp, AdaDelta, and Ftrl are used as optimization functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' We have considered about 20% of the learning data as validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' To determine the quality of the model, we determine the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The cost function that we have considered for the network is mean squared error(MSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The number of epochs for the network learning process is equal to 50 and the batch size is equal to 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Results of the cost function values for each learning process with different optimization functions and a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0005 are shown in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The results related to the loss value for learning and testing data with different optimization functions are reported in the TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' DATA AND NETWORK TOPOLOGY Using QuasarNET data we predict the SMBHs mass with the help of their redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' We use 3245 data for modelling, 2596 data for the network learning process, and 649 data for testing the network result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Data have a redshift range of 3 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In the first step, data are sorted in ascending order of their redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The reason is that 7 (a) (b) (c) (d) (e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 2: (a) shows model evaluation for SGD optimization function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Optimization function loss is illustrated by the blue line and orange lines represent validation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' (b) is the model evaluation using RMSProp whose optimization function loss and validation loss are shown in blue and orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' (c) illustrates the Adam model evaluation by comparing the Optimization function loss(blue line) and validation loss(orange line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The model evaluation for Ftel is shown in (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' loss of the optimization function is represented by the blue line and the validation function loss is shown by the orange line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' (e) shows model evaluation for the AdaDelta optimization function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Optimization function loss is illustrated by the blue line and orange lines represent validation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Optimization functions Train data MSE Test data MSE SGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='39 RMSProp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='38 Adam 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='23 AdaDelta 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='23 Ftrl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='27 TABLE I: This table shows the result of algorithm evaluation by SGD, RMSProp, Adam, Ftrl and AdaDelta optimization functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' redshift is a time series and LSTM has a recurrent archi- tecture which creates memory through time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Then, the learning and testing data are separated in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The network topology can be described by an LSTM layer as the dynamic layer of the network, a drop-out layer to prevent over-fitting, 3 dense layers as static lay- ers, and the output of the network which is printed by the last dense layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' We use the hyperbolic tangent which is an active function for the LSTM layer and the first dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Because the hyperbolic tangent is a non-linear function with a symmetric range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' It is a suitable option to control sudden changes when they are in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' For the second dense, we use the rectified linear unit(ReLU), to transfer the magnitude of the positive value to the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' For the third dense, which outputs the net- work as a continuous number, we use a linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' TABLE II shows the network structure based on the hy- perparameters of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Layers Neurons Computational Parameters Inputs LSTM (None,256) 264192 Dropout (None,256) 0 Dense (None,512) 131584 Dense (None,256) 131328 Dense (None,1) 257 Total Computational Parameters 527361 Trainable Computational Parameters 527361 Non-Trainable Computational Parameter 0 TABLE II: This table illustrates the network topology which includes each layer along with neurons and computational parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' One of the main challenges that always exists in ML and DL is the issue of transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Transparency is a dynamic issue and solving this problem is different for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' There is no specific method to solve this prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Many factors such as the design of an interpretable learning experience, the fundamental determination of SGD LoSS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='4 SGD Validation Loss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 SS0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 0 10 20 GE 40 50 EpochsRMSProp Loss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 RMSProp ValidationLoss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='8: SS0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 0 10 20 GE 40 50 Epochs0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='27 Adam Loss Adam Validation Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='25 SSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='22 0 10 20 30 40 50 EpochsFtel Loss 70 Ftrl Validation Loss 09 50 40 30 20 : 10 0 10 20 E 40 50 Epochs0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='30 AdaDelta Loss AdaDelta Validation Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='20 10 20 CE 40 50 Epochs8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 3: Model built using flux and volume-limited samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Corrected QuasarNET data are plotted with blue dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The black line represents our LSTM model best-fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In addition, red dotted lines represent our models that include 95 percent of all data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' hyperparameters by the task, the observance of the prin- ciples of feature selection, and the determination of the appropriate number of data based on characteristics can allow us to have a transparent model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Transparency in the structure of algorithms is also noteworthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In this paper, we investigate the trans- parency of the model built by the designed network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Trained data are also based on redshifts from 3 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' With the help of the built model, SMBHs mass at 0 < z < 3 and 7 < z < 10 are then predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' We can see the predicted changes of SMBHs mass through redshift in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 3 based on our built model with its 95 percent confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 4 compares the lin- ear best-fit with our LSTM model best-fit both before and after applying correction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' It can clearly be seen that stated correcting methods change our model significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' COMPARING WITH OTHER DATA Using corrected flux and volume-limited samples of QuasarNET data, we build a DL model for quasars’ cen- tral SMBH mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' By applying correction methods, only ≃ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='62% of QuasarNET data is accepted to use for mod- elling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 1 shows the accepted data along with the removed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Moreover, FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 3 illustrated our model whose best- fit contains 95 percent of corrected data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The model shows that SMBHs mass increases in 0 < z < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='72 and reaches its peak at z ≃ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The mass then falls exponentially with increasing redshift at z > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' It should be noted that our model yields a different result than what is shown in other recent works like [44], where the peak is z < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Nevertheless, in some studies which attempt to show quasars’ central SMBHs mass evolution, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 14 is used that does not include any peaks(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' see [59]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The model is then evaluated by using different data sets which are available in multiple tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' We use the results of the long-term spectroscopic monitoring of 15 PG quasars that have relatively strong Fe II emission to generate TABLE III [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Moreover, TABLE IV shows (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 4: (a) compares our model with the linear best-fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Blue dots indicate train data, the LSTM model prediction is showed the colour red, and the orange line is the linear best-fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' (b) illustrates our model and compares it with linear best-fit based on flux and volume-limited samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Train data is shown as blue dots, LSTM model prediction as red, and linear best-fit as an orange line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 5: (a) shows the examination of our model using multiple data sets in the redshift range of 0 < z < 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' An overview of the utilized data can be found in TABLE III and IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' (b) is also the model examination at 3 < z < 10 whose data is available in TABLE V and VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 11 LSTM Model Best-fit 95% CI QuasarNET Data 10 + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' (2021) M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Vestegaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' (2005) 9 log(MBH / Mo) 7 6 0 1 2 3 4 5 611 LSTM Model Best-fit 95% CI QuasarNETData Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='Aggarwal (2022) 10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' (2021) 9 g(MBH / M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=') 60 8 7 6 3 4 5 6 8 9 1011 LSTM Model Best-fit 95% CI QuasarNET Data 10- log( 8 - 7 - 6 0 1 2 3 5 6 711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0: TrainData LSTM Model Prediction Linear Best-Fit 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 log(M_bh) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content="5 0'6 B." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0Trained Data LSTM Model Best-fit 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 Linear Best-fit .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 log(MBH / Mo) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='59 relatively nearby quasars with redshifts obtained from the NED and central SMBHs mass determined through multi-epoch spectrophotometry and RM [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' A list of 69 high-redshift quasars is also available in TABLE V and TABLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' For each quasar, the most accurate es- timation of its central SMBH mass using Mg II emis- sion lines along with its uncertainty is shown [62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' While the model matches observational data quite well at 3 < z < 10, there is a minor deviation at lower redshifts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 0 < z < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The comparison between our model pre- dictions and the observational data for both low-redshift and high-redshift quasars can be seen in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In addition to gas being sucked into SMBHs, there is an alternative process that turns them into stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' There has been a comparison of SMBH accretion rate and SFR on a galactic scale in several observational studies [64– 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In our next work, we will address the SFR and its effects on the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Thus, it is possible to fix the minor deviation between the model and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Further, there are more data available for lower redshift quasars, compared to higher ones, whose reasons should be studied and may have an impact on the final results of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' CONCLUSIONS The question of how the SMBHs that have been ob- served in the universe came into being is one of the biggest questions in cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In recent years, it has been established that stellar BHs cannot accrete mass, resulting in such BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' If we want to consider these BHs as stellar BHs that have reached such incredible mass due to accretion, the age of the universe should have been much longer than it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' On the other hand, it is impossi- ble for a star to form a SMBH as a result of its collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In addition, there is another idea that states that these BHs are actually primordial BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Although this idea is very controversial, it has not been rejected yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' There are even hopes to prove such a thing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' One of the most interesting surveys available for quasars is the SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In this paper, we have used SDSS- DR16Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In particular, we have taken advantage of the QuasarNET research platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' QuasarNET specifically has focused on the study of SMBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Although 37648 data in redshifts between 3 and 7 have been reported in it, these data need accurate corrections to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' These corrections are flux and volume-limited, which makes the right conditions to work on SMBHs over time for training the machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' After applying these corrections, 3246 data remained and 34403 data were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' In FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 1 we have plotted accepted and removed data after correcting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Considering the remaining 3246 data of the mass of BHs in the center of quasars at redshifts between 3 and 7, we have modeled them over time with the help of the LSTM RNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' We have elaborated details of our used DL approach in several sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The model we have pre- sented with the help of QuasarNET data tries to predict the mass of the central massive BHs of quasars at red- shifts between 0 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Firstly, in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 4, we have compared our prediction with the linear best-fit of QuasarNET data before and after correcting data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Then, we illustrated the best-fit and a band that 95 percent of the QuasarNET data is within 2 standard deviations of the mean for our model in redshifts 0 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Eventually, we should have compared our model with other observational data at redshifts between 0 and 3 and also 7 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' This will enable us to see whether our model works or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' We have used four data sets for this comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Two of them are related to redshifts 0 to 3 and the other two are related to redshifts 7 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 5 demonstrates two redshift ranges, 0 to 7 and 3 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' As it is evident, at redshifts higher than 7, our model has a very good description of the data and can make a reliable prediction, but at redshifts below 3, it seems that there is a slight deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' This deviation can be due to not considering other pa- rameters describing quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' We have only used the esti- mation of the mass of the central SMBHs of quasars and their redshift in QuasarNET data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' However, data such as the Eddington ratio and bolometric luminosity are also available and can be used for subsequent modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Another thing that can improve the model is to con- sider star formation with the help of other observational data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Accurately obtaining the time of star forma- tion causes the redshift of the peak of the model we ob- tained to change to lower redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' This issue makes our model predict more massive central SMBHs at redshifts below 3, and as a result, it fits better with other data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Finally, we must state that this effort to model SMBHs at high redshifts will help us to find out when and how they have been formed and their role in the formation of the structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Furthermore, if the process of their growth through the accretion and merger of primordial BHs is also studied in future works, it will probably yield interesting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Because by going back through time, the initial masses of these central SMBHs can be exam- ined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Acknowledgement Authors thank Shant Baghram for the great discus- sions that helped us to model and correct the Quasar- NET data and Rahim Moradi for helpful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Data availability The catalogue underlying this paper is available in the Sloan Digital Sky Survey Quasar catalogue: 16th 10 data release (DR16Q) at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='org/dr16/ algorithms/qsocatalog/ [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The data that support the findings of this study are openly available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='com/ datasets/quasarnet/quasarnet, reference number [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [1] Inayoshi, Kohei, Eli Visbal, and Zolt´an Haiman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”The assembly of the first massive black holes.” arXiv preprint arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='05791 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [2] Blandford, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' McKee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Reverberation map- ping of the emission line regions of Seyfert galaxies and quasars.” The Astrophysical Journal 255 (1982): 419- 439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [3] Du, Pu, and Jian-Min Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”The radius–luminosity re- lationship depends on optical spectra in active galactic nuclei.” The Astrophysical Journal 886.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 (2019): 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [4] Antonucci, Robert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Unified models for active galactic nuclei and quasars.” Annual review of astronomy and astrophysics 31 (1993): 473-521.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [5] Shields, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Thermal continuum from accretion disks in quasars.” Nature 272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5655 (1978): 706-708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [6] Svensson, Roland, and Andrzej A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Zdziarski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Black hole accretion disks with coronae.” The Astrophysical Journal 436 (1994): 599-606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [7] Arnaud, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”EXOSAT observations of a strong soft X-ray excess in MKN 841.” Monthly Notices of the Royal Astronomical Society 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 (1985): 105-113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [8] Yang, Jinyi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”P¯oniu¯a ‘ena: A Luminous z= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 Quasar Hosting a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 Billion Solar Mass Black Hole.” The Astrophysical Journal Letters 897.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 (2020): L14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [9] Wise, John H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Formation of massive black holes in rapidly growing pre-galactic gas clouds.” Nature 566.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='7742 (2019): 85-88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [10] Kroupa, Pavel, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Very high redshift quasars and the rapid emergence of supermassive black holes.” Monthly Notices of the Royal Astronomical Society 498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='4 (2020): 5652-5683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [11] Bernal, Jos´e Luis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Signatures of primordial black holes as seeds of supermassive black holes.” Journal of Cosmology and Astroparticle Physics 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='05 (2018): 017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [12] Volonteri, Marta, Joseph Silk, and Guillaume Dubus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”The case for supercritical accretion onto massive black holes at high redshift.” The Astrophysical Journal 804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 (2015): 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [13] Lyke, Brad W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”The Sloan Digital Sky Survey Quasar Catalog: Sixteenth Data Release.” The Astro- physical Journal Supplement Series 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 (2020): 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [14] Busca, Nicolas, and Christophe Balland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”QuasarNET: Human-level spectral classification and redshifting with Deep Neural Networks.” arXiv preprint arXiv:1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='09955 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [15] Ball, Nicholas M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', and Robert J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Brunner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Data min- ing and machine learning in astronomy.” International Journal of Modern Physics D 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='07 (2010): 1049-1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [16] Baron, Dalya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Machine learning in astronomy: A prac- tical overview.” arXiv preprint arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='07248 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [17] Allen, Gabrielle, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Deep learning for multi- messenger astrophysics: A gateway for discovery in the big data era.” arXiv preprint arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='00522 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [18] Meher, Saroj K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', and Ganapati Panda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Deep learning in astronomy: a tutorial perspective.” The European Phys- ical Journal Special Topics 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='10 (2021): 2285-2317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [19] Nakoneczny, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Photometric selection and red- shifts for quasars in the Kilo-Degree Survey Data Release 4.” Astronomy and Astrophysics 649 (2021): A81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [20] Wenzl, Lukas, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Random forests as a viable method to select and discover high-redshift quasars.” The Astro- nomical Journal 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 (2021): 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [21] Burhanudin, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Light-curve classification with recurrent neural networks for GOTO: dealing with imbal- anced data.” Monthly Notices of the Royal Astronomical Society 505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='3 (2021): 4345-4361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [22] Vardoulaki, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”FR-type radio sources at 3 GHz VLA-COSMOS: Relation to physical properties and large-scale environment.” Astronomy and Astrophysics 648 (2021): A102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [23] Wang, Cunshi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”J-PLUS: Support vector machine applied to STAR-GALAXY-QSO classification.” Astron- omy and Astrophysics 659 (2022): A144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [24] Xiao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Efficient Fermi source identification with machine learning methods.” Astronomy and Com- puting 32 (2020): 100387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [25] Parkinson, PM Saz, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Classification and ranking of Fermi LAT gamma-ray sources from the 3FGL catalog using machine learning techniques.” The Astrophysical Journal 820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 (2016): 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [26] Li, Yin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”AI-assisted superresolution cosmological simulations.” Proceedings of the National Academy of Sciences 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='19 (2021): e2022038118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [27] Narendra, Aditya, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Predicting the redshift of gamma-ray loud quasars using Supervised Machine Learning: Part 2.” arXiv preprint arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='05374 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [28] York, Donald G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”The sloan digital sky survey: Technical summary.” The Astronomical Journal 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='3 (2000): 1579.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [29] Rastegarnia, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Deep learning in searching the spectroscopic redshift of quasars.” Monthly Notices of the Royal Astronomical Society 511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='3 (2022): 4490-4499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [30] Xie, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', Xie, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', Ma, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', and Zhou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Agn black hole masses and methods to esti- mate the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Publications of the Astronomical Society of Japan, 57(1):183–186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [31] Vestergaard, Marianne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Determining central black hole masses in distant active galaxies.” The Astrophysical Journal 571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 (2002): 733.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [32] Wandel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Peterson, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Malkan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Cen- tral masses and broad-line region sizes of active galac- tic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Comparing the photoionization and rever- beration techniques.” The Astrophysical Journal 526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 (1999): 579.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [33] Rodriguez-Pascual, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Steps toward determi- nation of the size and structure of the broad-Line region in active galactic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Ultraviolet observations of fairall 9.” The Astrophysical Journal Supplement Series 11 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 (1997): 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [34] Bentz, Misty C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”The Lick AGN monitoring project: Broad-line region radii and black hole masses from reverberation mapping of Hβ.” The Astrophysical Journal 705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 (2009): 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [35] Netzer, Hagai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The physics and evolution of active galac- tic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Cambridge university press, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [36] McLure, Ross J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', and James S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Dunlop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”The cosmo- logical evolution of quasar black hole masses.” Monthly Notices of the Royal Astronomical Society 352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='4 (2004): 1390-1404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [37] McLure, Ross J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', and Matt J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Jarvis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Measuring the black hole masses of high-redshift quasars.” Monthly No- tices of the Royal Astronomical Society 337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 (2002): 109-116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [38] Wills, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' J, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Browne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Relativistic beaming and quasar emission lines.” The Astrophysical Journal 302 (1986): 56-63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [39] Koss, Michael, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”BAT AGN spectroscopic survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Spectral measurements, derived quantities, and AGN demographics.” The Astrophysical Journal 850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 (2017): 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [40] Schutte, Zachary, Amy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Reines, and Jenny E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Greene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”The black hole–bulge mass relation including dwarf galaxies hosting active galactic nuclei.” The Astrophysi- cal Journal 887.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 (2019): 245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [41] Willott, Chris J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Eddington-limited accretion and the black hole mass function at redshift 6.” The Astro- nomical Journal 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 (2010): 546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [42] Fanidakis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”The evolution of active galactic nuclei across cosmic time: what is downsizing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='.” Monthly Notices of the Royal Astronomical Society 419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='4 (2012): 2797-2820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [43] Hopkins, Philip F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', Gordon T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Richards, and Lars Hern- quist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”An observational determination of the bolometric quasar luminosity function.” The Astrophysical Journal 654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 (2007): 731.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [44] Trakhtenbrot, Benny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”What do observations tell us about the highest-redshift supermassive black holes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='.” Proceedings of the International Astronomical Union 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='S356 (2019): 261-275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [45] Trakhtenbrot, Benny, Marta Volonteri, and Priyamvada Natarajan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”On the accretion rates and radiative efficien- cies of the highest-redshift quasars.” The Astrophysical Journal Letters 836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 (2017): L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [46] Natarajan, Priyamvada, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”QuasarNet: A new re- search platform for the data-driven investigation of black holes.” arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='13932 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [47] Salvato, Mara, Olivier Ilbert, and Ben Hoyle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”The many flavours of photometric redshifts.” Nature Astronomy 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='3 (2019): 212-222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [48] Tago, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Groups of galaxies in the SDSS Data Release 7-Flux-and volume-limited samples.” Astronomy and Astrophysics 514 (2010): A102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' lags.” The Astrophysical Journal 856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 (2018): 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [49] Staudemeyer, Ralf C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', and Eric Rothstein Morris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Understanding LSTM–a tutorial into long short-term memory recurrent neural networks.” arXiv preprint arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='09586 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [50] Williams, Ronald J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', and David Zipser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Gradient-based learning algorithms for recurrent connectionist networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Boston, MA: College of Computer Science, Northeastern University, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [51] Werbos, Paul J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Backpropagation through time: what it does and how to do it.” Proceedings of the IEEE 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='10 (1990): 1550-1560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [52] Bengio, Yoshua, Patrice Simard, and Paolo Frasconi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Learning long-term dependencies with gradient descent is difficult.” IEEE transactions on neural networks 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 (1994): 157-166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [53] Hochreiter, Sepp, and J¨urgen Schmidhuber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”LSTM can solve hard long time lag problems.” Advances in neural information processing systems 9 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [54] Gers, Felix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', J¨urgen Schmidhuber, and Fred Cummins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Learning to forget: Continual prediction with LSTM.” Neural computation 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='10 (2000): 2451-2471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [55] Gers, Felix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', Nicol N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Schraudolph, and J¨urgen Schmidhuber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Learning precise timing with LSTM re- current networks.” Journal of machine learning research 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='Aug (2002): 115-143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [56] Hochreiter, Sepp, and J¨urgen Schmidhuber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Long short- term memory.” Neural computation 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='8 (1997): 1735- 1780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [57] Graves, Alex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Long short-term memory.” Supervised se- quence labelling with recurrent neural networks (2012): 37-45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [58] Yao, Kaisheng, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Depth-gated LSTM.” arXiv preprint arXiv:1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='03790 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [59] Banados, Eduardo, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”An 800 million solar mass black hole in a significantly neutral universe at redshift 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5.” arXiv preprint arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='01860 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [60] Hu, Chen, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Supermassive Black Holes with High Accretion Rates in Active Galactic Nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' XII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Reverber- ation Mapping Results for 15 PG Quasars from a Long- duration High-cadence Campaign.” The Astrophysical Journal Supplement Series 253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 (2021): 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [61] Vestergaard, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Rejnmark, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Mosekilde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Rela- tive fracture risk in patients with diabetes mellitus, and the impact of insulin and oral antidiabetic medication on relative fracture risk.” Diabetologia 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='7 (2005): 1292- 1299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [62] Aggarwal, Yash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”New insights into the origins and growth of seeds of supermassive black holes.” [63] Yang, Jinyi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Probing Early Supermassive Black Hole Growth and Quasar Evolution with Near-infrared Spectroscopy of 37 Reionization-era Quasars at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='3 z 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='64.” The Astrophysical Journal 923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 (2021): 262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [64] Netzer, Hagai, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Spitzer quasar and ULIRG evo- lution study (QUEST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' The spectral energy distribu- tions of palomar-green quasars.” The Astrophysical Jour- nal 666.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 (2007): 806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [65] Wild, Vivienne, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Bursty stellar populations and ob- scured active galactic nuclei in galaxy bulges.” Monthly Notices of the Royal Astronomical Society 381.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 (2007): 543-572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [66] Wild, Vivienne, Timothy Heckman, and St´ephane Char- lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Timing the starburst–AGN connection.” Monthly Notices of the Royal Astronomical Society 405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 (2010): 933-947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [67] Rosario, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”The mean star formation rate of X-ray selected active galaxies and its evolution from z 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5: results from PEP-Herschel.” Astronomy and As- trophysics 545 (2012): A45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [68] QuasarNet (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' QuasarNet [Dataset].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='com/datasets/quasarnet/quasarnet [69] Wang, Feige, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”A luminous quasar at redshift 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='642.” The Astrophysical Journal Letters 907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 (2021): L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' a redshift of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5, Nature 553 473 (2017) 12 [70] Mortlock, Daniel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”A luminous quasar at a red- shift of z= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='085.” Nature 474.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='7353 (2011): 616-619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [71] Matsuoka, Yoshiki, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Discovery of the First Low- luminosity Quasar at z¿ 7.” The Astrophysical Journal Letters 872.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 (2019): L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [72] Wang, Feige, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”The discovery of a luminous broad absorption line quasar at a redshift of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='02.” The Astro- physical Journal Letters 869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 (2018): L9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [73] Wang, Feige, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”A Significantly Neutral Intergalactic Medium Around the Luminous z 7 Quasar J0252–0503.” The Astrophysical Journal 896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 (2020): 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [74] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Venemans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', Discovery of Three z 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 Quasars in the VISTA Kilo-Degree Infrared Galaxy (VIKING) Survey, ApJ 779 24 (2013) [75] Reed, Sophie L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Three new VHS–DES quasars at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='7¡ z¡ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='9 and emission line properties at z¿ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5.” Monthly Notices of the Royal Astronomical Society 487.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 (2019): 1874-1885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [76] Matsuoka, Yoshiki, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”SUBARU HIGH-z EXPLO- RATION OF LOW-LUMINOSITY QUASARS (SHEL- LQs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' DISCOVERY OF 15 QUASARS AND BRIGHT GALAXIES.” The Astrophysical Journal 828.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 (2016): 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [77] Mazzucchelli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Physical properties of 15 quasars .” The Astrophysical Journal 849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 (2017): 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [78] Eilers, Anna-Christina, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Detecting and character- izing young quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Systemic redshifts and proximity zone measurements.” The Astrophysical Journal 900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 (2020): 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [79] Onoue, Masafusa, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”No Redshift Evolution in the Broad-line-region Metallicity up to z= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='54: Deep Near- infrared Spectroscopy of ULAS J1342+ 0928.” The As- trophysical Journal 898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 (2020): 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' [80] Mortlock, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' ”Discovery of a redshift 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='13 quasar in the UKIRT infrared deep sky survey.” Astronomy and Astrophysics 505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 (2009): 97-104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 13 Object Redshift MBH(×107M⊙) PG 0003+199 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0259 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='50+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='18 PG 0804+761 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1005 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='14+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='98 PG 0838+770 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1316 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='89+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='01 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='13 PG 1115+407 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1542 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='76+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='23 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='95 PG 1322+659 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1678 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='35+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='92 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='71 PG 1402+261 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1643 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='41+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='28 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='51 PG 1404+226 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0972 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='68+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='23 PG 1415+451 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1132 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='75+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='32 PG 1440+356 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0770 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='49+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='49 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='55 PG 1448+273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0646 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='01+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='23 PG 1519+226 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1351 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='87+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='49 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='86 PG 1535+547 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0385 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='55+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='84 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='82 PG 1552+085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1187 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='30+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='68 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='65 PG 1617+175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1144 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='79+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='94 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='83 PG 1626+554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1316 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='17+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='98 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='73 TABLE III: This table contains 15 low redshift quasars at z < 1 with their central SMBH mass reported in [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' Object Redshift log(M/M⊙) (Hβ, rms) Mrk 335 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='02578 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='152+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='101 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='131 PG 0026+129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='14200 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='594+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='095 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='122 PG 0052+251 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='15500 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='567+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='081 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='100 Fairall 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='04702 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='407+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='086 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='108 Mrk 590 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='02638 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='677+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='063 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='074 3C 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='03301 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='744+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='195 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='226 Ark 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='03230 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='176+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='052 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='059 PG 0804+761 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='10000 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='841+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='049 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='055 PG 0844+349 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='06400 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='966+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='150 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='231 Mrk 110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='03529 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='400+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='094 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='121 PG 0953+414 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='23410 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='441+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='084 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='104 NGC 3783 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='00973 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='474+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='072 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='087 NGC 4151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='00332 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='124+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='129 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='184 PG 1226+023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='15830 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='947+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='083 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='103 PG 1229+204 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='06301 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='865+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='171 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='285 PG 1307+085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='15500 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='643+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='107 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='142 Mrk 279 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='03045 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='543+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='102 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='133 PG 1411+442 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='08960 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='646+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='124 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='174 NGC 5548 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='01717 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='827+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='017 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='017 PG 1426+015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='08647 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='113+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='113 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='153 Mrk 817 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='03145 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='694+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='063 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='074 PG 1613+658 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='12900 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='446+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='165 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='270 PG 1617+175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='11240 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='774+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='019 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='115 PG 1700+518 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='29200 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='893+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='091 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='103 3C 390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='05610 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='458+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='087 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='110 Mrk 509 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='03440 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='115+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='035 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='038 PG 2130+099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='06298 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='660+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='049 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='056 NGC 7469 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='01632 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='086+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='047 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='053 TABLE IV: This table contains 28 low redshift quasars at z < 1 with their central SMBH mass from [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 14 Object Redshift MBH(×109M⊙) Refence J0313-1806 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='16+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='4 [69] ULAS J1342+0928 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='541 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='91+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='14 [59] J100758.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='264+211529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='207 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='52 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 [8] ULAS J1120+0641 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='085 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='7 [70] J124353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='93+010038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='33+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 [71] J0038-1527 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='021 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='33+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='25 [72] DES J025216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='64–050331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='8 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='39+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='16 [73] ULAS J2348-3054 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='886 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 [74] VDES J0020-3653 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='834 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='67+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='32 [75] PSO J172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='3556+18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='7734 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='823 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='7+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='3 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 [74] ULAS J0109-3047 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='745 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 [74] HSC J1205-0000 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='73 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='15+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='39 [75] VDES J0244-5008 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='724 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='7+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='3 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 [74] PSO J338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2298 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='658 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='7+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='3 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 [74] ULAS J0305-3150 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='604 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 [74] PSO J323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1382 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='592 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='39+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='51 [77] PSO J231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='6575 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='587 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='05+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='44 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='24 [77] PSO J036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5078 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='527 3+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='92 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='77 [77] V DESJ0224 − 4711 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='526 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='12+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='42 [75] PSOJ167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='6415 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='508 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='3+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='012 [74] PSOJ261 + 19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='483 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='67+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='21 [78] PSOJ247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2970 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='476 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='25 [77] PSOJ011 + 09 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='458 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='51 [78] CFHQSJ0210 − 0456 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='438 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='08+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='055 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='04 [41] CFHQSJ2329 − 0301 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='417 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='4 [41] HSCJ0859 + 0022 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='388 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='038+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='001 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0018 [76] HSCJ2239 + 0207 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='245 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1+3 −2 [77] V DESJ0330–4025 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='239 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='87+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='89 [78] V DESJ0323–4701 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='55+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='126 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='126 [78] PSOJ359–06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='164 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='66+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='21 [78] CFHQSJ0221 − 0802 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='161 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='47 [41] HSCJ1208 − 0200 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='71+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='52 [79] ULASJ1319 + 0950 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='6 [80] CFHQSJ1509 − 1749 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='121 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='3 [41] PSOJ239–07 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='114 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='63+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 [78] HSCJ2216 − 0016 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='23 [79] CFHQSJ2100 − 1715 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='087 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='37+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='64 [41] PSOJ158–14 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='057 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='15+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='25 [78] CFHQSJ1641 + 3755 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='24+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='8 [41] CFHQSJ0055 + 0146 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='983 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='24+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='7 [41] PSOJ056–16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='75+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='007 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='007 [78] TABLE V: This table contains 41 high-redshift quasars at z > 5 with their central SMBH mass from different references which are identified in the fourth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content=' 15 Object Redshift MBH(×109M⊙) J002429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='77+391319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='620 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='02 J003836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='10-152723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='999 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='05 J004533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='57+090156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='441 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='02 J021847.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='04+000715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='766 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='07 J024655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='90-521949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='37 J025216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='64-050331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='09 J031343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='84-180636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='611 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='004 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='40 J031941.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='66-100846.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='816 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='03 J041128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='63-090749.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='827 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='09 J043947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='08+163415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='519 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='02 J052559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='68-240623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='543 ± 002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='29± J070626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='39+292105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5925 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0004 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='04 J082931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='97+411740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='384 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='004 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='16 J083737.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='84+492900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='773 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='18 J083946.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='88+390011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='702 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='02 J091054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='53-041406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='9046 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='671 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='003 J092120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='56+000722.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='610 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='03 J092347.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='12+040254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='719 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='01 J092359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='00+075349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='5654 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='02 J100758.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='26+211529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='682 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='15 J105807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='72+293041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='22 J110421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='59+213428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='585 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='03 J112001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='48+064124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='766 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='15 J112925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='34+184624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='070 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='003 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='04 J113508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='93+501133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='824 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='02 J121627.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='58+451910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='579 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='05 J131608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='14+102832.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='648 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='20 J134208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='10+092838.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='18 J153532.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='87+194320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='370 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='001 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='33 J172408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='74+190143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='480 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='08 J200241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='59-301321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='673 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='27 J210219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='22-145854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='652 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='11 J221100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='60-632055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='24 J223255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='15+293032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='655 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='003 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='36 J233807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='03+214358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='565 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} +page_content='03 TABLE VI: This table contains 35 high-redshift quasars at z > 6 with their central SMBH mass from [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQffvz_/content/2301.01459v1.pdf'} diff --git a/5dE0T4oBgHgl3EQfegDz/content/2301.02393v1.pdf b/5dE0T4oBgHgl3EQfegDz/content/2301.02393v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..edd0265fb0bb760aec9ee9e838cd54dec40e02a3 --- /dev/null +++ b/5dE0T4oBgHgl3EQfegDz/content/2301.02393v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:82d958f4aca2ffa47b0f232867d5007c09abe355dbdf20039e2147f25dc289aa +size 4796050 diff --git a/5dE1T4oBgHgl3EQfBAJm/vector_store/index.faiss b/5dE1T4oBgHgl3EQfBAJm/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..fb7309f7e7df036476a390223786d177c1b0c0bf --- /dev/null +++ b/5dE1T4oBgHgl3EQfBAJm/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:eee05b479da4dc12e7283f28994e2fdb5487c52ddac4d6960a06f76ae0b2614f +size 4325421 diff --git a/5dE2T4oBgHgl3EQfkQdW/content/tmp_files/2301.03976v1.pdf.txt b/5dE2T4oBgHgl3EQfkQdW/content/tmp_files/2301.03976v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a49852f662e3128fd24c306dfe28c421f87189ae --- /dev/null +++ b/5dE2T4oBgHgl3EQfkQdW/content/tmp_files/2301.03976v1.pdf.txt @@ -0,0 +1,1392 @@ +Semi-Supervised Learning with Pseudo-Negative Labels for Image +Classification +Hao Xua, Hui Xiaoa, Huazheng Haoa, Li Donga, Xiaojie Qiub and Chengbin Penga,∗ +aNingbo University, Ningbo, China +bZhejiang Keyongtai Automation Technology Co., Ltd., Ningbo, China +A R T I C L E I N F O +Keywords: +Semi-Supervised Learning +Image Classification +Mutual Learning +A B S T R A C T +Semi-supervised learning frameworks usually adopt mutual learning approaches with multiple +submodels to learn from different perspectives. To avoid transferring erroneous pseudo labels between +these submodels, a high threshold is usually used to filter out a large number of low-confidence +predictions for unlabeled data. However, such filtering can not fully exploit unlabeled data with +low prediction confidence. To overcome this problem, in this work, we propose a mutual learning +framework based on pseudo-negative labels. Negative labels are those that a corresponding data item +does not belong. In each iteration, one submodel generates pseudo-negative labels for each data item, +and the other submodel learns from these labels. The role of the two submodels exchanges after +each iteration until convergence. By reducing the prediction probability on pseudo-negative labels, +the dual model can improve its prediction ability. We also propose a mechanism to select a few +pseudo-negative labels to feed into submodels. In the experiments, our framework achieves state-of- +the-art results on several main benchmarks. Specifically, with our framework, the error rates of the +13-layer CNN model are 9.35% and 7.94% for CIFAR-10 with 1000 and 4000 labels, respectively. +In addition, for the non-augmented MNIST with only 20 labels, the error rate is 0.81% by our +framework, which is much smaller than that of other approaches. Our approach also demonstrates +a significant performance improvement in domain adaptation. +1. Introduction +Deep learning is widely used in many areas, and the +performance of deep learning models [10] heavily relies on +the amount of training data. However, in many real-world +scenarios [16, 24, 5, 37], labeled data are often limited, and +the annotation for unlabeled data can usually be expensive. +In such cases, a semi-supervised learning framework can be +adopted. +Semi-supervised learning frameworks include generative- +based models [18], graph-based models [25], consistency- +based regularization [20, 33, 27, 2, 35, 15, 4], self-training +with pseudo-labels [8, 32], and so on. +Among them, self-training methods can expand the +training set by producing pseudo labels for unlabeled data +to improve the model performance. Nevertheless, single +models are not robust to noisy data. Inspired by DML [40], +a natural idea is to simultaneously train two independently +initialized models, and predictions of one submodel can be +used as the learning target for the other submodel. +To avoid transferring erroneous predictions to each other +and alleviate parameter coupling between submodels in the +early stages of training, a dual student framework [15] is +proposed. +It prevents the mutual transfer of erroneous knowledge +by only passing high-confidence predictions to the other +learning model. However, such a mechanism can waste a +large amount of unlabeled data during training. +∗Corresponding author +pengchengbin@nbu.edu.cn (C. Peng) +ORCID(s): +Figure 1: The dual model on the left side represents general +mutual learning, i.e., the models pass strong information +to each other such as information about the category with +the highest prediction probability. The dual model near the +right side exchanges weak information between each other, +indicating which category the data does not belong to. +To address these problems, we propose a new semi- +supervised classification framework based on dual pseudo- +negative label learning. This framework comprises two +submodels, and each submodel generates pseudo-negative +labels as learning targets for the other submodel. Each sub- +model also provides pseudo-negative labels on augmented +data for self-training. The difference between our frame- +work and general mutual learning is shown in Figure 1. We +also propose a selection mechanism to identify the most +representative pseudo-negative labels for the other model. +The main contributions can be summarized as follows: +• We propose a Dual Negative Label Learning (DNLL) +framework, which not only improves the utiliza- +tion of unlabeled data but also significantly reduces +model parameter coupling compared to general mu- +tual learning methods. +Xu.eal: Preprint submitted to Elsevier +Page 1 of 10 +arXiv:2301.03976v1 [cs.CV] 10 Jan 2023 + +aM1 +M1' +Dog +Not Cat +Not Bird +M2 +Unlabeled Data +M2'• We propose a selection mechanism to help select +representative pseudo-negative labels and prove the +effectiveness of this approach theoretically. +• We demonstrate the effectiveness of the proposed +method experimentally on different benchmarks. +2. Related Work +2.1. Data Augmentation +Data augmentation plays a key role in model training, +which is widely used in classification or segmentation. Data +augmentation is used to expand the training set by applying +random perturbations to improve algorithm performance +and robustness. Simple augmentation methods include ran- +dom flips, horizontal or vertical transitions, geometric trans- +formations, changing the contrast of images, and so on. +There are also complex operations. Mixup randomly selects +two images and mixes them by a random proportion to +expand the data set. The Cutout method replaces randomly +selected image pixel values with zeros while leaving the +labels unchanged [7]. In order to maximize the effect of data +augmentation, strategies combining a range of augmentation +techniques are proposed, such as AutoAugmentation [38], +RandAugmentation [6], etc. We also employ data augmen- +tation methods similar to other semi-supervised learning +frameworks [2, 1]. +2.2. Semi-Supervised Learning +Semi-supervised learning has received a lot of attention +in recent years. The main task of semi-supervised learn- +ing is to utilize labeled and unlabeled data to train algo- +rithms. Many approaches based on consistency regularity, +Pi-Model, Temporal Ensembling Model [20], Mean Teacher +[33], Dual Student [15], and so on. Later, a series of holistic +analysis methods, such as MixMatch [2], ReMixMatch +[1], FixMatch [32], have been proposed. Alternatively, in +DMT, inconsistency between two models has also been +used to exploit the correctness of pseudo-labels [9]. In this +work, we propose an efficient semi-supervised classification +framework with dual negative label learning. +2.3. Learning with Noisy Labels +In this case, models are trained with correctly labeled +data and mistakenly labeled data. For example, based on +the recent memory effect of a neural network, co-teaching +[11] trains two models simultaneously, and each model can +help the other one to filter out samples with large losses. +Kim et al. [17] proposes a negative learning method for +training convolutional neural networks with noisy data. This +method provides feedback for input images about classes +to that they do not belong. In this work, we propose to +use low-confidence pseudo-labels as noisy labels for further +learning. +2.4. Learning from Complementary Labels +A category corresponding to the complementary label is +that a data item does not belong. Due to difficulties in col- +lecting labeled data, complementary-label learning is used +in fully supervised learning methods [14] and noisy-label +learning methods [17]. Complementary labels can be gen- +erated based on noisy labels [14, 17]. In our method, com- +plementary labels are generated based on model-generated +pseudo labels. +3. Methodology +3.1. Problem Definition +In traditional multi-model frameworks, learning models +under-fitted in the early stage of training are likely to pass +erroneous pseudo-labels to other models. Such errors can +be accumulated and need to be filtered out. In addition, +consistency loss on the same erroneous pseudo-labels can +also lead the multi-model framework to degenerate into a +self-training model. +Therefore, in this section, we propose a multi-model +semi-supervised learning framework to improve the utiliza- +tion of unlabeled data and alleviate degeneration. We first +describe the novel mutual learning framework called Dual +Negative Label Learning. That detailed framework is shown +in Figure 2, and then proposes an effective selection mech- +anism for choosing representative pseudo-negative labels. +In semi-supervised learning (SSL), the goal is to train a +model by utilizing a small amount of labeled data and a large +amount of unlabeled data. Formally, we define a training set +퐷 consisting of labeled data 퐷푙={(푋푖, 푌푖 +) ; 푖 ∈ (1, ..., 푁) +} +and unlabeled data 퐷푢={(푋푗 +) ; 푗 ∈ (1, ..., 푀) +}, and we use +a dual model to allow each submodel learning from the +other. The label 푌푖 of the 푖-th data item is a one-hot vector. +3.2. Supervised Learning +In supervised learning, labeled data are augmented by +different weak augmentations for different submodels. +푋(1) +푖 +=퐴(1) +푤 (푋푖), +(1) +푋(2) +푖 +=퐴(2) +푤 (푋푖). +(2) +where 퐴(1) +푤 , 퐴(2) +푤 denote different weak augmentation opera- +tions and 푋(1) +푖 , 푋(2) +푖 +denote weakly augmented data sets. +We use the cross-entropy (CE) function for the super- +vised loss. In classification tasks, the image-level CE loss is +as follows: +퐻(푌 , ̂푌 ) = − +∑ +푖 +푌푖푙표푔( ̂푌푖) +(3) +where ̂푌 is the predicted label, and 푌 is the ground truth. +The supervised losses of the two submodels are as +follows: +퓁(1) +푠푢푝 = 퐻(푓휃(푋(1) +푖 ), 푌푖), +(4) +퓁(2) +푠푢푝 = 퐻(푓휑(푋(2) +푖 ), 푌푖). +(5) +where 푓휃 and 푓휑 represent the operations of two submodels +respectively, and 휃 and 휑 represent parameters correspond- +ing submodels. +Xu.eal: Preprint submitted to Elsevier +Page 2 of 10 + +Figure 2: Overview of the DNLL framework. We use a small amount of labeled data and a large amount of unlabeled data +to train a dual model. Each submodel within the dual model has the same structure and is initialized independently. For each +labeled data, weak augmentations such as random cropping and random flipping are applied. A cross-entropy function is +used to calculate the supervised loss. For each unlabeled data, besides weak augmentations, strong augmentations such as +color jittering are applied. Each submodel generates pseudo-negative labels based on predictions of weakly augmented data, +and these labels are used to teach the other submodels when predicting strongly augmented data. +3.3. Unsupervised Learning +3.3.1. Dual pseudo-negative label Learning +Most unsupervised learning parts in semi-supervised +learning frameworks are realized by allowing each sub- +model to learn with pseudo-positive labels from other sub- +models. To avoid model degeneration and error accumula- +tion in this process, we propose a novel dual negative label +learning approach. In this approach, each submodel teaches +the other that a given data item should not belong to a +certain category. It allows model diversity and can reduce +transferring of erroneous information. +Pseudo-negative labels, namely, the labels that a corre- +sponding data item does not belong to, are generated by +taking complementary labels of the predicted label by a +submodel. In our approach, we also select a few pseudo- +negative labels as representative pseudo-negative labels. +For data item 푗, its pseudo label ̂푌푗 and its representative +pseudo-negative label 푌 푐 +푗 are randomly selected from all the +candidates with equal probability (EP) as follows: +̂푌푗 = 푓(푋푗), +(6) +푌 푐 +푗 ∈ 푧(푓(푋푗), 푚), +(7) +where 푚 is one by default, and 푧 is defined as follows: +푧(푓(푋푗), 푚) ={푣|푣 ∈ {0, 1}퐾, +∑ +푖 +푣푖 = 푚, +and 푣[arg max ̂푌푗] ≠ 1}. +(8) +Here, 퐾 is the number of categories, and {0, 1}퐾 represents +a vector of length 퐾 with elements equal to zero or one. To +increase the convergence rate, we can allow each submodel +to generate multiple representative pseudo-negative labels +for each weakly augmented data item for the other submodel +to learn. Thus, 푚 can also be positive integers larger than one +and less than 퐾. +By teaching each other with pseudo-negative labels +only, we reduce the coupling between submodels. The loss +function can be written as follows: +퐿( ̂푌 , 푌 푐) = − +∑ +푖 +푌 푐 +푖 log(1 − ̂푌푖) +(9) +where ̂푌 denotes the predictions from one submodel and 푌 푐 +is the representative pseudo-negative labels from the other +submodel. +We also use weak and strong data augmentations for un- +labeled data to improve the generalization ability of the dual +model. The weak augmentations can be random cropping, +random flipping, or simply outputting the original images. +The strong augmentation operations can be color dithering +or noise perturbations. Usually, predictions for weakly aug- +mented data by a submodel will be more accurate than that +for strongly augmented data. Thus, in our framework, the +predictions of weakly augmented data by one submodel are +used for generating pseudo-negative labels. We use these +labels as learning targets for the other submodel feed by +strongly augmented images. The augmentation process can +be written as follows: +푋(푤) +푗 +=퐴푤(푋푗), +(10) +푋(푠) +푗 +=퐴푠(푋푗), +(11) +where 퐴푤 and 퐴푠 denote the weak and strong augmentation +operations, respectively. 푋(푤) +푗 +and 푋(푠) +푗 +denote the weakly +Xu.eal: Preprint submitted to Elsevier +Page 3 of 10 + +Weak Augmentations +Labeled Prediction +Supervised Loss +Labeled Data +Unlabeled Prediction +Pseudo Label +Pseudo-Negative Labe +Net A +..... +Unsupervised Loss +GT +Unlabeled Data +NetB +Pseudo Label +Pseudo-Negative Labe +Supervised Loss +Strong Augmentations +Labeled Predictionand strongly augmented data items. Consequently, we have +푌 푐1 ∈ 푧(푓휃(푋(푤) +푗 +), 푚), +(12) +푌 푐2 ∈ 푧(푓휑(푋(푤) +푗 +), 푚). +(13) +Therefore, the loss of learning between submodels is as +follows: +퓁(1) +푐푟표푠푠 = 퐿(푓휃(푋(푠) +푗 ), 푌 푐2), +(14) +퓁(2) +푐푟표푠푠 = 퐿(푓휑(푋(푠) +푗 ), 푌 푐1). +(15) +To further utilize the augmented data, we also developed +a self-learning approach. In this approach, the generated +pseudo-negative labels with weakly augmented data are also +used by the same submodel to feed strong augmented data. +The loss function can be written as follows: +퓁(1) +푠푒푙푓 = 퐿(푓휃(푋(푠) +푗 ), 푌 푐1), +(16) +퓁(2) +푠푒푙푓 = 퐿(푓휑(푋(푠) +푗 ), 푌 푐2). +(17) +The unsupervised loss of the dual model is a combina- +tion of the previous loss functions: +퓁(1) +푢푛푠푢푝 = 퓁(1) +푐표푟푠푠 + 퓁(1) +푠푒푙푓, +(18) +퓁(2) +푢푛푠푢푝 = 퓁(2) +푐표푟푠푠 + 퓁(2) +푠푒푙푓. +(19) +The final total loss of the dual model in the DNLL is a +combination of the supervised loss and the unsupervised +one, as follows: +퓁(1) = 퓁(1) +푠푢푝 + 휆퓁(1) +푢푛푠푢푝, +(20) +퓁(2) = 퓁(2) +푠푢푝 + 휆퓁(2) +푢푛푠푢푝, +(21) +where 휆 is a hyperparameter to balance the supervised +loss item and the unsupervised loss item. The complete +algorithm is shown in Algorithm 1. +From this pseudo code, we can see that the running +time is proportional to the size of the input data. If the +size of unlabeled data, 푀, is much larger than that of the +labeled data, 푁, which usually happens in semi-supervised +learning, the running time is approximately proportional to +the size of the unlabeled data. Thus, the time complexity is +푂(푀). +3.3.2. Error Perception Mechanism for Selecting +Pseudo-Negative Labels +In the above section, for an unlabeled data item, a rep- +resentative pseudo-negative label is randomly selected from +all the candidates with equal probability. To incorporate the +performance of each submodel in different categories, we +propose an Error Perception Mechanism (EPM). +In this approach, for a given data item, if a submodel is +prone to misclassify it into the other category, the pseudo- +negative label generated by the other submodel should +include that misclassified category. Therefore, we compute +the probability of misclassification for each category of each +Figure 3: The generating process of pseudo-negative labels. +For an unlabeled data item, a submodel makes a prediction +to generate a pseudo label (3 in this example) and then +randomly selects two pseudo-negative labels according to +푅 of the other submodel. +submodel using labeled data. Formally, for a submodel, we +define a vector 푃푟푘 for category 푘 with its 푖-th element +defined as follows: +푃푟푘[푖] = +{∑푁푘 +푗=1 푝푖푗, +푖 ≠ 푘 +0 +푖 = 푘 +(22) +where 푁푘 denotes the total number of data with category +푘 being misclassified into category 푖, and 푝푖푗 represents the +confidence that the 푗-th misclassified sample belongs to the +푖-th category. We may also use EMA to update 푃푟푘 for +stability. +It is then normalized with a softmax function. +푅푘 = 푆표푓푡푚푎푥(푃푟푘). +(23) +We use superscripts to represent submodels, so 푅(1) +푘 and 푅(2) +푘 +are misclassification probabilities for the first and the second +submodels. An example of the 푅푘-based pseudo-negative +label generation process is shown in Figure 3. +Therefore, when computing 퓁2 +푐푟표푠푠, we sample 푌 푐1 from +푧(푓휃(푋(푤) +푗 +), 푚) such that the probability that 푌 푐2 +푗 [푘] = 1 +is proportional to 푅(2) +푘 . A similar approach applies when +computing 퓁1 +푐푟표푠푠. +3.4. Theoretical Analysis +First, we demonstrate that in the mutual learning frame- +work based on a dual model, passing pseudo-negative labels +between submodels is less likely to have error accumulation +than that of passing pseudo labels, especially at the early +stages of training. +Theorem 3.1. The error rate (ER) for transferring pseudo- +negative labels from one submodel to the other is expected +to be +푚 +퐾−1 of the ER when transferring pseudo labels, where +푚 is the number of selected pseudo-negative labels and 퐾 is +the number of categories for each data item. +Proof. Without loss of generality, we define that the pre- +diction accuracy of one submodel is 푞 for unlabeled data. +Xu.eal: Preprint submitted to Elsevier +Page 4 of 10 + +0.26 +0.12 +0.10 +0.17 +0.07 +0.05 +0.03 +0.09 +0.11 +2 +3 +4 +5 +6 +7 +8 +0 +9 +R for category 3 of Submodel B +2 +Pseudo label by SubmodeI A +Pseudo-negative labels for Submodel BAlgorithm 1 Pseudo code for the training process of DNLL. +Input:The labeled dataset 퐷푙={(푋푖, 푌푖 +) ; 푖 ∈ (1, ..., 푁) +} +and the unlabeled dataset 퐷푢={(푋푗 +) ; 푗 ∈ (1, ..., 푀) +}. +The two submodels are 푓휃 and 푓휑. +1: for each epoch do +2: +for each batch do +3: +(휒푙, 푌푙) ∶ select a batch of data from 퐷푙 +4: +(휒푢) ∶ select a batch of data from 퐷푢 +5: +휒(1) +푙 += 퐴(1) +푤 (휒푙) +6: +휒(2) +푙 += 퐴(2) +푤 (휒푙) +7: +휒(푤) +푢 += 퐴푤(휒푢) +8: +휒(푠) +푢 += 퐴푠(휒푢) +9: +퓁(1) +푠푢푝 = 퐻(푓휃(휒(1) +푙 ), 푌푙) +10: +퓁(2) +푠푢푝 = 퐻(푓휑(휒(2) +푙 ), 푌푙) +11: +푌 푐1 ∈ 푧(푓휃(휒(푤) +푢 +), 푚) +12: +푌 푐2 ∈ 푧(푓휑(휒(푤) +푢 +), 푚) +13: +퓁(1) +푢푛푠푢푝 = 퐿(푓휃(휒(푠) +푢 ), 푌 푐2) +14: +퓁(2) +푢푛푠푢푝 = 퐿(푓휑(휒(푠) +푢 ), 푌 푐1) +15: +푓휃 = arg min푓휃(퓁(1) +푠푢푝 + 휆퓁(1) +푢푛푠푢푝) +16: +푓휑 = arg min푓휑(퓁(2) +푠푢푝 + 휆퓁(2) +푢푛푠푢푝) +17: +end for +18: end for +return 푓휃, 푓휑 +Therefore, when transferring pseudo labels, the probability +that that submodel provides correct learning targets to the +other is 푞. +When transferring 푚 pseudo-negative labels, if the sub- +model predicts correctly, it transfers correct negative labels. +If the submodel predicts mistakenly, the chance of providing +correct negative labels is +퐶푚 +퐾−2 +퐶푚 +퐾−1 +, +(24) +where 퐶푚 +퐾−1 denotes the total number of combinations of +selecting 푚 pseudo-negative labels from all the 퐾 − 1 +pseudo-negative labels, and 퐶푚 +퐾−2 denotes the number of +combinations of selecting 푚 pseudo-negative labels from +퐾 − 2 truly negative labels. 퐾 − 2 is obtained by taking +all the 퐾 categories except two categories corresponding to +one pseudo label and one ground-truth label. Therefore, the +probability of providing the correct learning target is +푞 + (1 − 푞) +퐶푚 +퐾−2 +퐶푚 +퐾−1 += 1 − (1 − 푞)푚 +퐾 − 1 . +(25) +Therefore, the error rate of transferring pseudo-negative +labels is +1 − (1 − (1 − 푞)푚 +퐾 − 1 ) = (1 − 푞) +푚 +퐾 − 1. +(26) +As the error rate of transferring pseudo labels is 1 − 푞, the +error rate of transferring pseudo-negative labels is +푚 +퐾−1 of +that of transferring pseudo labels. Therefore, transferring +pseudo labels can provide a better learning target, and a +smaller 푚 and a larger 퐾 can further reduce the error +accumulation. +For two submodels with the same structure, when they +are converged to be the same, they can no longer be used for +semi-supervised learning. We need to avoid such scenarios, +especially in the early training stages. In the unsupervised +learning part, we demonstrate that when transferring knowl- +edge with pseudo-negative labels, it is unlikely to have two +submodels degenerate into the same. +Theorem 3.2. When transferring representative pseudo- +negative labels randomly, the probability that two submod- +els are optimized for different objectives is 1 − +√ +2휋푚 +푒퐾 ( 푚 +퐾 )푚 +approximately, where 푚 is the number of representative +pseudo-negative labels and 퐾 is the number of categories. +Proof. Without loss of generality, we assume that two sub- +models produce the same prediction with probability 푞 and +when they produce the same pseudo labels, the probability +that the two submodels can produce the same representative +pseudo-negative labels is +1 +퐶푚 +퐾−1 +. +(27) +Similarly, the probability that two submodels produce dif- +ferent predictions is 1 − 푞, and when they produce different +predictions, the probability that they produce the same +pseudo labels is +1 +퐶푚 +퐾−2 +. +(28) +Thus, the probability that the two submodels transfer- +ring the same representative pseudo-negative label is +푞 +퐶푚 +퐾−1 ++ 1 − 푞 +퐶푚 +퐾−2 +(29) +=푚!(퐾 − 2 − 푚)!(퐾 − 1 − 푞푚) +(퐾 − 1)! +(30) +≈(퐾 − 1 − 푞푚)× +√ +2휋푚( 푚 +푒 )푚√ +2휋(퐾 − 2 − 푚)( 퐾−2−푚 +푒 +)퐾−2−푚 +√ +2휋(퐾 − 1)( 퐾−1 +푒 )퐾−1 +(31) +≈ +√ +2휋푚 +푒퐾 +( 푚 +퐾 )푚 +(32) +where the approximation in Eq. (31) is obtained by the Stir- +ling’s approximation, and that in Eq. (32) is by considering +퐾 >> 푚. +4. Experiments +In this section, we first introduce benchmarks used +in experiments and briefly describe the details of the ex- +periments. Then we compare DNLL with other methods. +Xu.eal: Preprint submitted to Elsevier +Page 5 of 10 + +Finally, we evaluate the efficiency of DNLL from different +perspectives. +4.1. Benchmark datasets +In the classification task, we use the public bench- +mark datasets CIFAR-10 [19], SVHN [28], and MNIST as +many others. The CIFAR-10 dataset includes 50000 training +images and 10000 test images, and the total number of +categories is ten. We randomly select 500 images for each +category as the validation set. The total number of categories +of SVHN Dataset is ten, in which the training set contains +73257 images and the test set contains 26032 images. We +also randomly select 500 images for each category as the +validation set. The MNIST dataset includes 60000 training +images and 10000 test images, and the total number of +categories also is ten. We randomly select 50 images for +each category as the validation set. +4.2. Implementation Details +Our approach is implemented on Pytorch. For the train- +ing stage, the following configurations are used. The learn- +ing rate is 0.03, and the weight decay is 5 × 10−4. The +momentum is 0.9. We use the cosine annealing technique +with batch size 256. We report performances on the test +set averaged from three runnings. For dual models, we use +WideResNet-28-2 (WRN-28-2)[39] and 13-layer CNN as +other approaches [2, 15]. +We use data augmentation techniques in our experi- +ments. The data augmentation operation for each data set +is performed exactly following its corresponding literature +for fairness. Specifically, for the MNIST dataset, we do +not change the input data [25]. For the CIFAR-10 dataset, +when using the 13-layer CNN as the model [15], we make +the original image as a weakly augmented version and the +noise-processed image as a strongly augmented version. +When using WideResNet-28-2 as the model [9], the weak +augmentation operations we used include random cropping +and random flipping, and the strong augmentation operation +is random color jittering. For the SVHN dataset [20], we +only use the horizontal translation as the strong augmenta- +tion operation and the original image as the weakly aug- +mented version. +4.3. Comparison on Benchmarks +In experiments with the CIFAR-10 dataset, we randomly +select 1K, 2K, and 4K data items, respectively, as labeled +data and the rest as unlabeled data. +We compare our method with others: Π model, Tempo- +ral Ensembling [20], VAT[27] and Mean Teacher [33] based +on consistency regularization; Π+STNG [25], LP+SSDL +and LP-SSDL-MT [13] based on graph methods; Filtering +CCL, Temperature CCL [23], TSSDL, TSSDL-MT [31] and +TNAR-VAE [36] based on mean-teacher frameworks; Cur- +riculum Labeling (CL) [3] based self-training; MixMatch +[2] based on strong hybrid method. We also compare our ap- +proach with others based on dual models: Deep Co-Training +(DCT) [29], Dual student(DS) [15], Mutual Learning of +Complementary Networks(CCN) [34] and Dynamic Mutual +Table 1 +Accuracy on the Test Set of CIFAR-10 with the 13-layer CNN +as the backbone. +Method +1K +2K +4K +Π model† +68.35 +82.43 +87.64 +Temporal ensembling† +76.69 +84.36 +87.84 +Mean Teacher +81.78 +85.67 +88.59 +Π+SNTG† +78.77 +85.35 +88.64 +LP-SSDL† +77.98 +84.34 +87.31 +LP-SSDL-MT† +83.07 +86.78 +89.39 +Filtering CCL† +81.78 +85.67 +88.59 +Temperature CCL† +83.01 +87.43 +89.37 +TSSDL† +78.87 +85.35 +89.10 +TSSDL-MT† +81.59 +86.46 +90.70 +TNAR-VAE† +- +- +91.15 +DCT +- +- +90.97 +Dual Student +85.83 +89.28 +91.11 +CCN +87.95 +89.63 +91.2 +DNLL (Ours) +87.87 +90.65 +92.06 +Table 2 +Accuracy on the Test Set of CIFAR-10 with the WRN-28-2 as +the backbone. +Method +1K +4K +VAT† +81.36 +88.95 +Mean Teacher† +82.68 +89.64 +CL +90.61 +94.02 +MixMatch +92.25 +93.76 +DMT +91.51 +94.21 +DNLL (Ours) +92.03 +94.29 +Training (DMT) [9]. The symbol † indicates that the results +are reported in [4] and [12]. The symbol ’-’ indicates that +the corresponding results have not been reported in this +literature. +From Table 1 and Table 2, we can find that our method +performs relatively well with 1k labels and outperforms all +the other methods in other cases. From Table 1, the accuracy +of our approach ranges between 87.87% and 92.06%, which +outperforms most of the other methods using the dual +model, i.e., DCT, Dual Student, and CCN. From Table 2, +the MixMatch is 0.53% lower than our approach at the +accuracy with 4K labels. The DMT is 0.41% and 0.08% +lower than our approach at the accuracy with 1K and 4K +labels, respectively. Figure 4 demonstrates the performance +of DNLL during the training process on the test set. As the +epoch number increases, the training accuracy increases. +In the SVHN dataset, 1K and 4K items are also ran- +domly selected as labeled data. We compare our method +with others as follows: Π model [20], Pseudo-Labeling [21], +VAT [27] and Mean Teacher [33]. The symbol † indicates +that the results are reported in [12]. All the approaches +use WideResNet-28-2 as the backbone model. As shown in +Table 3, our method outperforms all the other approaches. +Xu.eal: Preprint submitted to Elsevier +Page 6 of 10 + +Table 3 +Accuracy on the Test Set of SVHN with the WRN-28-2 as the +backbone. +Method +1K +4K +Pseudo-Labeling +90.06 +- +Π model +92.46 +- +VAT† +94.02 +95.80 +Mean Teacher† +96.25 +96.61 +DNLL (Ours) +96.41 +96.84 +Table 4 +Accuracy on the Test Set of MNIST with the 13-layer CNN +as the backbone. +Method +20 +50 +100 +ImprovedGAN† +83.23 +97.79 +99.07 +Triple GAN† +95.19 +98.44 +99.09 +Π model† +93.68 +98.98 +99.11 +Π + SNTG† +98.64 +99.06 +93.34 +DNLL (Ours) +99.19 +99.32 +99.54 +Figure 4: Performance of DNLL on the test set during +training with the CIFAR-10 dataset of 1000 and 4000 labeled +data. +For the MNIST dataset, 20, 50, and 100 data items are +randomly selected as labeled data. We compare the DNLL +with other semi-supervised methods, i.e., ImprovedGAN +[30], Triple GAN [22], Π model [20] and Π + STNG [25]. +The symbol † indicates that the results are reported in [25]. +All the above methods use the 13-layer CNN as the model. +As shown in Table 4, the DNLL outperforms the other +approaches. +4.4. Sensitivity Analysis +We conduct a sensitivity analysis on the CIFAR-10 +dataset with 4K labeled data items to analyze the relation- +ship between representative pseudo-negative label number +푚 and the accuracy of the model under different selection +mechanisms that were introduced in the methodology sec- +tion: Equal Probability (EP) vs. Error Perception Mech- +anism (EPM). As the number of representative pseudo- +negative labels tends to be less than half of the total number +Table 5 +Accuracy under different choices of 푚 and different selection +mechanisms for representative pseudo-negative labels. +Selection Method +푚 = 1 +푚 = 2 +푚 = 3 +푚 = 4 +EP +92.9 +93.76 +94.01 +93.78 +EPM +93.12 +93.84 +94.29 +93.77 +Table 6 +Comparison of the performance of mutual learning (ML) and +self-learning (SL) with DNLL. +Method +4k labels +SL w/o EPM +92.78 +SL +93.03 +ML w/o EPM +94.01 +ML +94.29 +of categories, here we compare with 푚 ≤ 4. From Table. 5, +we can find that generally, the error perception mechanism +performs better than selecting with equal probability, and +moderately increasing 푚 is helpful to increase the perfor- +mance. When 푚 is too large, for example, close to half of +the total number of categories, pseudo labels are likely to be +selected, and the performance can be undermined. +4.5. Comparison with variants of DNLL +In this part, we demonstrate that using mutual learning +framework in DNLL is more efficient compared to a self- +learning framework. We compare the performance of these +two learning frameworks. We can see from Table. 6 that the +mutual learning framework under the dual model is better. +This is mainly because erroneous information can be filtered +out by each other with different capabilities, avoiding the +accumulation of errors. +4.6. Visualization of embeddings +We conduct experiments on MNIST with 20 labels +without augmentation [25]. We visualize the embeddings of +DNLL and a fully supervised learning method, respectively, +on testing data under the same settings. We use t-SNE [26] +to project the representations of the last hidden layer into +two dimensions. Figure 5 shows the results. Each point +corresponds to an item in the testing set, and different +ground-truth classes are encoded with different colors. It +demonstrates that the representations obtained from DNLL +can better identify each class in the embedding space. +4.7. Generalizability of DNLL +To verify the generalizability of DNLL, we combine the +ideology of DNLL method with the Dual Student method +and the Mean Teacher method. For Dual Student, we use +DNLL on the unstable samples discarded by the Dual +Student. As can be observed from the left side of Figure 6, +our approach can take advantage of the discarded unlabeled +data, which in turn improves the overall performance. In +addition, we combine DNLL with Mean Teacher to use all +Xu.eal: Preprint submitted to Elsevier +Page 7 of 10 + +Training process of CIFAl-10 with 1k/4k labels +90 +80 +Accurary(%) +70 +60 +ModelA with 1klabels +Model B with 1k labels +Model A with 4k labels +Model B with 4k labels +50 +0 +100 +200 +300 +400 +500 +600 +700 +EpochFigure 5: The t-SNE plot of the last hidden layer on the test +data of MNIST with 20 labels: the baseline model (left) and +our model (right). Our model can learn more discriminative +representation. +the unlabeled data together. From the right side of Figure 6, +we can see that DNLL contributes significantly to the overall +performance improvement. These experiments demonstrate +that DNLL can be used in combination with other semi- +supervised methods to jointly improve model performance. +Figure 6: The left side of the above figure shows the iteration +process of combining DNLL and Dual Student. The right side +shows the training process of combining DNLL and Mean +Teacher. +4.8. Domain Adaptation using DNLL +Figure 7: Test curves of domain adaptation from USPS to +MNIST versus the number of epochs. The DNLL avoids +overfitting and improves the result remarkably. +Domain adaptation is the closely related to semi-supervised +learning. It aims at knowledge transfer from the source +Table 7 +The execution time (seconds) of DNLL and other competitive +methods such as Mean Teacher (MT) and Dual Student +(DS). +MT +DS +DNLL +Train iteration time +0.072 +0.145 +0.143 +Inference iteration time +0.0183 +0.0189 +0.0184 +domain to the target domain. Zhan et al. [15] propose Dual +Student method to overcome the shortcomings of Mean +Teacher and demonstrate the effectiveness of a dual model +in domain adaptation tasks. In this section, we use DNLL for +adapting digital pattern recognition from USPS to MNIST. +We use USPS as the source domain and MNIST as the target +domain and show that the DNLL has advantages over the +Dual Student and Mean Teacher. +USPS and MNIST are both grayscale hand-written digi- +tal datasets, the difference is that the image size is 16x16 for +USPS and 28x28 for MNIST. The training set of USPS con- +tains 7291 images, and the training set of MNIST contains +60,000 images. And the test set for the experiments uses +the MNIST test set containing 10,000 images. We compare +DNLL with Dual Student, Mean Teacher, fully supervised +learning for the source domain and fully supervised learning +for the target domain with 7k balanced labels. Following +experiment settings in Dual Student [15], we use cubic +spline interpolation to match the resolution between the +two dataset images and employ a 3-layer CNN [15] as the +backbone, with random noise for data augmentation. +Figure 7 shows the test accuracy versus the number +of epochs. We can see that as the number of epochs in- +creases, overfitting occurs in both Mean Teacher and the +fully supervised learning for the source domain. From this +figure, we can see that DNLL not only avoids the overfitting +phenomenon but also is superior to Dual Student, and its +performance is very close to that of the target domain +supervision. +4.9. Execution time of DNLL +In this section, we conduct experiments to investigate +the execution time of DNLL. We report the average time +for each iteration during training and testing. We evaluate +the execution time with the CIFAR-10 dataset using 4000 +randomly selected training samples as labeled data. The +batch size is set to 100. The number of both labeled and +unlabeled data in a batch is 50. We compare DNLL with +Mean Teacher and Dual Student in the same settings in +terms of execution time. The experiment is performed on a +GTX 3060 GPU with Pytorch-1.10.2 software toolkit. The +system memory is 64 GB, and the CPU is Intel Core i5- +11400F. The experimental results are shown in Table 7 and +Figure 8. +From Table 7 and Figure 8, we can see that Mean +Teacher takes the shortest training time but produces the +lowest testing accuracy on the testing set. As both DNLL +Xu.eal: Preprint submitted to Elsevier +Page 8 of 10 + +100 +90 +80 +Accurary(%) +70 +60 +50 +MNIST Supervised +40 +DNLL +DS +MT +30 +USPS Supervised +20 +40 +60 +80 +100 +0 +Epoch80 +DS +DS+DNLL +75 +70 +Accurary(%) +65 +60 +55 +0 +20 +40 +60 +80 +100 +EpochMT +MT+DNLL +75 +70 +Accurary(%) +65 +60 +55 +0 +20 +40 +60 +80 +100 +Epochand Dual Student use a dual model structure, the train +time for each iteration is approximately twice that of Mean +Teacher, but both have higher accuracy. The training time +of DNLL and Dual Student are similar, but the performance +of DNLL is higher than that of Dual Student. The average +testing time of each iteration is shown in Table 7. Due to +the similarity in model architectures, the testing time of all +methods is similar. +Figure 8: The training time (seconds) for each iteration and +the testing accuracies of DNLL, Mean Teacher and Dual +Student. +5. Discussions +Our approach has several advantages over existing semi- +supervised algorithms. Firstly, in semi-supervised learning, +our approach outperforms state-of-the-art approaches on +benchmarks. Secondly, the unsupervised learning part of +our methods can easily be used as add-ons for other semi- +supervised learning methods to improve their performance. +Finally, our approach fits domain adaptation tasks as well. +We discuss the differences between DNLL and other meth- +ods that use a dual model. +Mean Teacher (MT): MT [33] has been proposed to +improve the temporal-ensembling model [20]. The frame- +work of MT consists of a student model and a teacher +model. The student model is trained by perturbing the input +data. The output of the student model is trained to be +consistent with the output of the teacher model. Different +from DNLL, in MT, the teacher model is only updated +by EMA. Thus, the predictions between the teacher model +and the student model converge to be the same relatively +fast during training. In addition, submodels in DNLL can +generate pseudo-negative labels to help each other filter +out erroneous information, while the student model and the +teacher model in MT cannot. +Dual Student (DS): DS [15] has been proposed to im- +prove MT. DS trains two submodels online simultaneously +with different initialization parameters in order to avoid +coupling between the two models in the early training +stages. To transfer reliable knowledge, submodels in DS fil- +ter unlabeled data with low prediction confidences or inter- +submodel consistency. This can lead to an underutilization +of a significant amount of unlabeled data. On the other +hand, in DNLL, most of the unlabeled data can be used +in the training process, and the transferring of erroneous +information is also reduced by using pseudo-negative labels. +Mutual Learning of Complementary Networks: This +method proposes a complementary correction network +(CCN) [34] based on Deep Mutual Learning (DML) [40]. +This method simultaneously trains three submodels, in- +cluding two submodels with the same structure and one +CCN. The CCN takes the output from one submodel and +the intermediate features extracted by another submodel as +input and is trained with labeled data only. This network +is then used to correct predictions by submodels. The +prediction is then used as pseudo-labels for one of the +submodels. The performance of the CCN can significantly +determine the quality of the pseudo label, which in turn +affects the training of the underlying submodel. On the other +hand, DNLL is trained in a much simpler and more effective +way. +Dynamic Mutual Training (DMT): DMT [9] uses a +weighted loss to control the selection of unlabeled data +items so that data items with inconsistent predictions by +submodels are filtered in the loss calculation. In addition, +this method uses a course learning strategy in which unla- +beled data are gradually used in the training process rather +than used as a whole from the beginning. Compared with +DNLL, this method also suffers from the underutilization +of unlabeled data, and it is also time-consuming to train +repetitively during course learning. +6. Conclusion +The paper analyzes submodel degeneration and under- +utilization problems suffered from traditional mutual learn- +ing approaches. To address these problems, we propose a +novel mutual learning method for semi-supervised learning. +Submodels in this approach provide each other with pseudo- +negative labels instead of traditional pseudo labels. It can +reduce error accumulation and promote unlabeled data uti- +lization and is justified theoretically and experimentally. We +also propose the error perception mechanism to help select +efficient pseudo-negative labels. This framework can also be +useful in different tasks. +Acknowledgements +This work was supported by the Natural Science Foun- +dation of Zhejiang Province (NO. LGG20F020011), Ningbo +Science and Technology Innovation Project (No. 2022Z075), +and Open Fund by Ningbo Institute of Materials Technology +& Engineering, the Chinese Academy of Sciences. +References +[1] Berthelot, D., Carlini, N., Cubuk, E.D., Kurakin, A., Sohn, K., Zhang, +H., Raffel, C., 2019a. Remixmatch: Semi-supervised learning with +Xu.eal: Preprint submitted to Elsevier +Page 9 of 10 + +93 +Mean Teacher +Dual Student +92 +DNLL +DNLL +91 +Dual Student +90 +Accurary(%) +89 +Mean Teacher +88 +87 +86 +85 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +train iteration time(seconds)distribution alignment and augmentation anchoring. arXiv preprint +arXiv:1911.09785 . +[2] Berthelot, D., Carlini, N., Goodfellow, I., Papernot, N., Oliver, A., +Raffel, C.A., 2019b. +Mixmatch: A holistic approach to semi- +supervised learning. +Advances in Neural Information Processing +Systems 32. +[3] Cascante-Bonilla, P., Tan, F., Qi, Y., Ordonez, V., 2020. Curriculum +labeling: Revisiting pseudo-labeling for semi-supervised learning. +arXiv preprint arXiv:2001.06001 . +[4] Chen, J., Yang, M., Gao, G., 2020. +Semi-supervised dual-branch +network for image classification. Knowledge-Based Systems 197, +105837. +[5] Chen, K., Yao, L., Zhang, D., Wang, X., Chang, X., Nie, F., 2019. A +semisupervised recurrent convolutional attention model for human +activity recognition. +IEEE Transactions on Neural Networks and +Learning Systems 31, 1747–1756. +[6] Cubuk, E.D., Zoph, B., Shlens, J., Le, Q.V., 2020. Randaugment: +Practical automated data augmentation with a reduced search space, +in: Proceedings of the IEEE/CVF Conference on Computer Vision +and Pattern Recognition Workshops, pp. 702–703. +[7] DeVries, T., Taylor, G.W., 2017. Improved regularization of convolu- +tional neural networks with cutout. arXiv preprint arXiv:1708.04552 +. +[8] Feng, Z., Zhou, Q., Cheng, G., Tan, X., Shi, J., Ma, L., 2020. +Semi-supervised semantic segmentation via dynamic self-training +and classbalanced curriculum. arXiv preprint arXiv:2004.08514 1, +5. +[9] Feng, Z., Zhou, Q., Gu, Q., Tan, X., Cheng, G., Lu, X., Shi, J., Ma, L., +2022. Dmt: Dynamic mutual training for semi-supervised learning. +Pattern Recognition , 108777. +[10] Gao, W., Wu, M., Lam, S.K., Xia, Q., Zou, J., 2022. Decoupled self- +supervised label augmentation for fully-supervised image classifica- +tion. Knowledge-Based Systems 235, 107605. +[11] Han, B., Yao, Q., Yu, X., Niu, G., Xu, M., Hu, W., Tsang, I., +Sugiyama, M., 2018. Co-teaching: Robust training of deep neural +networks with extremely noisy labels. Advances in Neural Informa- +tion Processing Systems 31. +[12] Hu, Z., Yang, Z., Hu, X., Nevatia, R., 2021. Simple: Similar pseudo +label exploitation for semi-supervised classification, in: Proceedings +of the IEEE/CVF Conference on Computer Vision and Pattern +Recognition, pp. 15099–15108. +[13] Iscen, A., Tolias, G., Avrithis, Y., Chum, O., 2019. Label propagation +for deep semi-supervised learning, in: Proceedings of the IEEE/CVF +Conference on Computer Vision and Pattern Recognition, pp. 5070– +5079. +[14] Ishida, T., Niu, G., Hu, W., Sugiyama, M., 2017. +Learning from +complementary labels. Advances in Neural Information Processing +Systems 30. +[15] Ke, Z., Wang, D., Yan, Q., Ren, J., Lau, R.W., 2019. Dual student: +Breaking the limits of the teacher in semi-supervised learning, in: +Proceedings of the IEEE/CVF International Conference on Computer +Vision, pp. 6728–6736. +[16] Khaki, S., Pham, H., Han, Y., Kuhl, A., Kent, W., Wang, L., +2021. Deepcorn: A semi-supervised deep learning method for high- +throughput image-based corn kernel counting and yield estimation. +Knowledge-Based Systems 218, 106874. +[17] Kim, Y., Yim, J., Yun, J., Kim, J., 2019. +Nlnl: Negative learning +for noisy labels, in: Proceedings of the IEEE/CVF International +Conference on Computer Vision, pp. 101–110. +[18] Kingma, D.P., Mohamed, S., Rezende, D.J., Welling, M., 2014. Semi- +supervised learning with deep generative models, in: Advances in +Neural Information Processing Systems, pp. 3581–3589. +[19] Krizhevsky, A., Hinton, G., et al., 2009. Learning multiple layers of +features from tiny images . +[20] Laine, S., Aila, T., 2016. Temporal ensembling for semi-supervised +learning. arXiv preprint arXiv:1610.02242 . +[21] Lee, D.H., et al., 2013. Pseudo-label: The simple and efficient semi- +supervised learning method for deep neural networks, in: Workshop +on Challenges in Representation Learning, ICML, p. 896. +[22] Li, C., Xu, T., Zhu, J., Zhang, B., 2017. Triple generative adversarial +nets. Advances in Neural Information Processing Systems 30. +[23] Li, Y., Liu, L., Tan, R.T., 2019. Certainty-driven consistency loss for +semi-supervised learning . +[24] Luo, M., Chang, X., Nie, L., Yang, Y., Hauptmann, A.G., Zheng, Q., +2017. An adaptive semisupervised feature analysis for video semantic +recognition. IEEE Transactions on Cybernetics 48, 648–660. +[25] Luo, Y., Zhu, J., Li, M., Ren, Y., Zhang, B., 2018. Smooth neighbors +on teacher graphs for semi-supervised learning, in: Proceedings of +the IEEE Conference on Computer Vision and Pattern Recognition, +pp. 8896–8905. +[26] Van der Maaten, L., Hinton, G., 2008. Visualizing data using t-sne. +Journal of Machine Learning Research 9. +[27] Miyato, T., Maeda, S.i., Koyama, M., Ishii, S., 2018. +Virtual +adversarial training: a regularization method for supervised and semi- +supervised learning. +IEEE Transactions on Pattern Analysis and +Machine Intelligence 41, 1979–1993. +[28] Netzer, Y., Wang, T., Coates, A., Bissacco, A., Wu, B., Ng, A.Y., +2011. Reading digits in natural images with unsupervised feature +learning . +[29] Qiao, S., Shen, W., Zhang, Z., Wang, B., Yuille, A., 2018. +Deep +co-training for semi-supervised image recognition, in: Proceedings of +the European Conference on Computer Vision (ECCV), pp. 135–152. +[30] Salimans, T., Goodfellow, I., Zaremba, W., Cheung, V., Radford, A., +Chen, X., 2016. Improved techniques for training gans. Advances in +Neural Information Processing Systems 29. +[31] Shi, W., Gong, Y., Ding, C., Tao, Z.M., Zheng, N., 2018. Transductive +semi-supervised deep learning using min-max features, in: Proceed- +ings of the European Conference on Computer Vision (ECCV), pp. +299–315. +[32] Sohn, K., Berthelot, D., Li, C.L., Zhang, Z., Carlini, N., Cubuk, E.D., +Kurakin, A., Zhang, H., Raffel, C., 2020. +Fixmatch: Simplifying +semi-supervised learning with consistency and confidence. +arXiv +preprint arXiv:2001.07685 . +[33] Tarvainen, A., Valpola, H., 2017. Mean teachers are better role mod- +els: Weight-averaged consistency targets improve semi-supervised +deep learning results. Advances in Neural Information Processing +Systems 30. +[34] Wu, S., Li, J., Liu, C., Yu, Z., Wong, H.S., 2019. Mutual learning +of complementary networks via residual correction for improving +semi-supervised classification, in: Proceedings of the IEEE/CVF +Conference on Computer Vision and Pattern Recognition, pp. 6500– +6509. +[35] Xie, Q., Dai, Z., Hovy, E., Luong, T., Le, Q., 2020. Unsupervised +data augmentation for consistency training. +Advances in Neural +Information Processing Systems 33, 6256–6268. +[36] Yu, B., Wu, J., Ma, J., Zhu, Z., 2019. Tangent-normal adversarial +regularization for semi-supervised learning, in: Proceedings of the +IEEE/CVF Conference on Computer Vision and Pattern Recognition, +pp. 10676–10684. +[37] Yu, E., Sun, J., Li, J., Chang, X., Han, X.H., Hauptmann, A.G., 2018. +Adaptive semi-supervised feature selection for cross-modal retrieval. +IEEE Transactions on Multimedia 21, 1276–1288. +[38] Yun, S., Han, D., Oh, S.J., Chun, S., Choe, J., Yoo, Y., 2019. Cutmix: +Regularization strategy to train strong classifiers with localizable +features, in: Proceedings of the IEEE/CVF International Conference +on Computer Vision, pp. 6023–6032. +[39] Zagoruyko, S., Komodakis, N., 2016. Wide residual networks. arXiv +preprint arXiv:1605.07146 . +[40] Zhang, Y., Xiang, T., Hospedales, T.M., Lu, H., 2018. Deep mutual +learning, in: Proceedings of the IEEE Conference on Computer +Vision and Pattern Recognition, pp. 4320–4328. +Xu.eal: Preprint submitted to Elsevier +Page 10 of 10 + diff --git a/5dE2T4oBgHgl3EQfkQdW/content/tmp_files/load_file.txt b/5dE2T4oBgHgl3EQfkQdW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eb3a2d5eb69bb124b1d10352ad0ddf730dd20580 --- /dev/null +++ b/5dE2T4oBgHgl3EQfkQdW/content/tmp_files/load_file.txt @@ -0,0 +1,856 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf,len=855 +page_content='Semi-Supervised Learning with Pseudo-Negative Labels for Image Classification Hao Xua, Hui Xiaoa, Huazheng Haoa, Li Donga, Xiaojie Qiub and Chengbin Penga,∗ aNingbo University, Ningbo, China bZhejiang Keyongtai Automation Technology Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Ningbo, China A R T I C L E I N F O Keywords: Semi-Supervised Learning Image Classification Mutual Learning A B S T R A C T Semi-supervised learning frameworks usually adopt mutual learning approaches with multiple submodels to learn from different perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' To avoid transferring erroneous pseudo labels between these submodels, a high threshold is usually used to filter out a large number of low-confidence predictions for unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' However, such filtering can not fully exploit unlabeled data with low prediction confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' To overcome this problem, in this work, we propose a mutual learning framework based on pseudo-negative labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Negative labels are those that a corresponding data item does not belong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In each iteration, one submodel generates pseudo-negative labels for each data item, and the other submodel learns from these labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The role of the two submodels exchanges after each iteration until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' By reducing the prediction probability on pseudo-negative labels, the dual model can improve its prediction ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We also propose a mechanism to select a few pseudo-negative labels to feed into submodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In the experiments, our framework achieves state-of- the-art results on several main benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Specifically, with our framework, the error rates of the 13-layer CNN model are 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='35% and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='94% for CIFAR-10 with 1000 and 4000 labels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In addition, for the non-augmented MNIST with only 20 labels, the error rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='81% by our framework, which is much smaller than that of other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Our approach also demonstrates a significant performance improvement in domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Introduction Deep learning is widely used in many areas, and the performance of deep learning models [10] heavily relies on the amount of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' However, in many real-world scenarios [16, 24, 5, 37], labeled data are often limited, and the annotation for unlabeled data can usually be expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In such cases, a semi-supervised learning framework can be adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Semi-supervised learning frameworks include generative- based models [18], graph-based models [25], consistency- based regularization [20, 33, 27, 2, 35, 15, 4], self-training with pseudo-labels [8, 32], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Among them, self-training methods can expand the training set by producing pseudo labels for unlabeled data to improve the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Nevertheless, single models are not robust to noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Inspired by DML [40], a natural idea is to simultaneously train two independently initialized models, and predictions of one submodel can be used as the learning target for the other submodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' To avoid transferring erroneous predictions to each other and alleviate parameter coupling between submodels in the early stages of training, a dual student framework [15] is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' It prevents the mutual transfer of erroneous knowledge by only passing high-confidence predictions to the other learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' However, such a mechanism can waste a large amount of unlabeled data during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' ∗Corresponding author pengchengbin@nbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='cn (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Peng) ORCID(s): Figure 1: The dual model on the left side represents general mutual learning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', the models pass strong information to each other such as information about the category with the highest prediction probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The dual model near the right side exchanges weak information between each other, indicating which category the data does not belong to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' To address these problems, we propose a new semi- supervised classification framework based on dual pseudo- negative label learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' This framework comprises two submodels, and each submodel generates pseudo-negative labels as learning targets for the other submodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Each sub- model also provides pseudo-negative labels on augmented data for self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The difference between our frame- work and general mutual learning is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We also propose a selection mechanism to identify the most representative pseudo-negative labels for the other model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The main contributions can be summarized as follows: We propose a Dual Negative Label Learning (DNLL) framework, which not only improves the utiliza- tion of unlabeled data but also significantly reduces model parameter coupling compared to general mu- tual learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='eal: Preprint submitted to Elsevier Page 1 of 10 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='03976v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content="CV] 10 Jan 2023 aM1 M1' Dog Not Cat Not Bird M2 Unlabeled Data M2'• We propose a selection mechanism to help select representative pseudo-negative labels and prove the effectiveness of this approach theoretically." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We demonstrate the effectiveness of the proposed method experimentally on different benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Data Augmentation Data augmentation plays a key role in model training, which is widely used in classification or segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Data augmentation is used to expand the training set by applying random perturbations to improve algorithm performance and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Simple augmentation methods include ran- dom flips, horizontal or vertical transitions, geometric trans- formations, changing the contrast of images, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' There are also complex operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Mixup randomly selects two images and mixes them by a random proportion to expand the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The Cutout method replaces randomly selected image pixel values with zeros while leaving the labels unchanged [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In order to maximize the effect of data augmentation, strategies combining a range of augmentation techniques are proposed, such as AutoAugmentation [38], RandAugmentation [6], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We also employ data augmen- tation methods similar to other semi-supervised learning frameworks [2, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Semi-Supervised Learning Semi-supervised learning has received a lot of attention in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The main task of semi-supervised learn- ing is to utilize labeled and unlabeled data to train algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Many approaches based on consistency regularity, Pi-Model, Temporal Ensembling Model [20], Mean Teacher [33], Dual Student [15], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Later, a series of holistic analysis methods, such as MixMatch [2], ReMixMatch [1], FixMatch [32], have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Alternatively, in DMT, inconsistency between two models has also been used to exploit the correctness of pseudo-labels [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In this work, we propose an efficient semi-supervised classification framework with dual negative label learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Learning with Noisy Labels In this case, models are trained with correctly labeled data and mistakenly labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' For example, based on the recent memory effect of a neural network, co-teaching [11] trains two models simultaneously, and each model can help the other one to filter out samples with large losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [17] proposes a negative learning method for training convolutional neural networks with noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' This method provides feedback for input images about classes to that they do not belong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In this work, we propose to use low-confidence pseudo-labels as noisy labels for further learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Learning from Complementary Labels A category corresponding to the complementary label is that a data item does not belong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Due to difficulties in col- lecting labeled data, complementary-label learning is used in fully supervised learning methods [14] and noisy-label learning methods [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Complementary labels can be gen- erated based on noisy labels [14, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In our method, com- plementary labels are generated based on model-generated pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Problem Definition In traditional multi-model frameworks, learning models under-fitted in the early stage of training are likely to pass erroneous pseudo-labels to other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Such errors can be accumulated and need to be filtered out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In addition, consistency loss on the same erroneous pseudo-labels can also lead the multi-model framework to degenerate into a self-training model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Therefore, in this section, we propose a multi-model semi-supervised learning framework to improve the utiliza- tion of unlabeled data and alleviate degeneration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We first describe the novel mutual learning framework called Dual Negative Label Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' That detailed framework is shown in Figure 2, and then proposes an effective selection mech- anism for choosing representative pseudo-negative labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In semi-supervised learning (SSL), the goal is to train a model by utilizing a small amount of labeled data and a large amount of unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Formally, we define a training set 퐷 consisting of labeled data 퐷푙={(푋푖, 푌푖 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 푖 ∈ (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 푁) } and unlabeled data 퐷푢={(푋푗 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 푗 ∈ (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 푀) }, and we use a dual model to allow each submodel learning from the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The label 푌푖 of the 푖-th data item is a one-hot vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Supervised Learning In supervised learning, labeled data are augmented by different weak augmentations for different submodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 푋(1) 푖 =퐴(1) 푤 (푋푖), (1) 푋(2) 푖 =퐴(2) 푤 (푋푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' (2) where 퐴(1) 푤 , 퐴(2) 푤 denote different weak augmentation opera- tions and 푋(1) 푖 , 푋(2) 푖 denote weakly augmented data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We use the cross-entropy (CE) function for the super- vised loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In classification tasks, the image-level CE loss is as follows: 퐻(푌 , ̂푌 ) = − ∑ 푖 푌푖푙표푔( ̂푌푖) (3) where ̂푌 is the predicted label, and 푌 is the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The supervised losses of the two submodels are as follows: 퓁(1) 푠푢푝 = 퐻(푓휃(푋(1) 푖 ), 푌푖), (4) 퓁(2) 푠푢푝 = 퐻(푓휑(푋(2) 푖 ), 푌푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' (5) where 푓휃 and 푓휑 represent the operations of two submodels respectively, and 휃 and 휑 represent parameters correspond- ing submodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='eal: Preprint submitted to Elsevier Page 2 of 10 Figure 2: Overview of the DNLL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We use a small amount of labeled data and a large amount of unlabeled data to train a dual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Each submodel within the dual model has the same structure and is initialized independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' For each labeled data, weak augmentations such as random cropping and random flipping are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' A cross-entropy function is used to calculate the supervised loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' For each unlabeled data, besides weak augmentations, strong augmentations such as color jittering are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Each submodel generates pseudo-negative labels based on predictions of weakly augmented data, and these labels are used to teach the other submodels when predicting strongly augmented data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Unsupervised Learning 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Dual pseudo-negative label Learning Most unsupervised learning parts in semi-supervised learning frameworks are realized by allowing each sub- model to learn with pseudo-positive labels from other sub- models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' To avoid model degeneration and error accumula- tion in this process, we propose a novel dual negative label learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In this approach, each submodel teaches the other that a given data item should not belong to a certain category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' It allows model diversity and can reduce transferring of erroneous information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Pseudo-negative labels, namely, the labels that a corre- sponding data item does not belong to, are generated by taking complementary labels of the predicted label by a submodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In our approach, we also select a few pseudo- negative labels as representative pseudo-negative labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' For data item 푗, its pseudo label ̂푌푗 and its representative pseudo-negative label 푌 푐 푗 are randomly selected from all the candidates with equal probability (EP) as follows: ̂푌푗 = 푓(푋푗), (6) 푌 푐 푗 ∈ 푧(푓(푋푗), 푚), (7) where 푚 is one by default, and 푧 is defined as follows: 푧(푓(푋푗), 푚) ={푣|푣 ∈ {0, 1}퐾, ∑ 푖 푣푖 = 푚, and 푣[arg max ̂푌푗] ≠ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' (8) Here, 퐾 is the number of categories, and {0, 1}퐾 represents a vector of length 퐾 with elements equal to zero or one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' To increase the convergence rate, we can allow each submodel to generate multiple representative pseudo-negative labels for each weakly augmented data item for the other submodel to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Thus, 푚 can also be positive integers larger than one and less than 퐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' By teaching each other with pseudo-negative labels only, we reduce the coupling between submodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The loss function can be written as follows: 퐿( ̂푌 , 푌 푐) = − ∑ 푖 푌 푐 푖 log(1 − ̂푌푖) (9) where ̂푌 denotes the predictions from one submodel and 푌 푐 is the representative pseudo-negative labels from the other submodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We also use weak and strong data augmentations for un- labeled data to improve the generalization ability of the dual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The weak augmentations can be random cropping, random flipping, or simply outputting the original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The strong augmentation operations can be color dithering or noise perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Usually, predictions for weakly aug- mented data by a submodel will be more accurate than that for strongly augmented data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Thus, in our framework, the predictions of weakly augmented data by one submodel are used for generating pseudo-negative labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We use these labels as learning targets for the other submodel feed by strongly augmented images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The augmentation process can be written as follows: 푋(푤) 푗 =퐴푤(푋푗), (10) 푋(푠) 푗 =퐴푠(푋푗), (11) where 퐴푤 and 퐴푠 denote the weak and strong augmentation operations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 푋(푤) 푗 and 푋(푠) 푗 denote the weakly Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='eal: Preprint submitted to Elsevier Page 3 of 10 Weak Augmentations Labeled Prediction Supervised Loss Labeled Data Unlabeled Prediction Pseudo Label Pseudo-Negative Labe Net A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Unsupervised Loss GT Unlabeled Data NetB Pseudo Label Pseudo-Negative Labe Supervised Loss Strong Augmentations Labeled Predictionand strongly augmented data items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Consequently, we have 푌 푐1 ∈ 푧(푓휃(푋(푤) 푗 ), 푚), (12) 푌 푐2 ∈ 푧(푓휑(푋(푤) 푗 ), 푚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' (13) Therefore, the loss of learning between submodels is as follows: 퓁(1) 푐푟표푠푠 = 퐿(푓휃(푋(푠) 푗 ), 푌 푐2), (14) 퓁(2) 푐푟표푠푠 = 퐿(푓휑(푋(푠) 푗 ), 푌 푐1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' (15) To further utilize the augmented data, we also developed a self-learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In this approach, the generated pseudo-negative labels with weakly augmented data are also used by the same submodel to feed strong augmented data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The loss function can be written as follows: 퓁(1) 푠푒푙푓 = 퐿(푓휃(푋(푠) 푗 ), 푌 푐1), (16) 퓁(2) 푠푒푙푓 = 퐿(푓휑(푋(푠) 푗 ), 푌 푐2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' (17) The unsupervised loss of the dual model is a combina- tion of the previous loss functions: 퓁(1) 푢푛푠푢푝 = 퓁(1) 푐표푟푠푠 + 퓁(1) 푠푒푙푓, (18) 퓁(2) 푢푛푠푢푝 = 퓁(2) 푐표푟푠푠 + 퓁(2) 푠푒푙푓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' (19) The final total loss of the dual model in the DNLL is a combination of the supervised loss and the unsupervised one, as follows: 퓁(1) = 퓁(1) 푠푢푝 + 휆퓁(1) 푢푛푠푢푝, (20) 퓁(2) = 퓁(2) 푠푢푝 + 휆퓁(2) 푢푛푠푢푝, (21) where 휆 is a hyperparameter to balance the supervised loss item and the unsupervised loss item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The complete algorithm is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' From this pseudo code, we can see that the running time is proportional to the size of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' If the size of unlabeled data, 푀, is much larger than that of the labeled data, 푁, which usually happens in semi-supervised learning, the running time is approximately proportional to the size of the unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Thus, the time complexity is 푂(푀).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Error Perception Mechanism for Selecting Pseudo-Negative Labels In the above section, for an unlabeled data item, a rep- resentative pseudo-negative label is randomly selected from all the candidates with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' To incorporate the performance of each submodel in different categories, we propose an Error Perception Mechanism (EPM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In this approach, for a given data item, if a submodel is prone to misclassify it into the other category, the pseudo- negative label generated by the other submodel should include that misclassified category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Therefore, we compute the probability of misclassification for each category of each Figure 3: The generating process of pseudo-negative labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' For an unlabeled data item, a submodel makes a prediction to generate a pseudo label (3 in this example) and then randomly selects two pseudo-negative labels according to 푅 of the other submodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' submodel using labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Formally, for a submodel, we define a vector 푃푟푘 for category 푘 with its 푖-th element defined as follows: 푃푟푘[푖] = {∑푁푘 푗=1 푝푖푗, 푖 ≠ 푘 0 푖 = 푘 (22) where 푁푘 denotes the total number of data with category 푘 being misclassified into category 푖, and 푝푖푗 represents the confidence that the 푗-th misclassified sample belongs to the 푖-th category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We may also use EMA to update 푃푟푘 for stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' It is then normalized with a softmax function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 푅푘 = 푆표푓푡푚푎푥(푃푟푘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' (23) We use superscripts to represent submodels, so 푅(1) 푘 and 푅(2) 푘 are misclassification probabilities for the first and the second submodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' An example of the 푅푘-based pseudo-negative label generation process is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Therefore, when computing 퓁2 푐푟표푠푠, we sample 푌 푐1 from 푧(푓휃(푋(푤) 푗 ), 푚) such that the probability that 푌 푐2 푗 [푘] = 1 is proportional to 푅(2) 푘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' A similar approach applies when computing 퓁1 푐푟표푠푠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Theoretical Analysis First, we demonstrate that in the mutual learning frame- work based on a dual model, passing pseudo-negative labels between submodels is less likely to have error accumulation than that of passing pseudo labels, especially at the early stages of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The error rate (ER) for transferring pseudo- negative labels from one submodel to the other is expected to be 푚 퐾−1 of the ER when transferring pseudo labels, where 푚 is the number of selected pseudo-negative labels and 퐾 is the number of categories for each data item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Without loss of generality, we define that the pre- diction accuracy of one submodel is 푞 for unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='eal: Preprint submitted to Elsevier Page 4 of 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='11 2 3 4 5 6 7 8 0 9 R for category 3 of Submodel B 2 Pseudo label by SubmodeI A Pseudo-negative labels for Submodel BAlgorithm 1 Pseudo code for the training process of DNLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Input:The labeled dataset 퐷푙={(푋푖, 푌푖 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 푖 ∈ (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 푁) } and the unlabeled dataset 퐷푢={(푋푗 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 푗 ∈ (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 푀) }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The two submodels are 푓휃 and 푓휑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 1: for each epoch do 2: for each batch do 3: (휒푙,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 푌푙) ∶ select a batch of data from 퐷푙 4: (휒푢) ∶ select a batch of data from 퐷푢 5: 휒(1) 푙 = 퐴(1) 푤 (휒푙) 6: 휒(2) 푙 = 퐴(2) 푤 (휒푙) 7: 휒(푤) 푢 = 퐴푤(휒푢) 8: 휒(푠) 푢 = 퐴푠(휒푢) 9: 퓁(1) 푠푢푝 = 퐻(푓휃(휒(1) 푙 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 푌푙) 10: 퓁(2) 푠푢푝 = 퐻(푓휑(휒(2) 푙 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 푌푙) 11: 푌 푐1 ∈ 푧(푓휃(휒(푤) 푢 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 푚) 12: 푌 푐2 ∈ 푧(푓휑(휒(푤) 푢 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 푚) 13: 퓁(1) 푢푛푠푢푝 = 퐿(푓휃(휒(푠) 푢 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 푌 푐2) 14: 퓁(2) 푢푛푠푢푝 = 퐿(푓휑(휒(푠) 푢 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 푌 푐1) 15: 푓휃 = arg min푓휃(퓁(1) 푠푢푝 + 휆퓁(1) 푢푛푠푢푝) 16: 푓휑 = arg min푓휑(퓁(2) 푠푢푝 + 휆퓁(2) 푢푛푠푢푝) 17: end for 18: end for return 푓휃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 푓휑 Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' when transferring pseudo labels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' the probability that that submodel provides correct learning targets to the other is 푞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' When transferring 푚 pseudo-negative labels, if the sub- model predicts correctly, it transfers correct negative labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' If the submodel predicts mistakenly, the chance of providing correct negative labels is 퐶푚 퐾−2 퐶푚 퐾−1 , (24) where 퐶푚 퐾−1 denotes the total number of combinations of selecting 푚 pseudo-negative labels from all the 퐾 − 1 pseudo-negative labels, and 퐶푚 퐾−2 denotes the number of combinations of selecting 푚 pseudo-negative labels from 퐾 − 2 truly negative labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 퐾 − 2 is obtained by taking all the 퐾 categories except two categories corresponding to one pseudo label and one ground-truth label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Therefore, the probability of providing the correct learning target is 푞 + (1 − 푞) 퐶푚 퐾−2 퐶푚 퐾−1 = 1 − (1 − 푞)푚 퐾 − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' (25) Therefore, the error rate of transferring pseudo-negative labels is 1 − (1 − (1 − 푞)푚 퐾 − 1 ) = (1 − 푞) 푚 퐾 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' (26) As the error rate of transferring pseudo labels is 1 − 푞, the error rate of transferring pseudo-negative labels is 푚 퐾−1 of that of transferring pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Therefore, transferring pseudo labels can provide a better learning target, and a smaller 푚 and a larger 퐾 can further reduce the error accumulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' For two submodels with the same structure, when they are converged to be the same, they can no longer be used for semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We need to avoid such scenarios, especially in the early training stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In the unsupervised learning part, we demonstrate that when transferring knowl- edge with pseudo-negative labels, it is unlikely to have two submodels degenerate into the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' When transferring representative pseudo- negative labels randomly, the probability that two submod- els are optimized for different objectives is 1 − √ 2휋푚 푒퐾 ( 푚 퐾 )푚 approximately, where 푚 is the number of representative pseudo-negative labels and 퐾 is the number of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Without loss of generality, we assume that two sub- models produce the same prediction with probability 푞 and when they produce the same pseudo labels, the probability that the two submodels can produce the same representative pseudo-negative labels is 1 퐶푚 퐾−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' (27) Similarly, the probability that two submodels produce dif- ferent predictions is 1 − 푞, and when they produce different predictions, the probability that they produce the same pseudo labels is 1 퐶푚 퐾−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' (28) Thus, the probability that the two submodels transfer- ring the same representative pseudo-negative label is 푞 퐶푚 퐾−1 + 1 − 푞 퐶푚 퐾−2 (29) =푚!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' (퐾 − 2 − 푚)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' (퐾 − 1 − 푞푚) (퐾 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' (30) ≈(퐾 − 1 − 푞푚)× √ 2휋푚( 푚 푒 )푚√ 2휋(퐾 − 2 − 푚)( 퐾−2−푚 푒 )퐾−2−푚 √ 2휋(퐾 − 1)( 퐾−1 푒 )퐾−1 (31) ≈ √ 2휋푚 푒퐾 ( 푚 퐾 )푚 (32) where the approximation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' (31) is obtained by the Stir- ling’s approximation, and that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' (32) is by considering 퐾 >> 푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Experiments In this section, we first introduce benchmarks used in experiments and briefly describe the details of the ex- periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Then we compare DNLL with other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='eal: Preprint submitted to Elsevier Page 5 of 10 Finally, we evaluate the efficiency of DNLL from different perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Benchmark datasets In the classification task, we use the public bench- mark datasets CIFAR-10 [19], SVHN [28], and MNIST as many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The CIFAR-10 dataset includes 50000 training images and 10000 test images, and the total number of categories is ten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We randomly select 500 images for each category as the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The total number of categories of SVHN Dataset is ten, in which the training set contains 73257 images and the test set contains 26032 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We also randomly select 500 images for each category as the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The MNIST dataset includes 60000 training images and 10000 test images, and the total number of categories also is ten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We randomly select 50 images for each category as the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Implementation Details Our approach is implemented on Pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' For the train- ing stage, the following configurations are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The learn- ing rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='03, and the weight decay is 5 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The momentum is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We use the cosine annealing technique with batch size 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We report performances on the test set averaged from three runnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' For dual models, we use WideResNet-28-2 (WRN-28-2)[39] and 13-layer CNN as other approaches [2, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We use data augmentation techniques in our experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The data augmentation operation for each data set is performed exactly following its corresponding literature for fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Specifically, for the MNIST dataset, we do not change the input data [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' For the CIFAR-10 dataset, when using the 13-layer CNN as the model [15], we make the original image as a weakly augmented version and the noise-processed image as a strongly augmented version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' When using WideResNet-28-2 as the model [9], the weak augmentation operations we used include random cropping and random flipping, and the strong augmentation operation is random color jittering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' For the SVHN dataset [20], we only use the horizontal translation as the strong augmenta- tion operation and the original image as the weakly aug- mented version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Comparison on Benchmarks In experiments with the CIFAR-10 dataset, we randomly select 1K, 2K, and 4K data items, respectively, as labeled data and the rest as unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We compare our method with others: Π model, Tempo- ral Ensembling [20], VAT[27] and Mean Teacher [33] based on consistency regularization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Π+STNG [25], LP+SSDL and LP-SSDL-MT [13] based on graph methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Filtering CCL, Temperature CCL [23], TSSDL, TSSDL-MT [31] and TNAR-VAE [36] based on mean-teacher frameworks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Cur- riculum Labeling (CL) [3] based self-training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' MixMatch [2] based on strong hybrid method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We also compare our ap- proach with others based on dual models: Deep Co-Training (DCT) [29], Dual student(DS) [15], Mutual Learning of Complementary Networks(CCN) [34] and Dynamic Mutual Table 1 Accuracy on the Test Set of CIFAR-10 with the 13-layer CNN as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Method 1K 2K 4K Π model† 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='35 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='43 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='64 Temporal ensembling† 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='69 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='36 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='84 Mean Teacher 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='78 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='67 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='59 Π+SNTG† 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='77 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='35 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='64 LP-SSDL† 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='98 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='34 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='31 LP-SSDL-MT† 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='07 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='78 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='39 Filtering CCL† 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='78 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='67 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='59 Temperature CCL† 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='01 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='43 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='37 TSSDL† 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='87 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='35 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='10 TSSDL-MT† 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='59 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='46 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='70 TNAR-VAE† 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='15 DCT 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='97 Dual Student 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='83 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='28 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='11 CCN 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='95 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='63 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='2 DNLL (Ours) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='87 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='65 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='06 Table 2 Accuracy on the Test Set of CIFAR-10 with the WRN-28-2 as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Method 1K 4K VAT† 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='36 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='95 Mean Teacher† 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='68 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='64 CL 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='61 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='02 MixMatch 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='25 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='76 DMT 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='51 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='21 DNLL (Ours) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='03 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='29 Training (DMT) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The symbol † indicates that the results are reported in [4] and [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The symbol ’-’ indicates that the corresponding results have not been reported in this literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' From Table 1 and Table 2, we can find that our method performs relatively well with 1k labels and outperforms all the other methods in other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' From Table 1, the accuracy of our approach ranges between 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='87% and 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='06%, which outperforms most of the other methods using the dual model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', DCT, Dual Student, and CCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' From Table 2, the MixMatch is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='53% lower than our approach at the accuracy with 4K labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The DMT is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='41% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='08% lower than our approach at the accuracy with 1K and 4K labels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Figure 4 demonstrates the performance of DNLL during the training process on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' As the epoch number increases, the training accuracy increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In the SVHN dataset, 1K and 4K items are also ran- domly selected as labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We compare our method with others as follows: Π model [20], Pseudo-Labeling [21], VAT [27] and Mean Teacher [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The symbol † indicates that the results are reported in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' All the approaches use WideResNet-28-2 as the backbone model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' As shown in Table 3, our method outperforms all the other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='eal: Preprint submitted to Elsevier Page 6 of 10 Table 3 Accuracy on the Test Set of SVHN with the WRN-28-2 as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Method 1K 4K Pseudo-Labeling 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='06 Π model 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='46 VAT† 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='02 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='80 Mean Teacher† 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='25 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='61 DNLL (Ours) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='41 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='84 Table 4 Accuracy on the Test Set of MNIST with the 13-layer CNN as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Method 20 50 100 ImprovedGAN† 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='23 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='79 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='07 Triple GAN† 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='19 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='44 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='09 Π model† 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='68 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='98 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='11 Π + SNTG† 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='64 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='06 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='34 DNLL (Ours) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='19 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='32 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='54 Figure 4: Performance of DNLL on the test set during training with the CIFAR-10 dataset of 1000 and 4000 labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' For the MNIST dataset, 20, 50, and 100 data items are randomly selected as labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We compare the DNLL with other semi-supervised methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', ImprovedGAN [30], Triple GAN [22], Π model [20] and Π + STNG [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The symbol † indicates that the results are reported in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' All the above methods use the 13-layer CNN as the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' As shown in Table 4, the DNLL outperforms the other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Sensitivity Analysis We conduct a sensitivity analysis on the CIFAR-10 dataset with 4K labeled data items to analyze the relation- ship between representative pseudo-negative label number 푚 and the accuracy of the model under different selection mechanisms that were introduced in the methodology sec- tion: Equal Probability (EP) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Error Perception Mech- anism (EPM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' As the number of representative pseudo- negative labels tends to be less than half of the total number Table 5 Accuracy under different choices of 푚 and different selection mechanisms for representative pseudo-negative labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Selection Method 푚 = 1 푚 = 2 푚 = 3 푚 = 4 EP 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='9 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='76 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='01 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='78 EPM 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='12 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='84 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='29 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='77 Table 6 Comparison of the performance of mutual learning (ML) and self-learning (SL) with DNLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Method 4k labels SL w/o EPM 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='78 SL 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='03 ML w/o EPM 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='01 ML 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='29 of categories, here we compare with 푚 ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' From Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 5, we can find that generally, the error perception mechanism performs better than selecting with equal probability, and moderately increasing 푚 is helpful to increase the perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' When 푚 is too large, for example, close to half of the total number of categories, pseudo labels are likely to be selected, and the performance can be undermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Comparison with variants of DNLL In this part, we demonstrate that using mutual learning framework in DNLL is more efficient compared to a self- learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We compare the performance of these two learning frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We can see from Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 6 that the mutual learning framework under the dual model is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' This is mainly because erroneous information can be filtered out by each other with different capabilities, avoiding the accumulation of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Visualization of embeddings We conduct experiments on MNIST with 20 labels without augmentation [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We visualize the embeddings of DNLL and a fully supervised learning method, respectively, on testing data under the same settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We use t-SNE [26] to project the representations of the last hidden layer into two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Figure 5 shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Each point corresponds to an item in the testing set, and different ground-truth classes are encoded with different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' It demonstrates that the representations obtained from DNLL can better identify each class in the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Generalizability of DNLL To verify the generalizability of DNLL, we combine the ideology of DNLL method with the Dual Student method and the Mean Teacher method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' For Dual Student, we use DNLL on the unstable samples discarded by the Dual Student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' As can be observed from the left side of Figure 6, our approach can take advantage of the discarded unlabeled data, which in turn improves the overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In addition, we combine DNLL with Mean Teacher to use all Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='eal: Preprint submitted to Elsevier Page 7 of 10 Training process of CIFAl-10 with 1k/4k labels 90 80 Accurary(%) 70 60 ModelA with 1klabels Model B with 1k labels Model A with 4k labels Model B with 4k labels 50 0 100 200 300 400 500 600 700 EpochFigure 5: The t-SNE plot of the last hidden layer on the test data of MNIST with 20 labels: the baseline model (left) and our model (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Our model can learn more discriminative representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' the unlabeled data together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' From the right side of Figure 6, we can see that DNLL contributes significantly to the overall performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' These experiments demonstrate that DNLL can be used in combination with other semi- supervised methods to jointly improve model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Figure 6: The left side of the above figure shows the iteration process of combining DNLL and Dual Student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The right side shows the training process of combining DNLL and Mean Teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Domain Adaptation using DNLL Figure 7: Test curves of domain adaptation from USPS to MNIST versus the number of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The DNLL avoids overfitting and improves the result remarkably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Domain adaptation is the closely related to semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' It aims at knowledge transfer from the source Table 7 The execution time (seconds) of DNLL and other competitive methods such as Mean Teacher (MT) and Dual Student (DS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' MT DS DNLL Train iteration time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='143 Inference iteration time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='0183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='0189 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='0184 domain to the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Zhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [15] propose Dual Student method to overcome the shortcomings of Mean Teacher and demonstrate the effectiveness of a dual model in domain adaptation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In this section, we use DNLL for adapting digital pattern recognition from USPS to MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We use USPS as the source domain and MNIST as the target domain and show that the DNLL has advantages over the Dual Student and Mean Teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' USPS and MNIST are both grayscale hand-written digi- tal datasets, the difference is that the image size is 16x16 for USPS and 28x28 for MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The training set of USPS con- tains 7291 images, and the training set of MNIST contains 60,000 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' And the test set for the experiments uses the MNIST test set containing 10,000 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We compare DNLL with Dual Student, Mean Teacher, fully supervised learning for the source domain and fully supervised learning for the target domain with 7k balanced labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Following experiment settings in Dual Student [15], we use cubic spline interpolation to match the resolution between the two dataset images and employ a 3-layer CNN [15] as the backbone, with random noise for data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Figure 7 shows the test accuracy versus the number of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We can see that as the number of epochs in- creases, overfitting occurs in both Mean Teacher and the fully supervised learning for the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' From this figure, we can see that DNLL not only avoids the overfitting phenomenon but also is superior to Dual Student, and its performance is very close to that of the target domain supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Execution time of DNLL In this section, we conduct experiments to investigate the execution time of DNLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We report the average time for each iteration during training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We evaluate the execution time with the CIFAR-10 dataset using 4000 randomly selected training samples as labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The batch size is set to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The number of both labeled and unlabeled data in a batch is 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We compare DNLL with Mean Teacher and Dual Student in the same settings in terms of execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The experiment is performed on a GTX 3060 GPU with Pytorch-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='2 software toolkit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The system memory is 64 GB, and the CPU is Intel Core i5- 11400F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The experimental results are shown in Table 7 and Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' From Table 7 and Figure 8, we can see that Mean Teacher takes the shortest training time but produces the lowest testing accuracy on the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' As both DNLL Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='eal: Preprint submitted to Elsevier Page 8 of 10 100 90 80 Accurary(%) 70 60 50 MNIST Supervised 40 DNLL DS MT 30 USPS Supervised 20 40 60 80 100 0 Epoch80 DS DS+DNLL 75 70 Accurary(%) 65 60 55 0 20 40 60 80 100 EpochMT MT+DNLL 75 70 Accurary(%) 65 60 55 0 20 40 60 80 100 Epochand Dual Student use a dual model structure, the train time for each iteration is approximately twice that of Mean Teacher, but both have higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The training time of DNLL and Dual Student are similar, but the performance of DNLL is higher than that of Dual Student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The average testing time of each iteration is shown in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Due to the similarity in model architectures, the testing time of all methods is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Figure 8: The training time (seconds) for each iteration and the testing accuracies of DNLL, Mean Teacher and Dual Student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Discussions Our approach has several advantages over existing semi- supervised algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Firstly, in semi-supervised learning, our approach outperforms state-of-the-art approaches on benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Secondly, the unsupervised learning part of our methods can easily be used as add-ons for other semi- supervised learning methods to improve their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Finally, our approach fits domain adaptation tasks as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We discuss the differences between DNLL and other meth- ods that use a dual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Mean Teacher (MT): MT [33] has been proposed to improve the temporal-ensembling model [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The frame- work of MT consists of a student model and a teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The student model is trained by perturbing the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The output of the student model is trained to be consistent with the output of the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Different from DNLL, in MT, the teacher model is only updated by EMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Thus, the predictions between the teacher model and the student model converge to be the same relatively fast during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In addition, submodels in DNLL can generate pseudo-negative labels to help each other filter out erroneous information, while the student model and the teacher model in MT cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Dual Student (DS): DS [15] has been proposed to im- prove MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' DS trains two submodels online simultaneously with different initialization parameters in order to avoid coupling between the two models in the early training stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' To transfer reliable knowledge, submodels in DS fil- ter unlabeled data with low prediction confidences or inter- submodel consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' This can lead to an underutilization of a significant amount of unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' On the other hand, in DNLL, most of the unlabeled data can be used in the training process, and the transferring of erroneous information is also reduced by using pseudo-negative labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Mutual Learning of Complementary Networks: This method proposes a complementary correction network (CCN) [34] based on Deep Mutual Learning (DML) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' This method simultaneously trains three submodels, in- cluding two submodels with the same structure and one CCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The CCN takes the output from one submodel and the intermediate features extracted by another submodel as input and is trained with labeled data only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' This network is then used to correct predictions by submodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The prediction is then used as pseudo-labels for one of the submodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' The performance of the CCN can significantly determine the quality of the pseudo label, which in turn affects the training of the underlying submodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' On the other hand, DNLL is trained in a much simpler and more effective way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Dynamic Mutual Training (DMT): DMT [9] uses a weighted loss to control the selection of unlabeled data items so that data items with inconsistent predictions by submodels are filtered in the loss calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' In addition, this method uses a course learning strategy in which unla- beled data are gradually used in the training process rather than used as a whole from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Compared with DNLL, this method also suffers from the underutilization of unlabeled data, and it is also time-consuming to train repetitively during course learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Conclusion The paper analyzes submodel degeneration and under- utilization problems suffered from traditional mutual learn- ing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' To address these problems, we propose a novel mutual learning method for semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Submodels in this approach provide each other with pseudo- negative labels instead of traditional pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' It can reduce error accumulation and promote unlabeled data uti- lization and is justified theoretically and experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' We also propose the error perception mechanism to help select efficient pseudo-negative labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' This framework can also be useful in different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Acknowledgements This work was supported by the Natural Science Foun- dation of Zhejiang Province (NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' LGG20F020011), Ningbo Science and Technology Innovation Project (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 2022Z075), and Open Fund by Ningbo Institute of Materials Technology & Engineering, the Chinese Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' References [1] Berthelot, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Carlini, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Cubuk, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Kurakin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Sohn, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Raffel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Remixmatch: Semi-supervised learning with Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='eal: Preprint submitted to Elsevier Page 9 of 10 93 Mean Teacher Dual Student 92 DNLL DNLL 91 Dual Student 90 Accurary(%) 89 Mean Teacher 88 87 86 85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='16 train iteration time(seconds)distribution alignment and augmentation anchoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' arXiv preprint arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='09785 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [2] Berthelot, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Carlini, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Goodfellow, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Papernot, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Oliver, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Raffel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Mixmatch: A holistic approach to semi- supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [3] Cascante-Bonilla, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Tan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Qi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Ordonez, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Curriculum labeling: Revisiting pseudo-labeling for semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' arXiv preprint arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='06001 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [4] Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Gao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Semi-supervised dual-branch network for image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Knowledge-Based Systems 197, 105837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [5] Chen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Yao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Chang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Nie, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' A semisupervised recurrent convolutional attention model for human activity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' IEEE Transactions on Neural Networks and Learning Systems 31, 1747–1756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [6] Cubuk, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Zoph, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Shlens, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Le, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Randaugment: Practical automated data augmentation with a reduced search space, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 702–703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [7] DeVries, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Taylor, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Improved regularization of convolu- tional neural networks with cutout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' arXiv preprint arXiv:1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='04552 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [8] Feng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Zhou, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Cheng, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Tan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Ma, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Semi-supervised semantic segmentation via dynamic self-training and classbalanced curriculum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' arXiv preprint arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='08514 1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [9] Feng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Zhou, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Gu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Tan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Cheng, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Lu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Ma, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Dmt: Dynamic mutual training for semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Pattern Recognition , 108777.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [10] Gao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Wu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Lam, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Xia, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Zou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Decoupled self- supervised label augmentation for fully-supervised image classifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Knowledge-Based Systems 235, 107605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [11] Han, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Yao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Yu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Niu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Xu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Hu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Tsang, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Sugiyama, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Co-teaching: Robust training of deep neural networks with extremely noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Advances in Neural Informa- tion Processing Systems 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [12] Hu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Hu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Nevatia, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Simple: Similar pseudo label exploitation for semi-supervised classification, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 15099–15108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [13] Iscen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Tolias, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Avrithis, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Chum, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Label propagation for deep semi-supervised learning, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 5070– 5079.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [14] Ishida, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Niu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Hu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Sugiyama, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Learning from complementary labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [15] Ke, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Yan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Ren, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Lau, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Dual student: Breaking the limits of the teacher in semi-supervised learning, in: Proceedings of the IEEE/CVF International Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 6728–6736.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [16] Khaki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Pham, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Han, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Kuhl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Kent, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Deepcorn: A semi-supervised deep learning method for high- throughput image-based corn kernel counting and yield estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Knowledge-Based Systems 218, 106874.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [17] Kim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Yim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Yun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Nlnl: Negative learning for noisy labels, in: Proceedings of the IEEE/CVF International Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 101–110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [18] Kingma, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Mohamed, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Rezende, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Welling, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Semi- supervised learning with deep generative models, in: Advances in Neural Information Processing Systems, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 3581–3589.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [19] Krizhevsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Hinton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Learning multiple layers of features from tiny images .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [20] Laine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Aila, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Temporal ensembling for semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' arXiv preprint arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='02242 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [21] Lee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Pseudo-label: The simple and efficient semi- supervised learning method for deep neural networks, in: Workshop on Challenges in Representation Learning, ICML, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [22] Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Xu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Triple generative adversarial nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [23] Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Tan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Certainty-driven consistency loss for semi-supervised learning .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [24] Luo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Chang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Nie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Hauptmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Zheng, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' An adaptive semisupervised feature analysis for video semantic recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' IEEE Transactions on Cybernetics 48, 648–660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [25] Luo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Ren, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Smooth neighbors on teacher graphs for semi-supervised learning, in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 8896–8905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [26] Van der Maaten, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Hinton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Visualizing data using t-sne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Journal of Machine Learning Research 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [27] Miyato, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Maeda, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Koyama, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Ishii, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Virtual adversarial training: a regularization method for supervised and semi- supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' IEEE Transactions on Pattern Analysis and Machine Intelligence 41, 1979–1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [28] Netzer, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Coates, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Bissacco, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Wu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Ng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Reading digits in natural images with unsupervised feature learning .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [29] Qiao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Shen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Yuille, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Deep co-training for semi-supervised image recognition, in: Proceedings of the European Conference on Computer Vision (ECCV), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 135–152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [30] Salimans, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Goodfellow, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Zaremba, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Cheung, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Radford, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Improved techniques for training gans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [31] Shi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Gong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Ding, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Tao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Zheng, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Transductive semi-supervised deep learning using min-max features, in: Proceed- ings of the European Conference on Computer Vision (ECCV), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 299–315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [32] Sohn, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Berthelot, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Carlini, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Cubuk, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Kurakin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Raffel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Fixmatch: Simplifying semi-supervised learning with consistency and confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' arXiv preprint arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='07685 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [33] Tarvainen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Valpola, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Mean teachers are better role mod- els: Weight-averaged consistency targets improve semi-supervised deep learning results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [34] Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Yu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Wong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Mutual learning of complementary networks via residual correction for improving semi-supervised classification, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 6500– 6509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [35] Xie, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Dai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Hovy, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Luong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Le, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Unsupervised data augmentation for consistency training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 33, 6256–6268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [36] Yu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Ma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Zhu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Tangent-normal adversarial regularization for semi-supervised learning, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 10676–10684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [37] Yu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Sun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Chang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Han, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Hauptmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Adaptive semi-supervised feature selection for cross-modal retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' IEEE Transactions on Multimedia 21, 1276–1288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [38] Yun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Han, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Oh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Chun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Choe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Yoo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Cutmix: Regularization strategy to train strong classifiers with localizable features, in: Proceedings of the IEEE/CVF International Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 6023–6032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [39] Zagoruyko, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Komodakis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Wide residual networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' arXiv preprint arXiv:1605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='07146 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' [40] Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Xiang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Hospedales, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', Lu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Deep mutual learning, in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' 4320–4328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content=' Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} +page_content='eal: Preprint submitted to Elsevier Page 10 of 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfkQdW/content/2301.03976v1.pdf'} diff --git a/5tE0T4oBgHgl3EQfvwEb/content/2301.02621v1.pdf b/5tE0T4oBgHgl3EQfvwEb/content/2301.02621v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c6328e092be2f235d4e7406205a7964fc2cd4e3f --- /dev/null +++ b/5tE0T4oBgHgl3EQfvwEb/content/2301.02621v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7e64857abc9d52bc4de3465d0058814f07487ca68f3441b91fffab3fba385363 +size 754967 diff --git a/5tE0T4oBgHgl3EQfvwEb/vector_store/index.pkl b/5tE0T4oBgHgl3EQfvwEb/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..8fb2ff310af72d9e3935a02c9438a62e859ed16f --- /dev/null +++ b/5tE0T4oBgHgl3EQfvwEb/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e96ed696f5c2ad85a420d22e7d952f22f3347b9695362b1a7a1ee0bfd6ce1982 +size 130597 diff --git a/5tE4T4oBgHgl3EQf1g0M/content/2301.05290v1.pdf b/5tE4T4oBgHgl3EQf1g0M/content/2301.05290v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..58cb2ee3393b17130043f7e71e2259b2d9ee0679 --- /dev/null +++ b/5tE4T4oBgHgl3EQf1g0M/content/2301.05290v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:da7dfe6e54aaf9d9c527e16c198355c9a745ea054f894121d175118db06a47ef +size 851754 diff --git a/5tE4T4oBgHgl3EQf1g0M/vector_store/index.faiss b/5tE4T4oBgHgl3EQf1g0M/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..dd69d1e79896a1feced0bcc7ec98bfe11c9ab127 --- /dev/null +++ b/5tE4T4oBgHgl3EQf1g0M/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a01098725d7069a0d810cf93acd1eb9c4ac7c19684c940d14ba2c5b512341497 +size 3932205 diff --git a/5tE4T4oBgHgl3EQf1g0M/vector_store/index.pkl b/5tE4T4oBgHgl3EQf1g0M/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..a083dc3ff404fd8256e73b7f0ba56d82019a164b --- /dev/null +++ b/5tE4T4oBgHgl3EQf1g0M/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4d773d602659a776f67e206f16430363bed00556bc2d1146102d122a675daaf8 +size 137663 diff --git a/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf b/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e663ff9cc6e9c21694f4c4db2f2d689d7f9945cf --- /dev/null +++ b/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6010a0279f2813a9f503cf8a06c4579efd6ade470fa82352cfe70986eb1e43cb +size 640082 diff --git a/6dE1T4oBgHgl3EQfTQOX/vector_store/index.faiss b/6dE1T4oBgHgl3EQfTQOX/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..8d653a9212b38bcf4a85b25214dcb284b7b56fd8 --- /dev/null +++ b/6dE1T4oBgHgl3EQfTQOX/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ccbe2e9a05eb43fbcfec9e1ff5d03d1c520796a446fea7cebdf1ea2b8f6678ea +size 1638445 diff --git a/6dE1T4oBgHgl3EQfTQOX/vector_store/index.pkl b/6dE1T4oBgHgl3EQfTQOX/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..108ef2bb54e38786ae8211ec659304c9a4b83ad7 --- /dev/null +++ b/6dE1T4oBgHgl3EQfTQOX/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:884231cb1e140f3edd3bf3488bc64b2e4842d67180061acdbdb0e41d448546a9 +size 54194 diff --git a/7tE4T4oBgHgl3EQfCgsd/content/tmp_files/2301.04860v1.pdf.txt b/7tE4T4oBgHgl3EQfCgsd/content/tmp_files/2301.04860v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..13966582c2573cdb1a32b7c390a1c029c0b2e87a --- /dev/null +++ b/7tE4T4oBgHgl3EQfCgsd/content/tmp_files/2301.04860v1.pdf.txt @@ -0,0 +1,1059 @@ +JOURNAL OF LATEX CLASS FILES, 2019 +1 +Edge Preserving Implicit Surface +Representation of Point Clouds +Xiaogang Wang, Yuhang Cheng, Liang Wang, Jiangbo Lu, Senior Member, IEEE , Kai Xu, Senior +Member, IEEE , Guoqiang Xiao +Abstract— Learning implicit surface directly from raw data recently has become a very attractive representation method for 3D +reconstruction tasks due to its excellent performance. However, as the raw data quality deteriorates, the implicit functions often lead +to unsatisfactory reconstruction results. To this end, we propose a novel edge-preserving implicit surface reconstruction method, which +mainly consists of a differentiable Laplican regularizer and a dynamic edge sampling strategy. Among them, the differential Laplican +regularizer can effectively alleviate the implicit surface unsmoothness caused by the point cloud quality deteriorates; Meanwhile, in order +to reduce the excessive smoothing at the edge regions of implicit suface, we proposed a dynamic edge extract strategy for sampling near +the sharp edge of point cloud, which can effectively avoid the Laplacian regularizer from smoothing all regions. Finally, we combine them +with a simple regularization term for robust implicit surface reconstruction. Compared with the state-of-the-art methods, experimental +results show that our method significantly improves the quality of 3D reconstruction results. Moreover, we demonstrate through several +experiments that our method can be conveniently and effectively applied to some point cloud analysis tasks, including point cloud edge +feature extraction, normal estimation,etc. +Index Terms—Implicit surface representation, Differential Laplacian regularizer, Dynamic edge sampling, Point cloud, Geometric +modeling, Shape analysis. +! +1 +INTRODUCTION +Recently, Implicit Neural Representations (INRs) has gained made +great strides in the field of 3D reconstruction [1]–[8]. In contrast +to traditional explicit representations such as point clouds +[9], +voxels [10], [11] and mesh +[12]–[15], implicit neural repre- +sentations represent surface function primarily through neural +networks, providing higher quality, flexibility, and fidelity without +discretization errors, and significantly save amounts of storage +space to store high-quality results. +However, most of these methods need ground truth data as +supervision [1]–[3], which have difficulty in generalizing well to +unseen shapes that are dissimilar to the training samples. Recently, +some methods [16]–[20] have been proposed to reconstruct im- +plicit neural representations directly from raw data (point clouds, +triangle soups, unoriented meshes, etc.). Compared to data-driven +approaches, building implicit neural representations directly from +raw data is obviously more appealing. Generally speaking, the +core idea of such methods is to impose explicit/implicit regularity +constraints to reduce reliance on dataset. SAL [18] proposed a +unsigned regression loss to a given unsigned distance function +to raw data, which can produce signed solutions of implicit +functions. Specifically, starting from raw data (e.g., point clouds, +real scanned grids, etc.), implicit neural representations learn in a +self-supervised manner and can be trained reliably relying only +on raw input data by minimizing unsigned regression. Subse- +quently, SALD +[17], a generalized version of SAL +[18] was +proposed, which can obtain higher quality reconstruction results +• +Xiaogang Wang, Yuhang Cheng, Liang Wang and Guoqiang Xiao are +with College of Computer and Information Science, Southwest University, +China. +• +Jiangbo Lu is with the SmartMore Co., Ltd. +• +Kai Xu is with the National University of Defense Technology, China. +Fig. 1: Effect of edge preserving differential Laplacian regularizer. +(b) are the optimization results of the edge preserving differential +Laplacian regularizer which is incorporated into the state-of-the- +art methods IGR [19] . (a) and (c) are the results of IGR and +Ground truth respectively. +arXiv:2301.04860v1 [cs.CV] 12 Jan 2023 + +(a) IGR +(b) Our +(c) GTJOURNAL OF LATEX CLASS FILES, 2019 +2 +by incorporating an explicit gradient constraint on SAL. Gropp et +al. [19] proposed a novel implicit geometric regularization (IGR) +method to directly learn an implicit neural representation from +raw data and achieved surprising results. Different from SAL [18] +and SALD [17], IGR only relies on implicit regularization con- +straints, without the need for a unsigned distance function. More +specifically, IGR proposes an implicit geometric regularization, +which amounts to solving a particular Eikonal boundary value +problem that constrains the norm of spatial gradients to be 1 +almost everywhere. Yet, when the normal information cannot be +available and the number of input points is not dense enough, +the above algorithms often lead to unsatisfactory reconstruction +results (See Figure 1(a)). +We observed that the main reason for the unsatisfactory re- +construction results is that the implicit function needs to fit the +input point cloud as much as possible, and the noise information +in the point cloud tends to cause the implicit surface to be very +unsmooth. In other words, the main reason for this phenomenon is +the inconsistency of normal in the local region of the reconstructed +surface. Therefore, it is an intuitive idea to keep the local normal +of the surface consistent as much as possible; Meanwhile, it +should be noted that not all regions are restricted in their normal +consistency, for example, obviously sharp edges often exist in the +surface (as shown in Figure 1). In the reconstruction process, +we hope that this part of the area will not be overly smoothed. +Therefore, The edge preserving local normal consistency is more +accurate for implicit surface representation . +In view of the above problem, it can be visually viewed +as a standard Laplacian minimization problem; Meanwhile, we +can also use the Laplacian operator to identify the edge region +effectively, which has achieved good results in many image +processing tasks. Therefore, in other words, we can design an +intuitive Laplacian regularization, which can effectively improve +the quality of reconstruction results. +However, in this task, the raw data type we consider is point +cloud data, and the difference method cannot be directly used to +approximate high-order derivatives, mainly because point cloud +data does not have a clear topological relationship like mesh +or image. If the algorithm similar to KNN is used, the nearest +neighbor points searched cannot guarantee the correct topology +structure (as shown in Figure 3), especially when the point cloud +is not dense and the normal are not available , such wrong nearest +neighbor results will easily lead to the anti-optimization results (as +shown in Figure 4(a)). +Recently, there is growing interest in differentiable optimiza- +tion of implicit neural representations that enable differential +nature as supervision in learning frameworks +[3], [19], [21]– +[25]. The advantage of differentiable implicit neural represen- +tations is that it can directly solve the higher derivative of the +input signal instead of discretization approximation, which greatly +improves its optimization performance and application range. +Thanks to the analytically-differentiable nature of implicit neural +representation, we can easily design a differentiable Laplacian +regularizer. Meanwhile, the differentiable Laplacian regularizer +can be easily and intuitively incorporated into implicit neural +surface representations (as shown in Figure 1). We show that it +significantly improve the quality of 3D reconstruction. Meanwhile, +in order to facilitate qualitative and quantitative comparisons in +this paper, unless otherwise stated, in this paper, all experimental +results are obtained by incorporating them into IGR +[19]. We +carefully evaluate its performance through a series of ablation +studies. Meanwhile, we demonstrate through several experiments +that our method can be conveniently and effectively applied to +some point cloud analysis tasks, including point cloud edge feature +extraction, normal estimation, etc. +In summary, we make the following contributions: In this pa- +per, we use the infinite differentiability property of implicit neural +representation to propose a novel edge-preserving implicit surface +reconstruction method, which mainly consists of a differentiable +Laplican regularizer and a dynamic edge sampling strategy. 1), +Among them, the differential Laplican regularizer can effectively +alleviate the implicit surface unsmoothness caused by the point +cloud quality deteriorates; 2), Meanwhile, in order to reduce +the excessive smoothing at the edge regions of implicit suface, +we proposed a dynamic edge extract strategy for sampling near +the sharp edge of point cloud, which can effectively avoid the +Laplacian regularizer from smoothing all regions. +2 +RELATED WORK +2.1 +Data-driven based Implicit surface reconstruction +3D surface reconstruction from raw data has gained significant +research progress in recent year, benefiting from the advances in +machine learning techniques +[1]–[8]. Early studies +[26]–[28] +most utilize predefined geometric priors (such as local linearity +and smoothness) towards specific tasks. These geometric priors +often encode statistical properties of raw data and are designed +to be optimized, such as poisson equation +[28], [29], radius +basis function [26], moving least squares [27]. Recently, implicit +neural representation has gained significant research progress for +geometry reconstruction +[1]–[3], [6], [7], [16], [30]–[34] and +object representation [3], [23], [35]–[45] due to their simplicity +and excellent performance, which learn an approximate implicit +function with multi-layer perceptron (MLP). Compared to the +traditional continuous and discrete representations (grid, point +cloud and voxel), implicit neural representations have many poten- +tial benefits, which can provide higher modeling quality without +discretization errors, flexibility and fidelity, and save storage +space. However, most of these methods need ground truth data +as supervision [1]–[3], which have difficulty in generalizing well +to unseen shapes that are dissimilar to the training samples. +In addition, there are hybridization-based methods [46]–[50] +that combine data-driven priors with optimization strategy that can +achieve state-of-the-art performance. However, the above methods +also require additional ground truth data as supervision, which +seriously limits their applicability. +2.2 +Sign Agnostic Implicit surface reconstruction +Recently, some methods [17]–[20] have been proposed to re- +construct implicit neural representations directly from raw data. +Compared to big data-driven approaches, building implicit neural +representations directly from raw data is obviously more appeal- +ing. These methods can avoid the need for a large number of +ground truth signed distance representation of training data as +supervision. SAL +[18] introduces a sign agnostic regression +loss to a given unsigned distance function to raw data, which +is the signed version of unsigned distance function. Meanwhile, +that avoids the use of surface normals by properly initializing +implicit decoder networks so that they can only produce signed +solutions of implicit functions using unsigned distance function. +Subsequently, SALD +[17], a generalized version of SAL +[18] + +JOURNAL OF LATEX CLASS FILES, 2019 +3 +was proposed, which can obtain higher quality reconstruction +results by incorporating an explicit gradient constraint on SAL. +Similarly, in this paper, our approach also uses implicit neural +representation to estimate level set functions directly from raw +data. The major difference is that our proposed regularization +terms are directly based on differentiable implicit optimization, +and does not explicitly enforce some regularization on the zero +level set, such constraints, when the normal information cannot be +available and the number of input point cloud is not dense enough, +the implicit neural representation often lead to unsatisfactory +reconstruction results. +2.3 +Differentiable implicit neural representation +Compared with general implicit neural representation, differen- +tiable implicit neural representation has the advantage that it +can directly use various properties of differential geometry in- +stead of discretization approximation, which can lead to more +stable solutions in many optimization problems. Recently, there is +growing interest in differentiable optimization of implicit neural +representation that enable differential nature as supervision in +learning frameworks [3], [19], [21], [21]–[25]. General numerical +optimization often uses the discrete approximation of differential +geometry, for example, finite difference method is often used to +enhance the smoothness between adjacent samples in space. But +thanks to the analytically-differentiable nature of implicit neural +representation, differentiable implicit neural representations can +make direct use of many properties in differential geometry, such +as gradients [19], [21], [23], curvatures [24], and the solution of +partial differential equations [22], [25]. Recently, Gropp et al. [19] +proposed to use the differentiable implicit neural representation +to directly reconstruct surface from raw data. More specifically, +it proposes an implicit regularization constraint, which amounts +to solving a particular Eikonal boundary value problem that +constrains the norm of spatial gradients to be 1 almost everywhere. +Similarly, Sitzmann et al. [21] uses the proposed a differentiable +periodic activation functions to represent signed distance fields +in a fully-differentiable manner. Both of these works [19], [21] +, however, when the normal information cannot be available and +the number of input points is not dense enough, often lead to +unsatisfactory reconstruction results. In this paper, our work is also +based on the differentiability of implicit neural representations +to optimize implicit level set function estimated directly from +the input point cloud. Specifically, we designed an implicit dif- +ferentiable Laplacian regularizer, which effectively alleviated the +problem of unsatisfactory reconstruction results caused by direct +fitting of input point cloud by implicit neural function. +3 +METHOD +We present a differentiable laplacian regularizer for neural implicit +representation directly from input point cloud without normal +supervision. Note that our differential Laplacian regularizer can +be incorporated into any implicit neural representation, such as +IGR [19],SAL [18],SALD [17]. In this paper, unless otherwise +noted, we incorporate it in the IGR, which use level sets of neural +network to represent 3D shape (Sec. 3.1). More specifically, IGR +proposes an implicit geometric regularization, which amounts to +solving a particular Eikonal boundary value problem that con- +strains the norm of spatial gradients to be 1 almost everywhere. +Yet, when the normal information cannot be available and the +number of input points is not dense enough, IGR often lead +Fig. 2: Illustrations of the local normal consistency. +to unsatisfactory reconstruction results (See Figure 1(a)). We +observed that the main reason for the unsatisfactory reconstruction +results is that the implicit function needs to fit the input point cloud +as much as possible, and the noise information in the point cloud +tends to cause the implicit surface to be very unsmooth. +To +overcome +this +problem, +we +use +the +analytically- +differentiable nature of implicit neural representation, to propose +a differential Laplacian regularizer, which can effectively alleviate +the unsatisfactory reconstruction results (Sec. 3.2). Meanwhile, in +order to reduce the excessive smoothing at the edge regions of +3D shape (such as man-made shapes), a dynamic edge extraction +strategy (Sec. 3.2) is introduced for sampling near the sharp edge +of input point cloud, which can effectively avoid the Laplacian +regularizer from smoothing all regions, so as to effectively im- +prove the quality of reconstruction results while maintaining the +edge. +3.1 +Background +A neural implicit representations is a continuous function that +approximate the signed distance function. The underlying surface +of 3D shape is implicitly represented by the zero level set of this +function, +fθ(x) = 0, ∀x ∈ X. +(1) +where θ indicates the parameters to be learned and X indicates the +set of input point cloud. In general, one parameterize this function +using a multi-layer perceptron (MLP). Meanwhile, in order to +conveniently use the analytically-differentiable (such as, gradi- +ents,etc.) nature of implicit neural representation, recent works +[19], [21] usually replace the commonly used ReLU activation +function with a non-linear differentiable activation functions, thus +transforming MLP into a continuous and infinitely differentiable +function. +In IGR, the training is done by minimizing the loss that +encourages f to vanish on X: +Lvanish = +1 +N(X) +� +x∈X +|fθ(x)| +(2) +where N(X) is the number of point set X, | • | indicates abso- +lute value. if the input point cloud includes normal information +ngt(x), the corresponding loss function can be designed to make +the predicted normal (the differentiable gradient ▽fθ(x) of the +implicit function) as close as possible to the ground truth normal +ngt(x): +Lnormal = +1 +N(X) +� +x∈X +||▽fθ(x) − ngt(x)||2 +(3) +In addition to the above two intuitive fitting loss terms, IGR +[19] based on the Eikonal partial differential equation presents +an additional loss (Eikonal loss), which is equivalent to solve + +(b) +a +CJOURNAL OF LATEX CLASS FILES, 2019 +4 +Fig. 3: Illustrations of two different N nearest neighbors of non- +topological preservation (b) and topological preservation (c) for +geometric structure (a). +boundary value problems of a particular Eikonal that constrains +the norm of spatial gradients ▽fθ(x) to be 1 almost everywhere: +Leikonal = +1 +N(X) +� +x∈X +(||▽fθ(x)||2 − 1)2 +(4) +Note that, in our approach, we do not consider normal infor- +mation as supervision, so we will not consider Lnormal term in +all subsequent experiments. More specifically, our approach builds +upon the above two items Lvanish and Leikonal. +3.2 +Differentiable laplace regularization +Neighborhood normal consistency. A high-quality result can be +generated based on the above two terms (Lvanish and Leikonal) +when the input point data is large enough, however, when the +normal information cannot be available and the number of input +points is not dense enough, often lead to unsatisfactory reconstruc- +tion results (See Figure 1(a)). +We observed that the main reason for the unsatisfactory re- +construction results is that the implicit function needs to fit the +input point cloud as much as possible, and the noise information +in the point cloud tends to cause the implicit surface to be +very unsmooth. More specifically, the optimization results are +not guaranteed to provide a high-quality reconstruction result, +which is intuitively reflected by the possibility that the normal +of reconstruction result is inconsistent in the neighborhood. +From another perspective, it is well known that 3D shapes +tend to be piecewise smooth, that is, flat surfaces are more +likely than high-frequency structures [51]. For this purpose, we +incorporate this prior into implicit neural function by encouraging +the geometric smoothness of the reconstructed results. Therefore, +an intuitive solution is to constrain the consistency of the neighbor- +hood normal of the reconstruction results (as shown in Figure 2): +Lneibor = +� +x∈X +� +xi∈nei(x) +||▽fθ(x) − ▽fθ(xi)||2 +(5) +where nei(x) indicates the neighbor point set of point x. +However, in this paper, the raw data type we consider is point +cloud data, which does not have a clear topological structure +like mesh or voxels. If the algorithm similar to KNN is used, +the nearest neighbor points searched cannot guarantee that they +maintain the correct topology structure, especially when the point +cloud is not dense and the normal are not available, as shown in +Figure 3(b) where the three points P4, P5 and P6 do not meet the +nearest neighbor result of N = 5 under the maintenance of the +topology structure, and the correct set of nearest neighbor points +Fig. 4: The comparison of Lneibor (a) and Llaplacian (b). +should be {P1, P2, P3, P8, P9}. Moreover, it is difficult to get a +reasonable value for this parameter nei(x) in practice. As shown +in Figure 4, we can easily see that the wrong reconstructed results, +which is mainly caused by the above reasons. +Differentiable Laplacian regularizer. In fact, the above con- +straint Lneibor is mainly used to constrain the normal consistency +in the local domain, which can be easily interpreted as a discrete +Laplace operator. The Laplacian operator △f is a second-order +differential operator in n-dimensional euclidean space, defined as +the divergence (▽ · f) of the gradient (▽f). Thanks to the infinite +differentiability of implicit neural representation, we can design a +simple but effective differentiable Laplacian regularizer: +Llaplacian = +� +x∈X +△fθ(x)2 +(6) +where △fθ(x) indicates the differentiable Laplace operator of +point x. +As shown in Figure 4(b), compared with the explicit regular- +ization constraint Lneibor based on the nearest neighbor normal +consistency, the differentiable Laplacian regularizer can obtain +more stable results without introducing hyperparameter nearest +neighbors N. +3.3 +Dynamic edge sampling +However, while the differentiable Laplacian regularizer restricts +the normal consistency, it also brings a new problem: It imposes +undifferentiated constraints on all 3D regions, even in the sharp- +edge regions, as shown in Figure 6. As we know, complex 3D +shapes are generally constructed by multiple piecewise smooth +surfaces, which may not be differentiable at the joints, and are +more likely to form sharp edges. Therefore, in essence, a complex +3D shape (piecewise smooth model with sharp edges) cannot +be accurately represented by an implicit function, because it is +obviously not differentiable at sharp edges, so if it is forced to be +represented by an implicit function, especially only sparse point +sets without normal information are used as supervision, it is easy +to form an overly smooth reconstruction at the sharp edges (as +shown in Figure 6). +The most intuitive solution is to implicitly represent each +piecewise smooth surface separately, but this is difficult to do in +practice because it first requires the segmentation of the input point +set, which is difficult to do accurately in unsupervised conditions. +Therefore, we propose a novel dynamic edge sampling +strategy to effectively extract sharp edge regions in the training + +P +p +P +3 +2 +p +p +4 +4 +P +p +D +P +5 +5 +P +P +9 +6 +P8 +p, +P +P, +D +8 +(a) +(b) +(c)(a) KNN=9 +(b) Our +(c) GTJOURNAL OF LATEX CLASS FILES, 2019 +5 +Fig. 5: Statistics of Laplacian operators |△fθ(x)| and edge thresh- +old τ selection. +process. In theory, the remaining regions not only satisfy the +differentiable property, but also conform to the normal consistency +constraint, which can effectively avoid the indifference smoothing +of all regions, including the edge regions, of the laplace regular- +izer. +Specifically, for each point p in the input point set, we may +quickly determine whether it is an edge point according to its +differentiable Laplacian operator △fθ(x). Essentially, Laplacian +is mainly used to describe the rate of change of gradient, and +is often used for edge detection in image processing. From the +perspective of differential geometry, it is used to describe the +change rate of spatial position normal. Therefore, the larger the +laplacian of the point, the stronger the possibility that the point is +an edge point. We threshold the Laplacian |△fθ(x)| < τ to obtain +a corresponding set of non-edge points X′. According to statistics +(as shown in Figure 5), we set the parameter τ = 20 throughout +our experiments. This operation is performed before the back- +propagation of each iteration, therefore, we call it dynamic edge +sampling. +Llaplacian = +� +x∈X′ +△fθ(x)2 +(7) +where X′ indicates the non-edge subset of the input point cloud +X. Finally, we optimize the total loss: +Ltotal = Lvanish + λ1Leikonal + λ2Llaplacian +(8) +In which, we set λ1 = 0.1 and λ2 = 0.001 throughout our +experiments. +4 +DETAILS, RESULTS AND EVALUATIONS +4.1 +Implementation details +Data preparation. To facilitate quantitative evaluation of our +method on multiple tasks, including reconstruction , edge ex- +traction and normal estimation, we selected 100 3D shapes with +rich geometric topologies to construct the evaluation dataset (See +Figure 8) from ABC dataset [52], which provides more than 1 +million standard 3D CAD models with multiple types of standard +CAD format files. In addition to 3D geometry and normal informa- +tion, the geometric edges information mentioned above does not +provide us explicitly. To this end, we have developed a tool that, +Fig. 6: The comparison of with (b) and without Dynamic Edge +Sampling (DES) (a). +for each 3D shape, can quickly and easily extract the geometric +edge information from the multiple CAD files, thus fully meeting +the needs of our method for multi-task quantitative evaluation. +Point sampling. For each model, we sample it into a point +cloud containing 16, 384 points by uniform point sampling. +Meanwhile, in order to simulate the real point cloud noise, we +added Gaussian noise with mean µ = 0 and standard devia- +tion δ = 0.005 to each sampling point. In each case, except +where otherwise stated, the network is trained on the noisy data +throughout our experiments. A few metrics on point cloud multi- +tasks accuracy are defined to support quantitative evaluation of our +approach; see the following subsections for details. +4.2 +Metrics +In our experiments, both qualitative and quantitative evaluations +are provided. We evaluate our approach via ablation studies +(Section 4.6), comparisons to state-of-the-art methods for 3D +reconstruction (Section 4.3) , edge detection (Section 4.4) and +normal estimation (Section 4.5). For the quantitative assessment of +the 3D reconstruction results, we used the two-sided Chamfer dC +and Hausdorff distances dH introduced by [19]. For the evaluation +of the normal estimation, we use the angle dangle between the +predicted normal and the groudtruth normal as the metric. To +evaluate edge detection, we measure precision/recall and the +IoU between predictions and ground truth, while to evaluate the +geometric accuracy of the reconstructed edges, we employ the +Edge Chamfer Distance (ECD) introduced by [1]. +4.3 +Reconstruction +Comparison with IGR [19]. To facilitate a fair comparison with +IGR [19], our network architecture is consistent with IGR [19]. In +all experiments, we used the default training procedure specified +in IGR to train our network, except that we did not use normal +information in the training and set iterations to 10000. We set the +loss parameters (see equation (8)) λ2 = 0.1 and λ3 = 0.001 +throughout our experiments. Qualitative and quantitative experi- +ments are reported in Table 1 and Figure 7 we can also see that +the performance of our method is significantly better. +Comparison with state-of-the-art methods SAL [18] and +SALD [17]. In addition to IGR [19], our method is also compared +with SAL [18] and SALD [17], two state-of-the-art sign agnostic +learning based methods from raw data. The results shown in +Table 1(row 1 and 2) are inferior to those of our method. As shown +in Figure 7, the results demonstrate the significant advantage of + +4096 +8192 +300 +16384 +250 +abs(Laplacian) +200 +150 +100 +50 +0 +0 +2500 +5000 +7500 +10000 +12500 +15000 +Points(a) Our (w/o DES +(b) Our +(c) GTJOURNAL OF LATEX CLASS FILES, 2019 +6 +Fig. 7: Qualitative comparison with state-of-the-art methods IGR [19], SAL [18] and SALD [17]. +Fig. 8: An overview of multi-task evaluation dataset. +dC +dH +Mean +Median +Mean +Median +SAL [18] +0.019 +0.016 +0.094 +0.050 +SALD [17] +0.016 +0.015 +0.053 +0.042 +IGR [19] +0.028 +0.011 +0.111 +0.034 +Our (Llaplace) +0.017 +0.009 +0.068 +0.026 +Our (Llaplace + DES) +0.007 +0.007 +0.021 +0.021 +TABLE 1: A quantitative comparison of our method and ablation +against IGR [19], SAL [18] and SALD +[17] on multi-task +evaluation dataset. +our approach, due to the fact that differential Laplacian regularizer +can effectively alleviate the unsatisfactory reconstruction results. +4.4 +Edge recognition +Specifically, for each point p in the input point set, we may quickly +determine whether it is an edge point according to its differentiable +laplace operator △fθ(x) . Essentially, laplace operator is mainly +used to describe the rate of change of gradient, and is often used +for edge detection in image processing. From the perspective of +differential geometry, it is used to describe the change rate of +spatial position normal. Therefore, the larger the laplace operator +of the point, the stronger the possibility that the point is an edge +point. We threshold the laplace operator |△fθ(x)| > τ to obtain a +corresponding set of non-edge points Xedge. We set the parameter +τ = 20 throughout our experiments, as shown in Figure 10. +In addition to IGR [19], we also choose two representative +classical non-learning based methods: Voronoi Covariance Mea- +sure (VCM) [53], and Edge-Aware Resampling (EAR) [54], as +both have been adopted in the point-set processing routines of the +well known CGAL library. As reported in Table 4, our method +completely outperforms these classical methods, This is mainly +because we use the differentiable Laplacian operator of each +sampling point as the metric, which can be approximate to the +average curvature in the implicit surface representation. Note that, +there are a large number of high-quality edge detection methods +based on data-driven. We do not use these methods as references +here, mainly because ours is a self-supervised learning approach. +4.5 +Normal estimation +Essentially, an implicitly represented MLP with softplus activation +funtion represents a differentiable Signed Distance Functions d = +fθ(x). According to the properties of differential geometry, the +gradient operator of each point on the implicit surface fθ(x) = 0 +can be regarded as the normal vector of the current point x. +Therefore, after the training, for each point in the input point +cloud, we can directly calculate the gradient operator ▽fθ(x) of +the differentiable function fθ(x) at the current point x, that is, the +normal vector of the current point x. The experimental results are +reported in Table 1. The comparison results demonstrate how our +method achieves significantly better performance; as immediately +quantified by the fact that dangle is larger than the one reported +for our method. + +(b) IGR +(c) SAL +(a) Input +(d) SALD +(e) Our +(f) GT(a) Point cloud +(b) Normal +(c) EdgeJOURNAL OF LATEX CLASS FILES, 2019 +7 +Fig. 9: Visualization normal estimation of differential Laplacian regularizer (c) and dynamic edge sampling strategy (d). +Fig. 10: Visualization edge recognition of differential Laplacian regularizer (c) and dynamic edge sampling strategy (d). +dC +dH +Mean +Median +Mean +Median +X = 0.010 +0.0102 +0.0108 +0.0509 +0.0543 +X = 0.005 +0.0069 +0.0069 +0.0206 +0.0209 +X = 0.000 +0.0055 +0.0057 +0.0148 +0.0153 +D = 4, 096 +0.0075 +0.0075 +0.0350 +0.0328 +D = 8, 192 +0.0071 +0.0072 +0.0352 +0.0269 +D = 16, 384 +0.0069 +0.0069 +0.0206 +0.0209 +TABLE 2: Algorithm performance with respect to noise X and +sampling density D. +4.6 +Analysis of parameters and networks +Effect of noise. We stress test Laplacian regularizer by increasing +the level of noise. Specifically, we randomly add a Gaussian noise +whose mean is 0 and variance is X to each sampling point on +the surface of the 3D shape, where we tested four values of +X = {0, 0.005, 0.01, 0.02}. In each case, the implicit neural +surface was trained with the noise-added data. Table 2 shows +the quantitative results. As we can observe that, the Laplacian +regularizer, even when trained with noisy data, can still out- +perform these state-of-the-art methods [17]–[19] when they are +tested on point cloud with 0.005 noise. +Effect of density. We also train our method on point clouds +at a reduced density. Specifically, for each 3D shape, we sam- +pled a different number D of points to verify whether our +network could handle the sparser point clouds, where D = + +(c) +Liaplacian +(a) Input +(b) IGR +(e) GT(c) +Liaplacian +(a) Input +(b) IGR +(e) GTJOURNAL OF LATEX CLASS FILES, 2019 +8 +Fig. 11: Effect of edge preserving differential Laplacian regu- +larizer. (b) are the optimization results of the edge preserving +differential Laplacian regularizer which is incorporated into the +state-of-the-art method SALD [17]. (a) and (c) are the results of +SALD and Ground truth respectively. +dC +dH +dangle +IGR [19] +0.028 +0.111 +0.514 +Our (+Llaplacian) +0.017 +0.068 +0.274 +Our (+Llaplacian + DES) +0.009 +0.036 +0.133 +TABLE 3: Ablation studies – We evaluate the quantitative per- +formance of our method with/without components Llaplacian and +dynamic edge sampling (DES). +{4, 096, 8, 192, 16, 384}. (Results in Table 2 reveal a similar +trend as from the previous stress test. Namely, our network, when +trained on sparser point clouds, can still outperform these state-of- +the-art methods [17]–[19] when they are tested on or trained on +data at full resolution (16,384 points). +Effect of Llaplacian. To evaluate the effectiveness of loss +Llaplacian, We incorporate this into another state-of-the-art +method, SALD [17], This qualitative result is shown in Figure 11, +we can find that, compared with the original algorithm, the re- +construction quality can be effectively improved by incorporating +Laplacian. This is mainly because the differentiable Laplacian reg- +ularizer can effectively alleviate the unsatisfactory reconstruction +results. +Dynamic edge sampling. We evaluate the effect of dynamic +edge sampling strategy on reconstruction quality. We experiment +with the dynamic edge sampling, while keeping all other +parameters the same. From Table 1 and Figure 7 and 12 , we can +see that at the sharp edges, we can effectively improve the quality +of modeling compared with state-of-the-art methods (Table 1 +(rows 1 3)) and the baseline method without dynamic edge +sampling, this is largely due to thedynamic edge sampling +strategy for sampling near the sharp edge of input point cloud, +which can effectively avoid the regularizer from smoothing all +regions. +ECD +IoU +Precision +Recall +VCM [53] +0.0017 +0.1925 +0.2238 +0.5998 +EAR [54] +0.0071 +0.1146 +0.2399 +0.1933 +IGR [19] +0.0063 +0.0880 +0.0958 +0.5620 +Our +0.0015 +0.2375 +0.2665 +0.6934 +TABLE 4: Comparison state-of-the-art edge recognition tech- +niques - VCM [53], EAR [54], and IGR [19]. +5 +CONCLUSION AND LIMITATION +We present a differential Laplacian regularizer for neural implicit +representation directly from input point cloud without normal +supervision. More specifically, we use the infinite differentiability +property of implicit neural representation to propose a differen- +tiable Laplacian regularizer, which can effectively alleviate the +unsatisfactory reconstruction results. Meanwhile, we propose a +dynamic edge sampling strategy for sampling near the sharp +edge of input point cloud, which can effectively avoid the Lapla- +cian regularizer from smoothing all regions, so as to effectively +improve the quality of reconstruction results while maintaining +the edge. Moreover, the differentiable Laplacian regularizer can +be easily and intuitively incorporated into implicit neural sur- +face representations. We carefully evaluate its generation quality +through a series of ablation studies, which show that our method +significantly improve the quality of 3D reconstruction. In addition +to 3D reconstruction, our method can also be conveniently applied +to other point cloud analysis tasks, including edge extraction and +normal estimation, etc. +Limitation. Our approach has a few limitations, which point +out the directions of future study. Some representative failure cases +are shown in Figure 13. First, our method is prone to problems +in the reconstruction of ultra-thin geometric structures, probably +because the point cloud data is noisy, resulting in the geometric +structure has been completely destroyed. Second, Our method +for extremely detailed structure may be overlooked, resulting in +incorrect reconstruction results. +ACKNOWLEDGEMENT +We thank the anonymous reviewers for their valuable comments. +This work was supported in part by Natural Science Foundation +of China (62102328), and Fundamental Research Funds for the +Central Universities (SWU120076). +REFERENCES +[1] +Z. Chen and H. Zhang, “Learning implicit fields for generative shape +modeling,” in Proceedings of the IEEE/CVF Conference on Computer +Vision and Pattern Recognition, 2019, pp. 5939–5948. +[2] +L. Mescheder, M. Oechsle, M. Niemeyer, S. Nowozin, and A. Geiger, +“Occupancy networks: Learning 3d reconstruction in function space,” in +Proceedings of the IEEE/CVF conference on computer vision and pattern +recognition, 2019, pp. 4460–4470. +[3] +J. J. Park, P. Florence, J. Straub, R. Newcombe, and S. Lovegrove, +“Deepsdf: Learning continuous signed distance functions for shape +representation,” in Proceedings of the IEEE/CVF conference on computer +vision and pattern recognition, 2019, pp. 165–174. +[4] +J. Chibane, T. Alldieck, and G. Pons-Moll, “Implicit functions in feature +space for 3d shape reconstruction and completion,” in Proceedings of +the IEEE/CVF Conference on Computer Vision and Pattern Recognition, +2020, pp. 6970–6981. +[5] +P. Erler, P. Guerrero, S. Ohrhallinger, N. J. Mitra, and M. Wimmer, +“Points2surf learning implicit surfaces from point clouds,” in European +Conference on Computer Vision. +Springer, 2020, pp. 108–124. + ++ DES +(a) SALD +(c) GTJOURNAL OF LATEX CLASS FILES, 2019 +9 +Fig. 12: Visualization examples of differential Laplacian regularizer (c) and dynamic edge sampling strategy (d). +Fig. 13: Failure cases. +[6] +S. Peng, M. Niemeyer, L. Mescheder, M. Pollefeys, and A. Geiger, “Con- +volutional occupancy networks,” in European Conference on Computer +Vision. +Springer, 2020, pp. 523–540. +[7] +S. Saito, Z. Huang, R. Natsume, S. Morishima, A. Kanazawa, and H. Li, +“Pifu: Pixel-aligned implicit function for high-resolution clothed human +digitization,” in Proceedings of the IEEE/CVF International Conference +on Computer Vision, 2019, pp. 2304–2314. +[8] +Q. Xu, W. Wang, D. Ceylan, R. Mech, and U. Neumann, “Disn: Deep +implicit surface network for high-quality single-view 3d reconstruction,” +Advances in Neural Information Processing Systems, vol. 32, 2019. +[9] +H. Fan, H. Su, and L. J. Guibas, “A point set generation network for 3d +object reconstruction from a single image,” in Proceedings of the IEEE +conference on computer vision and pattern recognition, 2017, pp. 605– +613. +[10] C. B. Choy, D. Xu, J. Gwak, K. Chen, and S. Savarese, “3d-r2n2: A +unified approach for single and multi-view 3d object reconstruction,” in +European conference on computer vision. +Springer, 2016, pp. 628–644. +[11] J. Wu, C. Zhang, T. Xue, B. Freeman, and J. Tenenbaum, “Learning a +probabilistic latent space of object shapes via 3d generative-adversarial +modeling,” Advances in neural information processing systems, vol. 29, +2016. +[12] T. Groueix, M. Fisher, V. Kim, B. Russell, and M. Aubry, “Atlasnet: A +papier-mˆach´e approach to learning 3d surface generation. arxiv 2018,” +arXiv preprint arXiv:1802.05384, 1802. +[13] H. Kato, Y. Ushiku, and T. Harada, “Neural 3d mesh renderer,” in +Proceedings of the IEEE conference on computer vision and pattern +recognition, 2018, pp. 3907–3916. +[14] J. Tang, X. Han, J. Pan, K. Jia, and X. Tong, “A skeleton-bridged deep +learning approach for generating meshes of complex topologies from +single rgb images,” in Proceedings of the ieee/cvf conference on computer +vision and pattern recognition, 2019, pp. 4541–4550. +[15] J. Tang, X. Han, M. Tan, X. Tong, and K. Jia, “Skeletonnet: A topology- +preserving solution for learning mesh reconstruction of object surfaces +from rgb images,” IEEE transactions on pattern analysis and machine +intelligence, 2021. +[16] M. Atzmon, N. Haim, L. Yariv, O. Israelov, H. Maron, and Y. Lipman, +“Controlling neural level sets,” Advances in Neural Information Process- +ing Systems, vol. 32, 2019. +[17] M. Atzmon and Y. Lipman, “Sald: Sign agnostic learning with deriva- +tives,” arXiv preprint arXiv:2006.05400, 2020. +[18] ——, “Sal: Sign agnostic learning of shapes from raw data,” 2020, pp. +2565–2574. +[19] A. Gropp, L. Yariv, N. Haim, M. Atzmon, and Y. Lipman, “Im- +plicit geometric regularization for learning shapes,” arXiv preprint +arXiv:2002.10099, 2020. +[20] W. Zhao, J. Lei, Y. Wen, J. Zhang, and K. Jia, “Sign-agnostic implicit +learning of surface self-similarities for shape modeling and reconstruc- +tion from raw point clouds,” in Proceedings of the IEEE/CVF Conference +on Computer Vision and Pattern Recognition, 2021, pp. 10 256–10 265. +[21] V. Sitzmann, J. Martel, A. Bergman, D. Lindell, and G. Wetzstein, “Im- +plicit neural representations with periodic activation functions,” Advances +in Neural Information Processing Systems, vol. 33, pp. 7462–7473, 2020. +[22] H. Chen, R. Wu, E. Grinspun, C. Zheng, and P. Y. Chen, “Implicit +neural spatial representations for time-dependent pdes,” arXiv preprint +arXiv:2210.00124, 2022. +[23] L. Yariv, J. Gu, Y. Kasten, and Y. Lipman, “Volume rendering of neural +implicit surfaces,” Advances in Neural Information Processing Systems, +vol. 34, pp. 4805–4815, 2021. +[24] T. Ehret, R. Mar´ı, and G. Facciolo, “Nerf, meet differential geometry!” +arXiv preprint arXiv:2206.14938, 2022. +[25] J. Zehnder, Y. Li, S. Coros, and B. Thomaszewski, “Ntopo: Mesh-free +topology optimization using implicit neural representations,” Advances +in Neural Information Processing Systems, vol. 34, pp. 10 368–10 381, +2021. +[26] J. C. Carr, R. K. Beatson, J. B. Cherrie, T. J. Mitchell, W. R. Fright, B. C. +McCallum, and T. R. Evans, “Reconstruction and representation of 3d +objects with radial basis functions,” in Proceedings of the 28th annual +conference on Computer graphics and interactive techniques, 2001, pp. +67–76. +[27] M. Alexa, J. Behr, D. Cohen-Or, S. Fleishman, D. Levin, and C. T. Silva, +“Computing and rendering point set surfaces,” IEEE Transactions on +visualization and computer graphics, vol. 9, no. 1, pp. 3–15, 2003. +[28] M. Kazhdan, M. Bolitho, and H. Hoppe, “Poisson surface reconstruc- + +(a) Input +(b) IGR +(e) GTa +bJOURNAL OF LATEX CLASS FILES, 2019 +10 +tion,” in Proceedings of the fourth Eurographics symposium on Geometry +processing, vol. 7, 2006. +[29] M. Kazhdan and H. Hoppe, “Screened poisson surface reconstruction,” +ACM Transactions on Graphics (ToG), vol. 32, no. 3, pp. 1–13, 2013. +[30] K. Genova, F. Cole, D. Vlasic, A. Sarna, W. T. Freeman, and +T. Funkhouser, “Learning shape templates with structured implicit func- +tions,” in Proceedings of the IEEE/CVF International Conference on +Computer Vision, 2019, pp. 7154–7164. +[31] M. Niemeyer, L. Mescheder, M. Oechsle, and A. Geiger, “Occupancy +flow: 4d reconstruction by learning particle dynamics,” in Proceedings +of the IEEE/CVF international conference on computer vision, 2019, pp. +5379–5389. +[32] M. Oechsle, S. Peng, and A. Geiger, “Unisurf: Unifying neural implicit +surfaces and radiance fields for multi-view reconstruction,” in Proceed- +ings of the IEEE/CVF International Conference on Computer Vision, +2021, pp. 5589–5599. +[33] M. Tancik, P. Srinivasan, B. Mildenhall, S. Fridovich-Keil, N. Raghavan, +U. Singhal, R. Ramamoorthi, J. Barron, and R. Ng, “Fourier features let +networks learn high frequency functions in low dimensional domains,” +Advances in Neural Information Processing Systems, vol. 33, pp. 7537– +7547, 2020. +[34] P. Wang, L. Liu, Y. Liu, C. Theobalt, T. Komura, and W. Wang, “Neus: +Learning neural implicit surfaces by volume rendering for multi-view +reconstruction,” arXiv preprint arXiv:2106.10689, 2021. +[35] V. Sitzmann, M. Zollh¨ofer, and G. Wetzstein, “Scene representation +networks: Continuous 3d-structure-aware neural scene representations,” +Advances in Neural Information Processing Systems, vol. 32, 2019. +[36] A. Bergman, P. Kellnhofer, and G. Wetzstein, “Fast training of neural +lumigraph representations using meta learning,” Advances in Neural +Information Processing Systems, vol. 34, pp. 172–186, 2021. +[37] J. Chibane, A. Bansal, V. Lazova, and G. Pons-Moll, “Stereo radiance +fields (srf): Learning view synthesis for sparse views of novel scenes,” +in Proceedings of the IEEE/CVF Conference on Computer Vision and +Pattern Recognition, 2021, pp. 7911–7920. +[38] C. Gao, Y. Shih, W.-S. Lai, C.-K. Liang, and J.-B. Huang, “Portrait neural +radiance fields from a single image,” arXiv preprint arXiv:2012.05903, +2020. +[39] Y. Jiang, D. Ji, Z. Han, and M. Zwicker, “Sdfdiff: Differentiable render- +ing of signed distance fields for 3d shape optimization,” in Proceedings +of the IEEE/CVF conference on computer vision and pattern recognition, +2020, pp. 1251–1261. +[40] P. Kellnhofer, L. C. Jebe, A. Jones, R. Spicer, K. Pulli, and G. Wetzstein, +“Neural lumigraph rendering,” in Proceedings of the IEEE/CVF Confer- +ence on Computer Vision and Pattern Recognition, 2021, pp. 4287–4297. +[41] L. Liu, J. Gu, K. Zaw Lin, T.-S. Chua, and C. Theobalt, “Neural +sparse voxel fields,” Advances in Neural Information Processing Systems, +vol. 33, pp. 15 651–15 663, 2020. +[49] M. Yang, Y. Wen, W. Chen, Y. Chen, and K. Jia, “Deep optimized +priors for 3d shape modeling and reconstruction,” in Proceedings of +[42] R. Martin-Brualla, N. Radwan, M. S. Sajjadi, J. T. Barron, A. Doso- +vitskiy, and D. Duckworth, “Nerf in the wild: Neural radiance fields +for unconstrained photo collections,” in Proceedings of the IEEE/CVF +Conference on Computer Vision and Pattern Recognition, 2021, pp. +7210–7219. +[43] B. Mildenhall, P. P. Srinivasan, M. Tancik, J. T. Barron, R. Ramamoorthi, +and R. Ng, “Nerf: Representing scenes as neural radiance fields for view +synthesis,” arXiv preprint arXiv:2003.08934, 2020. +[44] M. Niemeyer, L. Mescheder, M. Oechsle, and A. Geiger, “Differentiable +volumetric rendering: Learning implicit 3d representations without 3d +supervision,” in Proceedings of the IEEE/CVF Conference on Computer +Vision and Pattern Recognition, 2020, pp. 3504–3515. +[45] L. Yariv, Y. Kasten, D. Moran, M. Galun, M. Atzmon, B. Ronen, and +Y. Lipman, “Multiview neural surface reconstruction by disentangling +geometry and appearance,” Advances in Neural Information Processing +Systems, vol. 33, pp. 2492–2502, 2020. +[46] C. Jiang, A. Sud, A. Makadia, J. Huang, M. Nießner, T. Funkhouser et al., +“Local implicit grid representations for 3d scenes,” in Proceedings of +the IEEE/CVF Conference on Computer Vision and Pattern Recognition, +2020, pp. 6001–6010. +[47] E. Tretschk, A. Tewari, V. Golyanik, M. Zollh¨ofer, C. Stoll, and +C. Theobalt, “Patchnets: Patch-based generalizable deep implicit 3d +shape representations,” in European Conference on Computer Vision. +Springer, 2020, pp. 293–309. +[48] R. Chabra, J. E. Lenssen, E. Ilg, T. Schmidt, J. Straub, S. Lovegrove, +and R. Newcombe, “Deep local shapes: Learning local sdf priors for +detailed 3d reconstruction,” in European Conference on Computer Vision. +Springer, 2020, pp. 608–625. +the IEEE/CVF Conference on Computer Vision and Pattern Recognition, +2021, pp. 3269–3278. +[50] J. Tang, J. Lei, D. Xu, F. Ma, K. Jia, and L. Zhang, “Sa-convonet: +Sign-agnostic optimization of convolutional occupancy networks,” in +Proceedings of the IEEE/CVF International Conference on Computer +Vision, 2021, pp. 6504–6513. +[51] J. Huang, A. B. Lee, and D. Mumford, “Statistics of range images,” in +Proceedings IEEE Conference on Computer Vision and Pattern Recogni- +tion. CVPR 2000 (Cat. No. PR00662), vol. 1. +IEEE, 2000, pp. 324–331. +[52] S. Koch, A. Matveev, Z. Jiang, F. Williams, A. Artemov, E. Burnaev, +M. Alexa, D. Zorin, and D. Panozzo, “Abc: A big cad model dataset for +geometric deep learning,” in The IEEE Conference on Computer Vision +and Pattern Recognition (CVPR), June 2019. +[53] Q. M´erigot, M. Ovsjanikov, and L. J. Guibas, “Voronoi-based curvature +and feature estimation from point clouds,” IEEE Transactions on Visual- +ization and Computer Graphics, vol. 17, no. 6, pp. 743–756, 2010. +[54] H. Huang, S. Wu, M. Gong, D. Cohen-Or, U. Ascher, and H. Zhang, +“Edge-aware point set resampling,” ACM transactions on graphics +(TOG), vol. 32, no. 1, pp. 1–12, 2013. + diff --git a/7tE4T4oBgHgl3EQfCgsd/content/tmp_files/load_file.txt b/7tE4T4oBgHgl3EQfCgsd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8eb7f249eebe6ad5b74786d3cbe0976d532a93d1 --- /dev/null +++ b/7tE4T4oBgHgl3EQfCgsd/content/tmp_files/load_file.txt @@ -0,0 +1,761 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf,len=760 +page_content='JOURNAL OF LATEX CLASS FILES, 2019 1 Edge Preserving Implicit Surface Representation of Point Clouds Xiaogang Wang, Yuhang Cheng, Liang Wang, Jiangbo Lu, Senior Member, IEEE , Kai Xu, Senior Member, IEEE , Guoqiang Xiao Abstract— Learning implicit surface directly from raw data recently has become a very attractive representation method for 3D reconstruction tasks due to its excellent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' However, as the raw data quality deteriorates, the implicit functions often lead to unsatisfactory reconstruction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' To this end, we propose a novel edge-preserving implicit surface reconstruction method, which mainly consists of a differentiable Laplican regularizer and a dynamic edge sampling strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Among them, the differential Laplican regularizer can effectively alleviate the implicit surface unsmoothness caused by the point cloud quality deteriorates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Meanwhile, in order to reduce the excessive smoothing at the edge regions of implicit suface, we proposed a dynamic edge extract strategy for sampling near the sharp edge of point cloud, which can effectively avoid the Laplacian regularizer from smoothing all regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Finally, we combine them with a simple regularization term for robust implicit surface reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Compared with the state-of-the-art methods, experimental results show that our method significantly improves the quality of 3D reconstruction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Moreover, we demonstrate through several experiments that our method can be conveniently and effectively applied to some point cloud analysis tasks, including point cloud edge feature extraction, normal estimation,etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Index Terms—Implicit surface representation, Differential Laplacian regularizer, Dynamic edge sampling, Point cloud, Geometric modeling, Shape analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 1 INTRODUCTION Recently, Implicit Neural Representations (INRs) has gained made great strides in the field of 3D reconstruction [1]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' In contrast to traditional explicit representations such as point clouds [9], voxels [10], [11] and mesh [12]–[15], implicit neural repre- sentations represent surface function primarily through neural networks, providing higher quality, flexibility, and fidelity without discretization errors, and significantly save amounts of storage space to store high-quality results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' However, most of these methods need ground truth data as supervision [1]–[3], which have difficulty in generalizing well to unseen shapes that are dissimilar to the training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Recently, some methods [16]–[20] have been proposed to reconstruct im- plicit neural representations directly from raw data (point clouds, triangle soups, unoriented meshes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Compared to data-driven approaches, building implicit neural representations directly from raw data is obviously more appealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Generally speaking, the core idea of such methods is to impose explicit/implicit regularity constraints to reduce reliance on dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' SAL [18] proposed a unsigned regression loss to a given unsigned distance function to raw data, which can produce signed solutions of implicit functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Specifically, starting from raw data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=', point clouds, real scanned grids, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' ), implicit neural representations learn in a self-supervised manner and can be trained reliably relying only on raw input data by minimizing unsigned regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Subse- quently, SALD [17], a generalized version of SAL [18] was proposed, which can obtain higher quality reconstruction results Xiaogang Wang, Yuhang Cheng, Liang Wang and Guoqiang Xiao are with College of Computer and Information Science, Southwest University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Jiangbo Lu is with the SmartMore Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=', Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Kai Xu is with the National University of Defense Technology, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 1: Effect of edge preserving differential Laplacian regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' (b) are the optimization results of the edge preserving differential Laplacian regularizer which is incorporated into the state-of-the- art methods IGR [19] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' (a) and (c) are the results of IGR and Ground truth respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='04860v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='CV] 12 Jan 2023 (a) IGR (b) Our (c) GTJOURNAL OF LATEX CLASS FILES, 2019 2 by incorporating an explicit gradient constraint on SAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Gropp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [19] proposed a novel implicit geometric regularization (IGR) method to directly learn an implicit neural representation from raw data and achieved surprising results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Different from SAL [18] and SALD [17], IGR only relies on implicit regularization con- straints, without the need for a unsigned distance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' More specifically, IGR proposes an implicit geometric regularization, which amounts to solving a particular Eikonal boundary value problem that constrains the norm of spatial gradients to be 1 almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Yet, when the normal information cannot be available and the number of input points is not dense enough, the above algorithms often lead to unsatisfactory reconstruction results (See Figure 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' We observed that the main reason for the unsatisfactory re- construction results is that the implicit function needs to fit the input point cloud as much as possible, and the noise information in the point cloud tends to cause the implicit surface to be very unsmooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' In other words, the main reason for this phenomenon is the inconsistency of normal in the local region of the reconstructed surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Therefore, it is an intuitive idea to keep the local normal of the surface consistent as much as possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Meanwhile, it should be noted that not all regions are restricted in their normal consistency, for example, obviously sharp edges often exist in the surface (as shown in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' In the reconstruction process, we hope that this part of the area will not be overly smoothed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Therefore, The edge preserving local normal consistency is more accurate for implicit surface representation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' In view of the above problem, it can be visually viewed as a standard Laplacian minimization problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Meanwhile, we can also use the Laplacian operator to identify the edge region effectively, which has achieved good results in many image processing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Therefore, in other words, we can design an intuitive Laplacian regularization, which can effectively improve the quality of reconstruction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' However, in this task, the raw data type we consider is point cloud data, and the difference method cannot be directly used to approximate high-order derivatives, mainly because point cloud data does not have a clear topological relationship like mesh or image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' If the algorithm similar to KNN is used, the nearest neighbor points searched cannot guarantee the correct topology structure (as shown in Figure 3), especially when the point cloud is not dense and the normal are not available , such wrong nearest neighbor results will easily lead to the anti-optimization results (as shown in Figure 4(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Recently, there is growing interest in differentiable optimiza- tion of implicit neural representations that enable differential nature as supervision in learning frameworks [3], [19], [21]– [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' The advantage of differentiable implicit neural represen- tations is that it can directly solve the higher derivative of the input signal instead of discretization approximation, which greatly improves its optimization performance and application range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Thanks to the analytically-differentiable nature of implicit neural representation, we can easily design a differentiable Laplacian regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Meanwhile, the differentiable Laplacian regularizer can be easily and intuitively incorporated into implicit neural surface representations (as shown in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' We show that it significantly improve the quality of 3D reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Meanwhile, in order to facilitate qualitative and quantitative comparisons in this paper, unless otherwise stated, in this paper, all experimental results are obtained by incorporating them into IGR [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' We carefully evaluate its performance through a series of ablation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Meanwhile, we demonstrate through several experiments that our method can be conveniently and effectively applied to some point cloud analysis tasks, including point cloud edge feature extraction, normal estimation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' In summary, we make the following contributions: In this pa- per, we use the infinite differentiability property of implicit neural representation to propose a novel edge-preserving implicit surface reconstruction method, which mainly consists of a differentiable Laplican regularizer and a dynamic edge sampling strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 1), Among them, the differential Laplican regularizer can effectively alleviate the implicit surface unsmoothness caused by the point cloud quality deteriorates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 2), Meanwhile, in order to reduce the excessive smoothing at the edge regions of implicit suface, we proposed a dynamic edge extract strategy for sampling near the sharp edge of point cloud, which can effectively avoid the Laplacian regularizer from smoothing all regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='1 Data-driven based Implicit surface reconstruction 3D surface reconstruction from raw data has gained significant research progress in recent year, benefiting from the advances in machine learning techniques [1]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Early studies [26]–[28] most utilize predefined geometric priors (such as local linearity and smoothness) towards specific tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' These geometric priors often encode statistical properties of raw data and are designed to be optimized, such as poisson equation [28], [29], radius basis function [26], moving least squares [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Recently, implicit neural representation has gained significant research progress for geometry reconstruction [1]–[3], [6], [7], [16], [30]–[34] and object representation [3], [23], [35]–[45] due to their simplicity and excellent performance, which learn an approximate implicit function with multi-layer perceptron (MLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Compared to the traditional continuous and discrete representations (grid, point cloud and voxel), implicit neural representations have many poten- tial benefits, which can provide higher modeling quality without discretization errors, flexibility and fidelity, and save storage space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' However, most of these methods need ground truth data as supervision [1]–[3], which have difficulty in generalizing well to unseen shapes that are dissimilar to the training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' In addition, there are hybridization-based methods [46]–[50] that combine data-driven priors with optimization strategy that can achieve state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' However, the above methods also require additional ground truth data as supervision, which seriously limits their applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='2 Sign Agnostic Implicit surface reconstruction Recently, some methods [17]–[20] have been proposed to re- construct implicit neural representations directly from raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Compared to big data-driven approaches, building implicit neural representations directly from raw data is obviously more appeal- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' These methods can avoid the need for a large number of ground truth signed distance representation of training data as supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' SAL [18] introduces a sign agnostic regression loss to a given unsigned distance function to raw data, which is the signed version of unsigned distance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Meanwhile, that avoids the use of surface normals by properly initializing implicit decoder networks so that they can only produce signed solutions of implicit functions using unsigned distance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Subsequently, SALD [17], a generalized version of SAL [18] JOURNAL OF LATEX CLASS FILES, 2019 3 was proposed, which can obtain higher quality reconstruction results by incorporating an explicit gradient constraint on SAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Similarly, in this paper, our approach also uses implicit neural representation to estimate level set functions directly from raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' The major difference is that our proposed regularization terms are directly based on differentiable implicit optimization, and does not explicitly enforce some regularization on the zero level set, such constraints, when the normal information cannot be available and the number of input point cloud is not dense enough, the implicit neural representation often lead to unsatisfactory reconstruction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='3 Differentiable implicit neural representation Compared with general implicit neural representation, differen- tiable implicit neural representation has the advantage that it can directly use various properties of differential geometry in- stead of discretization approximation, which can lead to more stable solutions in many optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Recently, there is growing interest in differentiable optimization of implicit neural representation that enable differential nature as supervision in learning frameworks [3], [19], [21], [21]–[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' General numerical optimization often uses the discrete approximation of differential geometry, for example, finite difference method is often used to enhance the smoothness between adjacent samples in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' But thanks to the analytically-differentiable nature of implicit neural representation, differentiable implicit neural representations can make direct use of many properties in differential geometry, such as gradients [19], [21], [23], curvatures [24], and the solution of partial differential equations [22], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Recently, Gropp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [19] proposed to use the differentiable implicit neural representation to directly reconstruct surface from raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' More specifically, it proposes an implicit regularization constraint, which amounts to solving a particular Eikonal boundary value problem that constrains the norm of spatial gradients to be 1 almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Similarly, Sitzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [21] uses the proposed a differentiable periodic activation functions to represent signed distance fields in a fully-differentiable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Both of these works [19], [21] , however, when the normal information cannot be available and the number of input points is not dense enough, often lead to unsatisfactory reconstruction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' In this paper, our work is also based on the differentiability of implicit neural representations to optimize implicit level set function estimated directly from the input point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Specifically, we designed an implicit dif- ferentiable Laplacian regularizer, which effectively alleviated the problem of unsatisfactory reconstruction results caused by direct fitting of input point cloud by implicit neural function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 3 METHOD We present a differentiable laplacian regularizer for neural implicit representation directly from input point cloud without normal supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Note that our differential Laplacian regularizer can be incorporated into any implicit neural representation, such as IGR [19],SAL [18],SALD [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' In this paper, unless otherwise noted, we incorporate it in the IGR, which use level sets of neural network to represent 3D shape (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' More specifically, IGR proposes an implicit geometric regularization, which amounts to solving a particular Eikonal boundary value problem that con- strains the norm of spatial gradients to be 1 almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Yet, when the normal information cannot be available and the number of input points is not dense enough, IGR often lead Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 2: Illustrations of the local normal consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' to unsatisfactory reconstruction results (See Figure 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' We observed that the main reason for the unsatisfactory reconstruction results is that the implicit function needs to fit the input point cloud as much as possible, and the noise information in the point cloud tends to cause the implicit surface to be very unsmooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' To overcome this problem, we use the analytically- differentiable nature of implicit neural representation, to propose a differential Laplacian regularizer, which can effectively alleviate the unsatisfactory reconstruction results (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Meanwhile, in order to reduce the excessive smoothing at the edge regions of 3D shape (such as man-made shapes), a dynamic edge extraction strategy (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='2) is introduced for sampling near the sharp edge of input point cloud, which can effectively avoid the Laplacian regularizer from smoothing all regions, so as to effectively im- prove the quality of reconstruction results while maintaining the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='1 Background A neural implicit representations is a continuous function that approximate the signed distance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' The underlying surface of 3D shape is implicitly represented by the zero level set of this function, fθ(x) = 0, ∀x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' (1) where θ indicates the parameters to be learned and X indicates the set of input point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' In general, one parameterize this function using a multi-layer perceptron (MLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Meanwhile, in order to conveniently use the analytically-differentiable (such as, gradi- ents,etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=') nature of implicit neural representation, recent works [19], [21] usually replace the commonly used ReLU activation function with a non-linear differentiable activation functions, thus transforming MLP into a continuous and infinitely differentiable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' In IGR, the training is done by minimizing the loss that encourages f to vanish on X: Lvanish = 1 N(X) � x∈X |fθ(x)| (2) where N(X) is the number of point set X, | • | indicates abso- lute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' if the input point cloud includes normal information ngt(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' the corresponding loss function can be designed to make the predicted normal (the differentiable gradient ▽fθ(x) of the implicit function) as close as possible to the ground truth normal ngt(x): Lnormal = 1 N(X) � x∈X ||▽fθ(x) − ngt(x)||2 (3) In addition to the above two intuitive fitting loss terms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' IGR [19] based on the Eikonal partial differential equation presents an additional loss (Eikonal loss),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' which is equivalent to solve (b) a CJOURNAL OF LATEX CLASS FILES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 2019 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 3: Illustrations of two different N nearest neighbors of non- topological preservation (b) and topological preservation (c) for geometric structure (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' boundary value problems of a particular Eikonal that constrains the norm of spatial gradients ▽fθ(x) to be 1 almost everywhere: Leikonal = 1 N(X) � x∈X (||▽fθ(x)||2 − 1)2 (4) Note that, in our approach, we do not consider normal infor- mation as supervision, so we will not consider Lnormal term in all subsequent experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' More specifically, our approach builds upon the above two items Lvanish and Leikonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='2 Differentiable laplace regularization Neighborhood normal consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' A high-quality result can be generated based on the above two terms (Lvanish and Leikonal) when the input point data is large enough, however, when the normal information cannot be available and the number of input points is not dense enough, often lead to unsatisfactory reconstruc- tion results (See Figure 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' We observed that the main reason for the unsatisfactory re- construction results is that the implicit function needs to fit the input point cloud as much as possible, and the noise information in the point cloud tends to cause the implicit surface to be very unsmooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' More specifically, the optimization results are not guaranteed to provide a high-quality reconstruction result, which is intuitively reflected by the possibility that the normal of reconstruction result is inconsistent in the neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' From another perspective, it is well known that 3D shapes tend to be piecewise smooth, that is, flat surfaces are more likely than high-frequency structures [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' For this purpose, we incorporate this prior into implicit neural function by encouraging the geometric smoothness of the reconstructed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Therefore, an intuitive solution is to constrain the consistency of the neighbor- hood normal of the reconstruction results (as shown in Figure 2): Lneibor = � x∈X � xi∈nei(x) ||▽fθ(x) − ▽fθ(xi)||2 (5) where nei(x) indicates the neighbor point set of point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' However, in this paper, the raw data type we consider is point cloud data, which does not have a clear topological structure like mesh or voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' If the algorithm similar to KNN is used, the nearest neighbor points searched cannot guarantee that they maintain the correct topology structure, especially when the point cloud is not dense and the normal are not available, as shown in Figure 3(b) where the three points P4, P5 and P6 do not meet the nearest neighbor result of N = 5 under the maintenance of the topology structure, and the correct set of nearest neighbor points Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 4: The comparison of Lneibor (a) and Llaplacian (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' should be {P1, P2, P3, P8, P9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Moreover, it is difficult to get a reasonable value for this parameter nei(x) in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' As shown in Figure 4, we can easily see that the wrong reconstructed results, which is mainly caused by the above reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Differentiable Laplacian regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' In fact, the above con- straint Lneibor is mainly used to constrain the normal consistency in the local domain, which can be easily interpreted as a discrete Laplace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' The Laplacian operator △f is a second-order differential operator in n-dimensional euclidean space, defined as the divergence (▽ · f) of the gradient (▽f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Thanks to the infinite differentiability of implicit neural representation, we can design a simple but effective differentiable Laplacian regularizer: Llaplacian = � x∈X △fθ(x)2 (6) where △fθ(x) indicates the differentiable Laplace operator of point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' As shown in Figure 4(b), compared with the explicit regular- ization constraint Lneibor based on the nearest neighbor normal consistency, the differentiable Laplacian regularizer can obtain more stable results without introducing hyperparameter nearest neighbors N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='3 Dynamic edge sampling However, while the differentiable Laplacian regularizer restricts the normal consistency, it also brings a new problem: It imposes undifferentiated constraints on all 3D regions, even in the sharp- edge regions, as shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' As we know, complex 3D shapes are generally constructed by multiple piecewise smooth surfaces, which may not be differentiable at the joints, and are more likely to form sharp edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Therefore, in essence, a complex 3D shape (piecewise smooth model with sharp edges) cannot be accurately represented by an implicit function, because it is obviously not differentiable at sharp edges, so if it is forced to be represented by an implicit function, especially only sparse point sets without normal information are used as supervision, it is easy to form an overly smooth reconstruction at the sharp edges (as shown in Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' The most intuitive solution is to implicitly represent each piecewise smooth surface separately, but this is difficult to do in practice because it first requires the segmentation of the input point set, which is difficult to do accurately in unsupervised conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Therefore, we propose a novel dynamic edge sampling strategy to effectively extract sharp edge regions in the training P p P 3 2 p p 4 4 P p D P 5 5 P P 9 6 P8 p, P P, D 8 (a) (b) (c)(a) KNN=9 (b) Our (c) GTJOURNAL OF LATEX CLASS FILES, 2019 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 5: Statistics of Laplacian operators |△fθ(x)| and edge thresh- old τ selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' In theory, the remaining regions not only satisfy the differentiable property, but also conform to the normal consistency constraint, which can effectively avoid the indifference smoothing of all regions, including the edge regions, of the laplace regular- izer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Specifically, for each point p in the input point set, we may quickly determine whether it is an edge point according to its differentiable Laplacian operator △fθ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Essentially, Laplacian is mainly used to describe the rate of change of gradient, and is often used for edge detection in image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' From the perspective of differential geometry, it is used to describe the change rate of spatial position normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Therefore, the larger the laplacian of the point, the stronger the possibility that the point is an edge point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' We threshold the Laplacian |△fθ(x)| < τ to obtain a corresponding set of non-edge points X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' According to statistics (as shown in Figure 5), we set the parameter τ = 20 throughout our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' This operation is performed before the back- propagation of each iteration, therefore, we call it dynamic edge sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Llaplacian = � x∈X′ △fθ(x)2 (7) where X′ indicates the non-edge subset of the input point cloud X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Finally, we optimize the total loss: Ltotal = Lvanish + λ1Leikonal + λ2Llaplacian (8) In which, we set λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='1 and λ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='001 throughout our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 4 DETAILS, RESULTS AND EVALUATIONS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='1 Implementation details Data preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' To facilitate quantitative evaluation of our method on multiple tasks, including reconstruction , edge ex- traction and normal estimation, we selected 100 3D shapes with rich geometric topologies to construct the evaluation dataset (See Figure 8) from ABC dataset [52], which provides more than 1 million standard 3D CAD models with multiple types of standard CAD format files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' In addition to 3D geometry and normal informa- tion, the geometric edges information mentioned above does not provide us explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' To this end, we have developed a tool that, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 6: The comparison of with (b) and without Dynamic Edge Sampling (DES) (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' for each 3D shape, can quickly and easily extract the geometric edge information from the multiple CAD files, thus fully meeting the needs of our method for multi-task quantitative evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Point sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' For each model, we sample it into a point cloud containing 16, 384 points by uniform point sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Meanwhile, in order to simulate the real point cloud noise, we added Gaussian noise with mean µ = 0 and standard devia- tion δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='005 to each sampling point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' In each case, except where otherwise stated, the network is trained on the noisy data throughout our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' A few metrics on point cloud multi- tasks accuracy are defined to support quantitative evaluation of our approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' see the following subsections for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='2 Metrics In our experiments, both qualitative and quantitative evaluations are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' We evaluate our approach via ablation studies (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='6), comparisons to state-of-the-art methods for 3D reconstruction (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='3) , edge detection (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='4) and normal estimation (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' For the quantitative assessment of the 3D reconstruction results, we used the two-sided Chamfer dC and Hausdorff distances dH introduced by [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' For the evaluation of the normal estimation, we use the angle dangle between the predicted normal and the groudtruth normal as the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' To evaluate edge detection, we measure precision/recall and the IoU between predictions and ground truth, while to evaluate the geometric accuracy of the reconstructed edges, we employ the Edge Chamfer Distance (ECD) introduced by [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='3 Reconstruction Comparison with IGR [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' To facilitate a fair comparison with IGR [19], our network architecture is consistent with IGR [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' In all experiments, we used the default training procedure specified in IGR to train our network, except that we did not use normal information in the training and set iterations to 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' We set the loss parameters (see equation (8)) λ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='1 and λ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='001 throughout our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Qualitative and quantitative experi- ments are reported in Table 1 and Figure 7 we can also see that the performance of our method is significantly better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Comparison with state-of-the-art methods SAL [18] and SALD [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' In addition to IGR [19], our method is also compared with SAL [18] and SALD [17], two state-of-the-art sign agnostic learning based methods from raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' The results shown in Table 1(row 1 and 2) are inferior to those of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' As shown in Figure 7, the results demonstrate the significant advantage of 4096 8192 300 16384 250 abs(Laplacian) 200 150 100 50 0 0 2500 5000 7500 10000 12500 15000 Points(a) Our (w/o DES (b) Our (c) GTJOURNAL OF LATEX CLASS FILES, 2019 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 7: Qualitative comparison with state-of-the-art methods IGR [19], SAL [18] and SALD [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 8: An overview of multi-task evaluation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' dC dH Mean Median Mean Median SAL [18] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='050 SALD [17] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='042 IGR [19] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='111 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='034 Our (Llaplace) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='026 Our (Llaplace + DES) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='021 TABLE 1: A quantitative comparison of our method and ablation against IGR [19], SAL [18] and SALD [17] on multi-task evaluation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' our approach, due to the fact that differential Laplacian regularizer can effectively alleviate the unsatisfactory reconstruction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='4 Edge recognition Specifically, for each point p in the input point set, we may quickly determine whether it is an edge point according to its differentiable laplace operator △fθ(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Essentially, laplace operator is mainly used to describe the rate of change of gradient, and is often used for edge detection in image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' From the perspective of differential geometry, it is used to describe the change rate of spatial position normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Therefore, the larger the laplace operator of the point, the stronger the possibility that the point is an edge point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' We threshold the laplace operator |△fθ(x)| > τ to obtain a corresponding set of non-edge points Xedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' We set the parameter τ = 20 throughout our experiments, as shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' In addition to IGR [19], we also choose two representative classical non-learning based methods: Voronoi Covariance Mea- sure (VCM) [53], and Edge-Aware Resampling (EAR) [54], as both have been adopted in the point-set processing routines of the well known CGAL library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' As reported in Table 4, our method completely outperforms these classical methods, This is mainly because we use the differentiable Laplacian operator of each sampling point as the metric, which can be approximate to the average curvature in the implicit surface representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Note that, there are a large number of high-quality edge detection methods based on data-driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' We do not use these methods as references here, mainly because ours is a self-supervised learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='5 Normal estimation Essentially, an implicitly represented MLP with softplus activation funtion represents a differentiable Signed Distance Functions d = fθ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' According to the properties of differential geometry, the gradient operator of each point on the implicit surface fθ(x) = 0 can be regarded as the normal vector of the current point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Therefore, after the training, for each point in the input point cloud, we can directly calculate the gradient operator ▽fθ(x) of the differentiable function fθ(x) at the current point x, that is, the normal vector of the current point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' The experimental results are reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' The comparison results demonstrate how our method achieves significantly better performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' as immediately quantified by the fact that dangle is larger than the one reported for our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' (b) IGR (c) SAL (a) Input (d) SALD (e) Our (f) GT(a) Point cloud (b) Normal (c) EdgeJOURNAL OF LATEX CLASS FILES, 2019 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 9: Visualization normal estimation of differential Laplacian regularizer (c) and dynamic edge sampling strategy (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 10: Visualization edge recognition of differential Laplacian regularizer (c) and dynamic edge sampling strategy (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' dC dH Mean Median Mean Median X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0509 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0543 X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0206 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0209 X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0153 D = 4, 096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0328 D = 8, 192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0071 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0352 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0269 D = 16, 384 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0206 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0209 TABLE 2: Algorithm performance with respect to noise X and sampling density D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='6 Analysis of parameters and networks Effect of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' We stress test Laplacian regularizer by increasing the level of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Specifically, we randomly add a Gaussian noise whose mean is 0 and variance is X to each sampling point on the surface of the 3D shape, where we tested four values of X = {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='02}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' In each case, the implicit neural surface was trained with the noise-added data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Table 2 shows the quantitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' As we can observe that, the Laplacian regularizer, even when trained with noisy data, can still out- perform these state-of-the-art methods [17]–[19] when they are tested on point cloud with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='005 noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Effect of density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' We also train our method on point clouds at a reduced density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Specifically, for each 3D shape, we sam- pled a different number D of points to verify whether our network could handle the sparser point clouds, where D = (c) +Liaplacian (a) Input (b) IGR (e) GT(c) +Liaplacian (a) Input (b) IGR (e) GTJOURNAL OF LATEX CLASS FILES, 2019 8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 11: Effect of edge preserving differential Laplacian regu- larizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' (b) are the optimization results of the edge preserving differential Laplacian regularizer which is incorporated into the state-of-the-art method SALD [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' (a) and (c) are the results of SALD and Ground truth respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' dC dH dangle IGR [19] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='111 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='514 Our (+Llaplacian) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='274 Our (+Llaplacian + DES) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='133 TABLE 3: Ablation studies – We evaluate the quantitative per- formance of our method with/without components Llaplacian and dynamic edge sampling (DES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' {4, 096, 8, 192, 16, 384}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' (Results in Table 2 reveal a similar trend as from the previous stress test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Namely, our network, when trained on sparser point clouds, can still outperform these state-of- the-art methods [17]–[19] when they are tested on or trained on data at full resolution (16,384 points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Effect of Llaplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' To evaluate the effectiveness of loss Llaplacian, We incorporate this into another state-of-the-art method, SALD [17], This qualitative result is shown in Figure 11, we can find that, compared with the original algorithm, the re- construction quality can be effectively improved by incorporating Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' This is mainly because the differentiable Laplacian reg- ularizer can effectively alleviate the unsatisfactory reconstruction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Dynamic edge sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' We evaluate the effect of dynamic edge sampling strategy on reconstruction quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' We experiment with the dynamic edge sampling, while keeping all other parameters the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' From Table 1 and Figure 7 and 12 , we can see that at the sharp edges, we can effectively improve the quality of modeling compared with state-of-the-art methods (Table 1 (rows 1 3)) and the baseline method without dynamic edge sampling, this is largely due to thedynamic edge sampling strategy for sampling near the sharp edge of input point cloud, which can effectively avoid the regularizer from smoothing all regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' ECD IoU Precision Recall VCM [53] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='1925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='2238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='5998 EAR [54] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0071 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='1146 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='2399 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='1933 IGR [19] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0880 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='5620 Our 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='2375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='2665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='6934 TABLE 4: Comparison state-of-the-art edge recognition tech- niques - VCM [53], EAR [54], and IGR [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 5 CONCLUSION AND LIMITATION We present a differential Laplacian regularizer for neural implicit representation directly from input point cloud without normal supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' More specifically, we use the infinite differentiability property of implicit neural representation to propose a differen- tiable Laplacian regularizer, which can effectively alleviate the unsatisfactory reconstruction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Meanwhile, we propose a dynamic edge sampling strategy for sampling near the sharp edge of input point cloud, which can effectively avoid the Lapla- cian regularizer from smoothing all regions, so as to effectively improve the quality of reconstruction results while maintaining the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Moreover, the differentiable Laplacian regularizer can be easily and intuitively incorporated into implicit neural sur- face representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' We carefully evaluate its generation quality through a series of ablation studies, which show that our method significantly improve the quality of 3D reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' In addition to 3D reconstruction, our method can also be conveniently applied to other point cloud analysis tasks, including edge extraction and normal estimation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Our approach has a few limitations, which point out the directions of future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Some representative failure cases are shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' First, our method is prone to problems in the reconstruction of ultra-thin geometric structures, probably because the point cloud data is noisy, resulting in the geometric structure has been completely destroyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Second, Our method for extremely detailed structure may be overlooked, resulting in incorrect reconstruction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' ACKNOWLEDGEMENT We thank the anonymous reviewers for their valuable comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' This work was supported in part by Natural Science Foundation of China (62102328), and Fundamental Research Funds for the Central Universities (SWU120076).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' REFERENCES [1] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Chen and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Zhang, “Learning implicit fields for generative shape modeling,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 5939–5948.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Mescheder, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Oechsle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Niemeyer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Nowozin, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Geiger, “Occupancy networks: Learning 3d reconstruction in function space,” in Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 4460–4470.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Park, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Florence, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Straub, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Newcombe, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Lovegrove, “Deepsdf: Learning continuous signed distance functions for shape representation,” in Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 165–174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Chibane, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Alldieck, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Pons-Moll, “Implicit functions in feature space for 3d shape reconstruction and completion,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 6970–6981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [5] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Erler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Guerrero, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Ohrhallinger, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Mitra, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Wimmer, “Points2surf learning implicit surfaces from point clouds,” in European Conference on Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Springer, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 108–124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' + DES (a) SALD (c) GTJOURNAL OF LATEX CLASS FILES, 2019 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 12: Visualization examples of differential Laplacian regularizer (c) and dynamic edge sampling strategy (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 13: Failure cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Peng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Niemeyer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Mescheder, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Pollefeys, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Geiger, “Con- volutional occupancy networks,” in European Conference on Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Springer, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 523–540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Saito, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Huang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Natsume, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Morishima, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Kanazawa, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Li, “Pifu: Pixel-aligned implicit function for high-resolution clothed human digitization,” in Proceedings of the IEEE/CVF International Conference on Computer Vision, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 2304–2314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [8] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Ceylan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Mech, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Neumann, “Disn: Deep implicit surface network for high-quality single-view 3d reconstruction,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [9] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Fan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Su, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Guibas, “A point set generation network for 3d object reconstruction from a single image,” in Proceedings of the IEEE conference on computer vision and pattern recognition, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 605– 613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [10] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Choy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Gwak, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Chen, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Savarese, “3d-r2n2: A unified approach for single and multi-view 3d object reconstruction,” in European conference on computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Springer, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 628–644.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Xue, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Freeman, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Tenenbaum, “Learning a probabilistic latent space of object shapes via 3d generative-adversarial modeling,” Advances in neural information processing systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 29, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [12] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Groueix, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Fisher, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Kim, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Russell, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Aubry, “Atlasnet: A papier-mˆach´e approach to learning 3d surface generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' arxiv 2018,” arXiv preprint arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='05384, 1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [13] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Kato, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Ushiku, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Harada, “Neural 3d mesh renderer,” in Proceedings of the IEEE conference on computer vision and pattern recognition, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 3907–3916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Tang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Han, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Pan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Jia, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Tong, “A skeleton-bridged deep learning approach for generating meshes of complex topologies from single rgb images,” in Proceedings of the ieee/cvf conference on computer vision and pattern recognition, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 4541–4550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Tang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Han, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Tan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Tong, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Jia, “Skeletonnet: A topology- preserving solution for learning mesh reconstruction of object surfaces from rgb images,” IEEE transactions on pattern analysis and machine intelligence, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Atzmon, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Haim, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Yariv, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Israelov, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Maron, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Lipman, “Controlling neural level sets,” Advances in Neural Information Process- ing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Atzmon and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Lipman, “Sald: Sign agnostic learning with deriva- tives,” arXiv preprint arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='05400, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [18] ——, “Sal: Sign agnostic learning of shapes from raw data,” 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 2565–2574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Gropp, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Yariv, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Haim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Atzmon, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Lipman, “Im- plicit geometric regularization for learning shapes,” arXiv preprint arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='10099, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [20] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Zhao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Lei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Wen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Zhang, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Jia, “Sign-agnostic implicit learning of surface self-similarities for shape modeling and reconstruc- tion from raw point clouds,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 10 256–10 265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [21] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Sitzmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Martel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Bergman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Lindell, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Wetzstein, “Im- plicit neural representations with periodic activation functions,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 7462–7473, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [22] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Chen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Wu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Grinspun, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Zheng, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Chen, “Implicit neural spatial representations for time-dependent pdes,” arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='00124, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [23] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Yariv, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Gu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Kasten, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Lipman, “Volume rendering of neural implicit surfaces,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 34, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 4805–4815, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [24] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Ehret, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Mar´ı, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Facciolo, “Nerf, meet differential geometry!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='14938, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Zehnder, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Coros, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Thomaszewski, “Ntopo: Mesh-free topology optimization using implicit neural representations,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 34, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 10 368–10 381, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [26] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Carr, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Beatson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Cherrie, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Mitchell, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Fright, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' McCallum, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Evans, “Reconstruction and representation of 3d objects with radial basis functions,” in Proceedings of the 28th annual conference on Computer graphics and interactive techniques, 2001, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 67–76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Alexa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Behr, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Cohen-Or, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Fleishman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Levin, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Silva, “Computing and rendering point set surfaces,” IEEE Transactions on visualization and computer graphics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 3–15, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Kazhdan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Bolitho, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Hoppe, “Poisson surface reconstruc- (a) Input (b) IGR (e) GTa bJOURNAL OF LATEX CLASS FILES, 2019 10 tion,” in Proceedings of the fourth Eurographics symposium on Geometry processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 7, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Kazhdan and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Hoppe, “Screened poisson surface reconstruction,” ACM Transactions on Graphics (ToG), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 32, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 1–13, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [30] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Genova, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Cole, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Vlasic, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Sarna, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Freeman, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Funkhouser, “Learning shape templates with structured implicit func- tions,” in Proceedings of the IEEE/CVF International Conference on Computer Vision, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 7154–7164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [31] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Niemeyer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Mescheder, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Oechsle, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Geiger, “Occupancy flow: 4d reconstruction by learning particle dynamics,” in Proceedings of the IEEE/CVF international conference on computer vision, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 5379–5389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [32] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Oechsle, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Peng, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Geiger, “Unisurf: Unifying neural implicit surfaces and radiance fields for multi-view reconstruction,” in Proceed- ings of the IEEE/CVF International Conference on Computer Vision, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 5589–5599.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [33] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Tancik, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Srinivasan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Mildenhall, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Fridovich-Keil, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Raghavan, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Singhal, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Ramamoorthi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Barron, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Ng, “Fourier features let networks learn high frequency functions in low dimensional domains,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 7537– 7547, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [34] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Theobalt, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Komura, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Wang, “Neus: Learning neural implicit surfaces by volume rendering for multi-view reconstruction,” arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='10689, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [35] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Sitzmann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Zollh¨ofer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Wetzstein, “Scene representation networks: Continuous 3d-structure-aware neural scene representations,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [36] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Bergman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Kellnhofer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Wetzstein, “Fast training of neural lumigraph representations using meta learning,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 34, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 172–186, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [37] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Chibane, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Bansal, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Lazova, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Pons-Moll, “Stereo radiance fields (srf): Learning view synthesis for sparse views of novel scenes,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 7911–7920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [38] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Shih, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Lai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Liang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Huang, “Portrait neural radiance fields from a single image,” arXiv preprint arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='05903, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [39] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Jiang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Ji, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Han, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Zwicker, “Sdfdiff: Differentiable render- ing of signed distance fields for 3d shape optimization,” in Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 1251–1261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [40] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Kellnhofer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Jebe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Jones, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Spicer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Pulli, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Wetzstein, “Neural lumigraph rendering,” in Proceedings of the IEEE/CVF Confer- ence on Computer Vision and Pattern Recognition, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 4287–4297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [41] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Gu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Zaw Lin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Chua, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Theobalt, “Neural sparse voxel fields,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 15 651–15 663, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [49] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Wen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Chen, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Jia, “Deep optimized priors for 3d shape modeling and reconstruction,” in Proceedings of [42] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Martin-Brualla, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Radwan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Sajjadi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Barron, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Doso- vitskiy, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Duckworth, “Nerf in the wild: Neural radiance fields for unconstrained photo collections,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 7210–7219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [43] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Mildenhall, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Srinivasan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Tancik, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Barron, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Ramamoorthi, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Ng, “Nerf: Representing scenes as neural radiance fields for view synthesis,” arXiv preprint arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content='08934, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [44] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Niemeyer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Mescheder, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Oechsle, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Geiger, “Differentiable volumetric rendering: Learning implicit 3d representations without 3d supervision,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 3504–3515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [45] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Yariv, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Kasten, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Moran, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Galun, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Atzmon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Ronen, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Lipman, “Multiview neural surface reconstruction by disentangling geometry and appearance,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 2492–2502, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [46] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Jiang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Sud, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Makadia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Huang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Nießner, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Funkhouser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=', “Local implicit grid representations for 3d scenes,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 6001–6010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [47] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Tretschk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Tewari, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Golyanik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Zollh¨ofer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Stoll, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Theobalt, “Patchnets: Patch-based generalizable deep implicit 3d shape representations,” in European Conference on Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Springer, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 293–309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [48] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Chabra, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Lenssen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Ilg, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Schmidt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Straub, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Lovegrove, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Newcombe, “Deep local shapes: Learning local sdf priors for detailed 3d reconstruction,” in European Conference on Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Springer, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 608–625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 3269–3278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [50] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Tang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Lei, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Xu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Ma, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Jia, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Zhang, “Sa-convonet: Sign-agnostic optimization of convolutional occupancy networks,” in Proceedings of the IEEE/CVF International Conference on Computer Vision, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 6504–6513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [51] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Huang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Lee, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Mumford, “Statistics of range images,” in Proceedings IEEE Conference on Computer Vision and Pattern Recogni- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' CVPR 2000 (Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' PR00662), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' IEEE, 2000, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 324–331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [52] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Koch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Matveev, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Jiang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Williams, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Artemov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Burnaev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Alexa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Zorin, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Panozzo, “Abc: A big cad model dataset for geometric deep learning,” in The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [53] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' M´erigot, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Ovsjanikov, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Guibas, “Voronoi-based curvature and feature estimation from point clouds,” IEEE Transactions on Visual- ization and Computer Graphics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 17, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 743–756, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' [54] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Wu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Gong, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Cohen-Or, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Ascher, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' Zhang, “Edge-aware point set resampling,” ACM transactions on graphics (TOG), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 32, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} +page_content=' 1–12, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfCgsd/content/2301.04860v1.pdf'} diff --git a/8NE0T4oBgHgl3EQffgAo/content/tmp_files/2301.02404v1.pdf.txt b/8NE0T4oBgHgl3EQffgAo/content/tmp_files/2301.02404v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f122c7b3a787fadaac047da459e2a26b35a42e71 --- /dev/null +++ b/8NE0T4oBgHgl3EQffgAo/content/tmp_files/2301.02404v1.pdf.txt @@ -0,0 +1,2401 @@ +Multiplicative topological semimetals +Adipta Pal,1, 2 Joe H. Winter,1, 2, 3 and Ashley M. Cook1, 2 +1Max Planck Institute for Chemical Physics of Solids, N¨othnitzer Strasse 40, 01187 Dresden, Germany +2Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Strasse 38, 01187 Dresden, Germany +3SUPA, School of Physics and Astronomy, University of St. Andrews, North Haugh, St. Andrews KY16 9SS, UK +Exhaustive study of topological semimetal phases of matter in equilibriated electonic systems and myriad +extensions has built upon the foundations laid by earlier introduction and study of the Weyl semimetal, with +broad applications in topologically-protected quantum computing, spintronics, and optical devices. We extend +recent introduction of multiplicative topological phases to find previously-overlooked topological semimetal +phases of electronic systems in equilibrium, with minimal symmetry-protection. We show these multiplicative +topological semimetal phases exhibit rich and distinctive bulk-boundary correspondence and response signatures +that greatly expand understanding of consequences of topology in condensed matter settings, such as the limits +on Fermi arc connectivity and structure, and transport signatures such as the chiral anomaly. Our work therefore +lays the foundation for extensive future study of multiplicative topological semimetal phases. +I +Introduction +Topological semimetals are a vast family1,2 of topological +phases of matter studied in great depth experimentally3–11 in +the search for table-top, quasiparticle realizations of high- +energy physics12. At the simplest level, the topological de- +generacies of band structures in these topological semimetal +phases are realized quite generically if either time-reversal +symmetry13 or inversion symmetry14 are broken. +This is +the requirement for two-fold topological degeneracies char- +acteristic of the Weyl semimetal phase, although it is desire- +able to realize such degeneracies in the vicinity of the Fermi +level15,16, with minimal contributions to the Fermi surface +from other electronic states. In such cases, the key signa- +tures of Weyl semimetals are especially prominent, includ- +ing the distinguishing Fermi arc surface states17–20, and trans- +port signatures associated with the chiral anomaly21–27. Such +isolation of Weyl nodes in the vicinity of the Fermi level is +also facilitated—and the physics of topological semimetals +enriched—by systematic study of these topological phases in +compounds with wide-ranging phenomena, including super- +conductivity, strong spin-orbit coupling, and strong correla- +tions28–34. Much progress has also been made in identifying +other topological semimetals with more complex topological +degeneracies2,35–37 in electronic band structures, protected by +a large set of crystalline point group symmetries in combi- +nation with additional anti-unitary symmetries such as time- +reversal. +The present work returns to the foundations of topologi- +cal semimetal studies by introducing previously-unidentified +topological semimetal phases of matter of electronic systems +in equilibrium, which may then be generalized in the same +manner as outlined above. We do so by studying the first topo- +logical semimetal realizations of multiplicative topological +phases, a recently-identified set of topological phases of mat- +ter described by Bloch Hamiltonians in an infinitely-large, pe- +riodic bulk, which are symmetry-protected tensor products of +“parent” Bloch Hamiltonians. These multiplicative topolog- +ical semimetal (MTSM) phases are therefore straightforward +constructions described by tensor products of Weyl semimetal +Bloch Hamiltonians, yet exhibit rich phenomena distinct from +all other known topological semimetals. +We first review the Weyl semimetal phase and its canonical +models. We then construct the first examples of multiplicative +topological semimetal phases using these past results. The +multiplicative topological semimetals are then first character- +ized in the bulk, and their bulk-boundary correspondence es- +tablished. +II +Review of the Weyl semimetal phase and suitable +models for constructing multiplicative phases +The Weyl semimetal is a topologically non-trivial phase +of matter characterized by topologically-protected, doubly- +degenerate and linearly-dispersing band crossings in the Bril- +louin zone38. That is, these band-crossings, known as Weyl +points or nodes, cannot be removed from the electronic struc- +ture through smooth deformations of the Hamiltonian, but +rather only through mutual annihilation of the Weyl nodes, by +bringing two nodes of opposite topological charge to the same +point in the Brillouin zone to gap out these band-touchings. +When the Fermi level intersects only the Weyl nodes of this +semimetal phase, their low-energy physics dominates, yield- +ing a variety of intensely-studied exotic phenomena of interest +for applications. At the simplest level, the Weyl nodes serve +as quasiparticle, table-top realizations of Weyl fermions pre- +dicted in high-energy physics. However, they are also a start- +ing point in going well beyond high-energy physics, by tilting +the Weyl cone to realize a type-II Weyl semimetal phase39, +in which the low-energy physics of the Weyl nodes is not +Lorentz-invariant. +Weyl semimetal phases can be realized in effectively non- +interacting systems where certain discrete symmetries are bro- +ken rather than respected, in contrast to many other effectively +non-interacting topological phases. +They may be derived +through symmetry-breaking starting from the Dirac semimetal +state40,41, for instance, (which could be topologically-robust +or fine-tuned) by breaking either time-reversal symmetry T or +inversion symmetry I, which pulls the two Weyl nodes com- +prising the Dirac node away from one another in momentum- +space42. This phase, characterized by Weyl nodes in the Bril- +louin zone, is topologically stable so long as Weyl nodes of +opposite topological charge do not annihilate one another43. +I-breaking Weyl semimetal phases are of tremendous ex- +arXiv:2301.02404v1 [cond-mat.mes-hall] 6 Jan 2023 + +2 +perimental interest, but are described by Bloch Hamiltonian +models with four bands at minimum. A more natural starting +point in deriving multiplicative topological semimetal phases +is instead to use the minimal Weyl semimetal Bloch Hamil- +tonian achieved by breaking T , which possesses only two +bands. Such two-band models for the Weyl semimetal cor- +respond to the non-trivial homotopy group π3(S2) and, sim- +ilarly to the two-band Chern and Hopf insulators44 and the +two-band Kitaev chain model 45, may be combined using +known constructions to form a multiplicative counterpart of +the Weyl semimetal phase, the multiplicative Weyl semimetal +phase (MWSM). +We therefore consider a well-established two-band Bloch +Hamiltonian previously used in study of Weyl nodes, with +various instances of this model serving as the parents of the +MWSM. +HW SM(k) =t1 sin kxτ x + t2 sin kyτ y ++ t3(2 + γ − cos kx − cos ky − cos kz)τ z. +(1) +where the τ j (j = x, y, z) are the Pauli matrices in the orbital +basis. The two band spectrum, +E(k) = ± +� +t2 +1 sin2 kx + t2 +2 sin2 ky + ϵ(k)2, +ϵ(k) = t3(2 + γ − cos kx − cos ky − cos kz), +(2) +has two gapless nodes at k = (0, 0, ±k0), for cos k0 = γ. We +refer to these as the Weyl nodes. The equation of motion for +Bloch electrons in the k-space in the presence of Berry curva- +ture is represented by ˙r = vk + ˙k×F(k). For the equation of +motion to remain invariant under T -symmetry, one must have +the equality, F(k) = −F(−k). The breaking of T -symmetry, +then involves a minimum of two Weyl nodes with opposite +Berry curvature at opposite momenta. Therefore, close to the +Weyl nodes, we have, +H±(k) = ±t1kxτ x + t2kyτ y ± t3 sin k0kzτ z, +(3) +which in turn corresponds to the Berry curvatures, +F±(k)|0,0,±k0 = ± +t1t2t3 sin k0 +2[t1k2x + t2k2y + (t3 sin k0)2k2z]3/2 (kx, ky, kz). +(4) +The Chern number of the lower-energy band for the range, +kx = 0, ky = 0 and kz ∈ (−k0, k0) is C = ±1 depend- +ing on the direction of the magnetic field corresponding to the +monopoles at the two Weyl points. The Weyl nodes are in- +volved with exotic boundary states at surfaces perpendicular +to the z-axis, called the Fermi Arc surface states. For the case +where the surfaces are open in the x-direction, the surface dis- +persion is given by, +E(ky) = ±t2 sin ky, +(5) +and the arc-states, +Ψ(x, ky, kz) = e+ikyy+ikzz(e−λ1x − e−λ2x) 1 +√ +2 +� +1 +±i +� +. +In the k-space, this includes all contours cos ky + cos kz > +1 + cos k0. +III +Multiplicative Weyl Semimetal (MWSM) in the bulk +A protocol for constructing the child Hamiltonian for the +MWSM, Hc derived from Hp1 and Hp2 as first reported in +Cook and Moore46, is given as follows. Given two two-band +Bloch Hamiltonians Hp1 and Hp2 written in a general form, +with momentum-dependence suppressed, as +Hp1 = +� +a b +c d +� +; +Hp2 = +� +α β +γ δ +� +, +(6) +the multiplicative child Bloch Hamiltonian constructed +from these two parents can be written as Hc +12, where +Hc +12 = +� +� +� +aδ +−aγ +bδ +−bγ +−aβ +aα +−bβ +bα +cδ +−cγ +dδ +−dγ +−cβ +cα +−dβ +dα +� +� +� . +(7) +Expressing the two-band parent Bloch Hamiltonians +Hp1(k) and Hp2(k) more compactly as the following, +Hp1(k) = d1(k) · τ; +Hp2(k) = d2(k) · σ, +(8) +where d1(k) and d2(k) are momentum-dependent, three- +component vectors of scalar functions, and each of σ and τ is +the vector of Pauli matrices, the multiplicative child Hamilto- +nian may more compactly be written as, +Hc +12(k) = (d11, d21, d31) · τ ⊗ (−d12, d22, −d32) · σ, (9) +to highlight the tensor product structure of the child Hamil- +tonian, which can be symmetry-protected as discussed in ear- +lier work by Cook and Moore on multiplicative topological +phases, and therefore can describe phases of matter, even in +the presence of additional bands46. +The tensor-product structure guarantees that the energy +spectrum of the child Hamiltonian is a product of the energy +spectrum of Hp1(k), Ep1(k), and of Hp2(k), Ep2(k), respec- +tively, +Ec +12(k) = ±Ep1(k)Ep2(k). +(10) +This implies that bands of the child Hamiltonian dispersion +are at least doubly degenerate everywhere in the bulk Bril- +louin zone. +We will consider two cases in this work: (1) the Weyl +node separation of each parent is along one axis in the Bril- +louin zone, and (2) the axis along which Weyl nodes are +separated in one parent is perpendicular to the axis along +which Weyl nodes are separated in the other parent. Spectral +and magneto-transport properties differ significantly between +these two cases, as we will show, demonstrating the richness +of MTSM phases of matter. +A +Multiplicative Weyl Semimetal - parallel axis par- +ents +The construction of the MWSM for both parents along the +same axis is derived from two parent WSMs. As an example, + +3 +we consider the following parents and the resulting child: +Hp1(k) =t11 sin kxτ x + t21 sin kyτ y ++ t31(2 + γ1 − cos kx − cos ky − cos kz)τ z, +(11a) +Hp2(k) =t12 sin kxσx + t22 sin kyσy ++ t32(2 + γ2 − cos kx − cos ky − cos kz)σz, +(11b) +Hc(k) =[t11 sin kxτ x + t21 sin kyτ y ++ t31(2 + γ1 − cos kx − cos ky − cos kz)τ z] +⊗ [−t12 sin kxσx + t22 sin kyσy +− t32(2 + γ2 − cos kx − cos ky − cos kz)σz]. +(11c) +Each parent Hamiltonian realizes Weyl nodes at k += +� +0, 0, cos−1 γi +� +when −1 < γi < 1, (i = 1, 2). Examples of +such topologically non-trivial dispersion are shown in Fig. 1 +(a) and (b), respectively. +/2 0 +/2 +1 +0 +1 +E(k) +(a) WSM parent 1 +/2 0 +/2 +momentum kz +1 +0 +1 +E(k) +(b) WSM parent 2 +/2 +0 +/2 +momentum kz +1.0 +0.5 +0.0 +0.5 +1.0 +E(k) +(c) MWSM parallel child +band 1 +band 2 +band 3 +band 4 +FIG. 1: Dispersion E(k) for (a) WSM Parent Hamiltonian +with γ1 = 0.5 along kz and t11 = t21 = t31 = 1, (b) WSM +Parent Hamiltonian with γ2 = −0.5 along kz and +t12 = t22 = t32 = 1, and (c) the resulting MWSM parallel +Child Hamiltonian along kz. +From these parent Hamiltonian dispersions, we can find the +dispersion of the child. As given in Eq. 10, the bulk spectrum +is doubly degenerate and determined by the spectra of the par- +ent 1, Ep1(k), and parent 2, Ep2(k), respectively, which take +the following forms: +Ep1(k) = [t2 +11 sin2 kx + t2 +21 sin2 ky + ϵ1(k)2]1/2, +Ep2(k) = [t2 +12 sin2 kx + t2 +22 sin2 ky + ϵ2(k)2]1/2, +(12) +where ϵ1/2(k) = t31/2(2 + γ1/2 − cos kx − cos ky − cos kz). +For the sake of convenience, we refer to the MWSM with +Weyl node separation for each parent along the same axis in +the Brillouin zone (as in the case of parents given by Eq. 22a +and Eq. 22b) as MWSM||. For the MWSM|| bulk spectrum +given by Eq. 10 and Eq. 12, gapless points occur at the po- +sitions in the Brillouin zone where gapless points are present +for the parent systems. As γ1 and γ2 control separation of the +Weyl nodes in the Brillouin zone for the parents, they play a +major role in determining the number of nodes, the location +of the nodes, and the polynomial order of the nodes in the +Brillouin zone for the child. When γ1 = γ2, for instance, we +have two gapless points but the dispersion near the nodes is +quadratic. In contrast, for γ1 ̸= γ2 as for parents depicted +in Fig. 1 (a) and (b), the child MWSM|| has four nodes, and +bands disperse linearly in the vicinity of the nodes, as shown +in Fig. 1 (c). Each node is four-fold degenerate. +While such degeneracy naively suggests Dirac nodes or +Weyl nodes of higher charge, the multiplicative nodes are dis- +tinct in a number of ways. To examine this difference, we look +at the child Hamiltonian in the vicinity of each multiplicative +node for the case −1 < γ1 ̸= γ2 < 1. From the tensor product +structure, it easy to check that ∂E± +∂ki = const. which implies +that the dispersion is linear at each of the gapless nodes of the +MWSM. Therefore the possibility of a higher order Weyl node +is nullified. The position of each of the multiplicative nodes +are determined by the nodes in the respective parents. We re- +fer to (0, 0, ±k01) as the Weyl node positions derived from +the first parent, and (0, 0, ±k02) as the Weyl node positions +derived from the second parent. Here γi = cos k0i, (i = 1, 2). +If the gapless point is (0, 0, k02), then we define MWSM|| in +the vicinity as Hc +||,2, and, +Hc +||,2 = t31(γ1−γ2)τ z(−t12kxσx+t22kyσy−t32 sin k02¯kz,2σz), +(13) +where ¯kz,2 = (kz − k02). Surprisingly, this looks like a Dirac +semimetal Hamiltonian, whose Dirac node has been shifted in +k-space. Since it is no longer at the origin, the time-reversal +symmetry is broken. For the other node, γ1 = cos k01 for +(0, 0, k01), we define the multiplicative Hamiltonian in the +vicinity as Hc +||,1, so that, +Hc +||,1 = (t11kxτ x+t21kyτ y+t31 sin k01¯kz,1τ z)t32(γ1−γ2)σz, +(14) +where ¯kz,1d = (kz − k01) and contains off-diagonal terms for +the block Hamiltonian. But again, it is possible to perform +a similarity transformation on this Hamiltonian, in the form +U = R−1 +τ (θ, φ) ⊗ Rσ(θ, φ), so that we get another ‘shifted’ +Dirac semimetal type Hamiltonian, +¯Hc +||,1 = t32(γ1−γ2)τ z(t11kxσx+t21kyσy+t31 sin k01¯kz,1σz). +(15) +Again, the shift from the origin breaks the time-reversal sym- +metry of the original Dirac semimetal. It is therefore appropri- +ate to refer to the MWSM|| as possessing degeneracies con- +sisting of Weyl nodes, rather than possessing Dirac nodes, and +exhibit strikingly different physics as a result. +B +MWSM - perpendicular axis parents +Before characterizing bulk-boundary correspondence and +transport signatures of MTSMs, we explore further richness +of multiplicative constructions by considering cases where + +4 +parent Weyl nodes are separated along orthogonal axes in k- +space. As a specific case, we choose parent Hamiltonians such +that the first parent has Weyl node separation along the y-axis, +while the second one has Weyl node separation along the z- +axis, +Hp1(k) =t11 sin kxτ x + t21 sin kzτ y ++ t31(2 + γ1 − +� +i +cos ki)τ z, +(16a) +Hp2(k) =t12 sin kxσx + t22 sin kyσy ++ t32(2 + γ2 − +� +i +cos ki)σz. +(16b) +Again the bulk spectrum is derived from the tensor product +structure, +Ep1(k) = [t2 +11 sin2 kx + t2 +21 sin2 kz + ϵ2 +1(k)]1/2, +Ep2(k) = [t2 +12 sin2 kx + t2 +22 sin2 ky + ϵ2 +2(k)]1/2, +Ec +⊥k = ±Ep1(k)Ep2(k), +(17) +where ϵ1/2(k) = t31/32(2+γ1/2 −cos kx −cos ky −cos kz). +Examples of parent and child dispersion in this case are shown +in Fig. 2 for the values, γ1 = 0.5 and γ2 = −0.5. +We gain greater understanding of the multiplicative struc- +ture in this case by examining the low-energy expansion of +the Child Hamiltonian in the vicinity of its nodes. Taylor ex- +panding up to linear order around the point, (0, k0,1, 0) for +γ1 = cos k0,1, one gets, +Hc +⊥,1(k) =(t11kxτ x + t21kzτ y + t31 sin k0,1¯ky,1τ z) +⊗ (t22 sin k0,1σy − t32(γ2 − γ1)σz). +(18) +Similarly, expanding around (0, 0, k0,2) for γ2 = cos k0,2, we +get, +Hc +⊥,2(k) =(t21 sin k0,2τ y + t31(γ1 − γ2)τ z) +⊗ (−t12kxσx + t22kyσy − t32 sin k0,2¯kz,2σz). +(19) +One notices that Hc +⊥,2(k) is equivalent to a DSM when +γ1 = γ2. +C +Discrete Symmetries of the MWSM +The discrete symmetries satisfied by the parent WSMs include +invariance under particle-hole conjugation given by P = σxκ, +such that the Hamiltonian satisfies, +σxH∗ +1/2(k)σx = −H1/2(−k), +and invariance under spatial inversion given by I = σz, such +that the Hamiltonian satisfies, +σzH1/2(k)σz = H1/2(−k). +The MWSM|| or ⊥ child systems are instead invariant un- +der time reversal given by T = iτ xσxκ corresponding to the +transformation, +τ xσxH∗ +c(k)τ xσx = Hc(−k). +ky +1 +0 +1 +E +(a) +kz +1 +0 +1 +E +(b) +(0, 0, 0) +(0, 0, ) +(0, , ) +(0, , 0) +(0, 0, 0) +(0, , ) +k +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +E +(f) +ky +0 +kz +0 +E +5 +0 +5 +(e) +kz +0.5 +0.0 +0.5 +E +(c) +/20 +/2 +ky +0.5 +0.0 +0.5 +E +(d) +FIG. 2: Dispersion E(k) (t11 = t12 = 1, t21 = t22 = 1, +t31 = t32 = 1) for (a) WSM Parent Hamiltonian with +γ1 = 0.5 along ky, (b) WSM Parent Hamiltonian with +γ2 = −0.5 along kz and the resulting MWSM perpendicular +Child Hamiltonian along (c) kz and (d) kz. The energy +dispersion plotted along both ky and kz is shown in (e) and +the dispersion along a high-symmetry path in the first +quadrant of the two-dimensional (2d) BZ is shown in (f). +Inversion symmetry relates the nodes in the first quadrant to +those in the other quadrants, giving rise to four gapless nodes +in the 2d BZ. +They are also invariant under spatial inversion given by I = +τ zσz, corresponding to the transformation, +τ zσzHc(k)τ zσz = Hc(−k). +The MWSM should then satisfy the symmetry, T I, which +may also protect the Dirac semi-metal phase. Indeed, in some +cases, the Dirac Hamiltonian for the MWSM near the nodes +is reminiscent of the corresponding low-energy Hamiltonian +for a Dirac semi-metal. This invariance of the multiplicative +bulk Hamiltonian under products of transformations, which +leave each parent Hamiltonian invariant, is expected given the +multiplicative dependence of the child on the parents. + +5 +D +Bulk characterization of topology with Wilson loops +As calculated in Supplementary section S1, the Berry connec- +tion for the MWSM is given as +A = (A1,kx − A2,kx, A1,ky − A2,ky, A1,kz − A2,kz), (20) +where Aj,l = (i ⟨+j| ∂l |+j⟩ , i ⟨−j| ∂l |−j⟩). +Using this +expression for the Berry connection, we compute Wilson +loops and associated Wannier spectra by integrating over kx +for a given ky, as detailed in Alexandradinata et al47. +In +the parallel case illustrated in Fig. 3(a), the Wannier spectra +derived from Wilson loop calculations show that only in +regions where only one of the parent phases is non-trivial do +we get non-trivial Wannier spectra distinguished by π values +for Wannier charge centers. However, the Wannier spectra in +the region where each parent is topological appears trivial, +given the dependence of child Wannier spectra on parent +Wannier spectra distinctive of multiplicative topological +phases. We have referred to a pair of Weyl nodes of equal +and opposite topological charge as a ‘dipole’. We observe, +that the orientation of this dipole due to the two constituent +parents is important, as anti-parallel dipoles, as depicted +in Fig. +3(b), show non-trivial Wilson loop eigenvalues in +a region in the 2d BZ where neither of the parent systems +have non-trivial topological character. Analogous results for +the MWSM⊥ are shown in Fig. 4, although the Wannier +spectrum structure is far richer than in the parallel case. +/2 0 +/2 +kz +/2 +0 +/2 +ky +(a)WSM 1 x +/2 0 +/2 +kz +/2 +0 +/2 +ky +(b)WSM 2 x +/2 0 +/2 +kz +/2 +0 +/2 +ky +(c)MWSM pll +/2 0 +/2 +kz +/2 +0 +/2 +ky +(d)MWSM pll +0.25 +0.00 +0.25 +x +0.25 +0.00 +0.25 +x +0.50 +0.25 +0.00 +0.25 +0.50 +x, 1 +0.50 +0.25 +0.00 +0.25 +0.50 +x, 2 +(a) Each parent has Weyl node ‘dipole’ oriented in +ˆz direction. +/2 +0 +/2 +kz +/2 +0 +/2 +ky +(a)WSM 1 x +/2 +0 +/2 +kz +/2 +0 +/2 +ky +(b)WSM 2 x +/2 +0 +/2 +kz +/2 +0 +/2 +ky +(c)MWSM pll +/2 +0 +/2 +kz +/2 +0 +/2 +ky +(d)MWSM pll +0.25 +0.00 +0.25 +x +0.25 +0.00 +0.25 +x +0.50 +0.25 +0.00 +0.25 +0.50 +x, 1 +0.50 +0.25 +0.00 +0.25 +0.50 +x, 2 +(b) Parent 1 with dipole oriented along +ˆz direction and parent 2 +with dipole oriented in −ˆz direction. +FIG. 3: Wannier spectra in MWSM parallel for two filled +bands derived from Wilson loop around kx for parent 1 with +γ1 = −0.5 and parent 2 with γ2 = 0.5. The upper row (a) +and the lower row (b) have opposite orientation of the Weyl +node ’dipole’ for parent 2. Corresponding Wannier spectra of +the MWSM for the lowest-energy and second-lowest in +energy occupied bands is shown in (c) and (d), respectively. + +6 +/2 0 +/2 +kz +/2 +0 +/2 +ky +(a)WSM 1 x +/2 0 +/2 +kz +/2 +0 +/2 +ky +(b)WSM 2 x +/2 0 +/2 +kz +/2 +0 +/2 +ky +(c)MWSM perp +/2 0 +/2 +kz +/2 +0 +/2 +ky +(d)MWSM perp +0.25 +0.00 +0.25 +x +0.25 +0.00 +0.25 +x +0.50 +0.25 +0.00 +0.25 +0.50 +x, 1 +0.50 +0.25 +0.00 +0.25 +0.50 +x, 2 +(a) Parent 1 has Weyl node ‘dipole’ oriented along +ˆy direction and +parent 2 along −ˆz direction. +/2 0 +/2 +kz +/2 +0 +/2 +ky +(a)WSM 1 x +/2 0 +/2 +kz +/2 +0 +/2 +ky +(b)WSM 2 x +/2 0 +/2 +kz +/2 +0 +/2 +ky +(c)MWSM perp +/2 0 +/2 +kz +/2 +0 +/2 +ky +(d)MWSM perp +0.25 +0.00 +0.25 +x +0.25 +0.00 +0.25 +x +0.50 +0.25 +0.00 +0.25 +0.50 +x, 1 +0.50 +0.25 +0.00 +0.25 +0.50 +x, 2 +(b) Parent 1 has Weyl node ’dipole’ oriented in +haty direction and +parent 2 Weyl node dipole oriented in −ˆz direction. +FIG. 4: Wannier spectra in MWSM perpendicular for two +filled bands derived from Wilson loop around kx for parent 1 +with γ1 = −0.5 and parent 2 with γ2 = 0.5. The upper row +(a) and the lower row (b) have opposite orientation of the +Weyl node ’dipole’ for parent 2. Corresponding Wannier +spectra of the MWSM for the lowest-energy and +second-lowest in energy occupied bands is shown in (c) and +(d), respectively. +IV +MWSM with open-boundary conditions– +1 +Slab spectra of MWSM: +An important aspect of WSM physics is its distinctive +bulk-boundary correspondence: +Weyl nodes in the three- +dimensional bulk Brillouin zone serve as termination points of +topologically-protected boundary states known as Fermi arcs +when projected to a slab Brillouin zone corresponding to open +boundary conditions in one direction. We expect analogous +topologically-protected surface states in MTSMs and explore +possible realizations of these Fermi arc states in this section. +One might expect that the tensor product structure of the +multiplicative phases is visible in the surface spectrum of the +MWSM. Numerical simulations show that this is the case. For +the parent WSMs, the surface spectra is given as, E(ky) ∼ +sin(ky)(Fig. 5(a) and (b) and Fig. +6 2nd row) for nodes +along the z-axis and open boundaries along the x-direction +and E(kz) ∼ sin(kz)(Fig. 6 1st row) for nodes along the +y-axis and open boundaries along the x-direction. +Indeed, +corresponding surface spectra of child Hamiltonians depend +on these surface spectra in a multiplicative way. +Numeri- +cal simulation from Fig. 5 (c) shows that, for MWSM||, the +slab spectra disperses as E(ky) ∼ sin2(ky) for two parents +each with surface spectrum E(ky) ∼ sin(ky). In contrast, +Fig. 6 (c) shows that the surface spectrum instead disperses +as E(ky, kz) ∼ sin(ky) sin(kz) for MWSM⊥ when one par- +ent has the former surface spectrum and the other has the lat- +ter. We also show, for the case of each parent surface spec- +trum along kz, which exhibits flat bands between the two +Weyl nodes (Fig. 5(a) and (b)) corresponds to flat bands be- +tween all four gapless points in the MWSM parallel system +(Fig. 5(c)). However, fitting sin2(ky) curves to each of the +parallel and perpendicular MWSM spectra reveals that, except +in special cases when γ1 = γ2 where the fit is exact, the slab +spectra does not disperse as sin2(ky) and instead exhibits kz- +dependence. One can check this by comparing E vs. ky slab +spectra in the range −min(k0,1, k0,2) < kz < min(k0,1, k0,2) +and min(k0,1, k0,2) < kx < max(k0,1, k0,2). The spectra ap- +pears linear near zero in the latter case. +2 +Stability of surface states of MWSM +For the MWSM|| system, the low-energy expansion about +a node is reminiscent of a Dirac node, and it is therefore +possible to break apart the four-fold degeneracy at each +of the nodes by introducing an external magnetic field. +We introduce minimal coupling, ky → ky − eBx for the +MWSM|| to simulate the effect of applied magnetic field +on the spectral density of the Fermi arc surface states. We +observe that the Fermi arcs split but are not destroyed by the +applied field as in the case of the DSM. +3 +Fermi Arcs for the MWSM as a stack of MCIs: +WSMs can be interpreted as a set of Chern insula- +tors(CIs), +each +defined +in +a +2d +submanifold +of +the +3d BZ of the WSM (e.g., +each kx-ky +plane) for a +given value of kz, yielding a stack of CIs in the kz- +direction. +The Weyl nodes then correspond to topological + +7 +/2 +0 +/2 +ky +2 +1 +0 +1 +2 +E +(e)MWSM || +/2 +0 +/2 +ky +2 +1 +0 +1 +2 +E +(f)MWSM || +/2 +0 +/2 +kz +2 +1 +0 +1 +2 +E +(f)MWSM || +/2 +0 +/2 +ky +2 +0 +2 +E +(a)Parent 1 +/2 +0 +/2 +kz +2 +0 +2 +E +(b)Parent 1 +/2 +0 +/2 +ky +2 +0 +2 +E +(c)Parent 2 +/2 +0 +/2 +kz +2 +0 +2 +E +(d)Parent 2 +FIG. 5: Finite slab spectra(in x-direction, Lx = 80) along ky +(kz = 0) and kz(ky = 0) respectively for (a,b) WSM with +γ1 = −0.5, (c,d) WSM with γ2 = 0.5. In (e,f) the slab +spectra(Lx = 80) E vs. ky for the MWSM|| child created +from the above two parents for kz = 0 and kz = π +2 +respectively. (g) shows the slab spectra E vs. kz at ky = 0 +for the same MWSM|| child system. +phase +transitions—corresponding +to +gap-closings—in +the stack between intervals in kz +with topologically- +distinct CIs. +Specifically, +we use the SCZ model48, +/2 0 +/2 +kz +2.5 +0.0 +2.5 +E +(a) WSM parent 1 x +/2 0 +/2 +kz +2 +0 +2 +E +(b) WSM parent 2 +/2 0 +/2 +kz +2 +0 +2 +E += (c) MWSM perp. child +/2 0 +/2 +ky +2.5 +0.0 +2.5 +E +/2 0 +/2 +ky +2 +0 +2 +E +/2 0 +/2 +ky +2 +0 +2 +E +/2 0 +/2 +kz +2.5 +0.0 +2.5 +E +/2 0 +/2 +ky +2 +0 +2 +E +/2 0 +/2 +(kz + ky) +2 +0 +2 +E +FIG. 6: Finite slab spectra (in x direction, Nx = 80) with the +constituent parent Hamiltonians - WSM parent Hamiltonian +1 with γ1 = 0.5 and Weyl nodes along the ky-direction is +shown along column (a), WSM parent Hamiltonian 2 with +γ = −0.5 with Weyl nodes along the kz-direction along +column (b) and the MWSM perpendicular child Hamiltonian +along column (c). It is apparent how the surface spectra along +kz(for ky = 0) and ky(for kz = 0) combine multiplicatively +to create the surface spectra for the MWSM perpendicular +system. The lowest diagram along column (c) especially +shows the spectra along the diagonal kz + ky direction where +the component spectra sin(kz) and sin(ky) have combined to +produce sin(kz) sin(ky) as the leading term. +HCI = B(2+M −cos kx −cos ky)σz +sin kxσx +sin kyσy +in particle-hole space. In the WSM, the mass term is given +as M = γ − cos kz. Here, for the range, −1 < γ < 1, +kz ∈ [− cos−1 γ, cos−1 γ]. The Fermi arcs we observe in the +2d BZ defined in the ky − kz for open boundary conditions in +the x-direction are projections of the chiral edge states of the +slices of the corresponding CIs in the stack. +The multiplicative counterpart of a Chern insulator was +introduced recently by Cook and Moore46 as Multiplicative +Chern Insulators(MCIs). Here, one must notice that the MCI +has two mass terms derived from each of the parent systems, +one from each of the parent systems. Hence, there exists more +than one way to stack the MCIs in the kz direction. Either +parent mass term can be kz-dependent, for instance, or both +can be. Here, we have attached the momentum dependence +to both the mass terms, so that the difference in parent mass +parameters remains constant. We then characterize the multi- +plicative Fermi arc states by opening boundary conditions in +the x- and y-directions, and plotting the probability density for + +8 +the sum of 40 eigenstates nearest in energy to zero in Fig. 7 +for kz = 0 (a 2D submanifold of the BZ realizing an MCI) and +kz = π +2 (a 2D submanifold of the BZ that is topologically triv- +ial). For the former case shown in Fig. 7(a) and (b), the proba- +bility density in the corresponding child is localized at sites at +the boundary, but also at the sites adjacent to these sites. For +the latter case, parent 1 has edge states and parent 2 does not +as shown in Fig. 7(d) and (e). The resultant child probability +density shows low-energy states localize only at the boundary +sites as shown in Fig. 7(f). This localization behavior is sim- +ilar to that of the multiplicative Kitaev chain presented in a +second work by the present authors, where, if each parent is +topological, edge states are localized at lattice sites right at the +edge, but also at sites adjacent to these sites. We expect such +localization to protect the edge states from backscattering to +some extent, which we will explore in future work. +4 +Boundary states disconnected from bulk states— +The MCI introduced by Cook and Moore46 can exhibit topo- +logically robust yet floating edge states, which are separated +from the bulk by a finite energy gap. MTSMs constructed +from MCIs can inherit this exotic boundary state connectivity, +displaying boundary states disconnected from the bulk band +structure. +To realize such a MWSM, we first note when edge states +are disconnected from bulk states for the case of the MCI: +HCI,p1(k) =B1(2 + M1 − cos kx − cos ky)τ z ++ sin kxτ x + sin kyτ y, +(21a) +HCI,p2(k) =B2(2 + M2 − cos kx − cos ky)σz ++ sin kxσx + sin kyσy, +(21b) +HMCI,c(k) =[B1(2 + M1 − cos kx − cos ky)τ z ++ sin kxτ x + sin kyτ y] +⊗ [−B2(2 + M2 − cos kx − cos ky)σz +− sin kxσx + sin kyσy], +(21c) +the range of parameters over which this is possible is M1 ∈ +[−4, −2] and M2 ∈ [−2, 0] which corresponds to Chern num- +bers C = +1 and C = −1 respectively. We therefore con- +struct a MWSM for which the Weyl nodes of one parent WSM +are separated in k-space by a stack of Chern insulators, each +with total Chern number C = +1, and the Weyl nodes of the +other parent are separated by a stack of Chern insulators, each +with total Chern number C = −1. Comparing Eqn. (22a) +with (21a) and Eqn. (22b) with (21b), it is clear that, for each +Chern insulator in the stack, the following mapping holds, +Mi = γi − cos kz, i ∈ {1, 2}, and i labeling the parent. From +this mapping, it is not possible to have M2 ∈ (−4, −2) while +γi ∈ (−1, 1), i ∈ {1, 2}. We therefore generalize the map- +ping to the following form, Mi = γi − ri cos kz, i ∈ {1, 2}, +so that the parents and the child Hamiltonian for the MWSM +parallel are, +Hp1(k) =t11 sin kxτ x + t21 sin kyτ y ++ t31(2 + γ1 − cos kx − cos ky − r1 cos kz)τ z, +(22a) +Hp2(k) =t12 sin kxσx + t22 sin kyσy ++ t32(2 + γ2 − cos kx − cos ky − r2 cos kz)σz, +(22b) +Hc(k) =[t11 sin kxτ x + t21 sin kyτ y ++ t31(2 + γ1 − cos kx − cos ky − r1 cos kz)τ z] +⊗ [−t12 sin kxσx + t22 sin kyσy +− t32(2 + γ2 − cos kx − cos ky − r2 cos kz)σz]. +(22c) +To construct one parent with Chern number of this stack +non-trivial and opposite in sign to the Chern number of the +stack in the other parent, we first introduce some terminology. +We refer to the region between Weyl nodes including kz = 0 +as regular Weyl region (RWR) and the region including kz = +±π as the irregular Weyl region (IWR). The existence of Weyl +nodes requires |r1,2| ≥ 1 for |γ1,2| < 1. It is then possible to +realize a RWR with negative Chern number by varying r1,2, +so that γ1,2 − r1,2 cos kz ∈ (−4, −2). These RWRs—one of +each parent system—must then occur over the same interval in +kz, however, to realize topological floating surface states. We +set γ2 = 0 and r2 = 3, which means we have C = −1 for the +range [− cos−1( 2 +3), cos−1( 2 +3)] when M2 = γ2 − r2 cos kz ∈ +[−3, −2]. Then we must have γ1 = cos π +3 = 0.5 and r1 = 1 +so that in the region kz ∈ [− cos−1( 2 +3), cos−1( 2 +3)], we have +the same kind of MCI with edge states gapped from the bulk +as described in Cook and Moore46. These results are shown in +Fig. 8. +The MWSM⊥ case of topologically robust yet floating +Fermi arc surface states is constructed similarly, and we de- +fer thorough investigation of this case to later work. +V +Effect of Magnetic field on MWSM and Chiral +anomaly +We now investigate response signatures of MTSMs. +As +we consider MWSMs here, which may be constructed from +WSM parent systems, we focus in particular on the question +of whether there is a multiplicative generalization of the chi- +ral anomaly, one of the most important signatures of Weyl +semimetals: application of non-orthogonal electric and mag- +netic fields can pump electrons between Weyl nodes of oppo- +site chirality49. More specifically, applying an external mag- +netic field parallel to the axis along which Weyl nodes are +separated in k-space yields a single chiral Landau level near +each of the Weyl nodes. In Weyl semimetals, this suppresses +backscattering of electrons with opposite chirality, manifest- +ing as a negative magnetoresistance (MR). Weyl semimetals + +9 +FIG. 7: Probability densities of superposition of 40 edge state eigenvectors in a 30 × 30(Lx × Ly) square lattice at kz = 0 and +kz = π +2 for (a, d) Parent WSM 1 (γ1 = −0.5), (b, e) Parent WSM 2(γ2 = 0.5) and (c, f) MWSM || child (γ1 = −0.5 and +γ2 = 0.5) respectively. At kz = 0, both the parent systems are topological as seen from a visible edge state which results in +localization at both the edge and second last edge sites in the MWSM || child system. When kz = π +2 , the parent 1 is still +topological but the parent 2 is trivial as seen from the absence of edge states which results in localization only at the edge sites +of the MWSM || child system. +therefore serve as condensed matter platforms for study of the +chiral anomaly, also known as Adler-Bell-Jackiw anomaly, as- +sociated with the Standard Model of particle physics50. When +the external magnetic field is instead oriented perpendicular to +the k-space axis along which Weyl nodes are separated, semi- +classical calculations indicate the presence of quantum oscil- +lations in the density of states51, observable in magnetization, +magnetic torque, and MR measurements50. +To study the effects of external fields on the MWSM, +we first derive the Landau level structure for the the Weyl +semimetal in the cases of external magnetic fields applied par- +allel and perpendicular to the Weyl node axis. We can then +draw parallels between these results and their generalizations +in the case of the MWSM. +A +Chiral anomaly in WSM +To study the chiral anomaly in a WSM, we consider a par- +ticular Bloch Hamiltonian HW SM(k) characterizing a Weyl +semimetal phase and its expansion around the kz-axis, i.e. +k → (0, 0, kz) (up to 2nd order in kx and ky), +HW SM(k) =t(2 + γ − cos kx − cos ky − cos kz)σz ++ t′ sin kyσy + t′ sin kxσx, +≈t(Q + 1 +2(k2 +x + k2 +y))σz + t′kyσy + t′kxσx, +(23) +where Q = γ − cos kz. Applying the magnetic field, B = +Bˆz along the Weyl node axis, Peierls substitution changes the +momenta in the following way, kx → k′ +x = kx, ky → k′ +y = +ky + eBx, and kz → k′ +z = kz. The position-momentum +commutator, implies, [k′ +y, k′ +x] = ieB, so that, it is possible to +define bosonic ladder operators, +a = k′ +x − ik′ +y +√ +2eB +; +a† = k′ +x + ik′ +y +√ +2eB +; +[a, a†] = 1. +(24) + +10 +/2 +0 +/2 +ky +4 +2 +0 +2 +4 +E +(e)MWSM || +/2 +0 +/2 +kz +4 +2 +0 +2 +4 +E +(f)MWSM || +/2 +0 +/2 +ky +2.5 +0.0 +2.5 +E +(a)Parent 1 +/2 +0 +/2 +kz +2.5 +0.0 +2.5 +E +(b)Parent 1 +/2 +0 +/2 +ky +2.5 +0.0 +2.5 +E +(c)Parent 2 +/2 +0 +/2 +kz +2.5 +0.0 +2.5 +E +(d)Parent 2 +FIG. 8: Slab spectra along ky (subfigure a) and kz (subfigure +b) for WSM parent 1 with γ1 = 0, r1 = 3, and slab spectra +along ky (subfigure c) and kz (subfigure d) for WSM parent 2 +with γ2 = 2/3, r2 = 1, respectively. Corresponding slab +spectra for the MWSM|| with t11 = t12 = 1, t21 = t22 = 1, +t31 = t32 = 1 along (e) ky and (f) kz, respectively, with +edges separate from the bulk slab spectra along ky. +Applying Eqn.S18, after substituting k → k′, we get the fol- +lowing system which looks similar to the polariton conserving +Jaynes-Cummings Hamiltonian, +HW SM(k′) ≈ t(Q+eB(a†a+1 +2))σz+t′√ +2eB(aσ++a†σ−), +(25) +where σ± = +1 +2(σx ± iσy) are the spin ladder operators in +the basis {|+⟩ , |−⟩} of σz (σz |±⟩ = ± |±⟩). The ground +state from the above Hamiltonian is given by the eigenvec- +tor, |ψLLL⟩ = |0; −⟩ (states denoted as |n; s⟩ where n is the +bosonic number and s is the spin direction), which leads to the +lowest Landau level energy, +ELLL = −t(Q + 1 +2eB). +(26) +Near each of the Weyl nodes, it is easy to observe that |ψLLL⟩ +is chiral as shown in Fig. 9. The other Landau levels can be +derived by restricting to the two dimensional disjoint spaces, +{|n, −⟩ , |n − 1, +⟩}, parametrized by the bosonic number, n +so that in each such basis, the Hamiltonian is, +H(kz, n) = −teB +2 σ0 −t(Q+eBn)σz +t′√ +2eBnσx. (27) +The energy for the other Landau levels parametrized by n = +1, 2, ... is given by the eigenvalues of Eqn. 27, +EnLL = −teB +2 +± +� +t2(Q + eBn)2 + 2t′2eBn. +(28) +We have illustrated the analytically calculated Landau levels +in Fig. 9 and compared them to numerical calculations of Lan- +dau levels. The numerical computation involves plotting the +bands for the Peierls substituted Weyl semimetal with periodic +boundary conditions, say in the x-direction, and subjected to +magnetic field in integer multiples of 2π +L , where L is the size of +the lattice in the x-direction. We observe that the chiral Lan- +dau level from both analytical and numerical methods overlap, +with an approximate overlap of the other Landau levels since +we only considered till second order in kx and ky. +Next we consider the case when the magnetic field is di- +rected perpendicular to the Weyl node axis, say B = Bˆy. +Expanding the first line of Eqn. 23 around the Weyl node, +k = (0, 0, k0 = cos−1 γ) of positive chirality, and setting +t = t′ = 1, we get, +HW SM(k) ≈ sin k0(kz − k0)σz + kyσy + kxσx, +=⇒ H′ +W SM(k) ≈ − kyσz + kxσx + sin k0(kz − k0)σy, +(29) +where in the second line we have rotated the Hamiltonian to +a new basis via, σx → σz and σx → −σx. In the presence +of mentioned magnetic field perpendicular to the Weyl node +axis, the Peierls substitution is applied as kx → k′ +x = kx, +ky → k′ +y = ky and kz → k′ +z = kz − eBx. The commuta- +tion relation, [kx, sin k0(kz − k0 − eBx)] = ieB sin k0, then +constructs the bosonic ladder operators, +b = kz − k0 − eBx − ikx +√2eB sin k0 +; +b† = kz − k0 − eBx + ikx +√2eB sin k0 +. +(30) + +11 +2 +0 +2 +kz +4 +2 +0 +2 +4 +E +Numerical +Analytical +Chiral LLL(numerical) +Chiral LLL(analytical) +2 +0 +2 +ky +4 +2 +0 +2 +4 +E +Numerical +Analytical(near WN) +Chiral LLL(numerical) +Chiral LLL(analytical near WN) +FIG. 9: Landau Levels for the two-band Weyl Semimetal +calculated analytically from Eqn. 25 and numerically, with +t = 1 = t′, γ = 0 and B = 2π +51 ˆz(upper) and +B = 2π +51 ˆy(lower). The (black) bands indicate the numerically +calculated Landau levels and the (red) bands for the +analytically calculated Landau levels for n = 1, 2, ..., 19. The +(blue) band and the dotted (magenta) band is the Lowest +Landau Level(LLL) calculated numerically and analytically, +and is responsible for the Chiral anomaly in the upper figure +and Weyl orbits in the lower figure. +The system in Eqn. 29 then changes to, +HW SM(k′) ≈ −kyσz + +� +2eB sin k0(bσ+ + b†σ−). (31) +Similar +to +the +previous +case, +it +is +possible +to +re- +solve the Hamiltonian into the subspaces spanned by +{|n, −⟩ , |n − 1, +⟩}, where n is the eigenvalue of the num- +ber operator, b†b. We get two chiral lowest Landau levels with +energies, E = ±ky in the bulk, which are responsible for the +chiral anomaly50. +B +Chiral anomaly in the MWSM +We now study the response of the MWSM to external fields +for comparison with the signatures of the chiral anomaly in the +WSM reviewed in the previous section. We treat the MWSM +parallel and perpendicular cases separately, given the expected +sensitivity of the response to orientation of the axes of node +separation relative to the orientation of the external fields. +1 +Landau levels in the MWSM parallel system: +In Sec. III A we have derived the Dirac Hamiltonian for the +MWSM|| in the vicinity of each of its two nodes, (0, 0, k01) +and (0, 0, k02) derived respectively from each of its two par- +ents. +Hc +||,1(k) =(t′ +1kxτ x + t′ +1kyτ y ++ t1 sin k01¯kz,1τ z)t2(γ1 − γ2)σz, +Hc +||,2(k) =t1(γ1 − γ2)τ z(−t′ +2kxσx + t′ +2kyσy +− t2 sin k02¯kz,2σz) +In this section, we will only consider cases where γ1 ̸= γ2. To +investigate the response to external fields for the MWSM||, we +consider the effect of magnetic field along the Weyl node axis, +i.e., B = Bˆz. We use the exact Peierls substitution in Eqn. +S18, so that the two expressions above transform as follows, +Hc +||,1(k′) =t2(γ1 − γ2)(t1 sin k01¯kz,1τ z ++ t′ +1 +√ +2eB(aτ + + a†τ −))σz, +Hc +||,2(k′) =t1(γ1 − γ2)τ z(−t2 sin k01¯kz,2σz +− t′ +2 +√ +2eB(aσ− + a†σ+)). +(32) +Here τ ± = +1 +2(τ x ± iτ y) and σ± = +1 +2(σx ± iσy) are the +pseudo-spin ladder operators in the τ and σ spaces. The low- +est Landau levels from the above two expressions are given +below, +Hc +||,1 →E1,LLL = ±(γ1 − γ2)t1t2 sin k01¯kz,1, +|ψ1,LLL⟩ = |0; −, ±⟩ , +Hc +||,2 →E2,LLL = ∓(γ1 − γ2)t2t2 sin k02¯kz,2, +|ψ2,LLL⟩ = |0; ±, +⟩ . +(33) +One may notice that the eigenvector |0; −, +⟩ occurs in the +vicinity of each node. +Therefore, we calculate its energy +eigenvalue if one expands the MWSM parallel system in the +vicinity of the kz axis. The details of the calculation can be +found in the Supplementary Materials S3. We find the energy +is given as, +E|0;−,+⟩ = (Q1Q2 + 1 +2eB(Q1 + Q2)). +(34) +We show that this expression is consistent with the numer- +ically calculated Landau levels in Fig. 10. +The other chi- +ral Landau level consistent with the other two eigenvectors, +|0; −, −⟩ and |0; +, +⟩ near their respective Weyl nodes ap- +pears distinct from |0; −, +⟩ away from the Weyl nodes. + +12 +/2 +0 +/2 +kz +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +E +(a)Landau Levels for B = Bz in MWSM pll +/2 +0 +/2 +ky +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +E +(b)Landau Levels for B = By in MWSM pll +FIG. 10: The Landau Levels for the MWSM parallel +Hamiltonian with γ1 = −0.5, γ2 = 0.5, +t1 = t′ +1 = t2 = t′ +2 = 1 and B = 2π +80 . (a) and (b) show the +Landau levels for the magnetic field along the Weyl axis and +perpendicular to the Weyl axis (at Weyl node (0, 0, π +3 ) +respectively. The (red) bands refer to the lowest Landau +levels and the (black) bands form the bulk Landau levels. +In Fig. 10, for certain values of γ1 and γ2, it appears, at +first glance, as if there are two separate, chiral Landau lev- +els corresponding to |0; −, −⟩ and |1; , −, −⟩ respectively. All +four Weyl nodes are connected by each of these LLLs, how- +ever, and the two LLLs in combination furthermore account +for each chirality at each node. Although this is reminiscent of +the Dirac semimetal, there is potentially a distinction in char- +acter between the chiralities at each node. If each parent cor- +responds to a particular degree of freedom, for instance, and +these dofs are physically distinct from one another in some +sense, such as one parent corresponding to a two-fold valley +dof, and the other corresponding to a two-fold layer dof, the +chiral anomalies are inequivalent and do not compensate one +another as they would for a Dirac semimetal. +The two apparently ’separated’ LLLs seem to only scatter +between the Weyl nodes derived from their respective parents, +i.e. +intra-parent scattering. +Upon closer inspection, how- +ever, we see the intersection point between two apparently +separated Landau levels is actually a very small gap. +We +have verified in Supplementary Sec. S3, that the gap is fi- +nite in analytical calculations performed to second order in +momenta. The gap is an emergent feature of the multiplica- +tive chiral anomaly, with the single LLL reducing to |0; −, −⟩ +and |0; +, +⟩ at nodes associated with a particular parent. We +therefore interpret the multiplicative chiral anomaly as ex- +hibiting parent-graded features as well as emergent features +not associated with either individual parent. This is reminis- +cent of the topologically robust floating bands of the multi- +plicative Chern insulator46. +2 +Landau levels in the MWSM perpendicular system: +In Sec. III B, we had shown the linear expansion of the +MWSM⊥ Bloch Hamiltonian near each of the nodes corre- +sponding to one parent with Weyl nodes separated along the +ky axis and the other parent with Weyl nodes separated along +the kz axis in Eqn. 18 and 19. Without loss of generality, we +consider, t31 = t32 = t21 = t22 = 1 = t11 = t12. There +exists three separate cases one needs to check - (i) magnetic +field along the Weyl axis of the first parent, B = Bˆy, (ii) mag- +netic field along the Weyl axis of the second parent, B = Bˆz, +and (iii) magnetic field perpendicular to the Weyl axis of both +parents, B = Bˆx. +• Case 1 (B = Bˆy) : Substituting, kx → k′ +x = kx+eBz, +and using the bosonic ladder operators, a⊥,y = kz−ik′ +x +√ +2eB , +a† +⊥,y = kz+ik′ +x +√ +2eB , we have, from Eqn. 18, +H⊥,1(k′) =(sin k0,1(ky − k0,1)τ z + +√ +2eB(a⊥,yτ + + a† +⊥,yτ −) +⊗ (sin k0,1σy + (γ1 − γ2)σz). +(35) +For the expression from Eqn. +19, we instead con- +sider the following bosonic ladder operators, ˜a⊥,y = +˜ +kz−ik′ +x +√ +2eB sin k0,2 and ˜a† +⊥,y = +˜ +kz+ik′ +x +√ +2eB sin k0,2 , which gives us, +H⊥,2(k′) =(sin k0,2τ y + (γ1 − γ2)τ z) +⊗ (kyσy − +� +2eB sin k0,2(˜a⊥,yσ+ +y + ˜a† +⊥,yσ− +y )). +(36) +It is easy to find the lowest Landau level energies in +the vicinity of each node. From Eqn. 35 and 36, we +respectively have the LLL energies, +Ey,1,LLL = ± +� +sin2 k0,1 + (γ1 − γ2)2 sin k0,1(ky − k0,1), +Ey,2,LLL = ± +� +sin2 k0,2 + (γ1 − γ2)2ky. +(37) +We then find two ky-dependent chiral LLLs connecting +the nodes of the first parent, while we have two chiral +LLLs at ky = 0 due to the second parent, as shown in +Fig. 11 (a). The following result was expected if one +considers the Landau levels for the parents for different + +13 +directions of the magnetic field discussed in the previ- +ous subsection. For the MWSM perpendicular case, the +incident magnetic field in this case is both parallel to +the Weyl axis of parent 1 and perpendicular to the Weyl +axis of parent 2, so that we get both kinds of Landau +levels simultaneously. +• Case 2 (B = Bˆz): This produces results similar to Case +1, as shown in Fig. 11 (b). A similar calculation gives +us the lowest Landau level energies, +Ez,1,LLL = ± +� +sin2 k0,1 + (γ1 − γ2)2kz, +Ez,2,LLL = ± +� +sin2 k0,2 + (γ1 − γ2)2 sin k0,2(kz − k0,2). +(38) +VI +Discussion and Conclusion +In this work, we have introduced the previously-unidentified +multiplicative topological semimetal phases of matter, distin- +guished by Bloch Hamiltonians with a symmetry-protected +tensor product structure. Parent Bloch Hamiltonians, with ei- +ther one or both of the parents being topologically non-trivial, +may then be combined in the tensor product to realize mul- +tiplicative topological semimetal phases inheriting topology +from the parent states. +We consider foundational examples of multiplicative topo- +logical semimetals, with Bloch Hamiltonians constructed as +tensor products of two-band Bloch Hamiltonians, each char- +acterizing a Weyl semimetal phase. These multiplicative topo- +logical semimetal phases are protected by a combination of +symmetries of class DIII at the level of the child, and each +parent Bloch Hamiltonian in class D. Given the great vari- +ety of exotic crystalline point group symmetries considered to +protect most recently-identified topological semimetal phases, +it is remarkable that the symmetry-protection of these multi- +plicative semimetal phases is relatively simple, and suggests +many additional multiplicative semimetal phases may be iden- +tified by enforcing these many other symmetries on parent +Bloch Hamiltonians. +We first characterize multiplicative topological semimetal +phases in the bulk, showing the bulk spectrum of the child +Bloch Hamiltonian depends in a multiplicative way on the +spectra of the parent Bloch Hamiltonians: each eigenvalue of +the child, at a given point in k-space, is a product of eigen- +values, one from each parent. We furthermore consider two +different constructions of the multiplicative Weyl semimetal, +either for the case of each parent having a pair of Weyl nodes +separated along the same axis in k-space (parallel construc- +tion), or along perpendicular axes in k-space (perpendicu- +lar construction). For either construction, the multiplicative +symmetry-protected structure can then naturally yield nodal +degeneracies reminiscent of Dirac nodes or higher-charge +Weyl nodes. However, the multiplicative degeneracies are dis- +tinguished from these more familiar quasiparticles by distinc- +tive Wannier spectra signatures in the bulk, and exotic bulk- +boundary correspondence. Importantly, bulk characterization +/2 +0 +/2 +ky +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +E +(a)Landau Levels for B = By in MWSM perp +/2 +0 +/2 +kz +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +E +(b)Landau Levels for B = Bz in MWSM perp +FIG. 11: Landau levels for the MWSM perpendicular system +with γ1 = −0.5 and γ2 = 0.5 representing separation of +Weyl nodes along the ky and kz direction respectively. We +show two cases, (a) when magnetic field is along the +y-direction and (b) when magnetic field is along the +z-direction. Red lines indicate the chiral Landau levels. since +the magnetic field is paralle to one Weyl node separation and +perpendicular to another Weyl node separation, the above +behaviour is expected. +by Wannier spectra reveals a complex dependence of Berry +connection in the child Bloch Hamiltonian on Berry connec- +tion of each parent Bloch Hamiltonian, depending on whether +the parents are constructed with Weyl nodes separated along +the same axis in momentum-space (parallel) or not (perpen- +dicular). Additionally, the connectivity of Fermi arc surface +states for the multiplicative Weyl semimetal is far more com- +plex than in standard Dirac or Weyl semimetals, reflecting the +underlying dependence of the child topology on the topology +of the parents. An especially interesting example is the re- +alization of topologically-protected—yet floating—boundary +states. +Response signatures of the multiplicative Weyl semimetal + +14 +also inherit response signatures of the parents, with the po- +tential for emergent phenomena beyond that of either parent +individually. Here, we consider the multiplicative analog of +one of the defining response signatures of the Weyl semimetal, +the chiral anomaly, finding instead multiple co-existing chiral +anomalies graded by the parent degrees of freedom, as well as +emergent features in the Landau level structure not inherited +from a particular parent. In the case of parents correspond- +ing to effectively the same degree of freedom, the response +reduces to a signature reminiscent of a Dirac semimetal. This +brings up the possibility of controlled manipulation of partic- +ular properties of an electronic system similar to spintronics. +Future work will characterize other signatures of multi- +plicative topological semimetals anticipated given the exten- +sive characterization of Weyl and Dirac semimetals, particu- +larly optical and non-linear responses given the tremendous +interest in the bulk photovoltaic effect in Weyl semimetals, as +well as symmetry-protection of more exotic topological quasi- +particles, such as multiplicative generalizations of multifold +fermions or nodal lines. Given the immense body of work +on topological semimetals and the surprising consequences +of multiplicative topology for bulk-boundary correspondence, +nodal band structure, and Berry phase structure, our intro- +duction of previously-unidentified multiplicative topological +semimetals into the literature lays the foundation for consid- +erable future theoretical and experimental study, which will +greatly expand and deepen our understanding of topological +semimetal phases. +Acknowledgements - We gratefully acknowledge help- +ful discussions with J. E. Moore, I. A. Day, D. Varjas and +R. Calderon. +Correspondence +- +Correspondence +and +requests +for materials should be addressed to A.M.C. (email: +cooka@pks.mpg.de). +1 Alexey A. Soluyanov, Dominik Gresch, Zhijun Wang, QuanSheng +Wu, Matthias Troyer, Xi Dai, and B. Andrei Bernevig, “Type-II +Weyl semimetals,” Nature 527, 495–498 (2015). +2 Barry Bradlyn, Jennifer Cano, Zhijun Wang, M. G. Vergniory, +C. Felser, +R. J. Cava, +and B. Andrei Bernevig, “Be- +yond dirac and weyl fermions: +Unconventional quasiparti- +cles in conventional crystals,” Science 353, aaf5037 (2016), +https://www.science.org/doi/pdf/10.1126/science.aaf5037. +3 Shin-Ming Huang, Su-Yang Xu, Ilya Belopolski, Chi-Cheng Lee, +Guoqing Chang, BaoKai Wang, Nasser Alidoust, Guang Bian, +Madhab Neupane, Chenglong Zhang, Shuang Jia, Arun Bansil, +Hsin Lin, +and M. Zahid Hasan, “A Weyl Fermion semimetal +with surface Fermi arcs in the transition metal monopnictide TaAs +class,” Nature Communications 6, 7373 (2015). +4 B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, +P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, +T. Qian, and H. Ding, “Experimental discovery of weyl semimetal +taas,” Phys. Rev. X 5, 031013 (2015). +5 B. Q. Lv, N. Xu, H. M. Weng, J. Z. Ma, P. Richard, X. C. Huang, +L. X. Zhao, G. F. Chen, C. E. Matt, F. Bisti, V. N. Strocov, +J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, and H. Ding, “Ob- +servation of Weyl nodes in TaAs,” Nature Physics 11, 724–727 +(2015). +6 Su-Yang Xu, Nasser Alidoust, Ilya Belopolski, Zhujun Yuan, +Guang Bian, Tay-Rong Chang, Hao Zheng, Vladimir N. Strocov, +Daniel S. Sanchez, Guoqing Chang, Chenglong Zhang, Daixi- +ang Mou, Yun Wu, Lunan Huang, Chi-Cheng Lee, Shin-Ming +Huang, BaoKai Wang, Arun Bansil, Horng-Tay Jeng, Titus Neu- +pert, Adam Kaminski, Hsin Lin, Shuang Jia, and M. Zahid Hasan, +“Discovery of a Weyl fermion state with Fermi arcs in niobium ar- +senide,” Nature Physics 11, 748–754 (2015). +7 Su-Yang Xu, Ilya Belopolski, Nasser Alidoust, Madhab Neu- +pane, Guang Bian, Chenglong Zhang, Raman Sankar, Guoqing +Chang, Zhujun Yuan, Chi-Cheng Lee, Shin-Ming Huang, Hao +Zheng, Jie Ma, Daniel S. Sanchez, BaoKai Wang, Arun Ban- +sil, Fangcheng Chou, Pavel P. Shibayev, Hsin Lin, Shuang Jia, +and M. Zahid Hasan, “Discovery of a weyl fermion semimetal +and topological fermi arcs,” Science 349, 613–617 (2015), +https://www.science.org/doi/pdf/10.1126/science.aaa9297. +8 Sergey Borisenko, Quinn Gibson, Danil Evtushinsky, Volodymyr +Zabolotnyy, Bernd B¨uchner, and Robert J. Cava, “Experimental +realization of a three-dimensional dirac semimetal,” Phys. Rev. +Lett. 113, 027603 (2014). +9 Z. K. Liu, J. Jiang, B. Zhou, Z. J. Wang, Y. Zhang, H. M. Weng, +D. Prabhakaran, S-K. Mo, H. Peng, P. Dudin, T. Kim, M. Hoesch, +Z. Fang, X. Dai, Z. X. Shen, D. L. Feng, Z. Hussain, and Y. L. +Chen, “A stable three-dimensional topological Dirac semimetal +Cd3As2,” Nature Materials 13, 677–681 (2014). +10 Z. K. Liu, B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng, D. Prab- +hakaran, S.-K. Mo, Z. X. Shen, Z. Fang, X. Dai, Z. Hussain, and +Y. L. Chen, “Discovery of a three-dimensional topological dirac +semimetal, na¡sub¿3¡/sub¿bi,” Science 343, 864–867 (2014), +https://www.science.org/doi/pdf/10.1126/science.1245085. +11 Madhab Neupane, Su-Yang Xu, Raman Sankar, Nasser Ali- +doust, Guang Bian, Chang Liu, Ilya Belopolski, Tay-Rong Chang, +Horng-Tay Jeng, Hsin Lin, Arun Bansil, Fangcheng Chou, and +M. Zahid Hasan, “Observation of a three-dimensional topological +Dirac semimetal phase in high-mobility Cd3As2,” Nature Com- +munications 5, 3786 (2014). +12 Xiangang Wan, Ari M. Turner, Ashvin Vishwanath, and Sergey Y. +Savrasov, “Topological semimetal and fermi-arc surface states in +the electronic structure of pyrochlore iridates,” Phys. Rev. B 83, +205101 (2011). +13 A. A. Burkov and Leon Balents, “Weyl semimetal in a topological +insulator multilayer,” Phys. Rev. Lett. 107, 127205 (2011). +14 G´abor B. Hal´asz and Leon Balents, “Time-reversal invariant re- +alization of the weyl semimetal phase,” Phys. Rev. B 85, 035103 +(2012). +15 N. P. Armitage, E. J. Mele, and Ashvin Vishwanath, “Weyl and +dirac semimetals in three-dimensional solids,” Rev. Mod. Phys. +90, 015001 (2018). +16 Yan Sun, Shu-Chun Wu, Mazhar N. Ali, Claudia Felser, +and +Binghai Yan, “Prediction of weyl semimetal in orthorhombic +mote2,” Phys. Rev. B 92, 161107 (2015). +17 Yuval Baum, Erez Berg, S. A. Parameswaran, and Ady Stern, +“Current at a distance and resonant transparency in weyl semimet- +als,” Phys. Rev. X 5, 041046 (2015). +18 Andrew C Potter, Itamar Kimchi, +and Ashvin Vishwanath, +“Quantum oscillations from surface fermi arcs in weyl and dirac +semimetals,” Nature communications 5, 1–6 (2014). +19 Andr´as Gyenis, Hiroyuki Inoue, Sangjun Jeon, Brian B Zhou, +Benjamin E Feldman, Zhijun Wang, Jian Li, Shan Jiang, Quinn D + +15 +Gibson, Satya K Kushwaha, Jason W Krizan, Ni Ni, Robert J +Cava, B Andrei Bernevig, and Ali Yazdani, “Imaging electronic +states on topological semimetals using scanning tunneling mi- +croscopy,” New Journal of Physics 18, 105003 (2016). +20 Rajib Batabyal, +Noam Morali, +Nurit Avraham, +Yan Sun, +Marcus +Schmidt, +Claudia +Felser, +Ady +Stern, +Binghai +Yan, +and Haim Beidenkopf, “Visualizing weakly bound +surface +fermi +arcs +and +their +correspondence +to +bulk +weyl +fermions,” +Science +Advances +2, +e1600709 +(2016), +https://www.science.org/doi/pdf/10.1126/sciadv.1600709. +21 H.B. Nielsen and Masao Ninomiya, “The adler-bell-jackiw +anomaly and weyl fermions in a crystal,” Physics Letters B 130, +389–396 (1983). +22 D. T. Son and B. Z. Spivak, “Chiral anomaly and classical nega- +tive magnetoresistance of weyl metals,” Phys. Rev. B 88, 104412 +(2013). +23 S. A. Parameswaran, T. Grover, D. A. Abanin, D. A. Pesin, and +A. Vishwanath, “Probing the chiral anomaly with nonlocal trans- +port in three-dimensional topological semimetals,” Phys. Rev. X +4, 031035 (2014). +24 Xiaochun Huang, Lingxiao Zhao, Yujia Long, Peipei Wang, Dong +Chen, Zhanhai Yang, Hui Liang, Mianqi Xue, Hongming Weng, +Zhong Fang, Xi Dai, +and Genfu Chen, “Observation of the +chiral-anomaly-induced negative magnetoresistance in 3d weyl +semimetal taas,” Phys. Rev. X 5, 031023 (2015). +25 Cheng-Long Zhang, Su-Yang Xu, Ilya Belopolski, Zhujun Yuan, +Ziquan Lin, Bingbing Tong, Guang Bian, Nasser Alidoust, +Chi-Cheng Lee, Shin-Ming Huang, Tay-Rong Chang, Guoqing +Chang, Chuang-Han Hsu, Horng-Tay Jeng, Madhab Neupane, +Daniel S. Sanchez, Hao Zheng, Junfeng Wang, Hsin Lin, Chi +Zhang, Hai-Zhou Lu, Shun-Qing Shen, Titus Neupert, M. Za- +hid Hasan, and Shuang Jia, “Signatures of the Adler–Bell–Jackiw +chiral anomaly in a Weyl fermion semimetal,” Nature Communi- +cations 7, 10735 (2016). +26 Chandra Shekhar, Ajaya K. Nayak, Yan Sun, Marcus Schmidt, +Michael Nicklas, Inge Leermakers, Uli Zeitler, Yurii Skourski, +Jochen Wosnitza, Zhongkai Liu, Yulin Chen, Walter Schnelle, +Horst Borrmann, Yuri Grin, Claudia Felser, +and Binghai Yan, +“Extremely large magnetoresistance and ultrahigh mobility in the +topological Weyl semimetal candidate NbP,” Nature Physics 11, +645–649 (2015). +27 Frank Arnold, Chandra Shekhar, Shu-Chun Wu, Yan Sun, Ri- +cardo Donizeth dos Reis, Nitesh Kumar, Marcel Naumann, +Mukkattu O. Ajeesh, Marcus Schmidt, Adolfo G. Grushin, +Jens H. Bardarson, Michael Baenitz, Dmitry Sokolov, Horst Bor- +rmann, Michael Nicklas, Claudia Felser, Elena Hassinger, and +Binghai Yan, “Negative magnetoresistance without well-defined +chirality in the Weyl semimetal TaP,” Nature Communications 7, +11615 (2016). +28 Tanja Graf, Claudia Felser, and Stuart S.P. Parkin, “Simple rules +for the understanding of heusler compounds,” Progress in Solid +State Chemistry 39, 1–50 (2011). +29 N. P. Butch, P. Syers, K. Kirshenbaum, A. P. Hope, +and +J. Paglione, “Superconductivity in the topological semimetal +yptbi,” Phys. Rev. B 84, 220504 (2011). +30 F. F. Tafti, Takenori Fujii, A. Juneau-Fecteau, S. Ren´e de Cotret, +N. Doiron-Leyraud, Atsushi Asamitsu, +and Louis Taillefer, +“Superconductivity in the noncentrosymmetric half-heusler com- +pound luptbi: A candidate for topological superconductivity,” +Phys. Rev. B 87, 184504 (2013). +31 Y. Pan, A. M. Nikitin, T. V. Bay, Y. K. Huang, C. Paulsen, B. H. +Yan, and A. de Visser, “Superconductivity and magnetic order +in the noncentrosymmetric half-heusler compound erpdbi,” Euro- +physics Letters 104, 27001 (2013). +32 Yasuyuki Nakajima, Rongwei Hu, Kevin Kirshenbaum, Alex +Hughes, Paul Syers, Xiangfeng Wang, Kefeng Wang, Renx- +iong Wang, Shanta R. Saha, Daniel Pratt, Jeffrey W. Lynn, +and Johnpierre Paglione, “Topological ¡i¿r¡/i¿pdbi half-heusler +semimetals: +A new family of noncentrosymmetric mag- +netic superconductors,” Science Advances 1, e1500242 (2015), +https://www.science.org/doi/pdf/10.1126/sciadv.1500242. +33 Z. Fisk, P. C. Canfield, W. P. Beyermann, J. D. Thompson, M. F. +Hundley, H. R. Ott, E. Felder, M. B. Maple, M. A. Lopez de la +Torre, P. Visani, and C. L. Seaman, “Massive electron state in +ybbipt,” Phys. Rev. Lett. 67, 3310–3313 (1991). +34 Jason K. Kawasaki, Abhishek Sharan, Linda I. M. Johans- +son, Martin Hjort, Rainer Timm, Balasubramanian Thiagara- +jan, Brian D. Schultz, Anders Mikkelsen, Anderson Janotti, +and Chris J. Palmstrøm, “A simple electron counting model for +half-heusler surfaces,” Science Advances 4, eaar5832 (2018), +https://www.science.org/doi/pdf/10.1126/sciadv.aar5832. +35 Shuichi Murakami, “Phase transition between the quantum spin +hall and insulator phases in 3d: emergence of a topological gap- +less phase,” New Journal of Physics 9, 356 (2007). +36 Zhijun Wang, Yan Sun, Xing-Qiu Chen, Cesare Franchini, Gang +Xu, Hongming Weng, Xi Dai, and Zhong Fang, “Dirac semimetal +and topological phase transitions in A3bi (a = Na, k, rb),” Phys. +Rev. B 85, 195320 (2012). +37 S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele, and +A. M. Rappe, “Dirac semimetal in three dimensions,” Phys. Rev. +Lett. 108, 140405 (2012). +38 Xiangang Wan, Ari M Turner, Ashvin Vishwanath, and Sergey Y +Savrasov, “Topological semimetal and fermi-arc surface states in +the electronic structure of pyrochlore iridates,” Physical Review B +83, 205101 (2011). +39 Alexey A Soluyanov, Dominik Gresch, Zhijun Wang, QuanSheng +Wu, Matthias Troyer, Xi Dai, and B Andrei Bernevig, “Type-ii +weyl semimetals,” Nature 527, 495–498 (2015). +40 Zhijun Wang, Yan Sun, Xing-Qiu Chen, Cesare Franchini, Gang +Xu, Hongming Weng, Xi Dai, and Zhong Fang, “Dirac semimetal +and topological phase transitions in a 3 bi (a= na, k, rb),” Physical +Review B 85, 195320 (2012). +41 Steve M Young, Saad Zaheer, Jeffrey CY Teo, Charles L Kane, +Eugene J Mele, and Andrew M Rappe, “Dirac semimetal in three +dimensions,” Physical review letters 108, 140405 (2012). +42 AA Zyuzin, Si Wu, and AA Burkov, “Weyl semimetal with bro- +ken time reversal and inversion symmetries,” Physical Review B +85, 165110 (2012). +43 Takahiro Morimoto and Akira Furusaki, “Weyl and dirac +semimetals with z 2 topological charge,” Physical Review B 89, +235127 (2014). +44 Joel E. Moore, Ying Ran, and Xiao-Gang Wen, “Topological sur- +face states in three-dimensional magnetic insulators,” Phys. Rev. +Lett. 101, 186805 (2008). +45 A Yu Kitaev, “Unpaired Majorana fermions in quantum wires,” +Physics-Uspekhi 44, 131–136 (2001). +46 Ashley M. Cook and Joel E. Moore, “Multiplicative topological +phases,” Communications Physics 5, 262 (2022). +47 A. Alexandradinata, Xi Dai, and B. Andrei Bernevig, “Wilson- +loop characterization of inversion-symmetric topological insula- +tors,” Phys. Rev. B 89, 155114 (2014). +48 Xiao-Liang Qi, Yong-Shi Wu, and Shou-Cheng Zhang, “Topolog- +ical quantization of the spin hall effect in two-dimensional param- +agnetic semiconductors,” Physical Review B 74, 085308 (2006). +49 Shuang Jia, Su-Yang Xu, and M. Zahid Hasan, “Weyl semimet- +als, Fermi arcs and chiral anomalies,” Nature Materials 15, 1140– +1144 (2016). +50 Binghai Yan and Claudia Felser, “Topological materials: Weyl + +16 +semimetals,” Annual Review of Condensed Matter Physics +8, 337–354 (2017), https://doi.org/10.1146/annurev-conmatphys- +031016-025458. +51 Andrew C. Potter, Itamar Kimchi, +and Ashvin Vishwanath, +“Quantum oscillations from surface Fermi arcs in Weyl and Dirac +semimetals,” Nature Communications 5, 5161 (2014). +52 Yifei Guan, Adrien Bouhon, and Oleg V Yazyev, “Landau levels +of the euler class topology,” Physical Review Research 4, 023188 +(2022). + +17 +Supplemental material for “Multiplicative topological semimetals” +Adipta Pal1,2, Joe H. Winter1,2,3, and Ashley M. Cook1,2,∗ +1Max Planck Institute for Chemical Physics of Solids, N¨othnitzer Strasse 40, 01187 Dresden, Germany +2Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Strasse 38, 01187 Dresden, Germany +3SUPA, School of Physics and Astronomy, University of St. Andrews, North Haugh, St. Andrews KY16 9SS, UK +∗Electronic address: cooka@pks.mpg.de +S1 +Wilson loops for multiplicative Weyl semi-metal: +Labelling the parent Hamiltonians as Hp1 = (d1x, d1y, d1z) · τ and Hp2 = (d2x, d2y, d2z) · σ, with eigenvectors, {|+1⟩ |−1⟩} +and {|+2⟩ , |−2⟩} respectively, the child Hamiltonian is given by Hc = Hp1 ⊗ H′ +p2, where H′ +p2 = (−d2x, d2y, −d2z) · σ. The +ground state subspace of the child Hamiltonian is then spanned by, {|+1⟩ |−2⟩′ , |−1⟩ |+2⟩′} = {|+1⟩ |+2⟩, |−1⟩ |−2⟩}, where +|ψ⟩ denotes complex conjugation and ’ denotes an eigenstate of H′ +p2. The non-abelian Berry connection is then given as follows: +Aµ =i +� +⟨+1, +2| ∂µ |+1, +2⟩ +⟨−1, −2| ∂µ |−1, −2⟩ +� += i +� +⟨+1| ∂µ |+1⟩ +⟨−1| ∂µ |−1⟩ +� ++ i +� +⟨+2|∂µ|+2⟩ +⟨−2|∂µ|−2⟩ +� +, += +�A+ +1,µ − A+ +2,µ +A− +1,µ − A− +2,µ +� +, +(S1) +where Al +j,µ = i ⟨lj| ∂µ |lj⟩. For Berry connection around a loop in the Brillouin zone, the values of µ are {kx, ky, kz} for a 3d +Brillouin Zone. This clearly shows the difference between the parallel multiplicative phases and the perpendicular multiplicative +phases. For a 1d BZ, as shown in past work46, the connection for parallel MKC is A = (A1,kx − A2,kx, 0, 0), while for +the perpendicular MKC it is, A = (A1,kx, −A2,ky, 0). For 2d or 3d parent systems, it then becomes very straightforward to +extrapolate this trend such that the Berry connection looks qualitatively like the combination of the parallel and perpendicular +MKC connections based on which directions the parents have in common. This is particularly interesting for the case of parallel +and perpendicular Multiplicative Chern Insulators(MCIs), where parent CIs are each defined over a 2d BZ, and the parents can +share one or two axes. We illustrate the MCI parallel with two parent CIs on the x-y plane. The resultant Berry connection is +then A = (A1,kx − A2,kx, A1,ky − A2,ky, 0). The MCI perpendicular on the other hand is constructed with one parent in the x-y +plane and another in the x-z plane. The resulting Berry connection is then, A = (A1,kx − A2,kx, A1,ky, −A2,kz). The MWSM, +on the other hand, is a 3d system, so we instead consider parent Weyl nodes separated along parallel or perpendicular axes in +k-space. As explained in the main text, the parallel MWSM has parent Weyl nodes separated along the same axis in k-space +(the kz axis) while the perpendicular MWSM has parent 1 and parent 2 Weyl nodes separated along the ky-axis and kz-axis, +respectively. The resultant Berry connection is then, A = (A1,kx − A2,kx, A1,ky − A2,ky, A1,kz − A2,kz). +S2 +Calculation for the surface state spectrum of MWSM: +We write down here the derivation for the surface state energy for the MWSM parallel and MWSM perpendicular Hamiltonians, +for the case of open boundary conditions in the ˆx direction and periodic boundary conditions in the ˆy and ˆz directions. First, we +briefly specify how such a calculation should be done for the two band Weyl semi-metal. +A +Slab spectra for WSM: +We start by writing down the WSM Hamiltonian used, +HW SM(k) =t3(2 + γ − cos kx − cos ky − cos kz)σz + t2 sin kyσy + t1 sin kxσx, +=t3(f − cos kx)σz + t2 sin kyσy + t1 sin kxσx, +(S2) +where f = 2 + γ − cos ky − cos kz. Surface states decay into the bulk, so for open boundaries in the x-direction, we carry out +the transformation, kx → iq for edge states on the left side (x = 0), so that, +HW SM(iq, ky, kz) = t3(f − cosh q)σz + t2 sin kyσy + it1 sinh qσx. +(S3) +We claim that the determinant derived from the matrix due to the following limit must be zero, +lim +q1→q2 +H(iq1) − H(iq2) +2 sinh q− += −t3 sinh q+σz + it1 cosh q+σx, +(S4) +where q± = 1 +2(q1 ± q2). Carrying out the determinant, we get the following two conditions, +t3 sinh q+ = ±t1 cosh q+. +(S5) + +18 +Choosing the + sign, the RHS in Eqn. S4 becomes, −t1 cosh q+(σz − iσx), so that the null eigenvector derived from it is one +of the eigenvectors for the surface spectra, +|ψ+⟩ = +1 +√ +2 +� +1 +i +� +. +(S6) +The energy corresponding to this eigenvector can be found by solving the eigenvalue for the RHS in Eqn. S3 with the above +eigenvector. This gives us the eigen-energy, E, and the equation to determine the eigen-function, for the left boundary +E = t2 sin ky, +(S7a) +(t3 + t1)e−2q + 2fe−q + (t3 − t1) = 0, +=⇒ e−q± = −f ± +� +f 2 − (t2 +3 − t2 +1) +(t3 + t1) +, +Ψ+(x, y, z) ∼ (e−q+x − e−q−x)eikyy+ikzz |ψ+⟩ . +(S7b) +The eigen-function in the last line has the following form based on the boundary condition on the left edge, Ψ(x = 0) = 0. The +other edge can be derived similarly by shifting x → L + 1 − x where L is the length of the system along the x-direction. +B +Slab spectra for MWSM parallel: +We use the same method as in section S2A of the supplementary materials to derive surface states and spectra for the MWSM +parallel system. The Hamiltonian is given as follows, +HMW SM||(k) = [t31(f1 − cos kx)τ z + t21 sin kyτ y + t11 sin kxτ x] ⊗ [−t32(f2 − cos kx)σz + t22 sin kyσy − t12 sin kxσx], +(S8) +where f1/2 = 2 + γ1/2 − cos ky − cos kz. To ease our calculations, we carry out the following rotation on the four band basis, +τ z → τ y, τ y → −τ z and σz → −σy, σy → −σz. The Hamiltonian then becomes, +HMW SM||(k) =[t31(f1 − cos kx)τ y − t21 sin kyτ z + t11 sin kxτ x] ⊗ [−t32(f2 − cos kx)σy − t22 sin kyσz − t12 sin kxσx], +=[t31(f1 − cos kx)τ y + t11 sin kxτ x][−t32(f2 − cos kx)σy − t12 sin kxσx] +− t21 sin kyτ z[−t32(f2 − cos kx)σy − t12 sin kxσx] − t22 sin ky[t31(f1 − cos kx)τ y + t11 sin kxτ x]σz ++ t21t22 sin2 kyτ zσz. +(S9) +Again, without loss of generality, we set t11 = t21 = t31 = 1 = t32 = t22 = t12. The edge modes on the left edge (x = 0), +require we carry out the substitution, kx → iq, and the Hamiltonian is now, +HMW SM||(iq, ky, kz) =[(f1 − cosh q)τ y + i sinh qτ x][−(f2 − cosh q)σy − i sinh qσx] +− sin kyτ z[−(f2 − cosh q)σy − i sinh qσx] − sin ky[(f1 − cosh q)τ y + i sinh qτ x]σz ++ sin2 kyτ zσz. +(S10) +Carrying out our previous limit on the rotated Hamiltonian above, we get the following matrix, +lim +q1→q2 +HMW SM||(iq1) − HMW SM||(iq2) +2 sinh q− += +� +� +� +0 +i sin kyS+ +−i sin kyS+ +S+(−(f1 + f2) + 2S+) +i sin kyS− +0 +−f1S− + f2S+ +i sin kyS+ +−i sin kyS− +f1S+ − f2S− +0 +−i sin kyS+ +S−((f1 + f2) − 2S−) +i sin kyS− +−i sin kyS− +0 +� +� +� , +(S11) +where S± = cosh q+ ± sinh q+. The determinant of the RHS of Eqn. S11 must be zero, i.e., we have the condition, +S−S+[sin2 kyS−(f1 + f2 − 2S+)(f1 − f2)(S− + S+) − sin2 kyS+(f1 + f2 − 2S−)(f1 + f2)(S+ − S−) +− (f1 + f2 − 2S+)(f1 + f2 − 2S−)(f1S− − f2S+)(−f2S− + f1S+)] = 0 +(S12) +Let us start with the first condition, S− = 0. The RHS of Eqn. S11 then becomes, +lim +q1→q2 +HMW SM||(iq1) − HMW SM||(iq2) +2 sinh q− +=S+(−(f1 + f2) + 2S+)τ +σ+ + f2S+τ +σ− + f1S+τ −σ+ ++ i sin kyS+τ zσ+ − i sin kyS+τ +σz, +(S13) + +19 +where τ ± = 1 +2(τ x ±iτ y) and σ± = 1 +2(σx ±iσy) are the two level ladder operators. Here, if {|+⟩ , |−⟩} are eigen-vectors of τ z, +then τ + |−⟩ = |+⟩, τ + |+⟩ = 0, τ − |−⟩ = 0 and τ − |+⟩ = |−⟩. Similar relations exist for the σ counterpart. |ψ1⟩ = |+⟩⊗|+⟩ is +a null eigen-vector to the above expression on the RHS. We solve for the energy eigenvalue first for the special case γ1 = γ2.Then +from HMW SM||(iq, ky, kz) in Eqn. S10 due to the eigen-vector |ψ1⟩, we have the energy and the condition, +E = sin2 ky; +(S14a) +(f1 − cosh q + sinh q)(f2 − cosh q + sinh q) = 0. +(S14b) +S3 +Landau Level repulsion in the MWSM parallel system: +We start with the MWSM parallel case, +HMW SM,||(k) =[t1(2 + γ1 − cos kx − cos ky − cos kz)τ z + t′ +1 sin kyτ y + t′ +1 sin kxτ x] +⊗ [−t2(2 + γ2 − cos kx cos ky − cos kz)σz + t′ +2 sin kyσy − t′ +2 sin kxσx]. +(S15) +We expand the Bloch Hamiltonian near the z-axis i.e. k → (0, 0, kz), +HMW SM,||(k) ≈[t1(Q1 + 1 +2(k2 +x + k2 +y))τ z + t′ +1kyτ y + t′ +1kxτ x] +⊗ [−t2(Q2 + 1 +2(k2 +x + k2 +y))σz + t′ +2kyσy − t′ +2kxσx], +(S16) +where Qi = γi − cos kz (i=1,2). Expanding only up to second order in momenta, we have, +HMW SM,||(k) ≈ − t1t2(Q1Q2 + (Q1 + Q2)1 +2(k2 +x + k2 +y))τ zσz +− t1t′ +2Q1τ z(kxσx − kyσx) − t′ +1t2Q2(kxτ x + kyτ y)σz +− t′ +1t′ +2(k2 +xτ xσx − k2 +yτ yσy − 1 +2(kxky + kykx)τ xσy + 1 +2(kxky + kykx)τ yσx). +(S17) +/2 +0 +/2 +kz +0.4 +0.2 +0.0 +0.2 +0.4 +E +numerical +1st order +2nd order +FIG. S12: Comparison of the numerically calculated Landau Levels of the MWSM||. system with the analytically calculated +lower Landau levels for first order(blue dashed) and second order(red) expansion in momenta along the direction perpendicular +to kz. Level repulsion between two parent graded lowest Landau levels are only observed if one expands to second order in +momenta. +We consider B = Bˆz. After Peierls substitution, kx → k′ +x = kx, ky → k′ +y = ky + eBx, and kz → k′ +z = kz. The position- +momenta commutator leads to the commutator, [k′ +y, k′ +x] = ieB. Here, e is the charge of the particle in consideration. One can +therefore construct bosonic ladder operators of the form, +a = k′ +x − ik′ +x +√ +2eB +; +a† = k′ +x + ik′ +y +√ +2eB +; +[a, a†] = 1. +(S18) + +20 +We calculate some important identities via Eqn.S18 which we will be using in the next few lines, +1 +2(k′ +x +2 + k′ +y +2) = eB(a†a + 1 +2); +k′ +xσx + k′ +yσy = +√ +2eB(aσ+ + a†σ−); +k′ +xσx − k′ +yσy = +√ +2eB(aσ− + a†σ+), +k′ +x +2 − k′ +y +2 = eB(a2 + a†2); +i[k′ +xk′ +y + k′ +yk′ +x] = eB(a†2 − a2), +(S19) +where we have used τ ± = 1 +2(τ x ± iτ y) and σ± = 1 +2(σx ± iσy), which are spin ladder operators in the basis {|+⟩ , |−⟩} in +both the τ and σ spaces. Now, substituting k for k′ in Eqn. S17 and then transforming them via Eqn. S19, we get the following +expression, +HMW SM,||(k′) ≈ − t1t2(Q1Q2 + (Q1 + Q2)eB(a†a + 1 +2))τ zσz − t1t′ +2Q1 +√ +2eBτ z(aσ− + a†σ+) − t′ +1t2Q2 +√ +2eB(aτ + + a†τ −)σz +− t′ +1t′ +2(2eB)(a†a + 1 +2)(τ +σ+ + τ −σ−) − t′ +1t′ +2(2eB)(a2τ +σ− + a†2τ −σ+). +(S20) +Let us ignore the second order perturbations not in the mass term (i.e. τ zσz) and simplify the Hamiltonian, +HMW SM,||(k′) ≈ −(Q1Q2 +(Q1 +Q2)eB(a†a+ 1 +2))τ zσz −Q1 +√ +2eBτ z(aσ− +a†σ+)−Q2 +√ +2eB(aτ + +a†τ −)σz. (S21) +We obtain one of the lowest Landau levels, |ψ⟩1,LLL = |0; −, +⟩ with energy E1,LLL = (Q1Q2 + eB +2 (Q1 + Q2)) which match +exactly both numerically and analytically in first and second order expansions. For the other lowest Landau level, we observe an +amalgamation of chiral Landau levels obtained from each parent which cause level repulsion at the intersection point. +S4 +Euler space topology calculation +In the main text, we have already reported that the MWSM system possesses both time reversal, T and inversion symmetry, I and +hence the combined symmetry, T ′ denoted by τ yσyκ, where κ refers to complex conjugation. However, here T ′2 = 1, so that +a Z2 invariant is not possible. Instead, it is possible to find a basis, where T ′ = κ. Here we provide the unitary transformation +which makes this possible, +V = 1 +2[(1 + i)τ 0σ0 + (1 − i)τ yσy]. +(S22) +Based on the method provided in the appendix in a previous work52, the above unitary transformation satisfies, V τ yσyV T = 1, +so that we get a Hamiltonian, ˜H(k) = V H(k)V † which satisfies, ˜H(k) = ˜H∗(k), and is real and symmetric. Denoting the +MWSM in a condensed notation, +H = (M1τ z + Q1τ x + R1τ y) ⊗ (−M2σz − Q2σx + R2σy), +(S23) +we obtain after the transformation, +˜H = +M1(−M2τ zσz − Q2τ zσx + R2τ xσ0) +− Q1(M2τ xσz + Q2τ xσx + R2τ zσ0) +− R1(M2τ 0σx − Q2τ 0σz − R2τ yσy). +(S24) +Comparing with the method introduced in52, it is possible to view the real Hamiltonian as an element of a Real oriented Grass- +mannian, ˜GR +2,4 which is diffeomorphic to S2×S2. For a given kz, then it is possible to define a mapping from the 2d BZ spanned +by kx and ky (for MWSM ||) into (n1, n2) ∈ S2 × S2 and the topology of ˜H is then determined by the two skyrmion numbers, +Q[n1] = q1 and Q[n2] = q2 of parent 1 and parent 2, respectively. The Euler class topology is then found from these skyrmion +numbers as follows, +EI = q2 − q1; +EII = q2 + q1. +(S25) +The Euler numbers are unique up to the mapping (EI, EII) → (−EI, −EII). + diff --git a/8NE0T4oBgHgl3EQffgAo/content/tmp_files/load_file.txt b/8NE0T4oBgHgl3EQffgAo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..967ea1621cc374ea548a9496590f2819f1bcb8dd --- /dev/null +++ b/8NE0T4oBgHgl3EQffgAo/content/tmp_files/load_file.txt @@ -0,0 +1,978 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf,len=977 +page_content='Multiplicative topological semimetals Adipta Pal,1, 2 Joe H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Winter,1, 2, 3 and Ashley M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Cook1, 2 1Max Planck Institute for Chemical Physics of Solids, N¨othnitzer Strasse 40, 01187 Dresden, Germany 2Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Strasse 38, 01187 Dresden, Germany 3SUPA, School of Physics and Astronomy, University of St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Andrews, North Haugh, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Andrews KY16 9SS, UK Exhaustive study of topological semimetal phases of matter in equilibriated electonic systems and myriad extensions has built upon the foundations laid by earlier introduction and study of the Weyl semimetal, with broad applications in topologically-protected quantum computing, spintronics, and optical devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We extend recent introduction of multiplicative topological phases to find previously-overlooked topological semimetal phases of electronic systems in equilibrium, with minimal symmetry-protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We show these multiplicative topological semimetal phases exhibit rich and distinctive bulk-boundary correspondence and response signatures that greatly expand understanding of consequences of topology in condensed matter settings, such as the limits on Fermi arc connectivity and structure, and transport signatures such as the chiral anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Our work therefore lays the foundation for extensive future study of multiplicative topological semimetal phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' I Introduction Topological semimetals are a vast family1,2 of topological phases of matter studied in great depth experimentally3–11 in the search for table-top, quasiparticle realizations of high- energy physics12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' At the simplest level, the topological de- generacies of band structures in these topological semimetal phases are realized quite generically if either time-reversal symmetry13 or inversion symmetry14 are broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' This is the requirement for two-fold topological degeneracies char- acteristic of the Weyl semimetal phase, although it is desire- able to realize such degeneracies in the vicinity of the Fermi level15,16, with minimal contributions to the Fermi surface from other electronic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' In such cases, the key signa- tures of Weyl semimetals are especially prominent, includ- ing the distinguishing Fermi arc surface states17–20, and trans- port signatures associated with the chiral anomaly21–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Such isolation of Weyl nodes in the vicinity of the Fermi level is also facilitated—and the physics of topological semimetals enriched—by systematic study of these topological phases in compounds with wide-ranging phenomena, including super- conductivity, strong spin-orbit coupling, and strong correla- tions28–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Much progress has also been made in identifying other topological semimetals with more complex topological degeneracies2,35–37 in electronic band structures, protected by a large set of crystalline point group symmetries in combi- nation with additional anti-unitary symmetries such as time- reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The present work returns to the foundations of topologi- cal semimetal studies by introducing previously-unidentified topological semimetal phases of matter of electronic systems in equilibrium, which may then be generalized in the same manner as outlined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We do so by studying the first topo- logical semimetal realizations of multiplicative topological phases, a recently-identified set of topological phases of mat- ter described by Bloch Hamiltonians in an infinitely-large, pe- riodic bulk, which are symmetry-protected tensor products of “parent” Bloch Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' These multiplicative topolog- ical semimetal (MTSM) phases are therefore straightforward constructions described by tensor products of Weyl semimetal Bloch Hamiltonians, yet exhibit rich phenomena distinct from all other known topological semimetals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We first review the Weyl semimetal phase and its canonical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We then construct the first examples of multiplicative topological semimetal phases using these past results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The multiplicative topological semimetals are then first character- ized in the bulk, and their bulk-boundary correspondence es- tablished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' II Review of the Weyl semimetal phase and suitable models for constructing multiplicative phases The Weyl semimetal is a topologically non-trivial phase of matter characterized by topologically-protected, doubly- degenerate and linearly-dispersing band crossings in the Bril- louin zone38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' That is, these band-crossings, known as Weyl points or nodes, cannot be removed from the electronic struc- ture through smooth deformations of the Hamiltonian, but rather only through mutual annihilation of the Weyl nodes, by bringing two nodes of opposite topological charge to the same point in the Brillouin zone to gap out these band-touchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' When the Fermi level intersects only the Weyl nodes of this semimetal phase, their low-energy physics dominates, yield- ing a variety of intensely-studied exotic phenomena of interest for applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' At the simplest level, the Weyl nodes serve as quasiparticle, table-top realizations of Weyl fermions pre- dicted in high-energy physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' However, they are also a start- ing point in going well beyond high-energy physics, by tilting the Weyl cone to realize a type-II Weyl semimetal phase39, in which the low-energy physics of the Weyl nodes is not Lorentz-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Weyl semimetal phases can be realized in effectively non- interacting systems where certain discrete symmetries are bro- ken rather than respected, in contrast to many other effectively non-interacting topological phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' They may be derived through symmetry-breaking starting from the Dirac semimetal state40,41, for instance, (which could be topologically-robust or fine-tuned) by breaking either time-reversal symmetry T or inversion symmetry I, which pulls the two Weyl nodes com- prising the Dirac node away from one another in momentum- space42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' This phase, characterized by Weyl nodes in the Bril- louin zone, is topologically stable so long as Weyl nodes of opposite topological charge do not annihilate one another43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' I-breaking Weyl semimetal phases are of tremendous ex- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='02404v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='mes-hall] 6 Jan 2023 2 perimental interest, but are described by Bloch Hamiltonian models with four bands at minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' A more natural starting point in deriving multiplicative topological semimetal phases is instead to use the minimal Weyl semimetal Bloch Hamil- tonian achieved by breaking T , which possesses only two bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Such two-band models for the Weyl semimetal cor- respond to the non-trivial homotopy group π3(S2) and, sim- ilarly to the two-band Chern and Hopf insulators44 and the two-band Kitaev chain model 45, may be combined using known constructions to form a multiplicative counterpart of the Weyl semimetal phase, the multiplicative Weyl semimetal phase (MWSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We therefore consider a well-established two-band Bloch Hamiltonian previously used in study of Weyl nodes, with various instances of this model serving as the parents of the MWSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' HW SM(k) =t1 sin kxτ x + t2 sin kyτ y + t3(2 + γ − cos kx − cos ky − cos kz)τ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (1) where the τ j (j = x, y, z) are the Pauli matrices in the orbital basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The two band spectrum, E(k) = ± � t2 1 sin2 kx + t2 2 sin2 ky + ϵ(k)2, ϵ(k) = t3(2 + γ − cos kx − cos ky − cos kz), (2) has two gapless nodes at k = (0, 0, ±k0), for cos k0 = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We refer to these as the Weyl nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The equation of motion for Bloch electrons in the k-space in the presence of Berry curva- ture is represented by ˙r = vk + ˙k×F(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' For the equation of motion to remain invariant under T -symmetry, one must have the equality, F(k) = −F(−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The breaking of T -symmetry, then involves a minimum of two Weyl nodes with opposite Berry curvature at opposite momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Therefore, close to the Weyl nodes, we have, H±(k) = ±t1kxτ x + t2kyτ y ± t3 sin k0kzτ z, (3) which in turn corresponds to the Berry curvatures, F±(k)|0,0,±k0 = ± t1t2t3 sin k0 2[t1k2x + t2k2y + (t3 sin k0)2k2z]3/2 (kx, ky, kz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (4) The Chern number of the lower-energy band for the range, kx = 0, ky = 0 and kz ∈ (−k0, k0) is C = ±1 depend- ing on the direction of the magnetic field corresponding to the monopoles at the two Weyl points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The Weyl nodes are in- volved with exotic boundary states at surfaces perpendicular to the z-axis, called the Fermi Arc surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' For the case where the surfaces are open in the x-direction, the surface dis- persion is given by, E(ky) = ±t2 sin ky, (5) and the arc-states, Ψ(x, ky, kz) = e+ikyy+ikzz(e−λ1x − e−λ2x) 1 √ 2 � 1 ±i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' In the k-space, this includes all contours cos ky + cos kz > 1 + cos k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' III Multiplicative Weyl Semimetal (MWSM) in the bulk A protocol for constructing the child Hamiltonian for the MWSM, Hc derived from Hp1 and Hp2 as first reported in Cook and Moore46, is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Given two two-band Bloch Hamiltonians Hp1 and Hp2 written in a general form, with momentum-dependence suppressed, as Hp1 = � a b c d � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Hp2 = � α β γ δ � , (6) the multiplicative child Bloch Hamiltonian constructed from these two parents can be written as Hc 12, where Hc 12 = � � � aδ −aγ bδ −bγ −aβ aα −bβ bα cδ −cγ dδ −dγ −cβ cα −dβ dα � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (7) Expressing the two-band parent Bloch Hamiltonians Hp1(k) and Hp2(k) more compactly as the following, Hp1(k) = d1(k) · τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Hp2(k) = d2(k) · σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (8) where d1(k) and d2(k) are momentum-dependent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' three- component vectors of scalar functions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' and each of σ and τ is the vector of Pauli matrices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' the multiplicative child Hamilto- nian may more compactly be written as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Hc 12(k) = (d11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' d21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' d31) · τ ⊗ (−d12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' d22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' −d32) · σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (9) to highlight the tensor product structure of the child Hamil- tonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' which can be symmetry-protected as discussed in ear- lier work by Cook and Moore on multiplicative topological phases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' and therefore can describe phases of matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' even in the presence of additional bands46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The tensor-product structure guarantees that the energy spectrum of the child Hamiltonian is a product of the energy spectrum of Hp1(k), Ep1(k), and of Hp2(k), Ep2(k), respec- tively, Ec 12(k) = ±Ep1(k)Ep2(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (10) This implies that bands of the child Hamiltonian dispersion are at least doubly degenerate everywhere in the bulk Bril- louin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We will consider two cases in this work: (1) the Weyl node separation of each parent is along one axis in the Bril- louin zone, and (2) the axis along which Weyl nodes are separated in one parent is perpendicular to the axis along which Weyl nodes are separated in the other parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Spectral and magneto-transport properties differ significantly between these two cases, as we will show, demonstrating the richness of MTSM phases of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' A Multiplicative Weyl Semimetal - parallel axis par- ents The construction of the MWSM for both parents along the same axis is derived from two parent WSMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' As an example, 3 we consider the following parents and the resulting child: Hp1(k) =t11 sin kxτ x + t21 sin kyτ y + t31(2 + γ1 − cos kx − cos ky − cos kz)τ z, (11a) Hp2(k) =t12 sin kxσx + t22 sin kyσy + t32(2 + γ2 − cos kx − cos ky − cos kz)σz, (11b) Hc(k) =[t11 sin kxτ x + t21 sin kyτ y + t31(2 + γ1 − cos kx − cos ky − cos kz)τ z] ⊗ [−t12 sin kxσx + t22 sin kyσy − t32(2 + γ2 − cos kx − cos ky − cos kz)σz].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (11c) Each parent Hamiltonian realizes Weyl nodes at k = � 0, 0, cos−1 γi � when −1 < γi < 1, (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Examples of such topologically non-trivial dispersion are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 1 (a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' /2 0 /2 1 0 1 E(k) (a) WSM parent 1 /2 0 /2 momentum kz 1 0 1 E(k) (b) WSM parent 2 /2 0 /2 momentum kz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 E(k) (c) MWSM parallel child band 1 band 2 band 3 band 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 1: Dispersion E(k) for (a) WSM Parent Hamiltonian with γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 along kz and t11 = t21 = t31 = 1, (b) WSM Parent Hamiltonian with γ2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 along kz and t12 = t22 = t32 = 1, and (c) the resulting MWSM parallel Child Hamiltonian along kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' From these parent Hamiltonian dispersions, we can find the dispersion of the child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' As given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 10, the bulk spectrum is doubly degenerate and determined by the spectra of the par- ent 1, Ep1(k), and parent 2, Ep2(k), respectively, which take the following forms: Ep1(k) = [t2 11 sin2 kx + t2 21 sin2 ky + ϵ1(k)2]1/2, Ep2(k) = [t2 12 sin2 kx + t2 22 sin2 ky + ϵ2(k)2]1/2, (12) where ϵ1/2(k) = t31/2(2 + γ1/2 − cos kx − cos ky − cos kz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' For the sake of convenience, we refer to the MWSM with Weyl node separation for each parent along the same axis in the Brillouin zone (as in the case of parents given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 22a and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 22b) as MWSM||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' For the MWSM|| bulk spectrum given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 10 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 12, gapless points occur at the po- sitions in the Brillouin zone where gapless points are present for the parent systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' As γ1 and γ2 control separation of the Weyl nodes in the Brillouin zone for the parents, they play a major role in determining the number of nodes, the location of the nodes, and the polynomial order of the nodes in the Brillouin zone for the child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' When γ1 = γ2, for instance, we have two gapless points but the dispersion near the nodes is quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' In contrast, for γ1 ̸= γ2 as for parents depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 1 (a) and (b), the child MWSM|| has four nodes, and bands disperse linearly in the vicinity of the nodes, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Each node is four-fold degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' While such degeneracy naively suggests Dirac nodes or Weyl nodes of higher charge, the multiplicative nodes are dis- tinct in a number of ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' To examine this difference, we look at the child Hamiltonian in the vicinity of each multiplicative node for the case −1 < γ1 ̸= γ2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' From the tensor product structure, it easy to check that ∂E± ∂ki = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' which implies that the dispersion is linear at each of the gapless nodes of the MWSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Therefore the possibility of a higher order Weyl node is nullified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The position of each of the multiplicative nodes are determined by the nodes in the respective parents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We re- fer to (0, 0, ±k01) as the Weyl node positions derived from the first parent, and (0, 0, ±k02) as the Weyl node positions derived from the second parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Here γi = cos k0i, (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' If the gapless point is (0, 0, k02), then we define MWSM|| in the vicinity as Hc ||,2, and, Hc ||,2 = t31(γ1−γ2)τ z(−t12kxσx+t22kyσy−t32 sin k02¯kz,2σz), (13) where ¯kz,2 = (kz − k02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Surprisingly, this looks like a Dirac semimetal Hamiltonian, whose Dirac node has been shifted in k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Since it is no longer at the origin, the time-reversal symmetry is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' For the other node, γ1 = cos k01 for (0, 0, k01), we define the multiplicative Hamiltonian in the vicinity as Hc ||,1, so that, Hc ||,1 = (t11kxτ x+t21kyτ y+t31 sin k01¯kz,1τ z)t32(γ1−γ2)σz, (14) where ¯kz,1d = (kz − k01) and contains off-diagonal terms for the block Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' But again, it is possible to perform a similarity transformation on this Hamiltonian, in the form U = R−1 τ (θ, φ) ⊗ Rσ(θ, φ), so that we get another ‘shifted’ Dirac semimetal type Hamiltonian, ¯Hc ||,1 = t32(γ1−γ2)τ z(t11kxσx+t21kyσy+t31 sin k01¯kz,1σz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (15) Again, the shift from the origin breaks the time-reversal sym- metry of the original Dirac semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' It is therefore appropri- ate to refer to the MWSM|| as possessing degeneracies con- sisting of Weyl nodes, rather than possessing Dirac nodes, and exhibit strikingly different physics as a result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' B MWSM - perpendicular axis parents Before characterizing bulk-boundary correspondence and transport signatures of MTSMs, we explore further richness of multiplicative constructions by considering cases where 4 parent Weyl nodes are separated along orthogonal axes in k- space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' As a specific case, we choose parent Hamiltonians such that the first parent has Weyl node separation along the y-axis, while the second one has Weyl node separation along the z- axis, Hp1(k) =t11 sin kxτ x + t21 sin kzτ y + t31(2 + γ1 − � i cos ki)τ z, (16a) Hp2(k) =t12 sin kxσx + t22 sin kyσy + t32(2 + γ2 − � i cos ki)σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (16b) Again the bulk spectrum is derived from the tensor product structure, Ep1(k) = [t2 11 sin2 kx + t2 21 sin2 kz + ϵ2 1(k)]1/2, Ep2(k) = [t2 12 sin2 kx + t2 22 sin2 ky + ϵ2 2(k)]1/2, Ec ⊥k = ±Ep1(k)Ep2(k), (17) where ϵ1/2(k) = t31/32(2+γ1/2 −cos kx −cos ky −cos kz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Examples of parent and child dispersion in this case are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 2 for the values, γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 and γ2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We gain greater understanding of the multiplicative struc- ture in this case by examining the low-energy expansion of the Child Hamiltonian in the vicinity of its nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Taylor ex- panding up to linear order around the point, (0, k0,1, 0) for γ1 = cos k0,1, one gets, Hc ⊥,1(k) =(t11kxτ x + t21kzτ y + t31 sin k0,1¯ky,1τ z) ⊗ (t22 sin k0,1σy − t32(γ2 − γ1)σz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (18) Similarly, expanding around (0, 0, k0,2) for γ2 = cos k0,2, we get, Hc ⊥,2(k) =(t21 sin k0,2τ y + t31(γ1 − γ2)τ z) ⊗ (−t12kxσx + t22kyσy − t32 sin k0,2¯kz,2σz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (19) One notices that Hc ⊥,2(k) is equivalent to a DSM when γ1 = γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' C Discrete Symmetries of the MWSM The discrete symmetries satisfied by the parent WSMs include invariance under particle-hole conjugation given by P = σxκ, such that the Hamiltonian satisfies, σxH∗ 1/2(k)σx = −H1/2(−k), and invariance under spatial inversion given by I = σz, such that the Hamiltonian satisfies, σzH1/2(k)σz = H1/2(−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The MWSM|| or ⊥ child systems are instead invariant un- der time reversal given by T = iτ xσxκ corresponding to the transformation, τ xσxH∗ c(k)τ xσx = Hc(−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' ky 1 0 1 E (a) kz 1 0 1 E (b) (0, 0, 0) (0, 0, ) (0, , ) (0, , 0) (0, 0, 0) (0, , ) k 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 E (f) ky 0 kz 0 E 5 0 5 (e) kz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 E (c) /20 /2 ky 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 E (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 2: Dispersion E(k) (t11 = t12 = 1, t21 = t22 = 1, t31 = t32 = 1) for (a) WSM Parent Hamiltonian with γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 along ky, (b) WSM Parent Hamiltonian with γ2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 along kz and the resulting MWSM perpendicular Child Hamiltonian along (c) kz and (d) kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The energy dispersion plotted along both ky and kz is shown in (e) and the dispersion along a high-symmetry path in the first quadrant of the two-dimensional (2d) BZ is shown in (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Inversion symmetry relates the nodes in the first quadrant to those in the other quadrants, giving rise to four gapless nodes in the 2d BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' They are also invariant under spatial inversion given by I = τ zσz, corresponding to the transformation, τ zσzHc(k)τ zσz = Hc(−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The MWSM should then satisfy the symmetry, T I, which may also protect the Dirac semi-metal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Indeed, in some cases, the Dirac Hamiltonian for the MWSM near the nodes is reminiscent of the corresponding low-energy Hamiltonian for a Dirac semi-metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' This invariance of the multiplicative bulk Hamiltonian under products of transformations, which leave each parent Hamiltonian invariant, is expected given the multiplicative dependence of the child on the parents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 5 D Bulk characterization of topology with Wilson loops As calculated in Supplementary section S1, the Berry connec- tion for the MWSM is given as A = (A1,kx − A2,kx, A1,ky − A2,ky, A1,kz − A2,kz), (20) where Aj,l = (i ⟨+j| ∂l |+j⟩ , i ⟨−j| ∂l |−j⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Using this expression for the Berry connection, we compute Wilson loops and associated Wannier spectra by integrating over kx for a given ky, as detailed in Alexandradinata et al47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' In the parallel case illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 3(a), the Wannier spectra derived from Wilson loop calculations show that only in regions where only one of the parent phases is non-trivial do we get non-trivial Wannier spectra distinguished by π values for Wannier charge centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' However, the Wannier spectra in the region where each parent is topological appears trivial, given the dependence of child Wannier spectra on parent Wannier spectra distinctive of multiplicative topological phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We have referred to a pair of Weyl nodes of equal and opposite topological charge as a ‘dipole’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We observe, that the orientation of this dipole due to the two constituent parents is important, as anti-parallel dipoles, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 3(b), show non-trivial Wilson loop eigenvalues in a region in the 2d BZ where neither of the parent systems have non-trivial topological character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Analogous results for the MWSM⊥ are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 4, although the Wannier spectrum structure is far richer than in the parallel case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' /2 0 /2 kz /2 0 /2 ky (a)WSM 1 x /2 0 /2 kz /2 0 /2 ky (b)WSM 2 x /2 0 /2 kz /2 0 /2 ky (c)MWSM pll /2 0 /2 kz /2 0 /2 ky (d)MWSM pll 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='50 x, 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='50 x, 2 (a) Each parent has Weyl node ‘dipole’ oriented in +ˆz direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' /2 0 /2 kz /2 0 /2 ky (a)WSM 1 x /2 0 /2 kz /2 0 /2 ky (b)WSM 2 x /2 0 /2 kz /2 0 /2 ky (c)MWSM pll /2 0 /2 kz /2 0 /2 ky (d)MWSM pll 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='50 x, 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='50 x, 2 (b) Parent 1 with dipole oriented along +ˆz direction and parent 2 with dipole oriented in −ˆz direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 3: Wannier spectra in MWSM parallel for two filled bands derived from Wilson loop around kx for parent 1 with γ1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 and parent 2 with γ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The upper row (a) and the lower row (b) have opposite orientation of the Weyl node ’dipole’ for parent 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Corresponding Wannier spectra of the MWSM for the lowest-energy and second-lowest in energy occupied bands is shown in (c) and (d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 6 /2 0 /2 kz /2 0 /2 ky (a)WSM 1 x /2 0 /2 kz /2 0 /2 ky (b)WSM 2 x /2 0 /2 kz /2 0 /2 ky (c)MWSM perp /2 0 /2 kz /2 0 /2 ky (d)MWSM perp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='50 x, 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='50 x, 2 (a) Parent 1 has Weyl node ‘dipole’ oriented along +ˆy direction and parent 2 along −ˆz direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' /2 0 /2 kz /2 0 /2 ky (a)WSM 1 x /2 0 /2 kz /2 0 /2 ky (b)WSM 2 x /2 0 /2 kz /2 0 /2 ky (c)MWSM perp /2 0 /2 kz /2 0 /2 ky (d)MWSM perp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='50 x, 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='50 x, 2 (b) Parent 1 has Weyl node ’dipole’ oriented in +haty direction and parent 2 Weyl node dipole oriented in −ˆz direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 4: Wannier spectra in MWSM perpendicular for two filled bands derived from Wilson loop around kx for parent 1 with γ1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 and parent 2 with γ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The upper row (a) and the lower row (b) have opposite orientation of the Weyl node ’dipole’ for parent 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Corresponding Wannier spectra of the MWSM for the lowest-energy and second-lowest in energy occupied bands is shown in (c) and (d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' IV MWSM with open-boundary conditions– 1 Slab spectra of MWSM: An important aspect of WSM physics is its distinctive bulk-boundary correspondence: Weyl nodes in the three- dimensional bulk Brillouin zone serve as termination points of topologically-protected boundary states known as Fermi arcs when projected to a slab Brillouin zone corresponding to open boundary conditions in one direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We expect analogous topologically-protected surface states in MTSMs and explore possible realizations of these Fermi arc states in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' One might expect that the tensor product structure of the multiplicative phases is visible in the surface spectrum of the MWSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Numerical simulations show that this is the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' For the parent WSMs, the surface spectra is given as, E(ky) ∼ sin(ky)(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 5(a) and (b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 6 2nd row) for nodes along the z-axis and open boundaries along the x-direction and E(kz) ∼ sin(kz)(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 6 1st row) for nodes along the y-axis and open boundaries along the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Indeed, corresponding surface spectra of child Hamiltonians depend on these surface spectra in a multiplicative way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Numeri- cal simulation from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 5 (c) shows that, for MWSM||, the slab spectra disperses as E(ky) ∼ sin2(ky) for two parents each with surface spectrum E(ky) ∼ sin(ky).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' In contrast, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 6 (c) shows that the surface spectrum instead disperses as E(ky, kz) ∼ sin(ky) sin(kz) for MWSM⊥ when one par- ent has the former surface spectrum and the other has the lat- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We also show, for the case of each parent surface spec- trum along kz, which exhibits flat bands between the two Weyl nodes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 5(a) and (b)) corresponds to flat bands be- tween all four gapless points in the MWSM parallel system (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 5(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' However, fitting sin2(ky) curves to each of the parallel and perpendicular MWSM spectra reveals that, except in special cases when γ1 = γ2 where the fit is exact, the slab spectra does not disperse as sin2(ky) and instead exhibits kz- dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' One can check this by comparing E vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' ky slab spectra in the range −min(k0,1, k0,2) < kz < min(k0,1, k0,2) and min(k0,1, k0,2) < kx < max(k0,1, k0,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The spectra ap- pears linear near zero in the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 2 Stability of surface states of MWSM For the MWSM|| system, the low-energy expansion about a node is reminiscent of a Dirac node, and it is therefore possible to break apart the four-fold degeneracy at each of the nodes by introducing an external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We introduce minimal coupling, ky → ky − eBx for the MWSM|| to simulate the effect of applied magnetic field on the spectral density of the Fermi arc surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We observe that the Fermi arcs split but are not destroyed by the applied field as in the case of the DSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 3 Fermi Arcs for the MWSM as a stack of MCIs: WSMs can be interpreted as a set of Chern insula- tors(CIs), each defined in a 2d submanifold of the 3d BZ of the WSM (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=', each kx-ky plane) for a given value of kz, yielding a stack of CIs in the kz- direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The Weyl nodes then correspond to topological 7 /2 0 /2 ky 2 1 0 1 2 E (e)MWSM || /2 0 /2 ky 2 1 0 1 2 E (f)MWSM || /2 0 /2 kz 2 1 0 1 2 E (f)MWSM || /2 0 /2 ky 2 0 2 E (a)Parent 1 /2 0 /2 kz 2 0 2 E (b)Parent 1 /2 0 /2 ky 2 0 2 E (c)Parent 2 /2 0 /2 kz 2 0 2 E (d)Parent 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 5: Finite slab spectra(in x-direction, Lx = 80) along ky (kz = 0) and kz(ky = 0) respectively for (a,b) WSM with γ1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5, (c,d) WSM with γ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' In (e,f) the slab spectra(Lx = 80) E vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' ky for the MWSM|| child created from the above two parents for kz = 0 and kz = π 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (g) shows the slab spectra E vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' kz at ky = 0 for the same MWSM|| child system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' phase transitions—corresponding to gap-closings—in the stack between intervals in kz with topologically- distinct CIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Specifically, we use the SCZ model48, /2 0 /2 kz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 E (a) WSM parent 1 x /2 0 /2 kz 2 0 2 E (b) WSM parent 2 /2 0 /2 kz 2 0 2 E = (c) MWSM perp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' child /2 0 /2 ky 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 E /2 0 /2 ky 2 0 2 E /2 0 /2 ky 2 0 2 E /2 0 /2 kz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 E /2 0 /2 ky 2 0 2 E /2 0 /2 (kz + ky) 2 0 2 E FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 6: Finite slab spectra (in x direction, Nx = 80) with the constituent parent Hamiltonians - WSM parent Hamiltonian 1 with γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 and Weyl nodes along the ky-direction is shown along column (a), WSM parent Hamiltonian 2 with γ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 with Weyl nodes along the kz-direction along column (b) and the MWSM perpendicular child Hamiltonian along column (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' It is apparent how the surface spectra along kz(for ky = 0) and ky(for kz = 0) combine multiplicatively to create the surface spectra for the MWSM perpendicular system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The lowest diagram along column (c) especially shows the spectra along the diagonal kz + ky direction where the component spectra sin(kz) and sin(ky) have combined to produce sin(kz) sin(ky) as the leading term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' HCI = B(2+M −cos kx −cos ky)σz +sin kxσx +sin kyσy in particle-hole space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' In the WSM, the mass term is given as M = γ − cos kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Here, for the range, −1 < γ < 1, kz ∈ [− cos−1 γ, cos−1 γ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The Fermi arcs we observe in the 2d BZ defined in the ky − kz for open boundary conditions in the x-direction are projections of the chiral edge states of the slices of the corresponding CIs in the stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The multiplicative counterpart of a Chern insulator was introduced recently by Cook and Moore46 as Multiplicative Chern Insulators(MCIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Here, one must notice that the MCI has two mass terms derived from each of the parent systems, one from each of the parent systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Hence, there exists more than one way to stack the MCIs in the kz direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Either parent mass term can be kz-dependent, for instance, or both can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Here, we have attached the momentum dependence to both the mass terms, so that the difference in parent mass parameters remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We then characterize the multi- plicative Fermi arc states by opening boundary conditions in the x- and y-directions, and plotting the probability density for 8 the sum of 40 eigenstates nearest in energy to zero in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 7 for kz = 0 (a 2D submanifold of the BZ realizing an MCI) and kz = π 2 (a 2D submanifold of the BZ that is topologically triv- ial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' For the former case shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 7(a) and (b), the proba- bility density in the corresponding child is localized at sites at the boundary, but also at the sites adjacent to these sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' For the latter case, parent 1 has edge states and parent 2 does not as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 7(d) and (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The resultant child probability density shows low-energy states localize only at the boundary sites as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 7(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' This localization behavior is sim- ilar to that of the multiplicative Kitaev chain presented in a second work by the present authors, where, if each parent is topological, edge states are localized at lattice sites right at the edge, but also at sites adjacent to these sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We expect such localization to protect the edge states from backscattering to some extent, which we will explore in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 4 Boundary states disconnected from bulk states— The MCI introduced by Cook and Moore46 can exhibit topo- logically robust yet floating edge states, which are separated from the bulk by a finite energy gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' MTSMs constructed from MCIs can inherit this exotic boundary state connectivity, displaying boundary states disconnected from the bulk band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' To realize such a MWSM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' we first note when edge states are disconnected from bulk states for the case of the MCI: HCI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='p1(k) =B1(2 + M1 − cos kx − cos ky)τ z + sin kxτ x + sin kyτ y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (21a) HCI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='p2(k) =B2(2 + M2 − cos kx − cos ky)σz + sin kxσx + sin kyσy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (21b) HMCI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='c(k) =[B1(2 + M1 − cos kx − cos ky)τ z + sin kxτ x + sin kyτ y] ⊗ [−B2(2 + M2 − cos kx − cos ky)σz − sin kxσx + sin kyσy],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (21c) the range of parameters over which this is possible is M1 ∈ [−4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' −2] and M2 ∈ [−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 0] which corresponds to Chern num- bers C = +1 and C = −1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We therefore con- struct a MWSM for which the Weyl nodes of one parent WSM are separated in k-space by a stack of Chern insulators, each with total Chern number C = +1, and the Weyl nodes of the other parent are separated by a stack of Chern insulators, each with total Chern number C = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Comparing Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (22a) with (21a) and Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (22b) with (21b), it is clear that, for each Chern insulator in the stack, the following mapping holds, Mi = γi − cos kz, i ∈ {1, 2}, and i labeling the parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' From this mapping, it is not possible to have M2 ∈ (−4, −2) while γi ∈ (−1, 1), i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We therefore generalize the map- ping to the following form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Mi = γi − ri cos kz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 2},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' so that the parents and the child Hamiltonian for the MWSM parallel are,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Hp1(k) =t11 sin kxτ x + t21 sin kyτ y + t31(2 + γ1 − cos kx − cos ky − r1 cos kz)τ z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (22a) Hp2(k) =t12 sin kxσx + t22 sin kyσy + t32(2 + γ2 − cos kx − cos ky − r2 cos kz)σz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (22b) Hc(k) =[t11 sin kxτ x + t21 sin kyτ y + t31(2 + γ1 − cos kx − cos ky − r1 cos kz)τ z] ⊗ [−t12 sin kxσx + t22 sin kyσy − t32(2 + γ2 − cos kx − cos ky − r2 cos kz)σz].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (22c) To construct one parent with Chern number of this stack non-trivial and opposite in sign to the Chern number of the stack in the other parent, we first introduce some terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We refer to the region between Weyl nodes including kz = 0 as regular Weyl region (RWR) and the region including kz = ±π as the irregular Weyl region (IWR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The existence of Weyl nodes requires |r1,2| ≥ 1 for |γ1,2| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' It is then possible to realize a RWR with negative Chern number by varying r1,2, so that γ1,2 − r1,2 cos kz ∈ (−4, −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' These RWRs—one of each parent system—must then occur over the same interval in kz, however, to realize topological floating surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We set γ2 = 0 and r2 = 3, which means we have C = −1 for the range [− cos−1( 2 3), cos−1( 2 3)] when M2 = γ2 − r2 cos kz ∈ [−3, −2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Then we must have γ1 = cos π 3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 and r1 = 1 so that in the region kz ∈ [− cos−1( 2 3), cos−1( 2 3)], we have the same kind of MCI with edge states gapped from the bulk as described in Cook and Moore46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' These results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The MWSM⊥ case of topologically robust yet floating Fermi arc surface states is constructed similarly, and we de- fer thorough investigation of this case to later work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' V Effect of Magnetic field on MWSM and Chiral anomaly We now investigate response signatures of MTSMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' As we consider MWSMs here, which may be constructed from WSM parent systems, we focus in particular on the question of whether there is a multiplicative generalization of the chi- ral anomaly, one of the most important signatures of Weyl semimetals: application of non-orthogonal electric and mag- netic fields can pump electrons between Weyl nodes of oppo- site chirality49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' More specifically, applying an external mag- netic field parallel to the axis along which Weyl nodes are separated in k-space yields a single chiral Landau level near each of the Weyl nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' In Weyl semimetals, this suppresses backscattering of electrons with opposite chirality, manifest- ing as a negative magnetoresistance (MR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Weyl semimetals 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 7: Probability densities of superposition of 40 edge state eigenvectors in a 30 × 30(Lx × Ly) square lattice at kz = 0 and kz = π 2 for (a, d) Parent WSM 1 (γ1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5), (b, e) Parent WSM 2(γ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5) and (c, f) MWSM || child (γ1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 and γ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' At kz = 0, both the parent systems are topological as seen from a visible edge state which results in localization at both the edge and second last edge sites in the MWSM || child system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' When kz = π 2 , the parent 1 is still topological but the parent 2 is trivial as seen from the absence of edge states which results in localization only at the edge sites of the MWSM || child system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' therefore serve as condensed matter platforms for study of the chiral anomaly, also known as Adler-Bell-Jackiw anomaly, as- sociated with the Standard Model of particle physics50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' When the external magnetic field is instead oriented perpendicular to the k-space axis along which Weyl nodes are separated, semi- classical calculations indicate the presence of quantum oscil- lations in the density of states51, observable in magnetization, magnetic torque, and MR measurements50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' To study the effects of external fields on the MWSM, we first derive the Landau level structure for the the Weyl semimetal in the cases of external magnetic fields applied par- allel and perpendicular to the Weyl node axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We can then draw parallels between these results and their generalizations in the case of the MWSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' A Chiral anomaly in WSM To study the chiral anomaly in a WSM, we consider a par- ticular Bloch Hamiltonian HW SM(k) characterizing a Weyl semimetal phase and its expansion around the kz-axis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' k → (0, 0, kz) (up to 2nd order in kx and ky), HW SM(k) =t(2 + γ − cos kx − cos ky − cos kz)σz + t′ sin kyσy + t′ sin kxσx, ≈t(Q + 1 2(k2 x + k2 y))σz + t′kyσy + t′kxσx, (23) where Q = γ − cos kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Applying the magnetic field, B = Bˆz along the Weyl node axis, Peierls substitution changes the momenta in the following way, kx → k′ x = kx, ky → k′ y = ky + eBx, and kz → k′ z = kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The position-momentum commutator, implies, [k′ y, k′ x] = ieB, so that, it is possible to define bosonic ladder operators, a = k′ x − ik′ y √ 2eB ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' a† = k′ x + ik′ y √ 2eB ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' [a, a†] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (24) 10 /2 0 /2 ky 4 2 0 2 4 E (e)MWSM || /2 0 /2 kz 4 2 0 2 4 E (f)MWSM || /2 0 /2 ky 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 E (a)Parent 1 /2 0 /2 kz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 E (b)Parent 1 /2 0 /2 ky 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 E (c)Parent 2 /2 0 /2 kz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 E (d)Parent 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 8: Slab spectra along ky (subfigure a) and kz (subfigure b) for WSM parent 1 with γ1 = 0, r1 = 3, and slab spectra along ky (subfigure c) and kz (subfigure d) for WSM parent 2 with γ2 = 2/3, r2 = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Corresponding slab spectra for the MWSM|| with t11 = t12 = 1, t21 = t22 = 1, t31 = t32 = 1 along (e) ky and (f) kz, respectively, with edges separate from the bulk slab spectra along ky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Applying Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='S18, after substituting k → k′, we get the fol- lowing system which looks similar to the polariton conserving Jaynes-Cummings Hamiltonian, HW SM(k′) ≈ t(Q+eB(a†a+1 2))σz+t′√ 2eB(aσ++a†σ−), (25) where σ± = 1 2(σx ± iσy) are the spin ladder operators in the basis {|+⟩ , |−⟩} of σz (σz |±⟩ = ± |±⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The ground state from the above Hamiltonian is given by the eigenvec- tor, |ψLLL⟩ = |0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' −⟩ (states denoted as |n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' s⟩ where n is the bosonic number and s is the spin direction), which leads to the lowest Landau level energy, ELLL = −t(Q + 1 2eB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (26) Near each of the Weyl nodes, it is easy to observe that |ψLLL⟩ is chiral as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The other Landau levels can be derived by restricting to the two dimensional disjoint spaces, {|n, −⟩ , |n − 1, +⟩}, parametrized by the bosonic number, n so that in each such basis, the Hamiltonian is, H(kz, n) = −teB 2 σ0 −t(Q+eBn)σz +t′√ 2eBnσx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (27) The energy for the other Landau levels parametrized by n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' is given by the eigenvalues of Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 27, EnLL = −teB 2 ± � t2(Q + eBn)2 + 2t′2eBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (28) We have illustrated the analytically calculated Landau levels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 9 and compared them to numerical calculations of Lan- dau levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The numerical computation involves plotting the bands for the Peierls substituted Weyl semimetal with periodic boundary conditions, say in the x-direction, and subjected to magnetic field in integer multiples of 2π L , where L is the size of the lattice in the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We observe that the chiral Lan- dau level from both analytical and numerical methods overlap, with an approximate overlap of the other Landau levels since we only considered till second order in kx and ky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Next we consider the case when the magnetic field is di- rected perpendicular to the Weyl node axis, say B = Bˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Expanding the first line of Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 23 around the Weyl node, k = (0, 0, k0 = cos−1 γ) of positive chirality, and setting t = t′ = 1, we get, HW SM(k) ≈ sin k0(kz − k0)σz + kyσy + kxσx, =⇒ H′ W SM(k) ≈ − kyσz + kxσx + sin k0(kz − k0)σy, (29) where in the second line we have rotated the Hamiltonian to a new basis via, σx → σz and σx → −σx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' In the presence of mentioned magnetic field perpendicular to the Weyl node axis, the Peierls substitution is applied as kx → k′ x = kx, ky → k′ y = ky and kz → k′ z = kz − eBx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The commuta- tion relation, [kx, sin k0(kz − k0 − eBx)] = ieB sin k0, then constructs the bosonic ladder operators, b = kz − k0 − eBx − ikx √2eB sin k0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' b† = kz − k0 − eBx + ikx √2eB sin k0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (30) 11 2 0 2 kz 4 2 0 2 4 E Numerical Analytical Chiral LLL(numerical) Chiral LLL(analytical) 2 0 2 ky 4 2 0 2 4 E Numerical Analytical(near WN) Chiral LLL(numerical) Chiral LLL(analytical near WN) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 9: Landau Levels for the two-band Weyl Semimetal calculated analytically from Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 25 and numerically, with t = 1 = t′, γ = 0 and B = 2π 51 ˆz(upper) and B = 2π 51 ˆy(lower).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The (black) bands indicate the numerically calculated Landau levels and the (red) bands for the analytically calculated Landau levels for n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=', 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The (blue) band and the dotted (magenta) band is the Lowest Landau Level(LLL) calculated numerically and analytically, and is responsible for the Chiral anomaly in the upper figure and Weyl orbits in the lower figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The system in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 29 then changes to, HW SM(k′) ≈ −kyσz + � 2eB sin k0(bσ+ + b†σ−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (31) Similar to the previous case, it is possible to re- solve the Hamiltonian into the subspaces spanned by {|n, −⟩ , |n − 1, +⟩}, where n is the eigenvalue of the num- ber operator, b†b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We get two chiral lowest Landau levels with energies, E = ±ky in the bulk, which are responsible for the chiral anomaly50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' B Chiral anomaly in the MWSM We now study the response of the MWSM to external fields for comparison with the signatures of the chiral anomaly in the WSM reviewed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We treat the MWSM parallel and perpendicular cases separately, given the expected sensitivity of the response to orientation of the axes of node separation relative to the orientation of the external fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 1 Landau levels in the MWSM parallel system: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' III A we have derived the Dirac Hamiltonian for the MWSM|| in the vicinity of each of its two nodes, (0, 0, k01) and (0, 0, k02) derived respectively from each of its two par- ents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Hc ||,1(k) =(t′ 1kxτ x + t′ 1kyτ y + t1 sin k01¯kz,1τ z)t2(γ1 − γ2)σz, Hc ||,2(k) =t1(γ1 − γ2)τ z(−t′ 2kxσx + t′ 2kyσy − t2 sin k02¯kz,2σz) In this section, we will only consider cases where γ1 ̸= γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' To investigate the response to external fields for the MWSM||, we consider the effect of magnetic field along the Weyl node axis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=', B = Bˆz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We use the exact Peierls substitution in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' S18, so that the two expressions above transform as follows, Hc ||,1(k′) =t2(γ1 − γ2)(t1 sin k01¯kz,1τ z + t′ 1 √ 2eB(aτ + + a†τ −))σz, Hc ||,2(k′) =t1(γ1 − γ2)τ z(−t2 sin k01¯kz,2σz − t′ 2 √ 2eB(aσ− + a†σ+)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (32) Here τ ± = 1 2(τ x ± iτ y) and σ± = 1 2(σx ± iσy) are the pseudo-spin ladder operators in the τ and σ spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The low- est Landau levels from the above two expressions are given below, Hc ||,1 →E1,LLL = ±(γ1 − γ2)t1t2 sin k01¯kz,1, |ψ1,LLL⟩ = |0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' −, ±⟩ , Hc ||,2 →E2,LLL = ∓(γ1 − γ2)t2t2 sin k02¯kz,2, |ψ2,LLL⟩ = |0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' ±, +⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (33) One may notice that the eigenvector |0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' −, +⟩ occurs in the vicinity of each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Therefore, we calculate its energy eigenvalue if one expands the MWSM parallel system in the vicinity of the kz axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The details of the calculation can be found in the Supplementary Materials S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We find the energy is given as, E|0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='−,+⟩ = (Q1Q2 + 1 2eB(Q1 + Q2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (34) We show that this expression is consistent with the numer- ically calculated Landau levels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The other chi- ral Landau level consistent with the other two eigenvectors, |0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' −, −⟩ and |0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' +, +⟩ near their respective Weyl nodes ap- pears distinct from |0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' −, +⟩ away from the Weyl nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 12 /2 0 /2 kz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='6 E (a)Landau Levels for B = Bz in MWSM pll /2 0 /2 ky 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='6 E (b)Landau Levels for B = By in MWSM pll FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 10: The Landau Levels for the MWSM parallel Hamiltonian with γ1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5, γ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5, t1 = t′ 1 = t2 = t′ 2 = 1 and B = 2π 80 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (a) and (b) show the Landau levels for the magnetic field along the Weyl axis and perpendicular to the Weyl axis (at Weyl node (0, 0, π 3 ) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The (red) bands refer to the lowest Landau levels and the (black) bands form the bulk Landau levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 10, for certain values of γ1 and γ2, it appears, at first glance, as if there are two separate, chiral Landau lev- els corresponding to |0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' −, −⟩ and |1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' , −, −⟩ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' All four Weyl nodes are connected by each of these LLLs, how- ever, and the two LLLs in combination furthermore account for each chirality at each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Although this is reminiscent of the Dirac semimetal, there is potentially a distinction in char- acter between the chiralities at each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' If each parent cor- responds to a particular degree of freedom, for instance, and these dofs are physically distinct from one another in some sense, such as one parent corresponding to a two-fold valley dof, and the other corresponding to a two-fold layer dof, the chiral anomalies are inequivalent and do not compensate one another as they would for a Dirac semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The two apparently ’separated’ LLLs seem to only scatter between the Weyl nodes derived from their respective parents, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' intra-parent scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Upon closer inspection, how- ever, we see the intersection point between two apparently separated Landau levels is actually a very small gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We have verified in Supplementary Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' S3, that the gap is fi- nite in analytical calculations performed to second order in momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The gap is an emergent feature of the multiplica- tive chiral anomaly, with the single LLL reducing to |0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' −, −⟩ and |0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' +, +⟩ at nodes associated with a particular parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We therefore interpret the multiplicative chiral anomaly as ex- hibiting parent-graded features as well as emergent features not associated with either individual parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' This is reminis- cent of the topologically robust floating bands of the multi- plicative Chern insulator46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 2 Landau levels in the MWSM perpendicular system: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' III B, we had shown the linear expansion of the MWSM⊥ Bloch Hamiltonian near each of the nodes corre- sponding to one parent with Weyl nodes separated along the ky axis and the other parent with Weyl nodes separated along the kz axis in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 18 and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Without loss of generality, we consider, t31 = t32 = t21 = t22 = 1 = t11 = t12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' There exists three separate cases one needs to check - (i) magnetic field along the Weyl axis of the first parent, B = Bˆy, (ii) mag- netic field along the Weyl axis of the second parent, B = Bˆz, and (iii) magnetic field perpendicular to the Weyl axis of both parents, B = Bˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Case 1 (B = Bˆy) : Substituting, kx → k′ x = kx+eBz, and using the bosonic ladder operators, a⊥,y = kz−ik′ x √ 2eB , a† ⊥,y = kz+ik′ x √ 2eB , we have, from Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 18, H⊥,1(k′) =(sin k0,1(ky − k0,1)τ z + √ 2eB(a⊥,yτ + + a† ⊥,yτ −) ⊗ (sin k0,1σy + (γ1 − γ2)σz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (35) For the expression from Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 19, we instead con- sider the following bosonic ladder operators, ˜a⊥,y = ˜ kz−ik′ x √ 2eB sin k0,2 and ˜a† ⊥,y = ˜ kz+ik′ x √ 2eB sin k0,2 , which gives us, H⊥,2(k′) =(sin k0,2τ y + (γ1 − γ2)τ z) ⊗ (kyσy − � 2eB sin k0,2(˜a⊥,yσ+ y + ˜a† ⊥,yσ− y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (36) It is easy to find the lowest Landau level energies in the vicinity of each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' From Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 35 and 36, we respectively have the LLL energies, Ey,1,LLL = ± � sin2 k0,1 + (γ1 − γ2)2 sin k0,1(ky − k0,1), Ey,2,LLL = ± � sin2 k0,2 + (γ1 − γ2)2ky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (37) We then find two ky-dependent chiral LLLs connecting the nodes of the first parent, while we have two chiral LLLs at ky = 0 due to the second parent, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 11 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The following result was expected if one considers the Landau levels for the parents for different 13 directions of the magnetic field discussed in the previ- ous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' For the MWSM perpendicular case, the incident magnetic field in this case is both parallel to the Weyl axis of parent 1 and perpendicular to the Weyl axis of parent 2, so that we get both kinds of Landau levels simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Case 2 (B = Bˆz): This produces results similar to Case 1, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 11 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' A similar calculation gives us the lowest Landau level energies, Ez,1,LLL = ± � sin2 k0,1 + (γ1 − γ2)2kz, Ez,2,LLL = ± � sin2 k0,2 + (γ1 − γ2)2 sin k0,2(kz − k0,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (38) VI Discussion and Conclusion In this work, we have introduced the previously-unidentified multiplicative topological semimetal phases of matter, distin- guished by Bloch Hamiltonians with a symmetry-protected tensor product structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Parent Bloch Hamiltonians, with ei- ther one or both of the parents being topologically non-trivial, may then be combined in the tensor product to realize mul- tiplicative topological semimetal phases inheriting topology from the parent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We consider foundational examples of multiplicative topo- logical semimetals, with Bloch Hamiltonians constructed as tensor products of two-band Bloch Hamiltonians, each char- acterizing a Weyl semimetal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' These multiplicative topo- logical semimetal phases are protected by a combination of symmetries of class DIII at the level of the child, and each parent Bloch Hamiltonian in class D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Given the great vari- ety of exotic crystalline point group symmetries considered to protect most recently-identified topological semimetal phases, it is remarkable that the symmetry-protection of these multi- plicative semimetal phases is relatively simple, and suggests many additional multiplicative semimetal phases may be iden- tified by enforcing these many other symmetries on parent Bloch Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We first characterize multiplicative topological semimetal phases in the bulk, showing the bulk spectrum of the child Bloch Hamiltonian depends in a multiplicative way on the spectra of the parent Bloch Hamiltonians: each eigenvalue of the child, at a given point in k-space, is a product of eigen- values, one from each parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We furthermore consider two different constructions of the multiplicative Weyl semimetal, either for the case of each parent having a pair of Weyl nodes separated along the same axis in k-space (parallel construc- tion), or along perpendicular axes in k-space (perpendicu- lar construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' For either construction, the multiplicative symmetry-protected structure can then naturally yield nodal degeneracies reminiscent of Dirac nodes or higher-charge Weyl nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' However, the multiplicative degeneracies are dis- tinguished from these more familiar quasiparticles by distinc- tive Wannier spectra signatures in the bulk, and exotic bulk- boundary correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Importantly, bulk characterization /2 0 /2 ky 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='6 E (a)Landau Levels for B = By in MWSM perp /2 0 /2 kz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='6 E (b)Landau Levels for B = Bz in MWSM perp FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 11: Landau levels for the MWSM perpendicular system with γ1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 and γ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='5 representing separation of Weyl nodes along the ky and kz direction respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We show two cases, (a) when magnetic field is along the y-direction and (b) when magnetic field is along the z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Red lines indicate the chiral Landau levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' since the magnetic field is paralle to one Weyl node separation and perpendicular to another Weyl node separation, the above behaviour is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' by Wannier spectra reveals a complex dependence of Berry connection in the child Bloch Hamiltonian on Berry connec- tion of each parent Bloch Hamiltonian, depending on whether the parents are constructed with Weyl nodes separated along the same axis in momentum-space (parallel) or not (perpen- dicular).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Additionally, the connectivity of Fermi arc surface states for the multiplicative Weyl semimetal is far more com- plex than in standard Dirac or Weyl semimetals, reflecting the underlying dependence of the child topology on the topology of the parents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' An especially interesting example is the re- alization of topologically-protected—yet floating—boundary states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Response signatures of the multiplicative Weyl semimetal 14 also inherit response signatures of the parents, with the po- tential for emergent phenomena beyond that of either parent individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Here, we consider the multiplicative analog of one of the defining response signatures of the Weyl semimetal, the chiral anomaly, finding instead multiple co-existing chiral anomalies graded by the parent degrees of freedom, as well as emergent features in the Landau level structure not inherited from a particular parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' In the case of parents correspond- ing to effectively the same degree of freedom, the response reduces to a signature reminiscent of a Dirac semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' This brings up the possibility of controlled manipulation of partic- ular properties of an electronic system similar to spintronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Future work will characterize other signatures of multi- plicative topological semimetals anticipated given the exten- sive characterization of Weyl and Dirac semimetals, particu- larly optical and non-linear responses given the tremendous interest in the bulk photovoltaic effect in Weyl semimetals, as well as symmetry-protection of more exotic topological quasi- particles, such as multiplicative generalizations of multifold fermions or nodal lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Given the immense body of work on topological semimetals and the surprising consequences of multiplicative topology for bulk-boundary correspondence, nodal band structure, and Berry phase structure, our intro- duction of previously-unidentified multiplicative topological semimetals into the literature lays the foundation for consid- erable future theoretical and experimental study, which will greatly expand and deepen our understanding of topological semimetal phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Acknowledgements - We gratefully acknowledge help- ful discussions with J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Moore, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Day, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Varjas and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Calderon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Correspondence Correspondence and requests for materials should be addressed to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (email: cooka@pks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='de).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 1 Alexey A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Soluyanov, Dominik Gresch, Zhijun Wang, QuanSheng Wu, Matthias Troyer, Xi Dai, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Andrei Bernevig, “Type-II Weyl semimetals,” Nature 527, 495–498 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 2 Barry Bradlyn, Jennifer Cano, Zhijun Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Vergniory, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Felser, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Cava, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Andrei Bernevig, “Be- yond dirac and weyl fermions: Unconventional quasiparti- cles in conventional crystals,” Science 353, aaf5037 (2016), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='org/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='aaf5037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 3 Shin-Ming Huang, Su-Yang Xu, Ilya Belopolski, Chi-Cheng Lee, Guoqing Chang, BaoKai Wang, Nasser Alidoust, Guang Bian, Madhab Neupane, Chenglong Zhang, Shuang Jia, Arun Bansil, Hsin Lin, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Zahid Hasan, “A Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class,” Nature Communications 6, 7373 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Lv, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Weng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Fu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Miao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Ma, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Richard, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Huang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Zhao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Fang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Dai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Qian, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Ding, “Experimental discovery of weyl semimetal taas,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' X 5, 031013 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 5 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Lv, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Xu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Weng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Ma, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Richard, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Huang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Zhao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Matt, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Bisti, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Strocov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Mesot, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Fang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Dai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Qian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Shi, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Ding, “Ob- servation of Weyl nodes in TaAs,” Nature Physics 11, 724–727 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 6 Su-Yang Xu, Nasser Alidoust, Ilya Belopolski, Zhujun Yuan, Guang Bian, Tay-Rong Chang, Hao Zheng, Vladimir N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Strocov, Daniel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Sanchez, Guoqing Chang, Chenglong Zhang, Daixi- ang Mou, Yun Wu, Lunan Huang, Chi-Cheng Lee, Shin-Ming Huang, BaoKai Wang, Arun Bansil, Horng-Tay Jeng, Titus Neu- pert, Adam Kaminski, Hsin Lin, Shuang Jia, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Zahid Hasan, “Discovery of a Weyl fermion state with Fermi arcs in niobium ar- senide,” Nature Physics 11, 748–754 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 7 Su-Yang Xu, Ilya Belopolski, Nasser Alidoust, Madhab Neu- pane, Guang Bian, Chenglong Zhang, Raman Sankar, Guoqing Chang, Zhujun Yuan, Chi-Cheng Lee, Shin-Ming Huang, Hao Zheng, Jie Ma, Daniel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Sanchez, BaoKai Wang, Arun Ban- sil, Fangcheng Chou, Pavel P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Shibayev, Hsin Lin, Shuang Jia, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Zahid Hasan, “Discovery of a weyl fermion semimetal and topological fermi arcs,” Science 349, 613–617 (2015), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='org/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='aaa9297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 8 Sergey Borisenko, Quinn Gibson, Danil Evtushinsky, Volodymyr Zabolotnyy, Bernd B¨uchner, and Robert J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Cava, “Experimental realization of a three-dimensional dirac semimetal,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 113, 027603 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 9 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Jiang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Weng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Prabhakaran, S-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Mo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Peng, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Dudin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Kim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Hoesch, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Fang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Dai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Shen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Feng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Hussain, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Chen, “A stable three-dimensional topological Dirac semimetal Cd3As2,” Nature Materials 13, 677–681 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 10 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Liu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Weng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Prab- hakaran, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Mo, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Shen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Fang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Dai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Hussain, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Chen, “Discovery of a three-dimensional topological dirac semimetal, na¡sub¿3¡/sub¿bi,” Science 343, 864–867 (2014), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='org/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='1245085.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 11 Madhab Neupane, Su-Yang Xu, Raman Sankar, Nasser Ali- doust, Guang Bian, Chang Liu, Ilya Belopolski, Tay-Rong Chang, Horng-Tay Jeng, Hsin Lin, Arun Bansil, Fangcheng Chou, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Zahid Hasan, “Observation of a three-dimensional topological Dirac semimetal phase in high-mobility Cd3As2,” Nature Com- munications 5, 3786 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 12 Xiangang Wan, Ari M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Turner, Ashvin Vishwanath, and Sergey Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Savrasov, “Topological semimetal and fermi-arc surface states in the electronic structure of pyrochlore iridates,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' B 83, 205101 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 13 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Burkov and Leon Balents, “Weyl semimetal in a topological insulator multilayer,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 107, 127205 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 14 G´abor B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Hal´asz and Leon Balents, “Time-reversal invariant re- alization of the weyl semimetal phase,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' B 85, 035103 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 15 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Armitage, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Mele, and Ashvin Vishwanath, “Weyl and dirac semimetals in three-dimensional solids,” Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 90, 015001 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 16 Yan Sun, Shu-Chun Wu, Mazhar N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Ali, Claudia Felser, and Binghai Yan, “Prediction of weyl semimetal in orthorhombic mote2,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' B 92, 161107 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 17 Yuval Baum, Erez Berg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Parameswaran, and Ady Stern, “Current at a distance and resonant transparency in weyl semimet- als,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' X 5, 041046 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 18 Andrew C Potter, Itamar Kimchi, and Ashvin Vishwanath, “Quantum oscillations from surface fermi arcs in weyl and dirac semimetals,” Nature communications 5, 1–6 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 19 Andr´as Gyenis, Hiroyuki Inoue, Sangjun Jeon, Brian B Zhou, Benjamin E Feldman, Zhijun Wang, Jian Li, Shan Jiang, Quinn D 15 Gibson, Satya K Kushwaha, Jason W Krizan, Ni Ni, Robert J Cava, B Andrei Bernevig, and Ali Yazdani, “Imaging electronic states on topological semimetals using scanning tunneling mi- croscopy,” New Journal of Physics 18, 105003 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 20 Rajib Batabyal, Noam Morali, Nurit Avraham, Yan Sun, Marcus Schmidt, Claudia Felser, Ady Stern, Binghai Yan, and Haim Beidenkopf, “Visualizing weakly bound surface fermi arcs and their correspondence to bulk weyl fermions,” Science Advances 2, e1600709 (2016), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='org/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='1126/sciadv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='1600709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 21 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Nielsen and Masao Ninomiya, “The adler-bell-jackiw anomaly and weyl fermions in a crystal,” Physics Letters B 130, 389–396 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 22 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Son and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Spivak, “Chiral anomaly and classical nega- tive magnetoresistance of weyl metals,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' B 88, 104412 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 23 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Parameswaran, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Grover, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Abanin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Pesin, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Vishwanath, “Probing the chiral anomaly with nonlocal trans- port in three-dimensional topological semimetals,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' X 4, 031035 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 24 Xiaochun Huang, Lingxiao Zhao, Yujia Long, Peipei Wang, Dong Chen, Zhanhai Yang, Hui Liang, Mianqi Xue, Hongming Weng, Zhong Fang, Xi Dai, and Genfu Chen, “Observation of the chiral-anomaly-induced negative magnetoresistance in 3d weyl semimetal taas,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' X 5, 031023 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 25 Cheng-Long Zhang, Su-Yang Xu, Ilya Belopolski, Zhujun Yuan, Ziquan Lin, Bingbing Tong, Guang Bian, Nasser Alidoust, Chi-Cheng Lee, Shin-Ming Huang, Tay-Rong Chang, Guoqing Chang, Chuang-Han Hsu, Horng-Tay Jeng, Madhab Neupane, Daniel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Sanchez, Hao Zheng, Junfeng Wang, Hsin Lin, Chi Zhang, Hai-Zhou Lu, Shun-Qing Shen, Titus Neupert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Za- hid Hasan, and Shuang Jia, “Signatures of the Adler–Bell–Jackiw chiral anomaly in a Weyl fermion semimetal,” Nature Communi- cations 7, 10735 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 26 Chandra Shekhar, Ajaya K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Nayak, Yan Sun, Marcus Schmidt, Michael Nicklas, Inge Leermakers, Uli Zeitler, Yurii Skourski, Jochen Wosnitza, Zhongkai Liu, Yulin Chen, Walter Schnelle, Horst Borrmann, Yuri Grin, Claudia Felser, and Binghai Yan, “Extremely large magnetoresistance and ultrahigh mobility in the topological Weyl semimetal candidate NbP,” Nature Physics 11, 645–649 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 27 Frank Arnold, Chandra Shekhar, Shu-Chun Wu, Yan Sun, Ri- cardo Donizeth dos Reis, Nitesh Kumar, Marcel Naumann, Mukkattu O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Ajeesh, Marcus Schmidt, Adolfo G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Grushin, Jens H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Bardarson, Michael Baenitz, Dmitry Sokolov, Horst Bor- rmann, Michael Nicklas, Claudia Felser, Elena Hassinger, and Binghai Yan, “Negative magnetoresistance without well-defined chirality in the Weyl semimetal TaP,” Nature Communications 7, 11615 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 28 Tanja Graf, Claudia Felser, and Stuart S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Parkin, “Simple rules for the understanding of heusler compounds,” Progress in Solid State Chemistry 39, 1–50 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 29 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Butch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Syers, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Kirshenbaum, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Hope, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Paglione, “Superconductivity in the topological semimetal yptbi,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' B 84, 220504 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 30 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Tafti, Takenori Fujii, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Juneau-Fecteau, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Ren´e de Cotret, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Doiron-Leyraud, Atsushi Asamitsu, and Louis Taillefer, “Superconductivity in the noncentrosymmetric half-heusler com- pound luptbi: A candidate for topological superconductivity,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' B 87, 184504 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 31 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Pan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Nikitin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Bay, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Huang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Paulsen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Yan, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' de Visser, “Superconductivity and magnetic order in the noncentrosymmetric half-heusler compound erpdbi,” Euro- physics Letters 104, 27001 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 32 Yasuyuki Nakajima, Rongwei Hu, Kevin Kirshenbaum, Alex Hughes, Paul Syers, Xiangfeng Wang, Kefeng Wang, Renx- iong Wang, Shanta R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Saha, Daniel Pratt, Jeffrey W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Lynn, and Johnpierre Paglione, “Topological ¡i¿r¡/i¿pdbi half-heusler semimetals: A new family of noncentrosymmetric mag- netic superconductors,” Science Advances 1, e1500242 (2015), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='org/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='1126/sciadv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='1500242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 33 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Fisk, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Canfield, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Beyermann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Thompson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Hundley, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Ott, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Felder, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Maple, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Lopez de la Torre, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Visani, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Seaman, “Massive electron state in ybbipt,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 67, 3310–3313 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 34 Jason K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Kawasaki, Abhishek Sharan, Linda I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Johans- son, Martin Hjort, Rainer Timm, Balasubramanian Thiagara- jan, Brian D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Schultz, Anders Mikkelsen, Anderson Janotti, and Chris J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Palmstrøm, “A simple electron counting model for half-heusler surfaces,” Science Advances 4, eaar5832 (2018), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='org/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='1126/sciadv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='aar5832.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 35 Shuichi Murakami, “Phase transition between the quantum spin hall and insulator phases in 3d: emergence of a topological gap- less phase,” New Journal of Physics 9, 356 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 36 Zhijun Wang, Yan Sun, Xing-Qiu Chen, Cesare Franchini, Gang Xu, Hongming Weng, Xi Dai, and Zhong Fang, “Dirac semimetal and topological phase transitions in A3bi (a = Na, k, rb),” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' B 85, 195320 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 37 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Young, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Zaheer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Teo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Kane, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Mele, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Rappe, “Dirac semimetal in three dimensions,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 108, 140405 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 38 Xiangang Wan, Ari M Turner, Ashvin Vishwanath, and Sergey Y Savrasov, “Topological semimetal and fermi-arc surface states in the electronic structure of pyrochlore iridates,” Physical Review B 83, 205101 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 39 Alexey A Soluyanov, Dominik Gresch, Zhijun Wang, QuanSheng Wu, Matthias Troyer, Xi Dai, and B Andrei Bernevig, “Type-ii weyl semimetals,” Nature 527, 495–498 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 40 Zhijun Wang, Yan Sun, Xing-Qiu Chen, Cesare Franchini, Gang Xu, Hongming Weng, Xi Dai, and Zhong Fang, “Dirac semimetal and topological phase transitions in a 3 bi (a= na, k, rb),” Physical Review B 85, 195320 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 41 Steve M Young, Saad Zaheer, Jeffrey CY Teo, Charles L Kane, Eugene J Mele, and Andrew M Rappe, “Dirac semimetal in three dimensions,” Physical review letters 108, 140405 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 42 AA Zyuzin, Si Wu, and AA Burkov, “Weyl semimetal with bro- ken time reversal and inversion symmetries,” Physical Review B 85, 165110 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 43 Takahiro Morimoto and Akira Furusaki, “Weyl and dirac semimetals with z 2 topological charge,” Physical Review B 89, 235127 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 44 Joel E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Moore, Ying Ran, and Xiao-Gang Wen, “Topological sur- face states in three-dimensional magnetic insulators,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 101, 186805 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 45 A Yu Kitaev, “Unpaired Majorana fermions in quantum wires,” Physics-Uspekhi 44, 131–136 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 46 Ashley M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Cook and Joel E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Moore, “Multiplicative topological phases,” Communications Physics 5, 262 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 47 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Alexandradinata, Xi Dai, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Andrei Bernevig, “Wilson- loop characterization of inversion-symmetric topological insula- tors,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' B 89, 155114 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 48 Xiao-Liang Qi, Yong-Shi Wu, and Shou-Cheng Zhang, “Topolog- ical quantization of the spin hall effect in two-dimensional param- agnetic semiconductors,” Physical Review B 74, 085308 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 49 Shuang Jia, Su-Yang Xu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Zahid Hasan, “Weyl semimet- als, Fermi arcs and chiral anomalies,” Nature Materials 15, 1140– 1144 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 50 Binghai Yan and Claudia Felser, “Topological materials: Weyl 16 semimetals,” Annual Review of Condensed Matter Physics 8, 337–354 (2017), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='1146/annurev-conmatphys- 031016-025458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 51 Andrew C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Potter, Itamar Kimchi, and Ashvin Vishwanath, “Quantum oscillations from surface Fermi arcs in Weyl and Dirac semimetals,” Nature Communications 5, 5161 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 52 Yifei Guan, Adrien Bouhon, and Oleg V Yazyev, “Landau levels of the euler class topology,” Physical Review Research 4, 023188 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' 17 Supplemental material for “Multiplicative topological semimetals” Adipta Pal1,2, Joe H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Winter1,2,3, and Ashley M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Cook1,2,∗ 1Max Planck Institute for Chemical Physics of Solids, N¨othnitzer Strasse 40, 01187 Dresden, Germany 2Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Strasse 38, 01187 Dresden, Germany 3SUPA, School of Physics and Astronomy, University of St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Andrews, North Haugh, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Andrews KY16 9SS, UK ∗Electronic address: cooka@pks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='de S1 Wilson loops for multiplicative Weyl semi-metal: Labelling the parent Hamiltonians as Hp1 = (d1x, d1y, d1z) · τ and Hp2 = (d2x, d2y, d2z) · σ, with eigenvectors, {|+1⟩ |−1⟩} and {|+2⟩ , |−2⟩} respectively, the child Hamiltonian is given by Hc = Hp1 ⊗ H′ p2, where H′ p2 = (−d2x, d2y, −d2z) · σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The ground state subspace of the child Hamiltonian is then spanned by, {|+1⟩ |−2⟩′ , |−1⟩ |+2⟩′} = {|+1⟩ |+2⟩, |−1⟩ |−2⟩}, where |ψ⟩ denotes complex conjugation and ’ denotes an eigenstate of H′ p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The non-abelian Berry connection is then given as follows: Aµ =i � ⟨+1, +2| ∂µ |+1, +2⟩ ⟨−1, −2| ∂µ |−1, −2⟩ � = i � ⟨+1| ∂µ |+1⟩ ⟨−1| ∂µ |−1⟩ � + i � ⟨+2|∂µ|+2⟩ ⟨−2|∂µ|−2⟩ � , = �A+ 1,µ − A+ 2,µ A− 1,µ − A− 2,µ � , (S1) where Al j,µ = i ⟨lj| ∂µ |lj⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' For Berry connection around a loop in the Brillouin zone, the values of µ are {kx, ky, kz} for a 3d Brillouin Zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' This clearly shows the difference between the parallel multiplicative phases and the perpendicular multiplicative phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' For a 1d BZ, as shown in past work46, the connection for parallel MKC is A = (A1,kx − A2,kx, 0, 0), while for the perpendicular MKC it is, A = (A1,kx, −A2,ky, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' For 2d or 3d parent systems, it then becomes very straightforward to extrapolate this trend such that the Berry connection looks qualitatively like the combination of the parallel and perpendicular MKC connections based on which directions the parents have in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' This is particularly interesting for the case of parallel and perpendicular Multiplicative Chern Insulators(MCIs), where parent CIs are each defined over a 2d BZ, and the parents can share one or two axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We illustrate the MCI parallel with two parent CIs on the x-y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The resultant Berry connection is then A = (A1,kx − A2,kx, A1,ky − A2,ky, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The MCI perpendicular on the other hand is constructed with one parent in the x-y plane and another in the x-z plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The resulting Berry connection is then, A = (A1,kx − A2,kx, A1,ky, −A2,kz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The MWSM, on the other hand, is a 3d system, so we instead consider parent Weyl nodes separated along parallel or perpendicular axes in k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' As explained in the main text, the parallel MWSM has parent Weyl nodes separated along the same axis in k-space (the kz axis) while the perpendicular MWSM has parent 1 and parent 2 Weyl nodes separated along the ky-axis and kz-axis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The resultant Berry connection is then, A = (A1,kx − A2,kx, A1,ky − A2,ky, A1,kz − A2,kz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' S2 Calculation for the surface state spectrum of MWSM: We write down here the derivation for the surface state energy for the MWSM parallel and MWSM perpendicular Hamiltonians, for the case of open boundary conditions in the ˆx direction and periodic boundary conditions in the ˆy and ˆz directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' First, we briefly specify how such a calculation should be done for the two band Weyl semi-metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' A Slab spectra for WSM: We start by writing down the WSM Hamiltonian used, HW SM(k) =t3(2 + γ − cos kx − cos ky − cos kz)σz + t2 sin kyσy + t1 sin kxσx, =t3(f − cos kx)σz + t2 sin kyσy + t1 sin kxσx, (S2) where f = 2 + γ − cos ky − cos kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Surface states decay into the bulk, so for open boundaries in the x-direction, we carry out the transformation, kx → iq for edge states on the left side (x = 0), so that, HW SM(iq, ky, kz) = t3(f − cosh q)σz + t2 sin kyσy + it1 sinh qσx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (S3) We claim that the determinant derived from the matrix due to the following limit must be zero, lim q1→q2 H(iq1) − H(iq2) 2 sinh q− = −t3 sinh q+σz + it1 cosh q+σx, (S4) where q± = 1 2(q1 ± q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Carrying out the determinant, we get the following two conditions, t3 sinh q+ = ±t1 cosh q+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (S5) 18 Choosing the + sign, the RHS in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' S4 becomes, −t1 cosh q+(σz − iσx), so that the null eigenvector derived from it is one of the eigenvectors for the surface spectra, |ψ+⟩ = 1 √ 2 � 1 i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (S6) The energy corresponding to this eigenvector can be found by solving the eigenvalue for the RHS in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' S3 with the above eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' This gives us the eigen-energy, E, and the equation to determine the eigen-function, for the left boundary E = t2 sin ky, (S7a) (t3 + t1)e−2q + 2fe−q + (t3 − t1) = 0, =⇒ e−q± = −f ± � f 2 − (t2 3 − t2 1) (t3 + t1) , Ψ+(x, y, z) ∼ (e−q+x − e−q−x)eikyy+ikzz |ψ+⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (S7b) The eigen-function in the last line has the following form based on the boundary condition on the left edge, Ψ(x = 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The other edge can be derived similarly by shifting x → L + 1 − x where L is the length of the system along the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' B Slab spectra for MWSM parallel: We use the same method as in section S2A of the supplementary materials to derive surface states and spectra for the MWSM parallel system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The Hamiltonian is given as follows, HMW SM||(k) = [t31(f1 − cos kx)τ z + t21 sin kyτ y + t11 sin kxτ x] ⊗ [−t32(f2 − cos kx)σz + t22 sin kyσy − t12 sin kxσx], (S8) where f1/2 = 2 + γ1/2 − cos ky − cos kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' To ease our calculations, we carry out the following rotation on the four band basis, τ z → τ y, τ y → −τ z and σz → −σy, σy → −σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The Hamiltonian then becomes, HMW SM||(k) =[t31(f1 − cos kx)τ y − t21 sin kyτ z + t11 sin kxτ x] ⊗ [−t32(f2 − cos kx)σy − t22 sin kyσz − t12 sin kxσx], =[t31(f1 − cos kx)τ y + t11 sin kxτ x][−t32(f2 − cos kx)σy − t12 sin kxσx] − t21 sin kyτ z[−t32(f2 − cos kx)σy − t12 sin kxσx] − t22 sin ky[t31(f1 − cos kx)τ y + t11 sin kxτ x]σz + t21t22 sin2 kyτ zσz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (S9) Again, without loss of generality, we set t11 = t21 = t31 = 1 = t32 = t22 = t12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The edge modes on the left edge (x = 0), require we carry out the substitution, kx → iq, and the Hamiltonian is now, HMW SM||(iq, ky, kz) =[(f1 − cosh q)τ y + i sinh qτ x][−(f2 − cosh q)σy − i sinh qσx] − sin kyτ z[−(f2 − cosh q)σy − i sinh qσx] − sin ky[(f1 − cosh q)τ y + i sinh qτ x]σz + sin2 kyτ zσz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (S10) Carrying out our previous limit on the rotated Hamiltonian above, we get the following matrix, lim q1→q2 HMW SM||(iq1) − HMW SM||(iq2) 2 sinh q− = � � � 0 i sin kyS+ −i sin kyS+ S+(−(f1 + f2) + 2S+) i sin kyS− 0 −f1S− + f2S+ i sin kyS+ −i sin kyS− f1S+ − f2S− 0 −i sin kyS+ S−((f1 + f2) − 2S−) i sin kyS− −i sin kyS− 0 � � � , (S11) where S± = cosh q+ ± sinh q+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The determinant of the RHS of Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' S11 must be zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=', we have the condition, S−S+[sin2 kyS−(f1 + f2 − 2S+)(f1 − f2)(S− + S+) − sin2 kyS+(f1 + f2 − 2S−)(f1 + f2)(S+ − S−) − (f1 + f2 − 2S+)(f1 + f2 − 2S−)(f1S− − f2S+)(−f2S− + f1S+)] = 0 (S12) Let us start with the first condition, S− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The RHS of Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' S11 then becomes, lim q1→q2 HMW SM||(iq1) − HMW SM||(iq2) 2 sinh q− =S+(−(f1 + f2) + 2S+)τ +σ+ + f2S+τ +σ− + f1S+τ −σ+ + i sin kyS+τ zσ+ − i sin kyS+τ +σz, (S13) 19 where τ ± = 1 2(τ x ±iτ y) and σ± = 1 2(σx ±iσy) are the two level ladder operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Here, if {|+⟩ , |−⟩} are eigen-vectors of τ z, then τ + |−⟩ = |+⟩, τ + |+⟩ = 0, τ − |−⟩ = 0 and τ − |+⟩ = |−⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Similar relations exist for the σ counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' |ψ1⟩ = |+⟩⊗|+⟩ is a null eigen-vector to the above expression on the RHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We solve for the energy eigenvalue first for the special case γ1 = γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='Then from HMW SM||(iq, ky, kz) in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' S10 due to the eigen-vector |ψ1⟩, we have the energy and the condition, E = sin2 ky;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (S14a) (f1 − cosh q + sinh q)(f2 − cosh q + sinh q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (S14b) S3 Landau Level repulsion in the MWSM parallel system: We start with the MWSM parallel case, HMW SM,||(k) =[t1(2 + γ1 − cos kx − cos ky − cos kz)τ z + t′ 1 sin kyτ y + t′ 1 sin kxτ x] ⊗ [−t2(2 + γ2 − cos kx cos ky − cos kz)σz + t′ 2 sin kyσy − t′ 2 sin kxσx].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (S15) We expand the Bloch Hamiltonian near the z-axis i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' k → (0, 0, kz), HMW SM,||(k) ≈[t1(Q1 + 1 2(k2 x + k2 y))τ z + t′ 1kyτ y + t′ 1kxτ x] ⊗ [−t2(Q2 + 1 2(k2 x + k2 y))σz + t′ 2kyσy − t′ 2kxσx], (S16) where Qi = γi − cos kz (i=1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Expanding only up to second order in momenta, we have, HMW SM,||(k) ≈ − t1t2(Q1Q2 + (Q1 + Q2)1 2(k2 x + k2 y))τ zσz − t1t′ 2Q1τ z(kxσx − kyσx) − t′ 1t2Q2(kxτ x + kyτ y)σz − t′ 1t′ 2(k2 xτ xσx − k2 yτ yσy − 1 2(kxky + kykx)τ xσy + 1 2(kxky + kykx)τ yσx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (S17) /2 0 /2 kz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='4 E numerical 1st order 2nd order FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' S12: Comparison of the numerically calculated Landau Levels of the MWSM||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' system with the analytically calculated lower Landau levels for first order(blue dashed) and second order(red) expansion in momenta along the direction perpendicular to kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Level repulsion between two parent graded lowest Landau levels are only observed if one expands to second order in momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' We consider B = Bˆz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' After Peierls substitution, kx → k′ x = kx, ky → k′ y = ky + eBx, and kz → k′ z = kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The position- momenta commutator leads to the commutator, [k′ y, k′ x] = ieB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Here, e is the charge of the particle in consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' One can therefore construct bosonic ladder operators of the form, a = k′ x − ik′ x √ 2eB ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' a† = k′ x + ik′ y √ 2eB ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' [a, a†] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (S18) 20 We calculate some important identities via Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='S18 which we will be using in the next few lines, 1 2(k′ x 2 + k′ y 2) = eB(a†a + 1 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' k′ xσx + k′ yσy = √ 2eB(aσ+ + a†σ−);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' k′ xσx − k′ yσy = √ 2eB(aσ− + a†σ+), k′ x 2 − k′ y 2 = eB(a2 + a†2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' i[k′ xk′ y + k′ yk′ x] = eB(a†2 − a2), (S19) where we have used τ ± = 1 2(τ x ± iτ y) and σ± = 1 2(σx ± iσy), which are spin ladder operators in the basis {|+⟩ , |−⟩} in both the τ and σ spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Now, substituting k for k′ in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' S17 and then transforming them via Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' S19, we get the following expression, HMW SM,||(k′) ≈ − t1t2(Q1Q2 + (Q1 + Q2)eB(a†a + 1 2))τ zσz − t1t′ 2Q1 √ 2eBτ z(aσ− + a†σ+) − t′ 1t2Q2 √ 2eB(aτ + + a†τ −)σz − t′ 1t′ 2(2eB)(a†a + 1 2)(τ +σ+ + τ −σ−) − t′ 1t′ 2(2eB)(a2τ +σ− + a†2τ −σ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (S20) Let us ignore the second order perturbations not in the mass term (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' τ zσz) and simplify the Hamiltonian, HMW SM,||(k′) ≈ −(Q1Q2 +(Q1 +Q2)eB(a†a+ 1 2))τ zσz −Q1 √ 2eBτ z(aσ− +a†σ+)−Q2 √ 2eB(aτ + +a†τ −)σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (S21) We obtain one of the lowest Landau levels, |ψ⟩1,LLL = |0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' −, +⟩ with energy E1,LLL = (Q1Q2 + eB 2 (Q1 + Q2)) which match exactly both numerically and analytically in first and second order expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' For the other lowest Landau level, we observe an amalgamation of chiral Landau levels obtained from each parent which cause level repulsion at the intersection point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' S4 Euler space topology calculation In the main text, we have already reported that the MWSM system possesses both time reversal, T and inversion symmetry, I and hence the combined symmetry, T ′ denoted by τ yσyκ, where κ refers to complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' However, here T ′2 = 1, so that a Z2 invariant is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Instead, it is possible to find a basis, where T ′ = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Here we provide the unitary transformation which makes this possible, V = 1 2[(1 + i)τ 0σ0 + (1 − i)τ yσy].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (S22) Based on the method provided in the appendix in a previous work52, the above unitary transformation satisfies, V τ yσyV T = 1, so that we get a Hamiltonian, ˜H(k) = V H(k)V † which satisfies, ˜H(k) = ˜H∗(k), and is real and symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' Denoting the MWSM in a condensed notation, H = (M1τ z + Q1τ x + R1τ y) ⊗ (−M2σz − Q2σx + R2σy), (S23) we obtain after the transformation, ˜H = M1(−M2τ zσz − Q2τ zσx + R2τ xσ0) − Q1(M2τ xσz + Q2τ xσx + R2τ zσ0) − R1(M2τ 0σx − Q2τ 0σz − R2τ yσy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (S24) Comparing with the method introduced in52, it is possible to view the real Hamiltonian as an element of a Real oriented Grass- mannian, ˜GR 2,4 which is diffeomorphic to S2×S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' For a given kz, then it is possible to define a mapping from the 2d BZ spanned by kx and ky (for MWSM ||) into (n1, n2) ∈ S2 × S2 and the topology of ˜H is then determined by the two skyrmion numbers, Q[n1] = q1 and Q[n2] = q2 of parent 1 and parent 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' The Euler class topology is then found from these skyrmion numbers as follows, EI = q2 − q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' EII = q2 + q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} +page_content=' (S25) The Euler numbers are unique up to the mapping (EI, EII) → (−EI, −EII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQffgAo/content/2301.02404v1.pdf'} diff --git a/8dAzT4oBgHgl3EQfE_q4/content/tmp_files/2301.01004v1.pdf.txt b/8dAzT4oBgHgl3EQfE_q4/content/tmp_files/2301.01004v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a9df12eb7474092934172c21989c1d3f1445434 --- /dev/null +++ b/8dAzT4oBgHgl3EQfE_q4/content/tmp_files/2301.01004v1.pdf.txt @@ -0,0 +1,307 @@ +arXiv:2301.01004v1 [math.RT] 3 Jan 2023 +SPIN NORM AND LAMBDA NORM +CHAO-PING DONG AND DU CHENGYU +Abstract. Given a K-type π, it is known that its spin norm (due to first-named author) +is lower bounded by its lambda norm (due to Vogan). That is, ∥π∥spin ≥ ∥π∥lambda. This +note aims to describe for which π one can actually have equality. We apply the result to +tempered Dirac series. In the case of real groups, we obtain that the tempered Dirac series +are divided into #W 1 parts among all tempered modules with real infinitesimal characters. +1. Introduction +Let G be a linear real reductive Lie group which is in the Harish-Chandra class [5]. That +is, +• G has only a finite number of connected components; +• The derived group [G, G] has finite center; +• The adjoint action Ad(g) of any g ∈ G is an inner automorphism of g = (g0)C, where +g0 is the Lie algebra of G. +Let θ be a Cartan involution of G. We assume the subgroup K = Gθ of fixed points +of θ is a maximal compact subgroup of G. Let g0 = k0 ⊕ s0 be the corresponding Cartan +decomposition of g0. We drop the subscript for the complexification. +Let ˆGtemp,o denote the set of irreducible tempered representations with real infinitesimal +character (up to equivalence). Let ˆK denote the set of K-types. The following bijection was +noted by Trapa [10], after Vogan’s paper [12]. +Theorem 1.1. Let X be any irreducible tempered (g, K)-module with real infinitesimal char- +acter. Then X has a unique lowest K-type which occurs with multiplicity one. Moreover, +the map +φ : ˆGtemp,o → ˆK +defined by taking the lowest K-type, is a well-defined bijection. +Motivated by the lambda norm introduced by Vogan [11], the first-named author intro- +duced spin norm [4] for the classification of Dirac series (i.e., irreducible unitary representa- +tions of G with non-vanishing Dirac cohomology). We refer the reader to [6] and references +therein for the notion of Dirac cohomology. It was proven in [4] that the spin norm of a +K-type π is bounded below by its lambda norm. That is, +(1) +∥π∥spin ⩾ ∥π∥lambda. +2010 Mathematics Subject Classification. Primary 22E46. +Key words and phrases. lambda norm, spin norm, tempered representations. +1 + +2 +CHAO-PING DONG AND DU CHENGYU +This inequality turns out to have a nice interpretation in the setting of Theorem 1.1. Indeed, +let ˆGtemp,d collect the members of ˆGtemp,o with non-zero Dirac cohomology. Put +ˆKe := {π ∈ ˆK| ∥π∥spin = ∥π∥lambda}. +Theorem 1.2. ([2]) The map φ restricts to ˆGtemp,d is a bijection onto ˆKe. More precisely, +any member π ∈ ˆGtemp,o is a Dirac series if and only if the inequality (1) becomes an equality +on its unique lowest K-type. +Given an arbitrary K-type π, it is not easy to compute neither ∥π∥lambda nor ∥π∥spin. +Thus, it is subtle to detect whether the inequality (1) is strict or not. This note aims to +give a criterion on this aspect. Our main result is Theorem 3.4. The main idea is to insert +an intermediate value between ∥π∥lambda and ∥π∥spin. As an application, our result suggests +that the tempered Dirac series should be separated into #W 1 parts. See (5) for the definition +of W 1. +The note is outlined as follows: In Section 2, we recall lambda norm and spin norm. Then +we deduce our main result in Section 3. The last section considers tempered Dirac series. +2. Preliminaries +In this section, we briefly recall the definitions of the spin norm and the lambda norm. +2.1. The lambda norm. We keep the notations K, G, k, s, θ, etc as in the previous section. +Let T be a maximal torus of K and t0 be the Lie algebra of T. Recall that the analytic +Weyl group is defined by +W(k, t) = NK(T)/AK(T). +It acts on the root system ∆(k, t). Fix a choice of positive roots ∆+(k, t), and define +R(G) := {r ∈ W(k, t)|r∆+(k, t) = ∆+(k, t)}. +Given a K-type π, by Lemma 0.1 of [9], the collection of highest weights of π as k-module +is a single orbit of R(G) on ˆT ∈ it∗ +0, where ˆT is the abelian group of characters of T. +Now given any K-type π, take a highest weight µ of it. Then µ ∈ it∗ +0 is dominant integral +for ∆+(k, t). Denote by ρc the half sum of all roots in ∆+(k, t). Choose a positive root system +∆+(g, t) making µ + 2ρc dominant. Denote by ρ the half sum of all roots in ∆+(g, t). Let +P be the projection map to the dominant chamber C(g) corresponding to ∆+(g, t). Then +∥P(µ+2ρc −ρ)∥ is independent of the choices of µ and ∆+(g, t), cf. Section 1 and Corrollary +2.4 of [9]. Now we are ready to talk about the lambda norm. +Definition 2.1. ([11, 1]) For any π ∈ ˆK, the lambda norm of π is defined to be +(2) +∥π∥lambda := ∥P(µ + 2ρc − ρ)∥, +where µ is any highest weight of π. For any irreducible admissible (g, K)-module X, the +lambda norm of X is defined to be +(3) +∥X∥lambda := min +π ∥π∥lambda, +where π runs over all the K-types occurring in X. A K-type π is called a lowest K-type of +X if it occurs in X and ∥π∥lambda = ∥X∥lambda. + +SPIN NORM AND LAMBDA NORM +3 +2.2. The spin norm. Although the original definition of the spin norm involves the spin +module SG of the Clifford algebra C(s), our discussion here does not need a deep under- +standing of it. Our tool is mainly the root systems and their Weyl groups. +Definition 2.2. ([4]) For any π ∈ ˆK, its spin norm is defined to be +∥π∥spin := min ∥γ + ρc∥, +where γ runs over all the highest weights of the ˜K-types in π ⊗ SG. For any irreducible +admissible (g, K)-module X, its spin norm is defined to be +∥X∥spin := min +π ∥π∥spin, +where π runs over all the K-types occurring in X. We call π a spin lowest K-type of X if +it occurs in X and ∥π∥spin = ∥X∥spin. +3. When is the inequality (1) strict? +We fix a positive root system ∆+(k, t), and denote the half sum of roots in it by ρc. Let +W(g, t) (resp., W(k, t))) be the Weyl group of ∆(g, t) (resp., ∆(k, t)). Let C(k) be the closed +dominant Weyl chamber for ∆+(k, t). For any µ ∈ t∗, we use {µ} to denote the unique weight +in C(k) to which µ is conjugate under the action of W(k, t). Let ∆+(g, t) be a positive root +system of ∆(g, t) containing ∆+(k, t). +Lemma 3.1. ([4, Lemma 3.5]) For any K-type π with a highest weight µ ∈ t∗, we have +(4) +∥µ∥spin = min +w∈W 1 ∥{µ − wρ + ρc} + ρc∥, +where +(5) +W 1 := {w ∈ W(g, t)|wC(g) ⊆ C(k)}. +Lemma 3.2. ([7, §13.3, Lemma B]) Let λ ∈ C(k). Then +∥λ + ρc∥ ⩾ ∥wλ + ρc∥ +for any w ∈ W(k, t). Moreover, the equality holds if and only if λ = wλ. +Lemma 3.3. Let ∆ be a root system with Weyl group W. Fix a positive set ∆+ of roots +and denote by ρ the half sum of all positive roots. For any dominant weight λ, we have the +following inequality +(6) +∥λ − ρ∥ ⩽ ∥λ − wρ∥, ∀w ∈ W. +Moreover, if λ is dominant with respect to w∆+, we have +∥λ − ρ∥ = ∥λ − wρ∥ +Otherwise, the inequality (6) is strict. +Proof. We first prove the inequality. Compute the following difference +(∗) +∥λ − ρ∥2 − ∥λ − wρ∥2 = −2(λ, ρ − wρ). +A widely known fact is that ρ − wρ is a sum of positive roots. The weight λ is dominant by +assumption. Thus the pairing (λ, ρ − wρ) is non-negative, and (6) follows. + +4 +CHAO-PING DONG AND DU CHENGYU +Now suppose λ is dominant with respect to w∆+. Notice that the half sum of positive +roots with respect to w∆+ is wρ. Applying (6) to λ, w∆+ and wρ gives +∥λ − wρ∥ ⩽ ∥λ − w−1(wρ)∥ = ∥λ − ρ∥. +Therefore, (6) becomes an equality in the current setting. +Now suppose λ is not dominant with respect to the new positive set w∆+. Define +Dw := {γ ∈ ∆−|γ ∈ w∆+}, +where ∆− = −∆+. It is well-known that +ρ − wρ = +� +γ∈Dw +(−γ). +By assumption, λ is not dominant with respect to w∆+. There must exist β ∈ w∆+ such +that (λ, β) < 0. But it cannot live in ∆+, because λ is dominant with respect to ∆+. As a +consequence, β ∈ Dw. Continuing with (∗), we have that +−(λ, ρ − wρ) = − + +λ, +� +γ∈Dw +(−γ) + + = + +λ, +� +γ∈Dw +γ + + ⩽ (λ, β) < 0. +Thus (6) is strict in this case. +□ +Let us state the main result of this section. +Theorem 3.4. Let π be an irreducible representation of K with be a highest weight µ. +Choose a positive root system ∆+(g, t) making µ + 2ρc dominant. Let C(g) be the closed +dominant Weyl chamber corresponding to ∆+(g, t). Then the inequality (1) is strict if and +only if one of the following conditions holds: +(a) µ + 2ρc is irregular for ∆(g, t). +(b) µ − wρ + ρc /∈ C(k) for all w ∈ W(g, t) such that µ + 2ρc ∈ wC(g). +Proof. Let P(·) be the projection map to the cone C(g). It suffices to show that +(7) +∥π∥lambda = ∥P(µ + 2ρc − ρ)∥ ≤ ∥µ + 2ρc − ρ∥ ≤ ∥π∥spin, +that the first equality happens if and only if (a) holds, and that the second equality happens +if and only if (b) holds. +By the Pythagorean theorem, the first inequality in (7) holds, and it becomes an equality +if and only if P(µ + 2ρc − ρ) = µ + 2ρc − ρ, which is equivalent to µ + 2ρc − ρ ∈ C(g). The +latter is equivalent to (a) since µ + 2ρc ∈ C(g) and µ + 2ρc is integral. +Now let us consider the second inequality in (7). We collect all w ∈ W(g, t) such that +µ + 2ρc ∈ wC(g) as W 1(µ). Since µ + 2ρc ∈ C(k), it follows that W 1(µ) ⊆ W 1. Moreover, +the identity element e ∈ W 1(µ) due to µ + 2ρc ∈ C(g). +Using Lemma 3.1 and 3.2, we have that +(8) +∥π∥spin = min +w∈W 1 ∥{µ − wρ + ρc} + ρc∥ ⩾ min +w∈W 1 ∥µ − wρ + 2ρc∥. +Take λ = µ + 2ρc and ∆+ = ∆+(g, t) in Lemma 3.3. We have +(9) +∥µ − wρ + 2ρc∥ ≥ ∥µ − ρ + 2ρc∥. + +SPIN NORM AND LAMBDA NORM +5 +Furthermore, the inequality (9) is strict when w /∈ W 1(µ); yet it is an equality when w ∈ +W 1(µ). Now the second inequality in (7) follows from (8) and (9). +Assume (b) holds. For any w ∈ W 1 \ W 1(µ), one has that +∥{µ − wρ + ρc} + ρc∥ ⩾ ∥µ − wρ + 2ρc∥ > ∥µ − ρ + 2ρc∥. +On the other hand, for all w ∈ W 1(µ), one has that +∥{µ − wρ + ρc} + ρc∥ > ∥µ − wρ + 2ρc∥ = ∥µ − ρ + 2ρc∥. +The first strict inequality is due to the assumption that µ − wρ + ρc /∈ C(k) and Lemma 3.2. +Assume (b) does not hold. Then there exists some w0 ∈ W 1(µ) such that µ − w0ρ + ρc ∈ +C(k). Therefore, +∥{µ − w0ρ + ρc} + ρc∥ = ∥µ − w0ρ + 2ρc∥ = ∥µ − ρ + 2ρc∥. +Since we have proven that +min +w∈W 1 ∥{µ − wρ + ρc} + ρc∥ ≥ ∥µ − ρ + 2ρc∥, +we must have ∥π∥spin = ∥µ − ρ + 2ρc∥. +To sum up, the two inequalities in (7) are controlled by (a) and (b), respectively. Thus +∥π∥spin > ∥π∥lambda happens if and only if at least one of (a) and (b) holds. +□ +We record an interesting corollary from the above proof. +Corollary 3.5. If µ + ρc − wρ ∈ C(k) for some w ∈ W 1(µ). Then +(10) +∥µ∥spin = ∥µ + 2ρc − ρ∥. +4. Application to tempered Dirac series +We call an irreducible tempered representations with non-zero Dirac cohomology a tem- +pered Dirac series. Combining Theorems 3.4 and 1.2, we have the following. +Theorem 4.1. Let X be a tempered (g, K)-module with real infinitesimal character. Let π +be the unique lowest K-type of X which has a highest weight µ. Then HD(X) = 0 if and +only if +(a) µ + 2ρc is irregular for ∆(g, t); or +(b) µ − wρ + ρc is not dominant for ∆+(k, t) for any w ∈ W 1(µ). +Example 4.2. In the special case that W 1 = {e}, which is met for complex Lie groups, +SL(2n + 1, R), SL(n, H) and the linear E6(−26), we always have that µ + 2ρc is regular for +∆(g, t). Thus HD(X) = 0 if and only if µ − ρ + ρc is dominant for ∆+(k, t). +When #W 1 > 1, pick up two distinct elements w1, w2 from W 1 such that w1C(g)∩w2C(g) +is a codimension one facet of w1C(g). Then condition (a) holds for any µ such that µ+2ρc ∈ +w1C(g) ∩ w2C(g). This suggests that the tempered Dirac series of G should be divided into +#W 1 parts by those irreducible tempered X such that HD(X) vanishes. +From now on, we shall use a circle to stand for a K-type, and paint it if and only if (1) +is an equality. Let us see some concrete examples. + +6 +CHAO-PING DONG AND DU CHENGYU +-4 +0 +4 +Figure 1. Some K-types of SL(2, R) +(0,0) +(4,4) +(-4,-4) +(4,-4) +Figure 2. Some K-types of Sp(4, R) +Example 4.3. Consider SL(2, R), where ∆(g, t) = ∆(s, t) = {±2}. Then #W 1 = 2 and +C(g) ∩ sC(g) = {0}, where s is the non-trivial element in W 1. Condition (b) does not take +effect here since ∆(k, t) is empty. Thus µ = 0 is the unique K-type such that ∥µ∥spin > +∥µ∥lambda, and the tempered Dirac series of SL(2, R) are separated into two parts. +See +Figure 1. +Example 4.4. Consider G = Sp(4, R). Let K = U(2) and T = U(1) × U(1). Thus k has a +one-dimensional center. Fix +∆+(k, t) = {(1, −1)}, +∆+(g, t) = {(1, −1), (2, 0), (0, 2), (1, 1)}. +The corresponding simple roots are α1 = (1, −1) = 2ρc, and α2 = (0, 2). The highest weight +of a K-type is represented by a pair of integers (x, y) such that x ≥ y. +Condition (a) of says that the K-types on the three lines y = 1, x = −1 and y = −x +should not be painted. These lines intersect at the point (−1, 1), which is −2ρc. Condition +(b) further says that (1, 0) and (0, −1) should not be painted. Now Figure 2 suggests that +the tempered Dirac series of Sp(4, R) are separated into four parts. +Example 4.5. Let G be G2(2), the linear split G2, which is centerless, connected, but not +simply connected. We adopt the simple roots of ∆+(g, t) and ∆+(k, t) as in Knapp [8]. Let + +SPIN NORM AND LAMBDA NORM +7 +[0,0] +[0,4] +[22,0] +Figure 3. Some K-types of the linear split G2 +α1 be the short simple root and α2 be the long one. In this case, ∆(g, t) is of type G2, while +∆(k, t) is of type A1 × A1. We fix ∆+(k, t) = {γ1, γ2}, where γ1 := α1 and γ2 := 3α1 + 2α2. +Let ω1, ω2 be the fundamental weights for ∆(k, t) such that (ωi, α∨ +j ) = δij. The K-types are +parameterized via the highest weight theorem by [a, b] := aω1 + bω2, a, b ∈ Z⩾0 such that +a + b is even. +We show some of the K-types in Figure 3, where the a-coordinates of the bottom line are +0, 2, 4, 6, 8, . . . , and so are the b-coordinates of the left-most column. +Now condition (a) says that K-types on the two lines a = b and a = 3b + 4 should not +be painted. These two lines intersect at [−2, −2] = −2ρc. From Figure 3, one sees that the +tempered Dirac series are divided into three parts by the two lines. Condition (b) further +says that [2, 0] should not be painted. +To sum up, we have recovered Corollary 8.4 of [3]. +Funding +Dong is supported by the National Natural Science Foundation of China (grant 12171344). +References +[1] J. Carmona, Sur la classification des modules admissibles irr´eductibles, pp.11–34 in Noncommutative +Harmonic Analysis and Lie Groups, J. Carmona and M. Vergne, eds., Lecture Notes in Mathematics +1020, Springer-Verlag, New York, 1983. +[2] J. Ding and C.-P. Dong, Spin Norm, K-Types, and Tempered Representations, J. Lie Theory 26 (2016), +651–658. +[3] J. Ding, C.-P. Dong, and L. Yang, Dirac series for some real exceptional Lie groups, J. Algebra 559 +(2020) 379–407. +[4] C.-P. Dong, On the Dirac cohomology of complex Lie group representations, Transform. Groups 18 (2013), +61-79. [Erratum: Transform. Groups 18 (2013), 595–597.] +[5] Harish-Chandra, Harmonic analysis on real reductive Lie groups. I. The theory of the constant term J. +Funct. Anal. 19 (1975), 104–204. +[6] J.-S. Huang and P. Pandˇzi´c, Dirac cohomology, unitary representations and a proof of a conjecture of +Vogan, J. Amer. Math. Soc. 15 (2002), 185–202. + +8 +CHAO-PING DONG AND DU CHENGYU +[7] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, +1972. +[8] A. Knapp, Lie Groups, Beyond an Introduction, 2nd edition, Birkh¨auser, 2002. +[9] S. Salamanca-Riba and D. Vogan, On the classification of unitary representations of reductive Lie groups, +Ann. of Math. 148 (1998), 1067–1133. +[10] P. Trapa, A parametrization of ˆK (after Vogan), Notes from an AIM workshop, July 2004. +[11] D. Vogan, Representations of Real Reductive Groups, Birkh¨auser, 1981. +[12] D. Vogan, Unitarizability of certain series of representations, Ann. of Math. 120 (1984), 141–187. +(Dong) School of Mathematical Sciences, Soochow University, Suzhou 215006, P. R. China +Email address: chaopindong@163.com +(Du) School of Mathematical Sciences, Soochow University, Suzhou 215006, P. R. China +Email address: cydu0973@suda.edu.cn + diff --git a/8dAzT4oBgHgl3EQfE_q4/content/tmp_files/load_file.txt b/8dAzT4oBgHgl3EQfE_q4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..93849b46372f3fb84e7326ba2d644b8683470f26 --- /dev/null +++ b/8dAzT4oBgHgl3EQfE_q4/content/tmp_files/load_file.txt @@ -0,0 +1,303 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf,len=302 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='01004v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='RT] 3 Jan 2023 SPIN NORM AND LAMBDA NORM CHAO-PING DONG AND DU CHENGYU Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Given a K-type π, it is known that its spin norm (due to first-named author) is lower bounded by its lambda norm (due to Vogan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' That is, ∥π∥spin ≥ ∥π∥lambda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' This note aims to describe for which π one can actually have equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' We apply the result to tempered Dirac series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' In the case of real groups, we obtain that the tempered Dirac series are divided into #W 1 parts among all tempered modules with real infinitesimal characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Introduction Let G be a linear real reductive Lie group which is in the Harish-Chandra class [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' That is, G has only a finite number of connected components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' The derived group [G, G] has finite center;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' The adjoint action Ad(g) of any g ∈ G is an inner automorphism of g = (g0)C, where g0 is the Lie algebra of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let θ be a Cartan involution of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' We assume the subgroup K = Gθ of fixed points of θ is a maximal compact subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let g0 = k0 ⊕ s0 be the corresponding Cartan decomposition of g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' We drop the subscript for the complexification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let ˆGtemp,o denote the set of irreducible tempered representations with real infinitesimal character (up to equivalence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let ˆK denote the set of K-types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' The following bijection was noted by Trapa [10], after Vogan’s paper [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let X be any irreducible tempered (g, K)-module with real infinitesimal char- acter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Then X has a unique lowest K-type which occurs with multiplicity one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Moreover, the map φ : ˆGtemp,o → ˆK defined by taking the lowest K-type, is a well-defined bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Motivated by the lambda norm introduced by Vogan [11], the first-named author intro- duced spin norm [4] for the classification of Dirac series (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=', irreducible unitary representa- tions of G with non-vanishing Dirac cohomology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' We refer the reader to [6] and references therein for the notion of Dirac cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' It was proven in [4] that the spin norm of a K-type π is bounded below by its lambda norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' That is, (1) ∥π∥spin ⩾ ∥π∥lambda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Primary 22E46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' lambda norm, spin norm, tempered representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' 1 2 CHAO-PING DONG AND DU CHENGYU This inequality turns out to have a nice interpretation in the setting of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Indeed, let ˆGtemp,d collect the members of ˆGtemp,o with non-zero Dirac cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Put ˆKe := {π ∈ ˆK| ∥π∥spin = ∥π∥lambda}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' ([2]) The map φ restricts to ˆGtemp,d is a bijection onto ˆKe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' More precisely, any member π ∈ ˆGtemp,o is a Dirac series if and only if the inequality (1) becomes an equality on its unique lowest K-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Given an arbitrary K-type π, it is not easy to compute neither ∥π∥lambda nor ∥π∥spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Thus, it is subtle to detect whether the inequality (1) is strict or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' This note aims to give a criterion on this aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Our main result is Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' The main idea is to insert an intermediate value between ∥π∥lambda and ∥π∥spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' As an application, our result suggests that the tempered Dirac series should be separated into #W 1 parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' See (5) for the definition of W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' The note is outlined as follows: In Section 2, we recall lambda norm and spin norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Then we deduce our main result in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' The last section considers tempered Dirac series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Preliminaries In this section, we briefly recall the definitions of the spin norm and the lambda norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' The lambda norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' We keep the notations K, G, k, s, θ, etc as in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let T be a maximal torus of K and t0 be the Lie algebra of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Recall that the analytic Weyl group is defined by W(k, t) = NK(T)/AK(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' It acts on the root system ∆(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Fix a choice of positive roots ∆+(k, t), and define R(G) := {r ∈ W(k, t)|r∆+(k, t) = ∆+(k, t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Given a K-type π, by Lemma 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='1 of [9], the collection of highest weights of π as k-module is a single orbit of R(G) on ˆT ∈ it∗ 0, where ˆT is the abelian group of characters of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Now given any K-type π, take a highest weight µ of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Then µ ∈ it∗ 0 is dominant integral for ∆+(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Denote by ρc the half sum of all roots in ∆+(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Choose a positive root system ∆+(g, t) making µ + 2ρc dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Denote by ρ the half sum of all roots in ∆+(g, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let P be the projection map to the dominant chamber C(g) corresponding to ∆+(g, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Then ∥P(µ+2ρc −ρ)∥ is independent of the choices of µ and ∆+(g, t), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Section 1 and Corrollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='4 of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Now we are ready to talk about the lambda norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' ([11, 1]) For any π ∈ ˆK, the lambda norm of π is defined to be (2) ∥π∥lambda := ∥P(µ + 2ρc − ρ)∥, where µ is any highest weight of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' For any irreducible admissible (g, K)-module X, the lambda norm of X is defined to be (3) ∥X∥lambda := min π ∥π∥lambda, where π runs over all the K-types occurring in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' A K-type π is called a lowest K-type of X if it occurs in X and ∥π∥lambda = ∥X∥lambda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' SPIN NORM AND LAMBDA NORM 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' The spin norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Although the original definition of the spin norm involves the spin module SG of the Clifford algebra C(s), our discussion here does not need a deep under- standing of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Our tool is mainly the root systems and their Weyl groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' ([4]) For any π ∈ ˆK, its spin norm is defined to be ∥π∥spin := min ∥γ + ρc∥, where γ runs over all the highest weights of the ˜K-types in π ⊗ SG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' For any irreducible admissible (g, K)-module X, its spin norm is defined to be ∥X∥spin := min π ∥π∥spin, where π runs over all the K-types occurring in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' We call π a spin lowest K-type of X if it occurs in X and ∥π∥spin = ∥X∥spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' When is the inequality (1) strict?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' We fix a positive root system ∆+(k, t), and denote the half sum of roots in it by ρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let W(g, t) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=', W(k, t))) be the Weyl group of ∆(g, t) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=', ∆(k, t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let C(k) be the closed dominant Weyl chamber for ∆+(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' For any µ ∈ t∗, we use {µ} to denote the unique weight in C(k) to which µ is conjugate under the action of W(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let ∆+(g, t) be a positive root system of ∆(g, t) containing ∆+(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' ([4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='5]) For any K-type π with a highest weight µ ∈ t∗, we have (4) ∥µ∥spin = min w∈W 1 ∥{µ − wρ + ρc} + ρc∥, where (5) W 1 := {w ∈ W(g, t)|wC(g) ⊆ C(k)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' ([7, §13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='3, Lemma B]) Let λ ∈ C(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Then ∥λ + ρc∥ ⩾ ∥wλ + ρc∥ for any w ∈ W(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Moreover, the equality holds if and only if λ = wλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let ∆ be a root system with Weyl group W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Fix a positive set ∆+ of roots and denote by ρ the half sum of all positive roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' For any dominant weight λ, we have the following inequality (6) ∥λ − ρ∥ ⩽ ∥λ − wρ∥, ∀w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Moreover, if λ is dominant with respect to w∆+, we have ∥λ − ρ∥ = ∥λ − wρ∥ Otherwise, the inequality (6) is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' We first prove the inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Compute the following difference (∗) ∥λ − ρ∥2 − ∥λ − wρ∥2 = −2(λ, ρ − wρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' A widely known fact is that ρ − wρ is a sum of positive roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' The weight λ is dominant by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Thus the pairing (λ, ρ − wρ) is non-negative, and (6) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' 4 CHAO-PING DONG AND DU CHENGYU Now suppose λ is dominant with respect to w∆+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Notice that the half sum of positive roots with respect to w∆+ is wρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Applying (6) to λ, w∆+ and wρ gives ∥λ − wρ∥ ⩽ ∥λ − w−1(wρ)∥ = ∥λ − ρ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Therefore, (6) becomes an equality in the current setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Now suppose λ is not dominant with respect to the new positive set w∆+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Define Dw := {γ ∈ ∆−|γ ∈ w∆+}, where ∆− = −∆+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' It is well-known that ρ − wρ = � γ∈Dw (−γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' By assumption, λ is not dominant with respect to w∆+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' There must exist β ∈ w∆+ such that (λ, β) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' But it cannot live in ∆+, because λ is dominant with respect to ∆+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' As a consequence, β ∈ Dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Continuing with (∗), we have that −(λ, ρ − wρ) = − \uf8eb \uf8edλ, � γ∈Dw (−γ) \uf8f6 \uf8f8 = \uf8eb \uf8edλ, � γ∈Dw γ \uf8f6 \uf8f8 ⩽ (λ, β) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Thus (6) is strict in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' □ Let us state the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let π be an irreducible representation of K with be a highest weight µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Choose a positive root system ∆+(g, t) making µ + 2ρc dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let C(g) be the closed dominant Weyl chamber corresponding to ∆+(g, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Then the inequality (1) is strict if and only if one of the following conditions holds: (a) µ + 2ρc is irregular for ∆(g, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' (b) µ − wρ + ρc /∈ C(k) for all w ∈ W(g, t) such that µ + 2ρc ∈ wC(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let P(·) be the projection map to the cone C(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' It suffices to show that (7) ∥π∥lambda = ∥P(µ + 2ρc − ρ)∥ ≤ ∥µ + 2ρc − ρ∥ ≤ ∥π∥spin, that the first equality happens if and only if (a) holds, and that the second equality happens if and only if (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' By the Pythagorean theorem, the first inequality in (7) holds, and it becomes an equality if and only if P(µ + 2ρc − ρ) = µ + 2ρc − ρ, which is equivalent to µ + 2ρc − ρ ∈ C(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' The latter is equivalent to (a) since µ + 2ρc ∈ C(g) and µ + 2ρc is integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Now let us consider the second inequality in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' We collect all w ∈ W(g, t) such that µ + 2ρc ∈ wC(g) as W 1(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Since µ + 2ρc ∈ C(k), it follows that W 1(µ) ⊆ W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Moreover, the identity element e ∈ W 1(µ) due to µ + 2ρc ∈ C(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='2, we have that (8) ∥π∥spin = min w∈W 1 ∥{µ − wρ + ρc} + ρc∥ ⩾ min w∈W 1 ∥µ − wρ + 2ρc∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Take λ = µ + 2ρc and ∆+ = ∆+(g, t) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' We have (9) ∥µ − wρ + 2ρc∥ ≥ ∥µ − ρ + 2ρc∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' SPIN NORM AND LAMBDA NORM 5 Furthermore, the inequality (9) is strict when w /∈ W 1(µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' yet it is an equality when w ∈ W 1(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Now the second inequality in (7) follows from (8) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Assume (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' For any w ∈ W 1 \\ W 1(µ), one has that ∥{µ − wρ + ρc} + ρc∥ ⩾ ∥µ − wρ + 2ρc∥ > ∥µ − ρ + 2ρc∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' On the other hand, for all w ∈ W 1(µ), one has that ∥{µ − wρ + ρc} + ρc∥ > ∥µ − wρ + 2ρc∥ = ∥µ − ρ + 2ρc∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' The first strict inequality is due to the assumption that µ − wρ + ρc /∈ C(k) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Assume (b) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Then there exists some w0 ∈ W 1(µ) such that µ − w0ρ + ρc ∈ C(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Therefore, ∥{µ − w0ρ + ρc} + ρc∥ = ∥µ − w0ρ + 2ρc∥ = ∥µ − ρ + 2ρc∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Since we have proven that min w∈W 1 ∥{µ − wρ + ρc} + ρc∥ ≥ ∥µ − ρ + 2ρc∥, we must have ∥π∥spin = ∥µ − ρ + 2ρc∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' To sum up, the two inequalities in (7) are controlled by (a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Thus ∥π∥spin > ∥π∥lambda happens if and only if at least one of (a) and (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' □ We record an interesting corollary from the above proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' If µ + ρc − wρ ∈ C(k) for some w ∈ W 1(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Then (10) ∥µ∥spin = ∥µ + 2ρc − ρ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Application to tempered Dirac series We call an irreducible tempered representations with non-zero Dirac cohomology a tem- pered Dirac series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Combining Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='2, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let X be a tempered (g, K)-module with real infinitesimal character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let π be the unique lowest K-type of X which has a highest weight µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Then HD(X) = 0 if and only if (a) µ + 2ρc is irregular for ∆(g, t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' or (b) µ − wρ + ρc is not dominant for ∆+(k, t) for any w ∈ W 1(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' In the special case that W 1 = {e}, which is met for complex Lie groups, SL(2n + 1, R), SL(n, H) and the linear E6(−26), we always have that µ + 2ρc is regular for ∆(g, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Thus HD(X) = 0 if and only if µ − ρ + ρc is dominant for ∆+(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' When #W 1 > 1, pick up two distinct elements w1, w2 from W 1 such that w1C(g)∩w2C(g) is a codimension one facet of w1C(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Then condition (a) holds for any µ such that µ+2ρc ∈ w1C(g) ∩ w2C(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' This suggests that the tempered Dirac series of G should be divided into #W 1 parts by those irreducible tempered X such that HD(X) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' From now on, we shall use a circle to stand for a K-type, and paint it if and only if (1) is an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let us see some concrete examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' 6 CHAO-PING DONG AND DU CHENGYU 4 0 4 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Some K-types of SL(2, R) (0,0) (4,4) (-4,-4) (4,-4) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Some K-types of Sp(4, R) Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Consider SL(2, R), where ∆(g, t) = ∆(s, t) = {±2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Then #W 1 = 2 and C(g) ∩ sC(g) = {0}, where s is the non-trivial element in W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Condition (b) does not take effect here since ∆(k, t) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Thus µ = 0 is the unique K-type such that ∥µ∥spin > ∥µ∥lambda, and the tempered Dirac series of SL(2, R) are separated into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' See Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Consider G = Sp(4, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let K = U(2) and T = U(1) × U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Thus k has a one-dimensional center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Fix ∆+(k, t) = {(1, −1)}, ∆+(g, t) = {(1, −1), (2, 0), (0, 2), (1, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' The corresponding simple roots are α1 = (1, −1) = 2ρc, and α2 = (0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' The highest weight of a K-type is represented by a pair of integers (x, y) such that x ≥ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Condition (a) of says that the K-types on the three lines y = 1, x = −1 and y = −x should not be painted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' These lines intersect at the point (−1, 1), which is −2ρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Condition (b) further says that (1, 0) and (0, −1) should not be painted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Now Figure 2 suggests that the tempered Dirac series of Sp(4, R) are separated into four parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let G be G2(2), the linear split G2, which is centerless, connected, but not simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' We adopt the simple roots of ∆+(g, t) and ∆+(k, t) as in Knapp [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let SPIN NORM AND LAMBDA NORM 7 [0,0] [0,4] [22,0] Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Some K-types of the linear split G2 α1 be the short simple root and α2 be the long one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' In this case, ∆(g, t) is of type G2, while ∆(k, t) is of type A1 × A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' We fix ∆+(k, t) = {γ1, γ2}, where γ1 := α1 and γ2 := 3α1 + 2α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Let ω1, ω2 be the fundamental weights for ∆(k, t) such that (ωi, α∨ j ) = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' The K-types are parameterized via the highest weight theorem by [a, b] := aω1 + bω2, a, b ∈ Z⩾0 such that a + b is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' We show some of the K-types in Figure 3, where the a-coordinates of the bottom line are 0, 2, 4, 6, 8, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' , and so are the b-coordinates of the left-most column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Now condition (a) says that K-types on the two lines a = b and a = 3b + 4 should not be painted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' These two lines intersect at [−2, −2] = −2ρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' From Figure 3, one sees that the tempered Dirac series are divided into three parts by the two lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Condition (b) further says that [2, 0] should not be painted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' To sum up, we have recovered Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='4 of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Funding Dong is supported by the National Natural Science Foundation of China (grant 12171344).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Carmona, Sur la classification des modules admissibles irr´eductibles, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='11–34 in Noncommutative Harmonic Analysis and Lie Groups, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Carmona and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Vergne, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=', Lecture Notes in Mathematics 1020, Springer-Verlag, New York, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Ding and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Dong, Spin Norm, K-Types, and Tempered Representations, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Lie Theory 26 (2016), 651–658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Ding, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Dong, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Yang, Dirac series for some real exceptional Lie groups, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Algebra 559 (2020) 379–407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Dong, On the Dirac cohomology of complex Lie group representations, Transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Groups 18 (2013), 61-79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' [Erratum: Transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Groups 18 (2013), 595–597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='] [5] Harish-Chandra, Harmonic analysis on real reductive Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' The theory of the constant term J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' 19 (1975), 104–204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Huang and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Pandˇzi´c, Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' 15 (2002), 185–202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' 8 CHAO-PING DONG AND DU CHENGYU [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Knapp, Lie Groups, Beyond an Introduction, 2nd edition, Birkh¨auser, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Salamanca-Riba and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Vogan, On the classification of unitary representations of reductive Lie groups, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' 148 (1998), 1067–1133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' [10] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Trapa, A parametrization of ˆK (after Vogan), Notes from an AIM workshop, July 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' [11] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Vogan, Representations of Real Reductive Groups, Birkh¨auser, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' [12] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' Vogan, Unitarizability of certain series of representations, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' 120 (1984), 141–187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' (Dong) School of Mathematical Sciences, Soochow University, Suzhou 215006, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' China Email address: chaopindong@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='com (Du) School of Mathematical Sciences, Soochow University, Suzhou 215006, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content=' China Email address: cydu0973@suda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} +page_content='cn' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfE_q4/content/2301.01004v1.pdf'} diff --git a/A9E3T4oBgHgl3EQfswtu/content/tmp_files/2301.04670v1.pdf.txt b/A9E3T4oBgHgl3EQfswtu/content/tmp_files/2301.04670v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fb7debdbe5e06fe16dbdc5555c64b5e70ff01d50 --- /dev/null +++ b/A9E3T4oBgHgl3EQfswtu/content/tmp_files/2301.04670v1.pdf.txt @@ -0,0 +1,3263 @@ +MNRAS 000, 1–22 (2015) +Preprint 13 January 2023 +Compiled using MNRAS LATEX style file v3.0 +The effect of stellar encounters on the dark matter annihilation signal +from prompt cusps +Jens Stücker1★, Go Ogiya2, Simon D. M. White3, Raul E. Angulo1,4 +1Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain +2Institute for Astronomy, School of Physics, Zhejiang University, Hangzhou 310027, China +3Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany +4IKERBASQUE, Basque Foundation for Science, E-48013, Bilbao, Spain +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +Prompt cusps are the densest quasi-equilibrium dark matter objects; one forms at the instant of collapse within every isolated +peak of the initial cosmological density field. They have power-law density profiles, 𝜌 ∝ 𝑟−1.5 with central phase-space density +set by the primordial velocity dispersion of the dark matter. At late times they account for ∼ 1% of the dark matter mass but for +> 90% of its annihilation luminosity in all but the densest regions, where they are tidally disrupted. Here we demonstrate that +individual stellar encounters, rather than the mean galactic tide, are the dominant disruptors of prompt cusps within galaxies. +Their cumulative effect is fully (though stochastically) characterised by an impulsive shock strength 𝐵∗ = 2𝜋𝐺 +∫ +𝜌∗(x(𝑡)) d𝑡 +where 𝜌∗, the total mass density in stars, is integrated over a cusp’s entire post-formation trajectory. Stellar encounters and mean +tides have only a small effect on the halo annihilation luminosity seen by distant observers, but this is not true for the Galactic +halo because of the Sun’s position. For a 100 GeV WIMP, Earth-mass prompt cusps are predicted, and stellar encounters suppress +their mean annihilation luminosity by a factor of two already at 20 kpc, so that their annihilation emission is predicted to appear +almost uniform over the sky. The Galactic Center 𝛾-ray Excess is thus unaffected by cusps. If it is indeed dark matter annihilation +radiation, then prompt cusps in the outer Galactic halo and beyond must account for 20-80% of the observed isotropic 𝛾-ray +background in the 1 to 10 GeV range. +Key words: cosmology: dark matter – Galaxy: halo – gamma-rays: diffuse background +1 INTRODUCTION +The nature of dark matter is still unknown. One of the most popular +candidates for dark matter searches is a weakly interacting mas- +sive particle (WIMP) which could be produced thermally in the +early universe. If dark matter is a WIMP, it may have a signifi- +cant self-interaction cross section that allows dark matter to self- +annihilate in regions where the dark matter density is sufficiently +high (Roszkowski et al. 2018; Arcadi et al. 2018). +Detection of the secondary products of such self-annihilation – +also known as indirect dark matter detection – is one of the most +promising ways of learning more about the nature of dark matter. +Excitingly, the Fermi large area telescope (Fermi LAT) has detected +a galactic centre excess (GCE) in gamma ray radiation in the spectral +range of 1 − 10 GeV (Hooper & Goodenough 2011). Although there +has been a long-ongoing debate about the precise properties of the +signal (see Slatyer 2021, chapter 6, for a review), it seems so far that +the morphology and the spectrum of the signal could be consistent +with a dark matter self-annihilation signal from the central regions of +the Milky Way’s dark matter halo (e.g. Di Mauro 2021). Additionally, +recent high-resolution hydrodynamical simulations show that the +Milky Way likely exhibits a halo with the right spatial structure to +★ E-mail: jstuecker@dipc.org +explain the GCE (Grand & White 2022). However, there are also +other astrophysical processes that might explain the GCE so that it +cannot yet be clearly attributed to dark matter. +Recently, Delos & White (2022a) have highlighted a different +aspect of the interpretation of possible indirect detection signals from +the haloes of the Milky Way and of other galaxies. Recent advances +in the modelling of the formation of the smallest nonlinear objects +in a WIMP cosmology have led to a clearer understanding of their +origin, structure and late-time abundance, leading to a re-evaluation +of the expected dark matter self-annihilation signal at late times. +The primordial density field is smooth below the dark matter free- +streaming scale (e.g. ∼ 1 pc comoving for a typical WIMP) and so +exhibits a large number of density peaks at this scale (∼ 104 − 105 +per solar mass of dark matter). The first nonlinear structures begin to +form around redshift 30 through monolithic collapse of these peaks. +Their collapse history differs radically from that of traditional, more +massive haloes of the kind characterised by Navarro et al. (1996) +(hereafter: NFW). While NFW haloes assemble through hierarchical +accretion and merging over time-scales which are comparable to their +age, a prompt cusp forms almost instantaneously at the moment of +first collapse of a density peak and it contains dark particles with +orbital periods much shorter than the collapse time. As a result, the +density profiles of prompt cusps also differ from the NFW form, +following a steep 𝑟−1.5 density profile between an outer boundary +© 2015 The Authors +arXiv:2301.04670v1 [astro-ph.CO] 11 Jan 2023 + +2 +J. Stücker et al. +set by the curvature of the initial density peak and an inner core +determined by the physical nature of the dark matter. To distinguish +these dense “first” objects from traditional NFW haloes, we follow +Delos & White (2022a) in referring to them as “prompt cusps”. +Delos & White (2022b) argue that prompt cusps should still be +extremely abundant substructures today. In regions where their num- +ber is not significantly reduced by subsequent evolution, every solar +mass of dark matter should contain tens of thousands of them, and +we should expect ∼ 1016 cusps associated with the dark matter halo +of the Milky Way. Since they are denser in their centres than tra- +ditional NFW haloes, prompt cusps survive tidal effects better and +produce a substantially larger dark matter annihilation signal. In fact, +the annihilation signal from 𝑟−1.5-cusps is logarithmically divergent +with radius, limited by the inner and outer boundaries of the power- +law profile, which are set by the primordial dark matter phase-space +density (e.g. Macciò et al. 2012) and by the initial peak extent, re- +spectively. This raises the signal for indirect detection compared +to most previous studies. Thus Delos & White (2022b) predict that +these cusps should enhance the total annihilation luminosity of NFW +haloes by factors ranging between 100 and 2500, depending on halo +concentration, and additionally that they should significantly alter +the morphology of the signal which, except in the densest regions, is +proportional to the first power of the mean dark matter density, rather +than to its square, as usually assumed. +As Delos & White (2022b) note this “cusp”-component impacts +a possible annihilation interpretation of the GCE in two significant +ways. (1) If disruption of prompt cusps is ignored, their emission +dominates that of the smooth halo component beyond about 5 de- +grees from the Galactic Centre, resulting in a profile in disagreement +with observation. Accounting for truncation and disruption by Galac- +tic tides reduces cusp emission from the inner Galaxy but leaves it +still dominant beyond 10 degrees, reducing but not eliminating the +contradiction with observation. (2) If the GCE is nevertheless due to +annihilation, emission from prompt cusps in the outer Galactic halo +and external to the Milky Way must constitute a major contribution to +the isotropic 𝛾-ray background (IGRB). However, the IGRB appears +to be almost completely attributable to emission from star-forming +galaxies and AGN (Blanco & Hooper 2019) leaving little space to add +an additional component. The resulting upper limit on the prompt +cusp contribution to the IGRB allows Delos & White (2022b) to +strengthen constraints on the self-annihilation cross-section and the +thermal relic mass of a hypothetical WIMP dark matter particle – ef- +fectively excluding thermal relic WIMPs with masses 𝑀 ≤ 10 TeV1. +Delos & White (2022b) explicitly considered only the effect of the +smooth tidal field of the Milky Way on prompt cusps. However, stellar +encounters can also be important, inducing strong impulsive shocks +in dark matter substructures and potentially even disrupt them. Such +encounters should be very frequent in the central region of the galaxy +and therefore can affect both the predictions of the last paragraph. +The main goal of this paper is to evaluate quantitatively the ef- +fect of stellar encounters on the structure, survival and predicted +annihilation signal from prompt cusps. We will find that prediction +(1) is seriously affected. After accounting for the effect of stellar +encounters, the tension between the GCE emission profile and that +predicted including cusp emission vanishes. For prediction (2) we +will find that reduced emission from cusps in the inner Galactic halo +leads to slightly lower IGRB predictions for the annihilation cross- +1 This constraint is under the assumption of a bottom quark 𝑏𝑏 annihilation +channel and is independent of the possible annihilation interpretation of the +GCE. +section required to produce the GCE. These predictions remain in +tension with claims that the IGRB is almost entirely due to other +sources. +We note that there is already a large literature on the effect of stellar +encounters on NFW subhaloes (Goerdt et al. 2007; Angus & Zhao +2007; Green & Goodwin 2007; Schneider et al. 2010; Delos 2019a; +Kavanagh et al. 2021; Shen et al. 2022). Much of this is based on +the incorrect assumption that systems disrupt when the total injected +energy exceeds their initial binding energy, and hence needs to be +read with care (Aguilar & White 1985; van den Bosch et al. 2018). A +particularly clear and general treatment has been presented by Delos +(2019a) and we will often refer to this article as representative of +stellar encounters with NFW haloes. Such results cannot be applied to +prompt cusps, since their power-law structure dramatically enhances +their resilience to tidal effects in comparison to the inner regions of +NFW profiles (see Stücker et al. 2022, for a discussion of different +power-law profiles). The only previous study to consider the effect +of stellar encounters on prompt cusps is Ishiyama et al. (2010), +but unfortunately this presented a very limited treatment and used +incorrect assumptions when scaling with encounter strength, leading +to the erroneous conclusion that cusps would never be disrupted by +stellar encounters. We will discuss this in more detail below. +The structure of this paper is as follows. In Section 2 we briefly +introduce both the theoretical basis for predicting the distribution of +prompt cusp structural properties and the physical formalism needed +to describe the effects of their encounters with stars. We also present +a novel, simple and very general scheme for calculating the full distri- +bution of impulsive stellar shocks experienced by cusps (or normal +subhaloes) as they pass through a galaxy (e.g. through the Milky +Way’s disk or bulge). In Section 3 we numerically integrate cusp +orbits in a realistic Milky Way model, and we infer the parameters +describing their shock histories. In Section 4 we use idealized N-body +simulations to estimate the disruptive effects of stellar encounters, de- +veloping simple formulae that predict the structure and annihilation +signal of cusps that have gone through arbitrarily many encounters. +Additionally, we show how this can be supplemented to include the +effects of stripping by the smooth Galactic tidal field. In Section 5 +we present our main results, predictions for the profile of the prompt +cusp contribution to the annihilation signal of the Galactic halo both +as seen by a distant observer and as seen from the Earth, together +with an assessment of how prompt cusps affect the form and relative +amplitude of the GCE and the IGRB. In Section 6 we will discuss +the implications of these findings for indirect dark matter searches. +We make almost all codes used in this study available through an +online python repository2 so that our methods can easily be used +in future studies and the results of this paper can be reproduced +independently. +2 THEORY +As described in the introduction, the effect of stellar encounters +on NFW subhaloes has already been studied extensively, although +in many cases using an incorrect criterion for subhalo disruption. +There has, however, been no realistic study of the effects for power- +law cusps. Furthermore, even in the NFW case, most studies have not +realistically modelled the full distribution of encounter parameters. +Here we present the theoretical considerations necessary to treat full +encounter histories in an accurate, general but simple way. +2 https://github.com/jstuecker/cusp-encounters +MNRAS 000, 1–22 (2015) + +Prompt cusps & stellar encounters +3 +For this we present in Section 2.1 the distribution of initial cusp +properties as derived from the statistics of peaks in the initial gaussian +density field, in Section 2.2 how cusp structure can be modified by +an inner core to account for the upper limit on phase-space density, +in Section 2.3 the impulsive shocks induced by stellar encounters, in +Section 2.4 a calculation of the total number of encounters expected +on passing through a star distribution with arbitrary stellar mass and +velocity distributions, in Section 2.5 the statistical distribution of +shock histories that follows from these considerations. +2.1 Cusps +Several numerical studies have found that peaks on the dark matter +free-streaming scale collapse promptly to form dense cusps (Die- +mand et al. 2005; Ishiyama et al. 2010; Anderhalden & Diemand +2013; Ishiyama 2014; Polisensky & Ricotti 2015; Angulo et al. +2017; Ogiya & Hahn 2018; Delos et al. 2018a,b, 2019; Colombi +2021; Delos & White 2022a). These cusps have a density profile, +𝜌(𝑟) = 𝐴𝑟−3/2, +(1) +parameterised by a normalization 𝐴 and an outer radius 𝑟cusp which +limits the extent of the profile. Delos & White (2022a) find that both +parameters can be predicted well for a given cusp from the properties +of the initial density peak from which it forms. Specifically, +𝐴 = 24𝜌0𝑎−1.5 +col 𝑅1.5, +(2) +𝑟cusp = 0.11𝑎col𝑅, +(3) +where 𝜌0 is the mean dark matter density of the universe today, 𝑎col is +the scalefactor when the peak first collapses and 𝑅 is the Lagrangian +size of the initial density peak defined as 𝑅 = +√︁ +|𝛿/∇2𝛿|, where 𝛿(x) +is the linear overdensity field as a function of comoving position. +The time of collapse can be estimated to sufficient accuracy by an +ellipsoidal collapse model based on the triaxial structure of the initial +peak. A more detailed description can be found in Delos & White +(2022b). +We follow the descriptions of Delos & White (2022b) to sample +a distribution of cusps. For this we use a dark matter power spec- +trum generated by the Boltzmann code CLASS (Blas et al. 2011) +up to a resolved wavenumber of 𝑘 = 104 h Mpc−1 at 𝑧 = 30.6. +Beyond that scale we use the analytic prescriptions of Hu & +Sugiyama (1996), but normalized so that it matches the CLASS +spectrum at 𝑘 = 104 h Mpc−1. We multiply this spectrum using +the exponential power spectrum cutoff description of Bertschinger +(2006) for a WIMP with mass 𝑚WIMP = 100 GeV and decoupling +temperature 𝑇d = 30 MeV (corresponding to a decoupling scale- +factor of 𝑎d = 5.33 × 10−12). We use the free streaming length +𝑘FS = 1.06 pc−1 as calculated by Delos & White (2022b). The dis- +tribution of initial density peaks can be sampled analytically given +the initial power spectrum as described by Bardeen et al. (1986). We +have created our own implementation of the sampling of peaks which +is published in the code repository of this paper. However, we tested +it against the peak sampling implementation that was published by +Delos et al. (2019). We map the distribution of peaks onto a dis- +tribution of cusps by using equations (2) – (3) with the ellipsoidal +collapse correction for 𝑎col as explained by Delos & White (2022b) +based on the approximation from Sheth et al. (2001). +We show the resulting distribution of cusps in Figure 1 (dashed +contours). However, more relevant than the number distribution of +cusps is the annihilation weighted distribution. Under the assumption +of a velocity independent cross section, the annihilation rate of a cusp +10 +2 +rcusp [pc] +10 +5 +10 +4 +10 +3 +A [M + / pc3/2] +ann. weighted +num. weighted +90% +75% +50% +25% +10% +Figure 1. The distribution of the cusp normalization 𝐴versus the cusp trunca- +tion radius 𝑟cusp. Typical cusps that dominate the annihilation rate distribution +have 𝐴 ∼ 8 × 10−4 M⊙pc−3/2 and 𝑟cusp ∼ 5 × 10−3 pc. +should be proportional to +𝐽 = +∫ 𝑟cusp +𝑟core +𝜌2 d3𝑟 += 4𝜋𝐴2 log(𝑟cusp/𝑟core), +(4) +where we have neglected any contributions from 𝑟 < 𝑟core and 𝑟 > +𝑟cusp where 𝑟core is the core radius enforced by the phase-space +density constraint (Delos & White 2022a). We will explain in Section +2.2 how to calculate and treat the core radius more precisely. +The weighted distributions are shown as solid contours in Figure 1. +We can see that for this specific dark matter particle, the typical cusp +relevant for the annihilation rate has 𝐴 ∼ 8 × 10−4 M⊙pc−3/2 and +𝑟cusp ∼ 5 × 10−3 pc ≈ 103 AU (in physical units). It collapses at +redshift 𝑧 ∼ 30. +2.2 Phase-space cores +The fine-grained phase-space density of dark matter is constant as +a function of time (Liouville’s theorem). The coarse-grained phase- +space density – defined as the average over some finite phase-space +volume – can therefore never exceed the fine-grained value estab- +lished at dark matter freeze-out (Tremaine & Gunn 1979). Let us +denote the maximum of this fine-grained phase-space density by +𝑓max. The value of 𝑓max is set in the early universe and depends +strongly on the type of dark matter considered. For the WIMP model +considered above, for example, it is 𝑓max = 9.98 · 1013 +M⊙ +km3s−3pc3 , +whereas for thermal relic warm dark matter with 𝑚𝑋 = 3.5keV it is +𝑓max = 6.75 · 10−2 +M⊙ +km3s−3pc3 (see e.g. Delos & White 2022a)3. +The power-law profile of equation (1) corresponds, for an isotropic +3 Our values here differ slightly from the values of (Delos & White 2022a) +since we use Ω𝑥 = 0.26 instead of 0.3 for the dark matter density parameter. +MNRAS 000, 1–22 (2015) + +4 +J. Stücker et al. +velocity distribution, to a phase-space distribution function, +𝑓 (𝐸) = 𝑓0𝐸−9/2, +(5) +𝑓0 = 1120 +√ +2𝜋2𝐴4𝐺3 +9 +, +(6) +(see e.g. Stücker et al. 2022). This description must break down at +energies where 𝑓 (𝐸) > 𝑓max. To estimate the radial scale where +this happens we can consider at each radius the largest reachable +phase-space density – given by 𝑓 (𝜙(𝑟)) where 𝜙(𝑟) is the potential. +Inserting this into equation (5) and inverting for r we find +𝑟core = +3 +9√ +3 · 70 +4 +9 +64𝜋 +10 +9 𝐴 +2 +9 𝐺 +2 +3 𝑓 +4 +9max +≈ 1.03 × 10−5 pc +� +𝐴 +10−3 M⊙pc− 3 +2 +�− 2 +9 �� +� +𝑓max +1014 +M⊙ +km3s−3pc3 +�� +� +− 4 +9 +. +(7) +Thus in the above WIMP model, typical cusps relevant for the anni- +hilation calculation have thermal core size, 𝑟core ∼ 10−5 pc ∼ 2 AU. +The detailed shape of the density profile near the core radius +is unclear. It would be desirable to run simulations with the actual +primordial velocity distribution of a WIMP, so that phase-space cores +are created self-consistently, and then to measure their density profile. +Such simulations would be computationally demanding and have not +yet been performed, but they are not far beyond current capabilities. +(Consider that typically 𝑟cusp ∼ 500𝑟core so that simulations that +resolve the cusp profile well are typically only an order of magnitude +in linear scale from the core radius.) +To obtain a profile that has the desired maximum phase-space +density and connects smoothly to the appropriate power-law at larger +radii we make the following Ansatz for the phase-space density, +𝑓 (𝐸) = +𝑓0 +(𝐸 + 𝐸core)9/2 , +(8) +where 𝐸core is the core energy scale where the phase-space density +𝑓max would be reached in the fiducial power-law profile: +𝐸core = +� +𝑓0 +𝑓max +�2/9 +, +(9) +so that together with equation (6) the profile is fully specified through +𝐴 and 𝑓max. Note that for 𝑓max → ∞ the pure power-law profile is +recovered. +We can integrate over the velocity components of the phase-space +density to find the density, +𝜌(𝑟) = +∫ +𝑓 (𝐸) d3𝑣 += 4𝜋 +∫ ∞ +𝜙 +∫ 𝑟√ +2(𝐸−𝜙) +0 +𝐿 𝑓 (𝐸) +𝑟2 +√︃ +2𝐸 − 2𝜙 − 𝐿2 +𝑟2 +d𝐿 d𝐸 += 4𝜋 +∫ ∞ +𝜙 +𝑓 (𝐸) +√︁ +2(𝐸 − 𝜙) d𝐸 += +64𝜋 +√ +2 +105 (𝐸core + 𝜙(𝑟))3 , +(10) +as a function of the potential, 𝜙, which is normalized so that 𝜙(0) = 0 +and 𝜙(𝑟 → ∞) → ∞. Combined with Poisson’s equation this forms +a nonlinear second order differential equation for the potential, +𝜕𝑟 (𝑟2𝜕𝑟 𝜙) +4𝜋𝐺𝑟2 += +64𝜋 +√ +2 +105 (𝐸core + 𝜙(𝑟))3 . +(11) +10 +3 +10 +2 +10 +1 +100 +101 +f(E = +)/fmax +cored +powerlaw +fmax +10 +2 +10 +1 +100 +101 +(r)/ +max +cored +powerlaw +(r4 + r4 +core) +3/8 +10 +1 +100 +101 +r/rc +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +(r)/ +powerlaw(r) +cored +powerlaw +(r4 + r4 +core) +3/8 +Figure 2. The phase-space and density profiles of a 𝑟−3/2 cusp with a core +induced by the phase-space constraint 𝑓 < 𝑓max. The top panel shows the +phase-space density 𝑓 (𝜙(𝑟)) corresponding to the highest phase-space den- +sity present at each radius. The central panel shows the density profile and the +bottom panel shows the density profile divided by the fiducial power-law pro- +file. In each case, we see a rapid transition between the power-law behaviour +and a maximum central value. +We do not know how to solve this differential equation analytically +and therefore solve it through numerical integration starting at 𝑟 = 0 +using 𝜙(0) = 𝜕𝑟 𝜙(0) = 0. As a result we find 𝜌(𝑟) and 𝜙(𝑟) and we +display these and the phase-space density 𝑓 (𝜙(𝑟)) in Figure 2. +As expected, the density profile reaches a well defined maximum +at 𝑟 ≲ 𝑟core, given by +𝜌max = 𝐴𝑟−3/2 +core +∝ 𝐴4/3 𝑓 2/3 +max. +(12) +A good approximation to the density profile is given by +𝜌(𝑟) = 𝐴(𝑟4 + 𝑟4 +core)−3/8, +(13) +which we also show as an orange line in Figure 2. This deviates +MNRAS 000, 1–22 (2015) + +Prompt cusps & stellar encounters +5 +at most by 20% from the actual density around 𝑟 ∼ 3𝑟𝑐 where +the latter is slightly enhanced with respect to the fiducial power- +law. This enhancement is similar to the well-known behavior of the +isothermal sphere where the cored solution rises above the asymp- +totic 𝑟−2 power-law solution at radii comparable to the core radius +before aymptoting to constant density at smaller radii. (e.g. Binney +& Tremaine 2008). +We note that while there are no simulations which have a phase- +space density constraint consistent with the expected free-streaming +scale, there have been simulations by Macciò et al. (2012) with +artificially large initial velocity dispersion and hence a lowered upper +limit on phase-space density. Although it is difficult to compare these +directly with our profile, it seems that the final cores do saturate the +phase-space bound and that they transition relatively quickly to the +asymptotic power-law behaviour – both consistent with the profile +that we propose here. +We find that the annihilation radiation from the cored profile inside +radius 𝑟 can be approximated for 𝑟 > 10𝑟core by +𝐽(< 𝑟) = 4𝜋𝐴2(0.531 + log(𝑟/𝑟core)). +(14) +This is marginally larger than the annihilation radiation one obtains +when assuming the power-law profile down to 𝑟core (compare equa- +tion (4)) and a uniform density at smaller radii, in which case the +0.531 gets replaced by 0.333. This is due to the slight enhancement +of the profile around 3𝑟core. +When needed, we will use the numerical cored profile throughout +this study. +2.3 Impulsive encounters +We consider a star with mass 𝑀∗ passing a cusp on a linear orbit at +constant relative velocity 𝑣 and minimal distance 𝑏. We can approx- +imate the tidal forces acting on cusp particles through a multipole +expansion of the potential up to second order, +𝜙(𝒙, 𝑡) = 𝜙𝑠(𝒙 − 𝒙0, 𝑡) − T0(𝑡)(𝒙 − 𝒙0), +(15) +where 𝜙𝑠 denotes the self-potential of the cusp, 𝒙0 is the location of +centre of the cusp and T0(𝑡) is the tidal field of the star evaluated +at 𝒙0. Here we have neglected zeroth- and first-order terms, since +they do not affect the internal dynamics of the cusp (e.g. Stücker +et al. 2021). This “distant-tide” approximation is valid for particles +that are close enough to the centre, ||𝒙 − 𝒙0|| ≪ 𝑏 . We will see +that for encounters with stars with masses of order M⊙ the distant +tide approximation is excellent for all particles that remain bound +to the cusp (which typically have small Δ𝒙) and is only violated for +particles that are kicked so strongly that they will leave the system. +Thus, it is safe to adopt this approximation in all our calculations. +Further, we can assume the impulsive approximation which con- +siders the limit that particles move very little within the cusp during +the time of the encounter. In this case, the total change in the velocity +of a particle due to the encounter can be approximated by +Δ𝑣 = +�∫ +T0(𝑡)d𝑡 +� +(𝒙 − 𝒙0) += K(𝒙 − 𝒙0), +(16) +where we have defined the shock tensor 𝐾. +It is easy to see that the impulsive approximation is an excellent +approximation here. The dynamical time-scale at radius 𝑟 of our cusp +is given by +𝑡d(𝑟) = +𝑟 +𝑣circ(𝑟) += +√︄ +𝑟3 +𝐺𝑀(< 𝑟) , +(17) +where 𝑀(< 𝑟) is the enclosed mass profile. The internal dynamical +time-scale is shortest at the core-radius, where it is of order 2 × 104 yr +for the strongest cusps (which dominate the annihilation distribution). +The impact parameter of the weakest relevant encounters is of order +𝑏 ∼ 104 AU with 𝑣 ∼ 200 km s−1 which gives an encounter time- +scale of order 𝑡enc = 𝑏/𝑣 ∼ 200 yr which is two orders of magnitude +smaller than the dynamical time scale of the quickest particles in +the cusp. Stronger encounters (which are more relevant) will happen +on even shorter time-scales and most particles orbit on longer time +scales so that the ratio should be even larger in practice and we can +safely assume the impulsive limit for all of our calculations. +If we assume that the star is a point mass with tidal field +𝑇𝑖 𝑗 (𝑡) = −𝜕𝑖𝜕𝑗 +� +− +𝑀∗𝐺 +∥𝒙 − 𝑥∗(𝑡)∥ +� +, +(18) +and we assume its trajectory (without loss of generality) to be along +the y-direction with the closest encounter at the coordinate (𝑏, 0, 0), +then the shock tensor is given by +K = 𝐵 �� +� +1 +0 +0 +0 +0 +0 +0 +0 +−1 +�� +� +(19) +𝐵 = 2𝐺𝑀∗ +𝑣𝑏2 +(20) +(e.g. Aguilar & White 1985) where we have defined the shock pa- +rameter 𝐵. We note that 𝐵 has dimensions of the inverse of time, but +to simplify intuitive understanding we will typically state it in units +of km s−1 pc−1. +It is clear that the only relevant parameter for describing the effect +of a distant and impulsive stellar tidal shock on a prompt cusp is the +tidal shock parameter 𝐵. The individual values of 𝑏, 𝑀∗ and 𝑣 matter +only insofar that they determine the value of 𝐵. +We can get a feeling for what range of 𝐵 values will be relevant +for typical cusps by considering the values, +𝐵cusp = 𝑣circ(𝑟cusp) +𝑟cusp += +� +�8𝜋𝐺𝐴 +3𝑟3/2 +cusp +, +(21) +𝐵core = 𝑣circ(𝑟core) +𝑟core += +√︄ +8𝜋𝐺𝐴 +3𝑟3/2 +core +∝ 𝐴2/3 𝑓 1/3 +max. +(22) +𝐵cusp indicates the strength of a tidal shock that is needed to induce +a velocity change at the radius 𝑟cusp as large as the circular veloc- +ity and 𝐵core is the analogue quantity at the core radius. We show +the distributions of these two parameters for the fiducial 100 GeV +WIMP in Figure 3. We can expect that tidal shocks with 𝐵 ≳ 𝐵cusp +will significantly alter the profile of the cusp. Tidal shocks with +𝐵 ≳ 𝐵core may possibly lead to complete disruption. Shocks with +𝐵 ≪ 𝐵cusp will leave the cusp largely unaffected. Typical cusps that +are relevant for the annihilation rate have values of 𝐵cusp and 𝐵core +of order 0.3 km s−1 pc−1 and 30 km s−1 pc−1 respectively. The im- +pact parameters that are needed to reach such shock parameters for +𝑀∗ = 1 M⊙ and 𝑣 = 200 km s−1 are 1 × 10−2 pc ≈ 2000 AU and +1 × 10−3 pc ≈ 200 AU respectively. We will see that tidal shocks of +this order are not only possible, but also quite likely. It is therefore +important to make precise quantitative calculations to evaluate the +MNRAS 000, 1–22 (2015) + +6 +J. Stücker et al. +10 +2 +10 +1 +Bcusp [km/s / pc] +101 +Bcore [km/s / pc] +ann. weighted +num. weighted +90% +75% +50% +25% +10% +Figure 3. The distributions of 𝐵core and 𝐵cusp which indicate the resilience +of prompt cusps against tidal shocks. A tidal shock with 𝐵 ≳ 𝐵cusp will +likely affect the cusp’s profile significantly. Shocks with 𝐵 ≳ 𝐵core will lead +to disruption. +number of such encounters and the effect they have on the aggregate +annihilation luminosity from cusps. +Finally, it is useful to introduce the characteristic spatial scale +associated with the action of shocks on a cusp, +𝑟𝐵 = +� 8𝜋𝐺𝐴 +3𝐵2 +�2/3 += +� 2𝜋𝐴 +3𝐺 +�2/3 𝑏8/3𝑣4/3 +𝑀4/3 +∗ +, +(23) +which is the radius where the change in velocity induced by the +shock is of order the circular velocity. We will see in Section +4 that the majority of particles beyond 𝑟𝐵 will leave the sys- +tem. The distant tide approximation is a good approximation if +min(𝑟𝐵, 𝑟cusp) ≪ 𝑏 so that it holds for all particles that remain +bound. We find that typical encounter scenarios with stars for prompt +cusps have min(𝑟𝐵, 𝑟cusp) ≪ 0.1𝑏 for any impact parameter 𝑏, so +that the distant tide approximation is always valid. While we do not +focus on NFW haloes in this study, we note that many of our sub- +sequent derivations are of interest for studies of NFW substructures +as well. For these, we estimate that 𝑟𝐵,NFW < 0.1𝑏 (defined through +the circular velocity criterion for an NFW profile) holds for typical +closest impact parameters with 𝑏 ≲ 1000 AU for haloes with virial +masses 𝑚200c ≲ 1𝑀⊙ (and concentration 𝑐 ∼ 30). Many of our +results below can therefore also be applied to NFW subhaloes with +masses below 1.0𝑀⊙, but additional care must be taken if larger +masses are considered since the distant tide approximation may then +fail. +The goal of the remaining parts of this section is to determine the +distribution of shock parameters 𝐵 for a cusp that is moving through +an arbitrary stellar distribution (e.g. a component of the Milky Way). +We note that this calculation is both simpler and more accurate +than previous calculations in the literature which have focused on +estimating the distribution of the impact parameter 𝑏. +2.4 The expected number of encounters +We want to estimate the expected number of encounters with a tidal +shock parameter that is greater than 𝐵, 𝑁(> 𝐵), for a cusp that is +orbiting through an arbitrary stellar distribution. The stellar distri- +bution can be described by a mass dependent phase-space (number) +density +𝑓∗(𝒙, 𝒗, 𝑀) = +d𝑁 +d3𝑥 d3𝑣 d𝑀 , +(24) +which is normalized so that an integral over a phase-space volume +and over a stellar mass interval gives the number of stars within that +volume and mass range. An integral over phase space alone gives the +stellar mass function, which is allowed to vary spatially. +If our cusp was passing through a uniform medium without veloc- +ity dispersion and with number density 𝑛∗ then the expected number +of encounters in a time-interval d𝑡 with impact parameters in the +range [𝑏, 𝑏 + d𝑏] is given by +d𝑁 = 2𝜋𝑛∗𝑏𝑣 d𝑏 d𝑡, +(25) +where 𝑣 is the relative velocity with respect to the medium. Equation +(25) arises by considering stars when they get closest to our cusp, +which is when they pass through the plane orthogonal to the velocity +vector. The relevant velocity here is the relative velocity between the +stars and the cusp; higher velocities lead to more frequent encounters. +For arbitrary phase-space distributions we have to consider each +velocity and mass bin individually. The number of encounters within +time interval d𝑡, impact parameter range (𝑏, 𝑏 + d𝑏) and encounter +velocity bin d3𝑣 with stars that have masses in the range (𝑀, 𝑀+ d𝑀) +is given by +d𝑁 = (2𝜋𝑏 d𝑏)( 𝑓∗(𝒙, 𝒗 − 𝒗rel, 𝑀) d3𝑣 d𝑀)(𝑣 d𝑡), +(26) +where 𝒗 is the encounter velocity, and 𝑣rel is the relative velocity +between our cusp and the zero-point of the stellar phase-space dis- +tribution. Here, we have assumed that the phase space distribution +function does not vary significantly over the distance 𝑏. This is an ex- +cellent approximation, since typical encounters of interest will have +𝑏 ≪ 1 pc whereas the stellar distribution varies on much larger scales +(≫ 100 pc). +Now, it would be straightforward to estimate, for example, the +total number of encounters with impact parameters smaller than +some given 𝑏 by integrating (26) over the corresponding variables. +In general this would turn out to depend both on the phase-space +distribution of the stars and the stellar mass function. +However, as discussed in the previous subsection, we are actually +interested in the distribution of shock parameters 𝐵. This can be +evaluated by integrating (26) under the constraint that 𝐵 = 𝐺𝑀/𝑣𝑏2 +which leads us to +d𝑁 +d𝐵 = +∫ +6dim. +𝛿𝐷 +� 2𝑀𝐺 +𝑣𝑏2 − 𝐵 +� +d6𝑁 +d3𝑣 d𝑏 d𝑀 d𝑡 d3𝑣 d𝑏 d𝑀 d𝑡, +(27) +where 𝛿𝐷 is the Dirac delta function. We evaluate the integral, inte- +grating in 𝑏 first: +d𝑁 +d𝐵 = +∫ ∫ ∫ +𝑓∗(𝒙, 𝒗 − 𝒗rel, 𝑀)𝑣𝑔(𝑀, 𝐵, 𝑣) d3𝑣 d𝑡 d𝑀, +(28) +𝑔(𝑀, 𝐵, 𝑣) = +∫ ∞ +0 +2𝜋𝑏𝛿𝐷 +� 2𝑀𝐺 +𝑣𝑏2 − 𝐵 +� +d𝑏 += +∫ ∞ +0 +2𝜋 𝑀𝐺 +𝑣𝐵′2 𝛿𝐷 +�𝐵′ − 𝐵� d𝐵′ += 2𝜋𝑀𝐺 +𝑣𝐵2 , +(29) +MNRAS 000, 1–22 (2015) + +Prompt cusps & stellar encounters +7 +where we have used the substitution 𝐵′ = 2𝑀𝐺 +𝑣𝑏2 +to evaluate the +integral. +Sorting and evaluating individual terms gives us +d𝑁 +d𝐵 = 2𝜋𝐺 +𝐵2 +∫ ∫ ∫ +𝑀 𝑓∗(𝒙, 𝒗 − 𝒗rel, 𝑀) d3𝑣 d𝑀 d𝑡 += 2𝜋𝐺 +𝐵2 +∫ +𝜌∗(𝒙(𝑡)) d𝑡 += 2𝜋𝐺 +𝐵2 𝜒∗, +(30) +where we have used the fact that the mass weighted integral over the +phase-space number density gives the stellar mass density 𝜌∗. Further +we have defined the time integral of the stellar density along the cusp +trajectory 𝜒∗. It is further convenient to define the characteristic +shock parameter, +𝐵∗ = 2𝜋𝐺𝜒∗, +(31) +so that +d𝑁 +d𝐵 = 𝐵∗ +𝐵2 , +(32) +𝑁(> 𝐵) = 𝐵∗ +𝐵 . +(33) +Therefore, we expect on average one encounter with 𝐵 > 𝐵∗. +It is worth noting that this result is surprisingly independent of the +phase-space distribution function of stars and the stellar mass func- +tion, but depends only on the stellar mass density. The mass function +is irrelevant, because at a fixed mass density, a reduction in mass +leads to an increase in number density, thus making close encounters +more likely to the same degree that it decreases the strength of such +encounters. A similar coincidence holds for the velocity dependence. +The number of stars that are encountered increases with the velocity, +while at the same time the weight of each encounter decreases with +the velocity to such a degree that the two effects cancel exactly. We +call these effects the encounter conspiracy. +It is clear that these two simplifications could, in principle, break +down at some scale. Mass function independence breaks down if +the distant tide approximation fails – if we make our perturbers +less massive, the encounters get closer for a given value of 𝐵. For +very small masses e.g. 10−6 M⊙ the closest approach needed for a +significant perturbation would be of order one AU, smaller than core +radius of a cusp; the distant tide approximation would then certainly +fail. The integral over the mass function in equation (30) should +thus have a lower limit, in principle. The independence of velocity +breaks down for very small encounter velocities. If an encounter takes +longer than the orbital times within the cusp, then the cusp will react +adiabatically, with no long-termchanges inenergy except forparticles +that leave the system in the adiabatic limit. Thus our approximations +fail for stars that are moving almost at the same velocity as the cusp. +However, neither of these problems has a significant effect in practice, +since almost all stellar mass is in objects of mass within an order of +magnitude or so of 1.0 M⊙ and very few encounter velocities are +smaller than, say, 10 km s−1. +We note that the effects of encounters with other massive objects, +such as planets or other prompt cusps would be overestimated if the +calculation of this section were applied. However, the mass density +in planets is so much lower than that in stars that planets are quite +irrelevant in this context. The mass density in prompt cusps is non- +negligible at large halocentric radii, but when their extended profile +is taken into account, we find that the strongest possible shocks are +far below 𝐵 ≪ 10−4 km s−1 pc−1 so that cusps cannot significantly +shock other cusps (cf. Figure 3). Therefore, stars pose the only sig- +nificant contributor to the distribution of encounter shocks. +10 +1 +100 +101 +102 +103 +B/B * +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +dN / d log B +1st strongest +2nd strongest +3rd strongest +Beff [B > 10 +1B * ] +Beff [B > 10 +2B * ] +Beff [B > 10 +3B * ] +Beff [B > 10 +4B * ] +Figure 4. The distribution of shock parameters. The blue, orange and green +lines show the distribution of the strongest, 2nd strongest and 3rd strongest +shock. The dashed lines show the distribution of the effective shock parameter +according to equation (38) when truncating the (divergent) distribution at +different values of 𝐵min. +2.5 Shock Histories +We assume that all aspects of the problem follow Poisson statistics +– for example that stars are drawn through a Poisson process from +the continuous phase-space distribution and that stellar masses are +drawn through a Poisson process from the stellar mass function. Then +also the shock parameter distribution has to follow Poisson statistics. +That means the probability of having exactly 𝑘 encounters with shock +strength bigger than 𝐵 is given by +𝐹(𝑘, 𝐵) = (𝐵∗/𝐵)𝑘 exp(−𝐵∗/𝐵) +𝑘! +. +(34) +In particular, the probability of having at least one encounter with +shock strength bigger than 𝐵 is given by +𝐹(≥ 1, 𝐵) = 1 − 𝐹(0, 𝐵) += 1 − exp(−𝐵∗/𝐵). +(35) +The probability density function of the strongest encounter is there- +fore +𝑓1(𝐵) = d𝐹(≥ 1, 𝐵) +d𝐵 += 𝐵∗ +𝐵2 exp(−𝐵∗/𝐵). +(36) +It is straightforward to derive the corresponding functions for the +2nd, 3rd etc strongest shock. However, the distribution of e.g. the +strongest and the second strongest shock are not independent and +parameterising the joint distribution is rather cumbersome. When +considering a large number of encounters it is more convenient to +draw actual realisations. This can easily be done by mapping 𝐵 +onto another random variable that follows a uniform distribution +𝑥 := 𝐵∗/𝐵, +d𝑁 +d𝑥 = d𝑁 +d𝐵 +d𝐵 +d𝑥 = 1. +(37) +We can then create a realization of a history of shocks with 𝐵 > +𝐵min, by sampling 𝑘 uniformly distributed random variables 𝑥𝑖 on +the interval [0, 𝐵∗/𝐵min] and then transforming them as 𝐵 = 𝐵∗𝑥−1. +The number 𝑘 must itself be drawn from a Poisson distribution with +mean ⟨𝑘⟩ = 𝐵∗/𝐵min. We note that there will be infinitely many +MNRAS 000, 1–22 (2015) + +8 +J. Stücker et al. +encounters as 𝐵min → 0 so that it is in general not possible to +sample the full distribution, but only its truncated form. However, +this is sufficient, since encounters with very small shock parameters +become irrelevant. In particular, we will show in Section 4 that the +joint effect of 𝑘 encounters is more or less equivalent to a single +encounter with an effective shock strength, +𝐵eff = +� 𝑘 +∑︁ +𝑖=1 +𝐵𝑝 +𝑖 +�1/𝑝 +, +(38) +with 𝑝 = 1.2. +We sample a large number of shock histories and show the strongest +three encounters together with the distribution of 𝐵eff in Figure 4. The +distribution of 𝐵eff is already reasonably well approximated when +only considering shocks with 𝐵 > 0.1𝐵∗ i.e. approximately the 10 +strongest shocks. It is almost fully converged when considering all +shocks with 𝐵 > 10−3𝐵∗ i.e. the 1000 strongest shocks. Typically the +effective shock parameter is not too much larger than the strongest +shock. Its median lies at 4.5𝐵∗ whereas the median of the strongest +shock lies at 1.46𝐵∗. However, the low-end tail is shifted upwards +quite a bit so that there are almost no cases with 𝐵eff < 𝐵∗. The +high-end tail is virtually identical to the distribution of the strongest +shock. +3 SHOCK DISTRIBUTION FOR ORBITS IN THE MILKY +WAY +In this Section we evaluate numerically the quantities that are relevant +for describing the full distribution of shock parameters for a cusp that +is orbiting in the Milky Way. We showed in the last section that, for +a given orbit, the time integral of the stellar mass density along the +cusp’s trajectory, +𝜒∗ = +∫ +𝜌∗(𝒙(𝑡)) d𝑡, +(39) +is sufficient to parameterise the full distribution of encounter shocks. +Our aim is thus to infer realistic estimates of 𝜒∗ and of the corre- +sponding characteristic shock parameter 𝐵∗ = 2𝜋𝐺𝜒∗. +We assume that the orbital distribution of cusps follows that of +dark matter particles within the Milky Way’s halo. +3.1 Milky Way Potential +For the baryonic components of the Milky Way we assume the pre- +scriptions used in the Phat ELVIS simulations (Kelley et al. 2019) +at the specific time 𝑧 = 0. The observational parameters underly- +ing these simulations were taken from McMillan (2017) and Bland- +Hawthorn & Gerhard (2016). +The stellar disk and the gas disk are each modelled through a +superposition of three Miyamoto & Nagai (1975) (hereafter MN) +potentials as described by Smith et al. (2015). The parameters of the +MN potentials have been tuned to recreate the mass distribution of +an exponential disk up to 1% accuracy. For the stellar disk we use +a total mass of 𝑀d = 4.1 × 1010 M⊙ a scale radius 𝑅d = 2.5 kpc +and a height parameter of 𝑧d = 0.35 kpc. For the gas disk we use +𝑀d = 1.9 × 1010 M⊙, 𝑅d = 7.0 kpc and 𝑧d = 0.08 kpc. +We approximate the stellar bulge through a Hernquist (1990) +distribution with mass 𝑀B = 9 × 109 M⊙ and with scale-length +𝑟a = 500 pc. +For the Milky Way’s dark matter halo, we assume that in the +absence of baryonic components it would correspond to an NFW halo +100 +101 +102 +r [kpc] +109 +1010 +1011 +1012 +1013 +M( < r) [M +] +stellar disk +gas disk +bulge +all baryons +halo +total +NFW (uncontracted) +ISS V0 = 220km/s +r200c +Figure 5. Enclosed mass profiles for the different components of our Milky +Way model. Solid lines are used for components that contribute to the final +potential. The brown dashed line shows the uncontracted NFW halo profile +whereas the purple curve shows the profile after contraction due to the baryons +and is the curve actually used in our modelling. The dotted grey line indicates +the profile of a singular isothermal sphere with 𝑉0 = 220 km s−1 which +is sometimes used to approximate the spherically averaged potential of the +Milky Way (e.g. Errani & Navarro 2021). +with concentration 𝑐 = 8.7 and mass 𝑚200𝑐 = 1012 M⊙. However, +the baryons increase the dark matter density in the inner regions. We +model this contraction using the semi-analytic approach presented +in Cautun et al. (2020). Previously, we used the analytic procedure +of Sellwood & McGaugh (2005), but this produces a slightly denser +result in the central regions. We prefer the Cautun et al. (2020) +approach, since it has been tested in detail both against state-of-the- +art simulations and against observations of the Milky Way. +We show the result of the contraction of the halo together with +profiles of all the different components in Figure 5. We note that +the contracted halo (in purple) is significantly denser in the centre +than the uncontracted version (brown dashed line). This contraction +is only moderately relevant for getting the correct potential structure +of the Milky Way. However, it is very relevant for sampling self- +consistent orbits of the dark matter distribution. To sample orbits we +use the (Eddington 1916) inversion method on the density profile of +the contracted halo, but inside the joint potential of all components. +We have checked that if we sample particles from the contracted halo +profile and integrate their orbits in the joint potential, the density +profile is stable and evolves very little at later times. +3.2 Orbits +We create 105 test particles from the contracted halo up to 400 kpc. +We use an adaptive sampling method so that their initial number +density is proportional to 1/𝑟 times that of the halo. This way we +have a good sampling, both at small radii 𝑟 ≲ 1 kpc and close to +the virial radius 𝑟 ≳ 200 kpc. Throughout this paper, when showing +distributions, we always correct this non-uniform sampling using the +appropriate weights. We integrate particle orbits in the full 3d Milky +Way potential (including bulge, stellar disk, gas disk and contracted +halo) using 105 constant timesteps over a period of 10 Gyr. Along +each orbit we evaluate 𝜒∗ according to equation (39) using the stellar +density inferred from the Laplacian of the potential of the stellar +components (stellar disk and bulge) 𝜌∗ = Δ𝜙∗/(4𝜋𝐺). Note that this +can create negative densities in a (very) few locations, since the MN +MNRAS 000, 1–22 (2015) + +Prompt cusps & stellar encounters +9 +10 +3 +10 +2 +10 +1 +100 +101 +102 +103 +B * [km/s / pc] +100 +101 +r [kpc] +100 +101 +102 +103 +104 +105 +106 +107 +* +200km/s [M +/pc2] +median +Bcusp, Bcore +99.7% region +95% region +68% region +Figure 6. The distribution of 𝜒∗ as a function of radius. To facilitate inter- +pretation, the left y-axis gives 𝜒∗ multiplied by a velocity of 200 km s−1; this +estimates the total encountered stellar column density. An alternative y-axis +shows the values of 𝐵∗, with red dashed lines showing the critical values, +𝐵core and 𝐵cusp. It is clear that shocks will be relevant for almost all cusps +that orbit in the central 10 kpc of our Galaxy. +potential of the disk can create negative densities. We therefore clip +𝜒∗ after the integration at a minimum of 10−8 M⊙/pc2/(km/s). This +threshold has no impact on any of our results. The actual minimum +should be set by the diffuse stellar halo (and partially by encounters +with dark substructures). However, none of these aspects matter since, +as we will see in Section 4.6, the effect of the smooth tidal field is +much larger than this at radii where stellar encounters are infrequent. +We find the distributions of 𝜒∗ and 𝐵∗ shown in Figure 6. Here and +in later plots we assign each particle 1000 times with its final value +of 𝜒∗ at different radii chosen uniformly in time over its full orbit +history. To help with an intuitive understanding of the 𝜒∗ distribution, +we have multiplied the 𝜒∗ axis by a value of 200 km s−1 so that the +left y-axis would correspond to the total stellar column density if +the cusp always encountered stars at 200 km s−1. Figure 6 shows that +a typical cusps orbiting around the solar radius (8 kpc) encounters +a stellar column density of about 104 M⊙pc−2. These numbers can +easily be understood, as orbits at these radii will have passed through +the Galactic disk about 102 times and each disk passage adds a +column density of order 102 M⊙pc−2. +Further, we show as an alternative y-axis in Figure 6 the distri- +bution of 𝐵∗. Recall that 𝐵∗ indicates the value of 𝐵 such that we +expect on average one encounter with 𝐵 > 𝐵∗. Therefore we expect +typically one encounter with 𝐵 ≳ 1 km s−1 pc−1 for cusps orbiting +around the solar radius. +For each orbit we sample the strongest encounters corresponding +to the probability density in equation (36). In this way, we find the +distribution of strongest shocks shown in Figure 7. We also show the +corresponding encounter distance for encounters with mass 1 M⊙ +and velocity 200 km s−1. This Figure is not used in our final results, +but is useful to gain intuition for the relevant quantities. +We conclude that inside the central 10 kpc virtually all cusps will +have had at least one stellar encounter that might significantly de- +crease the annihilation luminosity and possibly even disrupt them. +For a reliable evaluation, we have to conduct numerical experiments +that evaluate how strongly cusps react to shocks of a given amplitude. +101 +102 +103 +104 +105 +closest encounter b [AU] +100 +101 +102 +r [kpc] +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +102 +103 +104 +Strongest Shock B [km/s / pc] +Bcusp, Bcore +median +99.7% region +95% region +68% region +Figure 7. The distribution of the strongest shock encountered by a cusp as a +function of its current Galactocentric radius. An alternative 𝑦-axis shows the +corresponding closest approach distance for encounters with a mass of 1 M⊙ +and velocity 200 km s−1. We see that within the central 10 kpc the strongest +shocks encountered will frequently affect the annihilation signal of cusps and +in some cases may disrupt them completely. +4 THE EFFECT OF IMPULSIVE ENCOUNTERS +To evaluate the effects of impulsive encounters on prompt cusps, we +ran several sets of N-body simulations addressing scenarios of in- +creasing complexity. The first set considers pure power-law profiles +experiencing a single encounter. The second set considers cored pro- +files also with a single encounter. The third set considers both cored +and power-law profiles experiencing multiple stellar encounters. For +each case we develop simple descriptions for the annihilation lumi- +nosity expected from the remnants. +Finally, at the end of this section we briefly discuss how the effect +of the smooth tidal field can be incorporated simply in this formalism. +4.1 Simulation Setup +We consider simulations starting from two different initial profiles. +Power-law simulations start from a Poisson realization of a pure𝑟−3/2 +density profile with the phase-space distribution from equation (5). +In this case there is no relevant length or mass scale so that the +results can be rescaled to any desired tidal shock strength. For cored +simulations we use the phase-space distribution from equation (8) +and the density profile that is self-consistently created from this +distribution, as described in Section 2.2. For these we set 𝑟core = 1 +so that length scales are given in units of the core radius. +To enhance the dynamic range that can be resolved in these sim- +ulations and to minimize the effects of the (numerically required) +outer truncation radius, we use a similar non-uniform mass sam- +pling strategy to Delos (2019b). Specifically, we arrange that the +same number of particles 𝑁split have pericentres in each of the radial +ranges (0, 1), (1, 10), (10, 100), (1000, 10000) which then have par- +ticle masses increasing by about a factor 30 between intervals. We +explain this in more detail in Appendix A1, where we also present +some numerical stability tests. An implementation of this method is +available in the code repository of this article. We do not find any ar- +tifacts due to the non-uniform mass sampling, most likely because we +use quite high resolution and because our pericentre criterion min- +imizes the intrusion of higher mass particles into higher resolution +regions. +For simulations with a single encounter (presented in Sections 4.2 +and 4.3) we apply at time 𝑡 = 0 a single kick according to equation +MNRAS 000, 1–22 (2015) + +10 +J. Stücker et al. +10 +3 +10 +2 +10 +1 +100 +101 +r/rB +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +(r)/ +0(r) +no kick +B (t = 2.5tB) +B × 4 (t = 10tB) +B × 4 (t = 50tB) +fit +Figure 8. The transfer function for pure 𝑟−3/2 power-law profiles exposed to +a tidal shock from a stellar encounter. When presented in units of 𝑟𝐵 (𝐵) the +transfer functions of simulations with different shock amplitudes (blue versus +orange and green) line up perfectly – as is expected from dimensional analysis. +The grey lines show the transfer functions of equivalent simulations without +tidal shocks evaluated at the same output times – proving the numerical +stability of our setup. The black line shows the fit to the transfer function +given in equation (43). +(16) using the tensor from equation (19) with shock amplitude 𝐵. +The shock introduces the characteristic spatial scale, +𝑟𝐵 = +� 8𝜋𝐺𝐴 +3𝐵2 +�2/3 +, +(40) +which is the radius where the tidal shock causes a (maximal) velocity +change equal to the circular velocity in a pure power-law profile. By +definition, 𝑟𝐵 equals 𝑟cusp and 𝑟core for shocks with strength 𝐵cusp +and 𝐵core respectively, so that realistic shocks can easily produce 𝑟𝐵 +values that lie at any point of the cusp (compare Figure 7). Further, +the kick introduces a characteristic time-scale – corresponding to the +dynamical time at this radius: +𝑡𝐵 = +𝑟𝐵 +𝑣circ(𝑟B) = 1 +𝐵 . +(41) +After the initial shock we evolve the simulation for several dynamical +times 𝑡𝐵 until the profile no longer evolves at the radii of interest +(e.g. ∼ 10𝑡𝐵 around 𝑟𝐵). Note that for typical shocks with 𝐵 ∼ +1 km s−1 pc−1, we have 𝑡𝐵 ∼ 1 Myr. +4.2 Truncation of Pure Power-law Profiles +We ran two high-resolution simulations of shocked power-law pro- +files which each have 𝑁split = 222 particles per radial interval, so that +they have in total about 2 × 107 particles. These two simulations used +different amplitudes for the shock parameter, varying by a factor of +4, leading to differing shock radii, 𝑟𝐵 = 20 and 𝑟𝐵 = 128. We remind +the reader that, in principle, the result of a shocked power-law can be +rescaled arbitrarily, so these two simulations differ only in how the +physical scale 𝑟𝐵 compares to resolution parameters. +We define the transfer function, +𝑇(𝑟) = 𝜌(𝑟) +𝜌0(𝑟) , +(42) +where 𝜌(𝑟) and 𝜌0(𝑟) are the final and initial density profiles, respec- +tively, and we present transfer functions for our simulations in Figure +8 as functions of 𝑟/𝑟𝐵. We show only the radial range that has reliably +converged to a final, stable post-shock profile, and we use different +output times to probe different parts of the transfer function. In Ap- +pendix A2 we provide convergence tests to determine these scales. +At the small end this is the largest radius where two-body relaxation +is irrelevant, while at the large end we limit to scales where at least +ten dynamical time-scales have passed. As additional evidence that +the simulations are converged, we also show as grey lines in Fig- +ure 8 reference simulations evolved to the same times but with no +shock. Clearly, these are stable and show no discernable numerical +evolution. +The simulations of different shock strengths line up very well if +radii are scaled by 𝑟𝐵. In turn, since 𝐵 ∝ 𝑏−2, 𝑟𝐵 scales with impact +parameter as 𝑏8/3. This is very different than the scaling of 𝑏8/11 +which Ishiyama et al. (2010) assumed (without further explanation) +to extrapolate their results and to argue that stellar encounters should +be irrelevant for the survival of cusps. Such a scaling is clearly +incorrect and is also inconsistent with the profiles of the simulations +that Ishiyama et al. (2010) presented themselves. These simulations, +in fact, seem consistent with a 𝑏8/3 scaling and agree qualitatively +with what we find here. +Inspired by Delos (2019b), we fit the simulated transfer functions +jointly using a function of the form, +𝑇(𝑟) = exp(−𝛼(𝑟/𝑟𝐵)𝛽), +(43) +where we find 𝛼 = 1.256 and 𝛽 = 0.639 as the best fitting parameters. +Interestingly the value of 𝛽 that we find here for our power-law profile +is not too far from the one that Delos (2019b) found (𝛽 = 0.78) for +an NFW initial structure. The small difference presumably reflects +the different central slopes, −1.5 and −1. +We note that our measured profiles disagree slightly with our fit +at larger radii 𝑟 ≳ 2𝑟𝐵. However, this is quite irrelevant for the +calculation of the annihilation luminosity where those contributions +are downweighted by a factor 𝑇2(𝑟). For our fitted function we find +that the annihilation rate is given by +𝐽pow(𝐵) = 4𝜋 +∫ 𝑟cusp +𝑟min +𝑇(𝑟)2𝜌2 +0(𝑟)𝑟2 d𝑟 += 4𝜋𝐴2 +𝛽 +� +Ei +� +−2𝛼 +�𝑟min +𝑟𝐵 +�𝛽� +− Ei +� +−2𝛼 +�𝑟cusp +𝑟𝐵 +�𝛽�� +(44) +≈ 4𝜋𝐴2 +𝛽 +Ei +� +−2𝛼 +�𝑟min +𝑟𝐵 +�𝛽� +, +where Ei is the exponential integral and 𝑟min is a lower limit of inte- +gration that is necessary to obtain a finite result. The approximation +that is used in the last line, 𝑟cusp → ∞, is already very accurate for +𝑟cusp ≳ 𝑟𝐵. Therefore, if 𝑟𝐵 ≤ 𝑟cusp, it is fine to neglect the ini- +tial boundary of the system, since the truncation through the shock +sets the actual boundary. Further it is insightful to consider the limit +𝑟min ≪ 𝑟B in which case +𝐽pow(𝐵) ≈ 4𝜋𝐴2 +𝛽 +� +−𝛾 − log +� +2𝛼 +�𝑟min +𝑟𝐵 +�𝛽�� += 4𝜋𝐴2 +� +log +� 𝑟𝐵 +𝑟min +� +− 2.35 +� += 4𝜋𝐴2 log +� 0.096𝑟𝐵 +𝑟min +� +, +(45) +where 𝛾 ≈ 0.577 is the Euler–Mascheroni constant. This approxi- +mation holds (at 5% accuracy or better) for radii 𝑟min < 0.1𝑟B. One +way to interpret this result is that the annihilation rate of the tidally +MNRAS 000, 1–22 (2015) + +Prompt cusps & stellar encounters +11 +𝐵/𝐵core +𝑟core/𝑟𝐵 +𝑡/𝑡𝐵 +𝐽/4𝜋𝐴2 +0.026 +0.008 +10 +3.044 +0.053 +0.020 +20 +2.213 +0.105 +0.050 +20 +1.457 +0.211 +0.125 +20 +0.723 +0.316 +0.215 +20 +0.338 +0.421 +0.316 +20 +0.134 +0.527 +0.425 +30 +0.037 +0.632 +0.543 +50 +0.003 +0.738 +0.666 +50 +0.000 +Table 1. Simulation parameters for the shocked cored profiles: The shock +parameter in units of the core scale, 𝐵/𝐵core, the ratio of core to shock +radius, 𝑟core/𝑟𝐵, the duration of the simulation in units of the dynamical +time, 𝑡/𝑡𝐵 and the post-shock annihilation rate. The last simulation shows +complete disruption. +truncated power-law corresponds to the annihilation rate of a power- +law profile that is sharply truncated at 10% of 𝑟𝐵 (compare Equation +(4)). This result is easily understood, since this is approximately the +radius where 𝑇2 reaches 50%. We expect this approximation to be +inaccurate if 𝑟core ≳ 0.1𝑟𝐵, since then shock effects are significant +at the core radius where they may be amplified. We will consider +such cases in the next subsection and derive a general formula for the +annihilation rate of shocked cusps. However, we can make a rough +estimate of the cored annihilation rate, by assuming the power-law ++ transfer profile up to the core radius and down-weighting the core +contribution by a factor of 𝑇2(𝑟core) +𝐽pow+core(𝐵) = 4𝜋𝐴2 +� +𝛽−1 Ei +� +−2𝛼 +�𝑟core +𝑟𝐵 +�𝛽� ++ 0.531𝑇2(𝑟core) +� +. +(46) +4.3 Truncation of cored Profiles +We also ran several simulations with cored initial conditions and with +different shock parameters. These are designed to probe the scales +where the shock starts affecting the core. Each uses 𝑁split = 220 and +therefore in total about 5 × 106 particles. We evaluate each simulation +at a time where the final profile has reached a stable result. This +requires a longer integration time for simulations where the central +density is reduced significantly. We give the simulation parameters, +durations and final annihilation luminosities in Table 1. +We show the transfer functions of these simulations in Figure 9. +We can see that as 𝑟core/𝑟𝐵 → 0, the pure power-law behaviour +is recovered; at large radii they have the same transfer function as +the power-law case. At small radii cored profiles are suppressed +additionally, and can even disrupt completely for sufficiently strong +shocks. +The simulation with 𝑟core/𝑟𝐵 = 0.666 is the first case which ex- +hibits complete disruption – this means that after 50𝑡𝐵 no stable rem- +nant is left, and densities keep decreasing everywhere. We checked +that this case still disrupts if a four times higher particle number and +a smaller softening are used, and that simulations with even larger +shocks disrupt also. In addition, we a provide a video of one non- +disrupting case and one disrupting case online.4. It seems clear that +although it is impossible to disrupt centrally divergent power-law +profiles, it is indeed possible to disrupt cored profiles. This agrees +with theoretical (Amorisco 2021; Stücker et al. 2022) and numerical +4 https://github.com/jstuecker/cusp-encounters +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +r/rB +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +(r)/ +0(r) +rcore/rB +0 +rcore/rB = 0.008 +rcore/rB = 0.020 +rcore/rB = 0.050 +rcore/rB = 0.125 +rcore/rB = 0.215 +rcore/rB = 0.316 +rcore/rB = 0.425 +rcore/rB = 0.543 +rcore/rB = 0.666 +Powerlaw T(r) +rrcore +Figure 9. The transfer functions of cored power-law profiles. Note that each +case has a different value of 𝑟core/𝑟𝐵 so that the initial profiles are not identical +as a function of 𝑟/𝑟𝐵. Each simulation is divided by its own initial profile in +making this plot. Core radii are indicated as vertical dotted lines. The transfer +functions all show enhanced suppression relative to the power-law transfer +function below and slightly above the core radius. +10 +3 +10 +2 +10 +1 +100 +101 +J/(4 A2) +Cored Sim. +Powerlaw Sim. +Powerlaw T(r) +Fit +Disruption +10 +3 +10 +2 +10 +1 +100 +B/Bcore +0.1 +0.0 +0.1 +J/J +Figure 10. Annihilation rates of cored power-law profiles that have been +exposed to shocks of varying amplitudes. The top panel shows the annihilation +rates and the bottom panel the residuals with respect to our fit (blue line). +The black line shows the estimate from equation (46) based on the power- +law transfer function. Clearly, the actual annihilation rates are additionally +suppressed with respect to the power-law result when shocks get close to the +core scale. For shocks with 𝐵 ≳ 0.65𝐵core we find complete disruption. +investigations (van den Bosch et al. 2018; Errani & Peñarrubia 2020; +Errani et al. 2022) in the context of NFW haloes. +For the cored profiles we do not attempt to fit a functional form +to the transfer functions, but instead directly calculate the annihi- +lation radiation from the spherically averaged density profiles. This +works very well numerically, because the part of the profile which +is responsible for the bulk of the annihilation radiation (most signifi- +cantly around 3𝑟core) is resolved very well.5 We describe the binning +and integration procedures in Appendix A3 and show that all ob- +tained annihilation luminosities are accurate at the 3% level – except +5 This is not the case for our power-law profiles, where a major fraction of +the radiation always comes from poorly resolved inner radii. +MNRAS 000, 1–22 (2015) + +12 +J. Stücker et al. +for the last non-disrupted case, 𝑟core/𝑟𝐵 = 0.543, where the relative +error is larger, but still less than 40%. +We list the annihilation rates we obtain in Table 1 and plot them in +Figure 10. In this Figure we also show the two power-law simulations +with annihilation rates estimated by individually fitting their transfer +functions and (arbitrarily) assuming a core radius, 𝑟core = 0.05, for +the annihilation estimate from equation (46). However, these two +simulations could be arbitrarily rescaled along the black line. +We are able to fit the post-shock annihilation rates of our cored +simulations using the function, +𝐽 = 4𝜋𝐴2 +� +0.531 + 1 +𝛽 log(𝑎) − 1 +𝛽 Ei +� +−2𝛼𝑎(𝑥core + 𝑏𝑥3 +core)4𝛽/3�� +, +(47) +𝑥core = +𝐵 +𝐵core += +�𝑟core +𝑟𝐵 +�3/4 +, +(48) +with free parameters 𝑎 and 𝑏. By construction, this function ap- +proaches the behaviour of the power-law truncation fit of equation +(46) for 𝐵 ≪ 𝐵core. However, it has two degrees of freedom 𝑎 and +𝑏 which can modify the behavior for large values of 𝐵. We find that +𝑎 = 0.708, 𝑏 = 5.98 gives a reasonable fit to all annihilation rates +at the 10% level – both in the power-law regime 𝐵 ≪ 𝐵core and +when the shock hits the core 𝐵 ≳ 0.1𝐵core. The function reaches +zero at 𝐵dis ≈ 0.65𝐵core. We assume that all cusps that had an en- +counter with 𝐵 > 𝐵dis are disrupted and have zero contribution to +the annihilation rate. +4.4 The effect of multiple encounters +As a final step, we need to understand what happens to the remnant if +it is exposed to multiple shocks. To test this, we set up a large number +of simulations (for both power-law and cored cases) with a variety +of different shock histories. For each of these simulations we apply +a shock approximately every ten dynamical time-scales 𝑡𝐵 and we +use up to 5 shocks. We always compute the annihilation rate when +10𝑡𝐵 have passed since the last shock. We note that the value of 𝑡𝐵 is +not very clearly defined in the case with different shock amplitudes, +we therefore choose a reference value 𝐵ref to define a 𝑡𝐵 that seems +appropriate for each individual history. We have checked that this +shock interval is large enough to ensure that the results would not +change by slightly decreasing it or by increasing it further. We list all +shock histories in 2. We have created several manually chosen shock +histories (e.g. equal shock strengths, descending shocks, ascending +shocks) and we have also sampled a few shock histories from the +full distribution of shock histories with 𝐵∗ = 𝐵ref, but only keeping +the 5 strongest shocks – see Section 2.5. We sorted some of these +histories accidentally, but we also created an additional set of four +cored simulations with unsorted shock histories. However, neither +the annihilation rate nor the transfer functions depend much on the +order in which shocks happen. +We present transfer functions for power-law simulations with mul- +tiple encounters in Appendix A4. Most importantly we note that +cases with multiple encounters are still well approximated by the +transfer function of equation (43), but with different cut-off radii and +we note that for the final transfer function, the order of shocks does +not matter, only their amplitudes. These results are both consistent +with findings in a previous study of NFW haloes (Delos 2019b). +We show the annihilation rates of cusps that have gone through +multiple shocks in Figure 11. The data points in this Figure are ob- +tained as follows. For power-law cases, we compute the annihilation +rate by first fitting a transfer function to each encounter individually +Type +𝐵ref +𝐵1 +𝐵2 +𝐵3 +𝐵4 +𝐵5 +𝐵eff +power-law +- +1 +1 +1 +1 +0 +3.2 +power-law +- +1 +0.5 +0.25 +0 +0 +1.5 +power-law +- +0.25 +0.5 +1 +0 +0 +1.5 +power-law 𝐵∗, S +- +1.1 +0.77 +0.51 +0.36 +0.29 +2.4 +power-law 𝐵∗, S +- +1.5 +0.41 +0.27 +0.19 +0.16 +2.1 +cored +0.11 +1.00 +1 +1 +1 +1 +3.8 +cored +0.11 +1.00 +0.5 +0.25 +0 +0 +1.5 +cored +0.11 +0.25 +0.5 +1 +0 +0 +1.5 +cored 𝐵∗, S +0.11 +1.08 +0.77 +0.51 +0.36 +0.29 +2.4 +cored 𝐵∗, S +0.11 +1.45 +0.41 +0.27 +0.19 +0.16 +2.1 +cored 𝐵∗, S +0.11 +9.05 +0.79 +0.68 +0.67 +0.43 +10 +cored 𝐵∗, S +0.11 +1.97 +1.6 +0.65 +0.46 +0.42 +4.0 +cored 𝐵∗, U +0.05 +3.43 +0.34 +1.4 +0.28 +1.6 +5.7 +cored 𝐵∗, U +0.11 +1.03 +0.57 +0.42 +0.65 +4.1 +5.6 +cored 𝐵∗, U +0.11 +0.35 +0.38 +0.44 +0.18 +0.28 +1.3 +cored 𝐵∗, U +0.26 +0.96 +0.15 +0.89 +0.28 +0.23 +2.0 +Table 2. The different shock histories simulated for the multiple encounters +scenario. 𝐵ref is given in units of 𝐵core and all other shocks parameters are +given in units of 𝐵ref. 𝐵1 - 𝐵5 indicate the strength of subsequent shocks and +𝐵eff is the final value of the effective shock parameter, defined in equation (49) +so that the effect of the whole shock history is equivalent to a single shock +with this parameter. Shock histories that were sampled from the distribution +of histories are indicated by a 𝐵∗. Those which were accidentally sorted in +descending order are marked with an ’S’ whereas unsorted ones are indicated +by a ’U’. +10 +3 +10 +2 +10 +1 +100 +101 +J/(4 A2) +Fit +Cored Sim. +Powerlaw Sim. +Powerlaw T(r) +Disruption +10 +2 +10 +1 +100 +Beff/Bcore +0.25 +0.00 +0.25 +J/J +0 +1 +2 +3 +4 +5 +Encounters +Figure 11. Annihilation rates for power-law and cored profiles after multiple +stellar encounters. The encounter histories are summarized through a single +effective parameter 𝐵eff so that their annihilation rate is approximately equiv- +alent to a single shock with 𝐵eff. The blue line shows our previous fit for +single encounters, and the bottom panel shows residuals with respect to this +fit. The effect of multiple encounters is captured to within 20% through a +single shock with the effective shock parameter 𝐵eff. +using equation (43) and then calculating the annihilation rate accord- +ing to equation (46). Here we rescale the power-law results to two +different core radii 𝑟core = 0.05 and 𝑟core = 0.2 so that each power- +law simulation appears twice in Figure 11. We note again that these +simulations could be rescaled to be anywhere along the black line. +For cored profiles we calculate the annihilation rates as explained +in the last section. For the 𝐵-axis we calculate an effective shock +parameter which summarizes the whole history of encounters of a +cusp through a single number, +𝐵eff = +𝑝√︃∑︁ +𝐵𝑝 +𝑖 , +(49) +which is the 𝑝-norm of the shock history. Different values of 𝑝 would +MNRAS 000, 1–22 (2015) + +Prompt cusps & stellar encounters +13 +give different importance to stronger versus weaker shocks. For 𝑝 = 1 +the value of 𝐵eff would correspond to the sum of all shock parameters. +For 𝑝 < 1 multiple shocks would have an enhanced effect, and for +𝑝 → ∞ only the strongest shock would matter. For reference, we +show, how such cases would appear in the Appendix A4. However, +we have found that 𝑝 = 1.2 gives excellent predictions, and this is the +value of 𝑝 that we use for calculating the 𝐵-values in Figure 11. We +note that some previous studies (of NFW subhaloes) have assumed +that multiple shocks can be treated by adding the changes in binding +energy (Shen et al. 2022). This would imply 𝑝 = 2 and would give +clearly wrong results for the case of prompt cusps and probably also +for NFW haloes. The results of Delos (2019b) indicate that multiple +shocks have also a significantly enhanced effect (𝑝 ≪ 2) for NFW +haloes. However, it is not clear that our effective description will +work equally well for NFW profiles, due to their more complicated +form. +The blue line in 11 shows the prediction obtained by treating the +shock histories as a single shock with effective shock parameter 𝐵eff +and inserting this into Equation (47) This predicts the annihilation +rates correctly to within 20%6. We show in Appendix A4 that this +is much more accurate than results that would be obtained by only +considering the strongest shock 𝑝 → ∞ (errors up to factor of a few) +or by considering the sum of shocks as the effective shock parameter +(errors up to 50%). +4.5 Summary of the effect of stellar encounters +We conclude that we can estimate the annihilation rate of cored +power-law profiles that have gone through complicated shock histo- +ries simply by using equation (47) with an effective shock parameter +calculated as the 1.2-norm of all the shocks. To account for the ini- +tial boundary of cusps when 𝐵 ≪ 𝐵cusp we additionally use the +boundary term from equation (44) so that we have +𝐽 = 4𝜋𝐴2𝛽−1 � +− Ei +� +−2𝛼𝑎(𝑥core + 𝑏𝑥3 +core)4/3𝛽� ++ Ei +� +−2𝛼𝑥4𝛽/3 +cusp +� ++ 0.531𝛽 − log(𝑎) +� +, +(50) +where we have defined +𝑥cusp = +𝐵 +𝐵cusp += +� 𝑟𝐵 +𝑟cusp +�3/4 +. +(51) +This is the main result of this section. Note that this function works +accurately in all regimes – it recovers the correct suppression for +𝐵 ≫ 𝐵cusp, but it also recovers equation (10) for weak shocks with +𝐵 ≪ 𝐵cusp. +We note that the results we find here suggest that the impact of +stellar encounters on the annihilation rate will be quite dramatic for +cusps in the inner part of the Milky Way. We show in Figure 12 the +distribution of the reduction in annihilation luminosity relative to the +initial luminosity as a function of 𝐵∗, the characteristic shock strength +of a cusp’s trajectory. Here we assume 𝐵core = 100𝐵cusp, which is +a typical ratio between these two parameters. We sample a large +number of shock histories, considering all shocks with 𝐵 > 10−3𝐵∗ +and evaluate the expected luminosities according to (50) using the +effective shock parameter. For comparison, we show also the lumi- +nosities that would be obtained by considering only the strongest +6 Only the last two data points have an error larger than this, but these +points also have the largest systematic uncertainty, since they are so close to +disruption. We would have needed to take additional care by choosing later +evaluation times to get more precise estimates. +10 +4 +10 +3 +10 +2 +10 +1 +100 +B * /Bcore +10 +3 +10 +2 +10 +1 +100 +J/J0 +median +only strongest shock +Bcusp, Bcore +99.7% region +95% region +68% region +Figure 12. Reduction in annihilation luminosity due to encounters along +a trajectory with a characteristic shock scale 𝐵∗ for a cusp with 𝐵core = +100𝐵cusp. The shaded regions show our predictions for the distribution of +luminosities using the effective shock parameter. Dotted lines show the results +that would have obtained by considering only the strongest shock. These have +a slight offset in color so that they can still be seen where they overlap with the +shaded regions. Clearly the effect of shocks with 𝐵 ≳ 𝐵cusp is quite dramatic, +and when 𝐵∗ gets close to the core scale, complete disruption is expected in +virtually all cases. +shock. Virtually all cusps with 𝐵∗ ≳ 0.3𝐵core get completely dis- +rupted and cusps with 𝐵∗ ≳ 𝐵cusp are already dramatically affected. +As expected (compare also Appendix A4) there is a significant differ- +ence between considering only the strongest shock and considering +the full history of shocks. However, even when considering only the +strongest shock the suppression is quite strong. +4.6 The effect of smooth tides +In addition to stellar encounters, the smooth tidal field of the Milky +Way can also induce mass-loss, and so a reduction in annihilation +luminosity. We do not discuss the effect of smooth tides in great +detail here, since this effect was already adequately incorporated +by Delos & White (2022b), based on the work of Stücker et al. +(2022). However, since the effect of smooth tides should be included +in addition to that of stellar encounters, we need to make a few +modifications to their approach. We will discuss these modifications +in Appendix B. In summary, we include the effect of smooth tides +by applying the adiabatic-tides model (Stücker et al. 2022) to the +cusps that remain after stellar shock truncation. We find that the +joint effect on annihilation radiation of a stellar shock with effective +strength 𝐵eff and the smooth tidal field is approximately equivalent +to a pure shock with an effective shock parameter, +𝐵eff,𝜆 = +√︃ +𝐵2 +eff + 42.2𝜆, +(52) +where 𝜆 is the largest eigenvalue of the tidal tensor of the spherically +averaged Galactic mass distribution at the pericentre of the cusp’s +orbit. +In this way we are able to incorporate the effect of smooth tides +simply through a redefinition of the shock parameter used in equa- +tion (50). In Figure 13 we compare the effective shock parameter +from the full history of shocks to the tidal contribution 𝐵𝜆 = +√ +42.2𝜆 +and to the total effect. At radii 𝑟 ≤ 20 kpc the effect of encounters +dominates, whereas at larger radii the effect of the smooth tide dom- +inates. Thus, viewed from the Earth’s position just 8 kpc from the +MNRAS 000, 1–22 (2015) + +14 +J. Stücker et al. +100 +101 +102 +r [kpc] +10 +2 +10 +1 +100 +101 +102 +103 +104 +Beff [km/s/pc] +Bcusp, Bcore +Beff [stars] +Beff [smooth tide] +Beff [total] +Figure 13. A comparison between the relative importance of the smooth tidal +field and of tidal shocks from stars. The shaded regions indicate the 68% +regions of the distributions, and the solid lines their medians. For cusps that +reach the vicinity of the disk, 𝑟 ≤ 20 kpc, encounters are dominant. In the +outskirts of the Milky Way halo the effect of smooth tides dominates, but is +too weak to affect the luminosity of most cusps. +Galactic centre, truncation by stellar encounters has a large effect on +the angular distribution of the prompt cusp annihilation signal. This +must be taken into account when evaluating whether these cusps af- +fect interpretation of the Galactic Center Excess measured by Fermi +LAT. On the other hand, stellar encounters have much less effect on +the radiation seen by a distant observer since most of the mass of the +Milky Way’s dark matter halo (and so most of its prompt cusps) are +at larger radius. +5 RESULTS +We combine results from Section 3, on the distribution of stellar +shock parameters, and from Section 4, on the effect of shocks and +smooth tides on prompt cusps, to estimate the spatial distribution of +the annihilation signal of cusps in the Milky Way. +For this we assume the cusp population expected for a WIMP with +mass 100 GeV and decoupling temperature 30 GeV. This leads to val- +ues 𝐵core and 𝐵cusp through equations (22) and to initial annihilation +rates through equation (14). We use the 105 orbits and the final 𝐵∗ +values inferred in Section 3, but we create a different realization of +a cusp and its shock history for each of 1000 different radii sampled +uniformly in time along each orbit’s trajectory. For the shock history +we consider all shocks stronger than 𝐵 > 10−4𝐵∗ (approximately the +10000 strongest shocks) when evaluating the effective shock param- +eter 𝐵eff. Additionally we keep track of the smallest radius 𝑟peri that +each cusp has reached and we evaluate the tidal field of the spher- +ically averaged mass distribution at this point to obtain the value +of 𝜆. With this we infer the effective shock parameter 𝐵eff,𝜆 as in +equation (52) and evaluate the expected final annihilation luminosity +according to equation (50). Thus, in total we obtain 108 pairs of radii +and annihilation luminosities. +We show the distribution of the ratio between initial and final +luminosities in the top panel of Figure 14 as a function of radius. +The percentiles of this distribution give an idea of how dramatic the +effect of stellar encounters on cusps is. Typical cusps inside of the +central 10 kpc are disrupted by stellar encounters. Within the central +3 kpc less than 2.5% of cusps survive, and almost all cusps reduce +100 +101 +102 +r [kpc] +10 +3 +10 +2 +10 +1 +100 +J/J0 +J / J0 (enc. + tide) +J / J0 (tide only) +J / J0 (only Bmax) +Figure 14. The distribution of the reduction in annihilation radiation from +prompt cusps as a function of their current Galactocentric radius. Shaded +regions indicate percentiles of the full distribution considering the joint effect +of all encounters (colours as in Figure 6). The mean (red line) gives the +resulting effective reduction of the contribution to the annihilation profile +from prompt cusps. The blue dashed line shows the mean reduction if only the +smooth tide is considered, while the orange dashed line shows the reduction +caused by the strongest shock alone. Clearly, stellar encounters have a dramatic +effect on the expected annihilation radiation from cusps in the inner Galaxy. +their luminosities by dramatic factors (e.g. 99.7% of cusps reduce +their luminosity at least by a factor 5). +However, more relevant than the percentiles of the distribution is +the annihilation weighted mean, since this gives the ratio between +the initial annihilation profile and the final one. We show this as +the red line in Figure 14. The mean is clearly dominated by the +least disrupted cusps. This is so, since cusps that contribute more +annihilation radiation are also more resilient to tides (compare Figure +3) and further, since the bulk of the distribution gets completely +disrupted, only the most resilient cusps with the least invasive shock +histories contribute to the mean. Even so, the mean is dramatically +suppressed with respect to the case where encounters are neglected. +At 8 kpc the average luminosity is already suppressed by a factor of +10. At smaller radii the effect is even more dramatic; only about 10−3 +of the luminosity expected in the absence of encounters remains near +the galactic bulge at about 1 kpc. We show as a blue dashed line the +mean annihilation reduction that would be obtained if only the effect +of the smooth tide were considered (as in Delos & White 2022b). +This greatly over-predicts the luminosity in the central regions, but +is accurate at larger radii 𝑟 ≳ 15 kpc. The orange dashed line shows +the mean if only the strongest shock is considered instead of the full +history. This leads to significantly smaller, although still substantial, +suppression. +In Figure 15 we show the annihilation profile as a function of +radius. Unlike the prediction when only the effect of smooth tides +is considered, stellar encounters lead to a non-monotonic profile +that decreases towards the centre. Its maximum is slightly outside +the solar radius at around 10 kpc. At larger radii the effect of stellar +encounters becomes irrelevant so that the original profile is recovered +at 𝑟 ≳ 40 kpc. +Let us briefly consider, how these effects alter the annihilation +luminosity of the Milky Way, as seen by a distant observer. For +this, we simply have to sum up the luminosity of all cusps out to +some truncation radius which we assume to be 𝑟200b = 340 kpc (the +radius where the enclosed mean density is 200 times the background +MNRAS 000, 1–22 (2015) + +Prompt cusps & stellar encounters +15 +100 +101 +102 +r [kpc] +10 +7 +10 +5 +10 +3 +10 +1 +101 +dJ/dV [M2 /pc3/pc3] +Smooth halo +cusps (unperturbed) +cusps (tides) +cusps (enc. + tides) +Figure 15. The radial distribution of annihilation radiation from cusps in the +Milky Way including the effects of both stellar encounters and the mean tide +(red line). For reference, the black line shows the radiation from the smooth +halo, the green line the cusp contribution ignoring all disruptive effects, and +the blue dashed line including only the effect of smooth tides. When all effects +are included the cusp contribution peaks at around 10 kpc. +initial +only tides +only enc. +tides + enc. +𝐽-factor +2.32e+11 +1.90e+11 +2.02e+11 +1.79e+11 +fraction +100% +82% +87% +77% +Table 3. Total dark matter annihilation luminosity of the Milky Way when +considering different tidal effects. 𝐽-factors are in units of M⊙pc−3. These +numbers are proportional to the luminosity of the Milky Way as seen by +a distant observer. Modelling tidal stripping or stellar encounters has only +moderate effects on the total luminosity. +density). Additionally we add the contribution of the smooth halo, +which is however very small (∼ 2%). We list the resulting total +luminosities in Table 3. Clearly, the joint effect of tidal stripping and +stellar encounters onto the total luminosity is relatively small, only +reducing the estimated luminosity by 23%. Stellar encounters do not +much affect the total luminosity, since most of the cusps do not get +close to the stars. As a result, it is not necessary to consider stellar and +tidal disruption effects on cusps when inferring the total contribution +of extragalactic sources to the IGRB. However, we observe the dark +matter distribution of our own Galaxy from a highly biased view- +point which greatly increases the sensitivity to what happens in its +inner regions. +We show in Figure 16 the radiation profile of the Milky Way as it +would be observed from the solar radius𝑟 = 8.2 kpc if dark matter has +a significant self-annihilation cross-section. Here we have inferred +for each component the line-of-sight J-factor column-densities, +𝑛𝐽 = +∫ ∞ +0 +𝜌2 d𝑙, +(53) +and then multiplied them by a normalization factor, +dΦ𝐸 +dΩ = 𝑛𝐽 × 1.44 × 10−5 MeVcm−2sr−1s−1pc5M−2 +⊙ , +(54) +that makes the smooth halo profile agree with the Galactic Centre +excess (this is the same factor that Delos & White (2022b) have used). +Additionally we have overplotted the data points of the observed GCE +as given by Di Mauro (2021). Many error bars are too small to see and +100 +101 +102 + (deg.) +10 +4 +10 +3 +10 +2 +10 +1 +d +E +d [MeV cm +2 s +1 sr +1] +Smooth halo +cusps (unperturbed) +cusps (tide only) +cusps (enc. + tide) +total +Di Mauro (2021) +Ackermann e.a. (2015) +sky average +Figure 16. Left: The annihilation flux as a function of Galactocentric angle. +When including the effect of stellar encounters the angular dependence of +the prompt cusp contribution vanishes almost completely, resulting in a total +signal which is compatible with the observed Galactic Centre excess (Di +Mauro 2021). Right: The sky average is significantly reduced due to stellar +encounters, but cusps still contribute a significant and potentially detectable +fraction of the isotropic 𝛾-ray background. +component +𝑛𝐽 +Flux +Fraction of IGRB +smooth halo +1.3e+00 +1.9e-05 +2.7% +cusps (no tides) +2.6e+01 +3.8e-04 +54.8% +cusps (tides) +1.5e+01 +2.2e-04 +31.3% +cusps (tides + enc.) +7.6e+00 +1.1e-04 +15.8% +cusps (extragal.) +1.2e+01 +1.7e-04 +25.0% +total DM +2.1e+01 +3.0e-04 +43.5% +total DM (previous) +2.8e+01 +4.1e-04 +59.0% +Table 4. Average J-factor column densities (in units of M2 +⊙pc−5) and sky- +averaged fluxes of 𝛾-ray radiation (in units of MeV cm−2sr−1s−1) for different +components when normalizing fluxes so that the Galactic Centre excess would +be explained by the smooth halo signal. Fluxes have been averaged over the +sky with 𝜃 > 20◦. If the GCE is due to dark matter, annihilation radiation +should constitute about 43% of the isotropic 𝛾-ray background. +some data points are only upper limits indicated by arrows.7 Stellar +encounters affect the profile so strongly that there is only a very small +variation with observation angle left. While the unperturbed profile +and the profile including tides alone both seem in tension with the +observed shape of the GCE, stellar encounters alleviate this tension +and the shape of the GCE is again compatible with dark matter as +an explanation. The signal from cusps in the Galactic halo makes an +almost isotropic contribution – only deviating by about 30% from its +sky-average at its brightest angle (40◦). When making conclusions +from the shape of the signal profile, including the effects of stellar +encounters is crucial. +However, if the GCE is due to dark matter we still expect a signif- +icant contribution to the 𝛾-ray background from prompt cusps. We +list the sky-averaged flux that we predict for each component in Table +4. It is most interesting to compare these numbers to the observed +isotropic 𝛾-ray background (IGRB) +dΦIGRB +dΩ += 6.9+3.1 +−2.8 × 10−4 MeVcm−2sr−1s−1 +(55) +where the error indicates the systematic uncertainty due to foreground +7 Our halo profile does not fit the GCE perfectly for the innermost angles. +We did not fit the profile to the GCE, but simply use the one that we have also +used for orbital modelling in Section 3. +MNRAS 000, 1–22 (2015) + +16 +J. Stücker et al. +modelling which we have assumed to accumulate linearly when in- +tegrating the spectrum as measured by Ackermann et al. (2015). The +observed IGRB only considers contributions from galactic latitudes +𝑏 > 20◦. To approximately mimic this, we have only included con- +tributions from the Galactocentric angles larger than 𝜃 > 20◦ in the +sky-averages listed in Table 4. Additionally we have indicated what +fraction of the IGRB each component would comprise. Finally, we +have also listed the extra-galactic flux here which we estimate in +the same manner as Delos & White (2022b) – neglecting the effect +from tidal fields, which might reduce the number by around 20% as +indicated by Table 3. +In comparison to the smooth tide prediction (equivalent to Delos +& White 2022b), the predicted background signal due to cusps in +our own Milky Way goes down by a factor 2.0 and the signal is +by a factor 3.5 smaller than the unperturbed one. After taking this +correction into account, the total dark matter signal (smooth halo + +Milky Way cusps + extra-galactic cusps) is dominated by the extra- +galactic signal and goes down by roughly one third. Delos & White +(2022b) argue that the morphology of the signal that is expected +from annihilation from prompt cusps and from dark matter decay +are essentially the same. Therefore, they use constraints on the de- +cay of dark matter from Blanco & Hooper (2019) and rescale them +to obtain constraints on the dark matter annihilation cross-section. +These constraints are proportional to the predicted background, thus +the upper limit on the cross-section has to be increased by about one +third. However, we note that these constraints depend quite strongly +on how well the astrophysical contributions to the diffuse 𝛾-ray back- +ground are understood (Blanco & Hooper 2019) and the assessment +of uncertainties has not been very rigorous so far. For example, the +constraints by Blanco & Hooper (2019) vary by a factor of about 4 +simply by considering different assumptions about how correlated +the error-bars are. Therefore, to infer reliable constraints it will nec- +essary to reanalyse the IGRB with a more sophisticated treatment of +the systematic and statistical uncertainties. +Perhaps even more intriguing than the implications for constrain- +ing dark matter, the predicted contribution to the IGRB offers an +independent test for the dark matter interpretation of the Galactic +Centre excess. If the GCE is due to annihilation, then we predict an +additional approximately isotropic 𝛾-ray signal that would comprise +about 40% of the observed 1 to 10 GeV background for a 100 GeV +WIMP. If we additionally consider that different WIMP models can +lead to a factor ∼ 2 variation in predicted cusp J-factors (consider +Delos & White 2022b, Figure 7 and Equation 2), the total dark mat- +ter contribution should range between 20% and 80% of the observed +IGRB. Given current uncertainties in the (apparently dominant) con- +tribution of star-forming galaxies and AGN to the observed signal, +it is unclear, whether there is room for such a component. A careful +reevaluation of these other contributions to the 𝛾-ray background is +clearly well motivated. Our work here can be used to infer templates +for the spatial and spectral shapes of the predicted annihilation signal +from prompt cusps. +The detection or exclusion of this additional component would +confirm or contradict the dark matter interpretation of the GCE, and +so substantially advance our understanding of dark matter itself. Firm +exclusion could rule out an annihilation interpretation of the GCE, +while robust detection would strongly support this interpretation, +since it would be a remarkable coincidence to find the GCE and the +additional IGRB contribution at the right relative level yet due to +different astrophysics. +6 DISCUSSION +In this article we have modeled the effect of stellar encounters on the +expected dark matter annihilation signal from prompt cusps orbiting +within the Milky Way’s halo, presenting several advances in the +treatment of such encounters. +Firstly, we have developed a new method for inferring the full +history of the impulsive shocks experienced by a dark substructure +as it moves through the Galaxy. This method is both simpler and more +general than previous approaches. While we have focused on prompt +cusps in this article, our results on the distribution of shock histories +(Sections 2 and 3) could be applied to conventional NFW subhaloes +also, as long as the dominant shocks are in the distant-encounter +regime (masses ≲ 1𝑀⊙). +Secondly, we have performed idealized N-body simulations to in- +fer how stellar encounters affect prompt cusps. Here, we have for +the first time considered the phase-space core that any (otherwise) +centrally divergent profile must exhibit. Only when this core is con- +sidered can a prompt cusp disrupt. Our simulations allow us to derive +accurate formula to describe the structural effect of encounters with +arbitrary combinations of shock history, core radius, truncation ra- +dius, characteristic density and smooth tidal field. As a result, we are +able to account for the joint effect of smooth tides and of any number +of stellar encounters along the entire trajectory of any cusp. +While our model is relatively complete and comprehensive, a few +of its assumptions could still be improved. We have assumed a static +Milky Way potential, and simply integrated cusp orbits for 10 Gyr +within it. A more accurate treatment would consider evolution of the +host potential and of the stellar population it contains, and would +follow cusps from their initial formation and growth through their +accretion onto precursor objects, and finally onto the Milky Way +itself. While the early stages of this process remain quite uncertain, +we believe that our procedure should be relatively accurate and should +be conservative for effects at later times, since the great majority of +Milky Way stars formed (approximately) in situ in the disc and the +bulge, and most of them are less than 10 Gyr old. +Another uncertain point is the precise profile of the phase-space +core of prompt cusps. Here, we used a simple heuristic approach to +obtain a stable profile that is consistent with the phase-space density +constraint, but a rigorous investigation of the central profiles with N- +body simulations would be desirable. However, we expect the results +of our study to be quite robust to this uncertainty, since annihilation +rates depend relatively weakly on the precise shape of the core. +When we apply our modeling to the prompt cusp population in +the Milky Way’s halo, we find that stellar encounters have a dra- +matic effect on any cusps that enter the star-dominated regions – the +vast majority get disrupted within the central 10 kpc, leading to a +radial annihilation radiation profile that peaks near 10 kpc, dropping +strongly at smaller radii. While this has little effect on the lumi- +nosity of the Milky Way as seen by a distant observer (≲ 20%), it +strongly affects the annihilation flux observable from Earth. Stellar +encounters destroy cusps so efficiently in the inner Galactic halo, that +the surface brightness of cusp annihilation radiation is predicted to +vary only slightly between the centre and anticentre directions. As +a result, it has no noticeable effect on the Galactic Centre Excess, +which therefore remains compatible with production by annihilation +of the smooth dark matter distribution in the inner few kpc. This +removes the issue raised by Delos & White (2022b) who included +cusp disruption due to the smooth Galactic tide but not that due to +stellar encounters, and hence found a total annihilation profile ap- +parently incompatible with the GCE profile of Di Mauro (2021, see +Figure 14). +MNRAS 000, 1–22 (2015) + +Prompt cusps & stellar encounters +17 +This opens up an intriguing new possibility. If the GCE is in- +deed due to dark matter annihilation, this implies an approximately +isotropic annihilation signal from prompt cusps that would have an +amplitude in the range 20% − 80% of the observed isotropic 𝛾-ray +background, depending on the mass and decoupling temperature of +the WIMP. The results of Blanco & Hooper (2019) suggest that a +signal of this amplitude is inconsistent with the observed IGRB, +since the latter appears to be explained entirely by emission from +star-forming galaxies and AGN. We think, however, that our current +results warrant a careful reevaluation of those of Blanco & Hooper +(2019) since it seems conceivable that some of the fitted templates +might absorb a near-isotropic dark matter annihilation signal, or that +the statistical treatment of systematic errors is not yet robust enough +to exclude such a signal. If such a signal is firmly ruled out, it becomes +unlikely that the GCE can be ascribed to annihilation radiation. On +the other hand, if it were robustly detected, this would confirm the +annihilation interpretation of the GCE. +ACKNOWLEDGEMENTS +JS thanks Sten Delos for answering quickly and clearly all questions +regarding the distribution and properties of cusps. JS thanks Mattia +Di Mauro for providing data related to the GCE. JS and RA thank all +members of the cosmology group at Donostia International Physics +Center for daily discussions and for the motivating research environ- +ment. JS and RA acknowledge the support of the European Research +Council through grant number ERC-StG/716151. GO was supported +by the National Key Research and Development Program of China +(No. 2022FYA1602903) and the Fundamental Research Fund for +Chinese Central Universities (No. 226-2022-00216). +DATA AVAILABILITY +The code used to generate all results of this study other than +the simulation-based analysis of Section 4 is available in an on- +line repository under https://github.com/jstuecker/cusp-encounters. +Some results are based on the adiabatic-tides code which is also +publicly available under https://github.com/jstuecker/adiabatic-tides. +The simulation data from Section 4 will be shared on reasonable re- +quest to the corresponding author. +REFERENCES +Ackermann M., et al., 2015, ApJ, 799, 86 +Aguilar L. A., White S. D. M., 1985, ApJ, 295, 374 +Aguirre-Santaella A., Sánchez-Conde M. A., Ogiya G., Stücker J., Angulo +R. E., 2022, arXiv e-prints, p. arXiv:2207.08652 +Amorisco N. C., 2021, Cold dark matter subhaloes at arbitrarily low masses +(arXiv:2111.01148) +Anderhalden D., Diemand J., 2013, J. Cosmology Astropart. Phys., 2013, 009 +Angulo R. E., Hahn O., Ludlow A. D., Bonoli S., 2017, MNRAS, 471, 4687 +Angus G. W., Zhao H., 2007, MNRAS, 375, 1146 +Arcadi G., Dutra M., Ghosh P., Lindner M., Mambrini Y., Pierre M., Profumo +S., Queiroz F. S., 2018, European Physical Journal C, 78, 203 +Bardeen J. M., Bond J. R., Kaiser N., Szalay A. S., 1986, ApJ, 304, 15 +Bertschinger E., 2006, Phys. Rev. D, 74, 063509 +Binney J., Tremaine S., 2008, Galactic Dynamics: Second Edition +Blanco C., Hooper D., 2019, Journal of Cosmology and Astroparticle Physics, +2019, 019 +Bland-Hawthorn J., Gerhard O., 2016, ARA&A, 54, 529 +Blas D., Lesgourgues J., Tram T., 2011, J. Cosmology Astropart. Phys., 2011, +034 +Cautun M., et al., 2020, Monthly Notices of the Royal Astronomical Society, +494, 4291 +Colombi S., 2021, A&A, 647, A66 +Delos M. S., 2019a, Physical Review D, 100 +Delos M. S., 2019b, Phys. Rev. D, 100, 083529 +Delos M. S., White S. D. M., 2022a, arXiv e-prints, p. arXiv:2207.05082 +Delos M. S., White S. D. M., 2022b, arXiv e-prints, p. arXiv:2209.11237 +Delos M. S., Erickcek A. L., Bailey A. P., Alvarez M. A., 2018a, Phys. Rev. +D, 97, 041303 +Delos M. S., Erickcek A. L., Bailey A. P., Alvarez M. A., 2018b, Phys. Rev. +D, 98, 063527 +Delos M. S., Bruff M., Erickcek A. L., 2019, Phys. Rev. D, 100, 023523 +Di Mauro M., 2021, Phys. Rev. D, 103, 063029 +Diemand J., Moore B., Stadel J., 2005, Nature, 433, 389 +Eddington A. S., 1916, MNRAS, 76, 572 +Errani R., Navarro J. F., 2021, Monthly Notices of the Royal Astronomical +Society, 505, 18–32 +Errani R., Peñarrubia J., 2020, MNRAS, 491, 4591 +Errani R., Navarro J. F., Peñarrubia J., Famaey B., Ibata R., 2022, MNRAS, +Goerdt T., Gnedin O. Y., Moore B., Diemand J., Stadel J., 2007, MNRAS, +375, 191 +Grand R. J. J., White S. D. M., 2022, Monthly Notices of the Royal Astro- +nomical Society: Letters, 511, L55 +Green A. M., Goodwin S. P., 2007, MNRAS, 375, 1111 +Hernquist L., 1990, ApJ, 356, 359 +Hooper D., Goodenough L., 2011, Physics Letters B, 697, 412 +Hu W., Sugiyama N., 1996, The Astrophysical Journal, 471, 542 +Ishiyama T., 2014, ApJ, 788, 27 +Ishiyama T., Makino J., Ebisuzaki T., 2010, ApJ, 723, L195 +Kavanagh B. J., Edwards T. D. P., Visinelli L., Weniger C., 2021, Phys. Rev. D, +104, 063038 +Kelley T., Bullock J. S., Garrison-Kimmel S., Boylan-Kolchin M., Pawlowski +M. S., Graus A. S., 2019, MNRAS, 487, 4409 +Macciò A. V., Paduroiu S., Anderhalden D., Schneider A., Moore B., 2012, +Monthly Notices of the Royal Astronomical Society, 424, 1105 +McMillan P. J., 2017, MNRAS, 465, 76 +Miyamoto M., Nagai R., 1975, PASJ, 27, 533 +Navarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 462, 563 +Ogiya G., Hahn O., 2018, MNRAS, 473, 4339 +Polisensky E., Ricotti M., 2015, MNRAS, 450, 2172 +Roszkowski L., Sessolo E. M., Trojanowski S., 2018, Reports on Progress in +Physics, 81, 066201 +Schneider A., Krauss L., Moore B., 2010, Phys. Rev. D, 82, 063525 +Sellwood J. A., McGaugh S. S., 2005, ApJ, 634, 70 +Shen X., Xiao H., Hopkins P. F., Zurek K. M., 2022, arXiv e-prints, p. +arXiv:2207.11276 +Sheth R. K., Mo H. J., Tormen G., 2001, Monthly Notices of the Royal +Astronomical Society, 323, 1 +Slatyer T. R., 2021, arXiv e-prints, p. arXiv:2109.02696 +Smith R., Flynn C., Candlish G. N., Fellhauer M., Gibson B. K., 2015, +MNRAS, 448, 2934 +Stücker J., Angulo R. E., Busch P., 2021, MNRAS, 508, 5196 +Stücker J., Ogiya G., Angulo R. E., Aguirre-Santaella A., Sánchez-Conde +M. A., 2022, arXiv e-prints, p. arXiv:2207.00604 +Tremaine S., Gunn J. E., 1979, Phys. Rev. Lett., 42, 407 +van den Bosch F. C., Ogiya G., Hahn O., Burkert A., 2018, MNRAS, 474, +3043 +APPENDIX A: NUMERICAL CONVERGENCE +A1 Non-uniform mass initial conditions +To simulate the behaviour of our cusps we have to truncate them at +some radius 𝑟max. We first tried to run numerical experiments with +uniform mass sampling and a sharp truncation radius. With particle +MNRAS 000, 1–22 (2015) + +18 +J. Stücker et al. +numbers of order 106 one can typically resolve scales down to 10−3- +10−2 𝑟max before two-body relaxation effects become substantial. +However, at the same time a sharply truncated 𝑟−3/2 power-law is far +from equilibrium around the truncation radius. We found that trunca- +tion still has significant numerical effects below 10−1𝑟max (depend- +ing on how many dynamical times are simulated). This behaviour is +much worse than for profiles that are steeper in their outer regions, +such as Hernquist or NFW profiles. As a result, only a small range +of radii is reliably resolved and a lot of care has to be taken to get +numerically converged results over a substantial radial range. Un- +fortunately, the transfer function we wish to measure spans several +orders of magnitude in radius. +We have therefore decided to abolish the uniform mass sampling +and instead divide the profile into several populations of particles +with differing mass. This allows us to get reliably converged results +over a dynamical range of 104 in radius. +We define five populations of particles where the 𝑖th population is +defined so that all of its particles have their orbital pericenters in the +range, +𝑟peri ∈ [𝑟split,i, 𝑟split,i+1], +(A1) +and we choose 𝒓split = (0, 1, 10, 100, 1000, 10000)𝑇 . +In each interval we choose to use the same number of particles +𝑁split. In principle one can create such a realisation by setting up +several full profiles at different resolutions and then discarding all +particles that do not fulfill the criterion of each population. However, +this would be very inefficient for generating the innermost popula- +tions. +Instead, we can directly sample particles from the cumulative en- +ergy, pericentre distribution function. The density contribution at +radius 𝑟 from particles which have pericentres in a radial range +𝑟1 < 𝑟peri < 𝑟2 and energy smaller than 𝐸 is given by +𝜌(𝑟,𝑟1 < 𝑟peri < 𝑟2, < 𝐸) += 4𝜋 +∫ 𝐸 +𝜙∗ +∫ 𝐿2 +𝐿1 +𝐿 𝑓 (𝐸) +𝑟2 +√︃ +2𝐸 − 2𝜙 − 𝐿2 +𝑟2 +d𝐿 d𝐸, +(A2) +where the boundaries 𝐿1 and 𝐿2 are chosen so that the integral goes +only over particles which fulfill the pericentre criterion: +𝐿1 = max(r1 +√︁ +2(E − 𝜙(r1)), 0), +(A3) +𝐿2 = min(r2 +√︁ +2(E − 𝜙(r2)), r +√︁ +2(E − 𝜙(r))). +(A4) +We find that +𝜌(𝑟,𝑟1 < 𝑟peri < 𝑟2, < 𝐸) += 𝐹(𝐸, 𝜙∗1) +√︄ +1 − +𝑟2 +1 +𝑟2 − 𝐹(𝐸, 𝜙∗2) +√︄ +1 − +𝑟2 +2 +𝑟2 , +(A5) +𝜙∗1(𝑟) = +𝜙(𝑟) − 𝜙(𝑟1) 𝑟2 +1 +𝑟2 +1 − +𝑟2 +1 +𝑟2 +, +(A6) +𝜙∗2(𝑟) = +𝜙(𝑟) − 𝜙(𝑟2) 𝑟2 +2 +𝑟2 +1 − +𝑟2 +2 +𝑟2 +, +(A7) +with +𝐹(𝐸, 𝜙) = +8 +√ +2𝜋 𝑓0 (𝐸 − 𝜙) +3 +2 +105 (𝐸 + 𝐸𝑐) +7 +2 (𝐸𝑐 + 𝜙)3 +� +8𝐸2 + 28𝐸𝐸𝑐 + 12𝐸𝜙 ++35𝐸2 +𝑐 + 42𝐸𝑐𝜙 + 15𝜙2� +, +(A8) +where the terms containing 𝑟1 have to be set to zero if 𝑟 < 𝑟1 and the +terms containing 𝑟2 have to be set to zero for 𝑟 < 𝑟2. +Knowledge of this density function is enough to sample radii and +energies of particles in each group. Radii can be sampled through +inverse-distribution function sampling with the cumulative mass pro- +file that can be obtained by integrating 𝜌 and using the limit 𝐸 → ∞, +but maintaining a finite pericentre range. Energies can be sampled +by inverse-distribution function sampling among energies using the +normalized version of 𝜌(𝑟, 𝑟1 < 𝑟peri < 𝑟2, < 𝐸). Finally, angu- +lar momenta can be sampled by considering the cumulative angular +momentum distribution function, +𝐹(< 𝐿|𝑟, 𝐸) = +�√︃ +2𝐸 − 2𝜙 − 𝐿2/𝑟2 +� 𝐿2 +𝐿1 +. +(A9) +An implementation of this scheme for creating initial conditions for +cored cusps is also publicly available in the code repository of this +paper. +We show the radial density profiles of the initial conditions that +have been created in this manner for a profile with 𝑟core = 1 and +𝑁split = 218 as solid lines in Figure A1. The individual components +are shown as coloured lines and the sum is shown as the black lines. +Clearly this gives an excellent initial representation of the density +profile. The dashed lines in the same Figure show the density profile +after the simulation has been evolved for 100 times the dynamical +time-scale at the core radius. The final profile still agrees excellently +with the initial one and we can see that particles of different masses +have mixed very little. Note that the particle number here is still a +factor 4 − 16 lower than for the simulations that we actually use in +the main text so that we can be fairly confident that all simulations +are stable and well converged. +A2 Time evolution and convergence of shocked power-law +profiles +We briefly discuss the convergence aspects of the shocked power-law +profiles here. We show the profile of the 𝑟𝐵 = 20 simulation with +𝑁split = 222 at different output times as solid lines in Figure A2. The +profile is very stable with time over large radial ranges, but there are +differences at large radii where at early times the profile has not yet +had enough time to relax to its final form. We find that a conservative +criterion for the largest radius 𝑟max where the profile has relaxed to +its final form is given by +𝑡sim = 10𝑡dyn(𝑟max) += 10 +√︄ +𝑟3max +𝑀(< 𝑟max)𝐺 , +(A10) +where 𝑡sim is the run-time of the simulation and we use the mass +profile at the output time (not the initial time) for 𝑀(< 𝑟). We invert +this equation numerically to find 𝑟max at each time and mark this +point in Figure A2. Clearly this is a very conservative criterion and +cuts out all parts of the profile that are yet to reach their final form. +The innermost radius 𝑟min where we can trust the simulations is +MNRAS 000, 1–22 (2015) + +Prompt cusps & stellar encounters +19 +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +(r)/ +max +M = 1Mhr +M = 34Mhr +M = 962Mhr +M = 20934Mhr +M = 141945Mhr +sum +t = 0 +t = 100tcore +10 +1 +100 +101 +102 +103 +104 +r/rcore +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +(r)/ +true(r) +Figure A1. A cored density profile that has been set up through our non- +uniform mass sampling technique (top) and the ratio to the true profile (bot- +tom). Black lines show the full profile whereas colored lines show the individ- +ual components made of particles with different masses that were separated +by their initial pericentre radii. The legend indicates the particle mass of each +population in units of the mass of the highest resolution particles. Dashed +lines show the profiles after evolving the simulation for 100 dynamical times. +Clearly our non-uniform mass sampling creates very stable profiles over a +large dynamical range. +given by the two-body relaxation radius which we can approximate +through +𝑡sim = 𝑡relax(𝑟min) ≈ 0.1𝑁(< 𝑟min) +log(𝑟0/𝜖) +𝑡dyn(𝑟min) +(A11) +(e.g. Binney & Tremaine 2008) where 𝑟0 is an estimate of the largest +two-body encounter radius and 𝜖 is the softening. Since, the depen- +dence on 𝑟0 is rather weak we just use 𝑟0 = 𝑟min. This formula only +holds strictly for systems that are made of uniform mass particles. +Therefore, to be extra conservative, we use here not the actual par- +ticle number here, but the particle number as if all mass inside of +𝑟min was made up from particles of the second highest resolution +level (from the pericentre interval [1, 10]). Since most particles in +the region are actually much lower mass (and none are higher mass), +the profiles are actually converged at substantially smaller radii than +the thus determined 𝑟min, which we indicate in Figure A2. Further, +we show two lower resolution simulations with 4 and 16 times fewer +particles to demonstrate that the simulations are actually very well +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(r)/ +0(r) +t = 40tB +t = 20tB +t = 10tB +t = 2.5tB +t = 10tB, N/4 +t = 10tB, N/16 +10 +1 +100 +101 +102 +103 +r +0.2 +0.0 +0.2 +/ +ref +1 +Figure A2. Convergence of the transfer function of a shocked power-law +profile. The top panel shows the density transfer functions, and the bottom +panel the residuals with respect to the 𝑡 = 40𝑡B reference case. Different lines +indicate different times and/or simulations with different particle numbers. +The dots and triangles indicate the inner and outer convergence radii which are +given on the inside by 𝑟min, a conservative estimate of the two-body relaxation +radius, and on the outside by 𝑟max, the radius where about 10 dynamical times +have passed since the shock. (The outer orange and red triangles are overlayed +by the purple one.) The simulations are very well converged inside this range +and the inferred convergence radii are very conservative estimates. +converged in the indicated regimes. In Figure 8 we show only the +reliable regions inside the range [𝑟min, 𝑟max]. +A3 Annihilation Rates +We describe here how we compute the 𝐽-factors of the cored power- +law profiles that we presented in Section 4.3. In many cases it is +numerically problematic to directly infer annihilation rates from N- +body simulations of centrally divergent profiles. This is so since +annihilation rates scale as the density squared, and are therefore par- +ticularly sensitive to the central regions of the profile which are, +however, in general also the regions that suffer the most from numer- +ical inaccuracies caused by two-body relaxation, sampling noise and +the effects of force softening. +However, for the simulated cored profiles none of these are prob- +lematic, since most of the annihilation radiation comes from radii +𝑟 ≳ 𝑟core which are resolved extremely well in our simulations. +Therefore, we estimate the annihilation radiation in the following +manner: We infer the radial density profile by assigning particles to +50 logarithmically spaced radial bins in the range [0.1 𝑟core, 104𝑟core] +(i.e. 10 bins per dex). We then numerically integrate the 𝐽-factor +by combining third-order spline interpolation with a large number +of integration points which we place between 𝑟min = 0.2𝑟core and +𝑟max = 6𝑟𝐵. Finally, we add the contribution from 𝑟 ≲ 𝑟min by as- +suming that the density is uniform in this range with a value given by +the average density within that radius. We show the resulting annihi- +lation rates as the blue line in Figure A3. We then consider different +variations of the numerical procedure: using a two times larger value +of 𝑟min = 0.4𝑟core; using a smaller value for 𝑟max = 3𝑟𝐵; using two +MNRAS 000, 1–22 (2015) + +20 +J. Stücker et al. +10 +2 +10 +1 +100 +J/(4 A2) +base +rrmin × 2 +rmax/2 +bins ×2 +5tB earlier +10 +1 +100 +B/Bcore +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +|1 +J/Jbase| +3% +Figure A3. Convergence of the annihilation rates. The top panel shows the +annihilation rates, whereas the bottom shows the difference relative to the +reference case. Different lines show variations of numerical parameters with +respect to the fiducial ones. All except the last data point have systematic +errors well below 3%. +times the number of bins (20 per dex); and by evaluating the profile +at a slightly different simulation time (5𝑡𝐵 earlier). Note that the last +variation tests that the final profile is both stable and robust to shot +noise. We show these cases in Figure A3, together with the resid- +uals with respect to the reference case in the bottom panel. Clearly +none of these deviations has a significant impact on the annihilation +radiation. In all cases the associated relative error is less than 3%, +except for the case of the strongest shock considered where the error +is significantly larger, but still less than 40%. These results are more +than accurate enough for the purposes of this paper. +A4 Transfer functions and multiple encounters +Here, we present the transfer functions obtained for power-law pro- +files that have gone through up to 5 shocks in different sequences. +Each shock is applied from a random direction. We use the power- +law simulations that are listed in Table 2 and show their transfer +functions in Figure A4. Additionally, we show the transfer function +that is predicted by treating the full history as a single encounter +with effective shock parameter 𝐵eff, given by the 𝑝 = 1.2 norm of +the shock history. Clearly this effective shock parameter predicts the +transfer function of multiple encounters reasonably well. +Finally, we present alternative versions of Figure 11 that use differ- +ent values of 𝑝 for calculating the norm of the shock history. The top +panel of Figure A5 assumes 𝑝 = 1, so that 𝐵eff corresponds to the lin- +ear sum of 𝐵. In this case the prediction (blue line) overestimates the +reduction in annihilation radiation by up to 50%. The bottom panel +adopts the value 𝑝 = 100 – effectively selecting only the strongest +shock to define 𝐵eff. Clearly, considering only the strongest shock can +dramatically underpredict the reduction in annihilation luminosity. +For example, at 𝐵eff/𝐵core ∼ 0.1 there are cases where the annihi- +lation luminosity based on the full shock history is 10 times smaller +than predicted from just the strongest encounter. +10 +3 +10 +2 +10 +1 +100 +101 +r/rB +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +(r)/ +0(r) +[1] +[1,1] +[1,1,1] +[1,1,1,1] +[1,0.5,0.25] r × 10 +[0.25,0.5,1.] r × 10 +his. A, r × 0.1 +his. B, r × 0.1 +Figure A4. Transfer functions for power-law profiles after experiencing mul- +tiple encounters. Dotted lines indicate predictions using an effective shock +parameter defined as the 𝑝-norm with 𝑝 = 1.2. To avoid clutter, some lines +have been offset by a factor 10 up or down in radius. For most cases the labels +indicate the shock histories and for the last two cases the shock histories can +be found in Table 2. +APPENDIX B: THE EFFECT OF THE SMOOTH TIDAL +FIELD +While the main text of this paper discusss the effect of stellar encoun- +ters in great detail, the smooth tidal field also affects the annihilation +luminosity of prompt cusps orbiting in the Milky Way (see e.g. De- +los & White 2022b). As discussed in Stücker et al. (2022), the most +important parameter determining this effect is the largest eigenvalue +𝜆 of the tidal tensor at the orbital pericentre of a prompt cusp’s or- +bit. After a sufficient amount of time (≳ 10 orbits) the orbiting cusp +approaches an asymptotic structure which changes little in subse- +quent evolution (Errani & Navarro 2021). Stücker et al. (2022) show +that the asymptotic remnant can be reasonably approximated by the +adiabatic-tides model. This is an analytic description of an object’s +reaction when a tidal field is applied very slowly (in the adiabatic +limit) and in a spherical approximation. The corresponding code can +be found online.8 While actual N-body simulations in the Galac- +tic context would of course be more accurate, the adiabatic-tides +model allows easy exploration of a variety of different scenarios. +We note that most studies of tidal stripping have focused on NFW +subhaloes which are much less resilient to tides than prompt cusps. +As a result, most previously published results cannot be applied to +the case of interest here. +For a pure power-law profile with slope −1.5 the truncation radius +due to smooth tides scales with a characteristic length 𝑟𝜆, which may +be defined by equating the cusp’s attractive force and the disruptive +force due to the tidal field: +𝜆 · 𝑟𝜆 = +���� +𝜕𝜙 +𝜕𝑟 (𝑟𝜆) +���� , +(B1) +𝑟𝜆 = +� 8𝜋𝐺𝐴 +3𝜆 +�2/3 +. +(B2) +For a pure power-law initial profile, the final profile reaches zero +density at the tidal radius 𝑟𝑡 = 0.24𝑟𝜆 (Stücker et al. 2022). If we set +up a pure power-law profile and evaluate the structure of the rem- +nant using the adiabatic-tides model, we find that its annihilation +8 https://github.com/jstuecker/adiabatic-tides +MNRAS 000, 1–22 (2015) + +Prompt cusps & stellar encounters +21 +10 +3 +10 +2 +10 +1 +100 +101 +J/(4 A2) +p = 1 +Fit +Cored Sim. +Powerlaw Sim. +Powerlaw T(r) +Disruption +10 +2 +10 +1 +100 +Beff/Bcore +0.25 +0.00 +0.25 +J/J +0 +1 +2 +3 +4 +5 +Encounters +10 +3 +10 +2 +10 +1 +100 +101 +J/(4 A2) +p = 100 +Fit +Cored Sim. +Powerlaw Sim. +Powerlaw T(r) +Disruption +10 +2 +10 +1 +100 +Beff/Bcore +0.25 +0.00 +0.25 +J/J +0 +1 +2 +3 +4 +5 +Encounters +Figure A5. Multiple encounter annihilation luminosity predictions using dif- +ferent values of 𝑝 to calculate the 𝑝-norm of the shock history (for comparison +with Figure 11) Top: 𝑝 = 1 corresponding to a linear sum of the shock pa- +rameters. This does not work well, giving annihilation rates off by more than +50% for shocks with 𝐵/𝐵core ≳ 0.2. Bottom: 𝑝 = 100 – approximately +corresponding to the ∞-norm, thus selecting only the strongest shock. This +gives very poor predictions (off by factors up to 5). It is clearly inadequate to +consider only the effect of the strongest shock. +luminosity is given by +𝐽(𝑟 > 𝑟min) = 4𝜋𝐴 log +� 7.759 × 10−3𝑟𝜆 +𝑟min +� +, +(B3) +so that in this case the effect of smooth tide on the annihilation +luminosity is equivalent to a sharp truncation of the initial profile at +0.78% of 𝑟𝜆. +To model accurately the joint effect of stellar encounters and tidal +stripping, the whole encounter and tidal history should, in principle, +be considered. Here, we will simply assume that we can model the +joint effect approximately by first considering the effect of all stellar +encounters – by truncating the prompt cusp as described in Section +4 – and thereafter applying the pericentre tidal field 𝜆. We choose +this order because it is technically simpler for us, but we expect that +consideration of the effects in reverse order would be just as valid +and would give similar results. +We set up a profile following the shocked power-law form, +𝜌(𝑟) = 𝐴𝑟−3/2 exp(−1.256(𝑟/𝑟𝐵)0.639), +(B4) +and we apply various tidal fields 𝜆 using the adiabatic-tides model. +We note that this set-up has one relevant parameter – the ratio between +the two truncation scales: +𝑟𝜆 +𝑟𝐵 += +� 𝐵2 +𝜆 +�2/3 +. +(B5) +We note that the annihilation luminosity in equation (B3) is equiva- +lent to that in equation (45) for a single stellar encounter with shock +parameter +𝐵𝜆 = +√ +43.4𝜆 . +(B6) +Thus, in the limit 𝐵𝜆 ≫ 𝐵 (where the tidal field sets the truncation +scale) we expect the annihilation luminosity to be equivalent to that +after a single shock with strength 𝐵𝜆, while in the limit 𝐵𝜆 ≪ 𝐵, it +should be equivalent to a single shock of strength 𝐵. A reasonable +guess for intermediate cases is for the joint effect to be given by a +weighted average of the two scales, +𝐵eff,𝜆 = +√︁ +𝐵2 + 42.2𝜆. +(B7) +In principle any 𝑝-norm of 𝐵 and 𝐵𝜆 would be a reasonable guess: +we find that 𝑝 = 2 works very well.9 In Figure B1 we compare the +transfer functions obtained from the adiabatic-tides calculations to +an effective description assuming that the joint effect of encounters +and tides is equivalent to a single shock with 𝐵eff,𝜆. This works +reasonably well in the regime where 𝜌/𝜌powerlaw > 0.5. In the regime +where this ratio is smaller, the approximation is worse; the tidal +truncation is complete at the tidal radius, whereas the encounter +truncation has a much longer tail. However, the tail of the profile is +almost irrelevant for the annihilation luminosity, and is unlikely to +be correct in the adiabatic-tides description (Stücker et al. 2022). +The bottom panel of Figure B1 shows the J-factors obtained by +integrating 𝜌2 for the adiabatic remnants over the range [10−6𝑟𝐵, ∞]. +Note that we chose the lower bound so that it is well in the power-law +regime of the profile for all the cases considered. Apart from that, +it is, of course, arbitrary, and so the absolute offset on the 𝐽-axis is +also arbitrary. We show additionally the two limiting cases of a pure +encounter truncation and a pure tidal truncation, together with the +effective description by a single shock with strength 𝐵eff,𝜆. Clearly +this recovers the asymptotic limiting cases as well as the intermediate +regime very well. +To verify that this gives a reasonable description of the joint effect +of encounters and of the smooth tidal field in all relevant cases, we +must also check that it applies to cored initial profiles. Here, we only +consider the case where 𝐵𝜆 ≫ 𝐵 so that we do not have to deal with +two scales at the same time, but only with a single scale 𝐵𝜆/𝐵core. +We set up cored profiles as described in Section 2.2 and apply tidal +fields of varying amplitudes through the adiabatic-tides model. We +show the corresponding 𝐽-factors in Figure B2. +Clearly the effective description works very well for cored pro- +files also, as long as 𝐵eff,𝜆 ≲ 0.2𝐵core. Beyond that scale the cored +profiles reach an earlier disruption threshold of 𝐵dis,𝜆 = 0.33𝐵core +which is a factor two smaller than the encounter disruption threshold. +While it would certainly be possible to improve our effective model +further to incorporate this reduced disruption threshold, this is unnec- +essary, since such large tidal fields are not reached in the Milky Way +– especially not at the large radii where smooth tides dominate over +stellar encounters and 𝐵𝜆 ≪ 1 km s−1 pc−1 typically. We conclude +that treating the joint effect of stellar encounters and smooth tides as +that due to a single encounter with a shock parameter 𝐵eff,𝜆 is an +excellent approximation within the adiabatic-tides framework. We +note that this may slightly overestimate the effects of smooth tides, +9 𝑝 = 1.8 actually works slightly better, but we stick to 𝑝 = 2 for simplicity. +MNRAS 000, 1–22 (2015) + +22 +J. Stücker et al. +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +r/rB +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +/ +powerlaw +/B2 = 0.001 +/B2 = 0.01 +/B2 = 0.1 +/B2 = 1 +/B2 = 10 +/B2 = 100 +initial profile +Beff, approx. +10 +3 +10 +2 +10 +1 +100 +/B2 +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +12.0 +J(r > 10 +6rB)/(4 A2) +A.T. models +Beff, description +encounter limit +tidal limit +Figure B1. Top: Transfer functions for shocked power-law profiles that are +subsequently adiabatically exposed to a tidal field with amplitude 𝜆. The +dashed line shows the initial (post-encounter) profile and the dotted lines +show the effective description that we propose. Bottom: 𝐽-factors for the +corresponding profiles. The two limiting cases apply when either encounter +truncation or tidal truncation dominate. The intermediate regime is described +by an effective shock parameter 𝐵eff,𝜆. +10 +2 +10 +1 +100 +Beff, /Bcore +0 +1 +2 +3 +4 +5 +J/4 A2 +A.T. Models +Beff, description +predicted disruption +actual disruption +Figure B2. 𝐽-factors of cored prompt cusps that were adiabatically exposed to +tidal fields of different amplitudes. The 𝐽-factors are very well approximated +by the 𝐵eff,𝜆-description, except for the regime 𝐵eff,𝜆 ≳ 0.2𝐵core. However, +such large tidal fields are anyways not found in the Milky Way. +since the adiabatic-tides predictions are often a slight overpredic- +tion in practice (Stücker et al. 2022; Aguirre-Santaella et al. 2022). +However, since the effects on cusp annihilation luminosity are in any +case rather weak, this seems acceptable. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–22 (2015) + diff --git a/A9E3T4oBgHgl3EQfswtu/content/tmp_files/load_file.txt b/A9E3T4oBgHgl3EQfswtu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f94391cf673c4f88146f9ed098884c453d827eeb --- /dev/null +++ b/A9E3T4oBgHgl3EQfswtu/content/tmp_files/load_file.txt @@ -0,0 +1,1543 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf,len=1542 +page_content='MNRAS 000, 1–22 (2015) Preprint 13 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 The effect of stellar encounters on the dark matter annihilation signal from prompt cusps Jens Stücker1★, Go Ogiya2, Simon D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' White3, Raul E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Angulo1,4 1Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain 2Institute for Astronomy, School of Physics, Zhejiang University, Hangzhou 310027, China 3Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 1, 85741 Garching, Germany 4IKERBASQUE, Basque Foundation for Science, E-48013, Bilbao, Spain Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' in original form ZZZ ABSTRACT Prompt cusps are the densest quasi-equilibrium dark matter objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' one forms at the instant of collapse within every isolated peak of the initial cosmological density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' They have power-law density profiles, 𝜌 ∝ 𝑟−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 with central phase-space density set by the primordial velocity dispersion of the dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' At late times they account for ∼ 1% of the dark matter mass but for > 90% of its annihilation luminosity in all but the densest regions, where they are tidally disrupted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Here we demonstrate that individual stellar encounters, rather than the mean galactic tide, are the dominant disruptors of prompt cusps within galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Their cumulative effect is fully (though stochastically) characterised by an impulsive shock strength 𝐵∗ = 2𝜋𝐺 ∫ 𝜌∗(x(𝑡)) d𝑡 where 𝜌∗, the total mass density in stars, is integrated over a cusp’s entire post-formation trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Stellar encounters and mean tides have only a small effect on the halo annihilation luminosity seen by distant observers, but this is not true for the Galactic halo because of the Sun’s position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For a 100 GeV WIMP, Earth-mass prompt cusps are predicted, and stellar encounters suppress their mean annihilation luminosity by a factor of two already at 20 kpc, so that their annihilation emission is predicted to appear almost uniform over the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The Galactic Center 𝛾-ray Excess is thus unaffected by cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' If it is indeed dark matter annihilation radiation, then prompt cusps in the outer Galactic halo and beyond must account for 20-80% of the observed isotropic 𝛾-ray background in the 1 to 10 GeV range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Key words: cosmology: dark matter – Galaxy: halo – gamma-rays: diffuse background 1 INTRODUCTION The nature of dark matter is still unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' One of the most popular candidates for dark matter searches is a weakly interacting mas- sive particle (WIMP) which could be produced thermally in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' If dark matter is a WIMP, it may have a signifi- cant self-interaction cross section that allows dark matter to self- annihilate in regions where the dark matter density is sufficiently high (Roszkowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Arcadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Detection of the secondary products of such self-annihilation – also known as indirect dark matter detection – is one of the most promising ways of learning more about the nature of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Excitingly, the Fermi large area telescope (Fermi LAT) has detected a galactic centre excess (GCE) in gamma ray radiation in the spectral range of 1 − 10 GeV (Hooper & Goodenough 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Although there has been a long-ongoing debate about the precise properties of the signal (see Slatyer 2021, chapter 6, for a review), it seems so far that the morphology and the spectrum of the signal could be consistent with a dark matter self-annihilation signal from the central regions of the Milky Way’s dark matter halo (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Di Mauro 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Additionally, recent high-resolution hydrodynamical simulations show that the Milky Way likely exhibits a halo with the right spatial structure to ★ E-mail: jstuecker@dipc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='org explain the GCE (Grand & White 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, there are also other astrophysical processes that might explain the GCE so that it cannot yet be clearly attributed to dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Recently, Delos & White (2022a) have highlighted a different aspect of the interpretation of possible indirect detection signals from the haloes of the Milky Way and of other galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Recent advances in the modelling of the formation of the smallest nonlinear objects in a WIMP cosmology have led to a clearer understanding of their origin, structure and late-time abundance, leading to a re-evaluation of the expected dark matter self-annihilation signal at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The primordial density field is smooth below the dark matter free- streaming scale (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' ∼ 1 pc comoving for a typical WIMP) and so exhibits a large number of density peaks at this scale (∼ 104 − 105 per solar mass of dark matter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The first nonlinear structures begin to form around redshift 30 through monolithic collapse of these peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Their collapse history differs radically from that of traditional, more massive haloes of the kind characterised by Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (1996) (hereafter: NFW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' While NFW haloes assemble through hierarchical accretion and merging over time-scales which are comparable to their age, a prompt cusp forms almost instantaneously at the moment of first collapse of a density peak and it contains dark particles with orbital periods much shorter than the collapse time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' As a result, the density profiles of prompt cusps also differ from the NFW form, following a steep 𝑟−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 density profile between an outer boundary © 2015 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='04670v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='CO] 11 Jan 2023 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' set by the curvature of the initial density peak and an inner core determined by the physical nature of the dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' To distinguish these dense “first” objects from traditional NFW haloes, we follow Delos & White (2022a) in referring to them as “prompt cusps”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Delos & White (2022b) argue that prompt cusps should still be extremely abundant substructures today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In regions where their num- ber is not significantly reduced by subsequent evolution, every solar mass of dark matter should contain tens of thousands of them, and we should expect ∼ 1016 cusps associated with the dark matter halo of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Since they are denser in their centres than tra- ditional NFW haloes, prompt cusps survive tidal effects better and produce a substantially larger dark matter annihilation signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In fact, the annihilation signal from 𝑟−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5-cusps is logarithmically divergent with radius, limited by the inner and outer boundaries of the power- law profile, which are set by the primordial dark matter phase-space density (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Macciò et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2012) and by the initial peak extent, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This raises the signal for indirect detection compared to most previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Thus Delos & White (2022b) predict that these cusps should enhance the total annihilation luminosity of NFW haloes by factors ranging between 100 and 2500, depending on halo concentration, and additionally that they should significantly alter the morphology of the signal which, except in the densest regions, is proportional to the first power of the mean dark matter density, rather than to its square, as usually assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' As Delos & White (2022b) note this “cusp”-component impacts a possible annihilation interpretation of the GCE in two significant ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (1) If disruption of prompt cusps is ignored, their emission dominates that of the smooth halo component beyond about 5 de- grees from the Galactic Centre, resulting in a profile in disagreement with observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Accounting for truncation and disruption by Galac- tic tides reduces cusp emission from the inner Galaxy but leaves it still dominant beyond 10 degrees, reducing but not eliminating the contradiction with observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (2) If the GCE is nevertheless due to annihilation, emission from prompt cusps in the outer Galactic halo and external to the Milky Way must constitute a major contribution to the isotropic 𝛾-ray background (IGRB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, the IGRB appears to be almost completely attributable to emission from star-forming galaxies and AGN (Blanco & Hooper 2019) leaving little space to add an additional component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The resulting upper limit on the prompt cusp contribution to the IGRB allows Delos & White (2022b) to strengthen constraints on the self-annihilation cross-section and the thermal relic mass of a hypothetical WIMP dark matter particle – ef- fectively excluding thermal relic WIMPs with masses 𝑀 ≤ 10 TeV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Delos & White (2022b) explicitly considered only the effect of the smooth tidal field of the Milky Way on prompt cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, stellar encounters can also be important, inducing strong impulsive shocks in dark matter substructures and potentially even disrupt them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Such encounters should be very frequent in the central region of the galaxy and therefore can affect both the predictions of the last paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The main goal of this paper is to evaluate quantitatively the ef- fect of stellar encounters on the structure, survival and predicted annihilation signal from prompt cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We will find that prediction (1) is seriously affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' After accounting for the effect of stellar encounters, the tension between the GCE emission profile and that predicted including cusp emission vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For prediction (2) we will find that reduced emission from cusps in the inner Galactic halo leads to slightly lower IGRB predictions for the annihilation cross- 1 This constraint is under the assumption of a bottom quark 𝑏𝑏 annihilation channel and is independent of the possible annihilation interpretation of the GCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' section required to produce the GCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' These predictions remain in tension with claims that the IGRB is almost entirely due to other sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We note that there is already a large literature on the effect of stellar encounters on NFW subhaloes (Goerdt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Angus & Zhao 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Green & Goodwin 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Delos 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Kavanagh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Much of this is based on the incorrect assumption that systems disrupt when the total injected energy exceeds their initial binding energy, and hence needs to be read with care (Aguilar & White 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' van den Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' A particularly clear and general treatment has been presented by Delos (2019a) and we will often refer to this article as representative of stellar encounters with NFW haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Such results cannot be applied to prompt cusps, since their power-law structure dramatically enhances their resilience to tidal effects in comparison to the inner regions of NFW profiles (see Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2022, for a discussion of different power-law profiles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The only previous study to consider the effect of stellar encounters on prompt cusps is Ishiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (2010), but unfortunately this presented a very limited treatment and used incorrect assumptions when scaling with encounter strength, leading to the erroneous conclusion that cusps would never be disrupted by stellar encounters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We will discuss this in more detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The structure of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In Section 2 we briefly introduce both the theoretical basis for predicting the distribution of prompt cusp structural properties and the physical formalism needed to describe the effects of their encounters with stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We also present a novel, simple and very general scheme for calculating the full distri- bution of impulsive stellar shocks experienced by cusps (or normal subhaloes) as they pass through a galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' through the Milky Way’s disk or bulge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In Section 3 we numerically integrate cusp orbits in a realistic Milky Way model, and we infer the parameters describing their shock histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In Section 4 we use idealized N-body simulations to estimate the disruptive effects of stellar encounters, de- veloping simple formulae that predict the structure and annihilation signal of cusps that have gone through arbitrarily many encounters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Additionally, we show how this can be supplemented to include the effects of stripping by the smooth Galactic tidal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In Section 5 we present our main results, predictions for the profile of the prompt cusp contribution to the annihilation signal of the Galactic halo both as seen by a distant observer and as seen from the Earth, together with an assessment of how prompt cusps affect the form and relative amplitude of the GCE and the IGRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In Section 6 we will discuss the implications of these findings for indirect dark matter searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We make almost all codes used in this study available through an online python repository2 so that our methods can easily be used in future studies and the results of this paper can be reproduced independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2 THEORY As described in the introduction, the effect of stellar encounters on NFW subhaloes has already been studied extensively, although in many cases using an incorrect criterion for subhalo disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' There has, however, been no realistic study of the effects for power- law cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Furthermore, even in the NFW case, most studies have not realistically modelled the full distribution of encounter parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Here we present the theoretical considerations necessary to treat full encounter histories in an accurate, general but simple way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='com/jstuecker/cusp-encounters MNRAS 000, 1–22 (2015) Prompt cusps & stellar encounters 3 For this we present in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1 the distribution of initial cusp properties as derived from the statistics of peaks in the initial gaussian density field, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 how cusp structure can be modified by an inner core to account for the upper limit on phase-space density, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='3 the impulsive shocks induced by stellar encounters, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='4 a calculation of the total number of encounters expected on passing through a star distribution with arbitrary stellar mass and velocity distributions, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 the statistical distribution of shock histories that follows from these considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1 Cusps Several numerical studies have found that peaks on the dark matter free-streaming scale collapse promptly to form dense cusps (Die- mand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Ishiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Anderhalden & Diemand 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Ishiyama 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Polisensky & Ricotti 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Angulo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Ogiya & Hahn 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Delos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2018a,b, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Colombi 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Delos & White 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' These cusps have a density profile, 𝜌(𝑟) = 𝐴𝑟−3/2, (1) parameterised by a normalization 𝐴 and an outer radius 𝑟cusp which limits the extent of the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Delos & White (2022a) find that both parameters can be predicted well for a given cusp from the properties of the initial density peak from which it forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Specifically, 𝐴 = 24𝜌0𝑎−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 col 𝑅1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5, (2) 𝑟cusp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='11𝑎col𝑅, (3) where 𝜌0 is the mean dark matter density of the universe today, 𝑎col is the scalefactor when the peak first collapses and 𝑅 is the Lagrangian size of the initial density peak defined as 𝑅 = √︁ |𝛿/∇2𝛿|, where 𝛿(x) is the linear overdensity field as a function of comoving position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The time of collapse can be estimated to sufficient accuracy by an ellipsoidal collapse model based on the triaxial structure of the initial peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' A more detailed description can be found in Delos & White (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We follow the descriptions of Delos & White (2022b) to sample a distribution of cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For this we use a dark matter power spec- trum generated by the Boltzmann code CLASS (Blas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2011) up to a resolved wavenumber of 𝑘 = 104 h Mpc−1 at 𝑧 = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Beyond that scale we use the analytic prescriptions of Hu & Sugiyama (1996), but normalized so that it matches the CLASS spectrum at 𝑘 = 104 h Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We multiply this spectrum using the exponential power spectrum cutoff description of Bertschinger (2006) for a WIMP with mass 𝑚WIMP = 100 GeV and decoupling temperature 𝑇d = 30 MeV (corresponding to a decoupling scale- factor of 𝑎d = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='33 × 10−12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We use the free streaming length 𝑘FS = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='06 pc−1 as calculated by Delos & White (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The dis- tribution of initial density peaks can be sampled analytically given the initial power spectrum as described by Bardeen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We have created our own implementation of the sampling of peaks which is published in the code repository of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, we tested it against the peak sampling implementation that was published by Delos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We map the distribution of peaks onto a dis- tribution of cusps by using equations (2) – (3) with the ellipsoidal collapse correction for 𝑎col as explained by Delos & White (2022b) based on the approximation from Sheth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We show the resulting distribution of cusps in Figure 1 (dashed contours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, more relevant than the number distribution of cusps is the annihilation weighted distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Under the assumption of a velocity independent cross section, the annihilation rate of a cusp 10 2 rcusp [pc] 10 5 10 4 10 3 A [M / pc3/2] ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' weighted num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' weighted 90% 75% 50% 25% 10% Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The distribution of the cusp normalization 𝐴versus the cusp trunca- tion radius 𝑟cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Typical cusps that dominate the annihilation rate distribution have 𝐴 ∼ 8 × 10−4 M⊙pc−3/2 and 𝑟cusp ∼ 5 × 10−3 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' should be proportional to 𝐽 = ∫ 𝑟cusp 𝑟core 𝜌2 d3𝑟 = 4𝜋𝐴2 log(𝑟cusp/𝑟core), (4) where we have neglected any contributions from 𝑟 < 𝑟core and 𝑟 > 𝑟cusp where 𝑟core is the core radius enforced by the phase-space density constraint (Delos & White 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We will explain in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 how to calculate and treat the core radius more precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The weighted distributions are shown as solid contours in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We can see that for this specific dark matter particle, the typical cusp relevant for the annihilation rate has 𝐴 ∼ 8 × 10−4 M⊙pc−3/2 and 𝑟cusp ∼ 5 × 10−3 pc ≈ 103 AU (in physical units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' It collapses at redshift 𝑧 ∼ 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 Phase-space cores The fine-grained phase-space density of dark matter is constant as a function of time (Liouville’s theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The coarse-grained phase- space density – defined as the average over some finite phase-space volume – can therefore never exceed the fine-grained value estab- lished at dark matter freeze-out (Tremaine & Gunn 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Let us denote the maximum of this fine-grained phase-space density by 𝑓max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The value of 𝑓max is set in the early universe and depends strongly on the type of dark matter considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For the WIMP model considered above, for example, it is 𝑓max = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='98 · 1013 M⊙ km3s−3pc3 , whereas for thermal relic warm dark matter with 𝑚𝑋 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5keV it is 𝑓max = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='75 · 10−2 M⊙ km3s−3pc3 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Delos & White 2022a)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The power-law profile of equation (1) corresponds, for an isotropic 3 Our values here differ slightly from the values of (Delos & White 2022a) since we use Ω𝑥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='26 instead of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='3 for the dark matter density parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' MNRAS 000, 1–22 (2015) 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' velocity distribution, to a phase-space distribution function, 𝑓 (𝐸) = 𝑓0𝐸−9/2, (5) 𝑓0 = 1120 √ 2𝜋2𝐴4𝐺3 9 , (6) (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This description must break down at energies where 𝑓 (𝐸) > 𝑓max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' To estimate the radial scale where this happens we can consider at each radius the largest reachable phase-space density – given by 𝑓 (𝜙(𝑟)) where 𝜙(𝑟) is the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Inserting this into equation (5) and inverting for r we find 𝑟core = 3 9√ 3 · 70 4 9 64𝜋 10 9 𝐴 2 9 𝐺 2 3 𝑓 4 9max ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='03 × 10−5 pc � 𝐴 10−3 M⊙pc− 3 2 �− 2 9 �� � 𝑓max 1014 M⊙ km3s−3pc3 �� � − 4 9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (7) Thus in the above WIMP model, typical cusps relevant for the anni- hilation calculation have thermal core size, 𝑟core ∼ 10−5 pc ∼ 2 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The detailed shape of the density profile near the core radius is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' It would be desirable to run simulations with the actual primordial velocity distribution of a WIMP, so that phase-space cores are created self-consistently, and then to measure their density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Such simulations would be computationally demanding and have not yet been performed, but they are not far beyond current capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (Consider that typically 𝑟cusp ∼ 500𝑟core so that simulations that resolve the cusp profile well are typically only an order of magnitude in linear scale from the core radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=') To obtain a profile that has the desired maximum phase-space density and connects smoothly to the appropriate power-law at larger radii we make the following Ansatz for the phase-space density, 𝑓 (𝐸) = 𝑓0 (𝐸 + 𝐸core)9/2 , (8) where 𝐸core is the core energy scale where the phase-space density 𝑓max would be reached in the fiducial power-law profile: 𝐸core = � 𝑓0 𝑓max �2/9 , (9) so that together with equation (6) the profile is fully specified through 𝐴 and 𝑓max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Note that for 𝑓max → ∞ the pure power-law profile is recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We can integrate over the velocity components of the phase-space density to find the density, 𝜌(𝑟) = ∫ 𝑓 (𝐸) d3𝑣 = 4𝜋 ∫ ∞ 𝜙 ∫ 𝑟√ 2(𝐸−𝜙) 0 𝐿 𝑓 (𝐸) 𝑟2 √︃ 2𝐸 − 2𝜙 − 𝐿2 𝑟2 d𝐿 d𝐸 = 4𝜋 ∫ ∞ 𝜙 𝑓 (𝐸) √︁ 2(𝐸 − 𝜙) d𝐸 = 64𝜋 √ 2 105 (𝐸core + 𝜙(𝑟))3 , (10) as a function of the potential, 𝜙, which is normalized so that 𝜙(0) = 0 and 𝜙(𝑟 → ∞) → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Combined with Poisson’s equation this forms a nonlinear second order differential equation for the potential, 𝜕𝑟 (𝑟2𝜕𝑟 𝜙) 4𝜋𝐺𝑟2 = 64𝜋 √ 2 105 (𝐸core + 𝜙(𝑟))3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (11) 10 3 10 2 10 1 100 101 f(E = )/fmax cored powerlaw fmax 10 2 10 1 100 101 (r)/ max cored powerlaw (r4 + r4 core) 3/8 10 1 100 101 r/rc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 (r)/ powerlaw(r) cored powerlaw (r4 + r4 core) 3/8 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The phase-space and density profiles of a 𝑟−3/2 cusp with a core induced by the phase-space constraint 𝑓 < 𝑓max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The top panel shows the phase-space density 𝑓 (𝜙(𝑟)) corresponding to the highest phase-space den- sity present at each radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The central panel shows the density profile and the bottom panel shows the density profile divided by the fiducial power-law pro- file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In each case, we see a rapid transition between the power-law behaviour and a maximum central value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We do not know how to solve this differential equation analytically and therefore solve it through numerical integration starting at 𝑟 = 0 using 𝜙(0) = 𝜕𝑟 𝜙(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' As a result we find 𝜌(𝑟) and 𝜙(𝑟) and we display these and the phase-space density 𝑓 (𝜙(𝑟)) in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' As expected, the density profile reaches a well defined maximum at 𝑟 ≲ 𝑟core, given by 𝜌max = 𝐴𝑟−3/2 core ∝ 𝐴4/3 𝑓 2/3 max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (12) A good approximation to the density profile is given by 𝜌(𝑟) = 𝐴(𝑟4 + 𝑟4 core)−3/8, (13) which we also show as an orange line in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This deviates MNRAS 000, 1–22 (2015) Prompt cusps & stellar encounters 5 at most by 20% from the actual density around 𝑟 ∼ 3𝑟𝑐 where the latter is slightly enhanced with respect to the fiducial power- law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This enhancement is similar to the well-known behavior of the isothermal sphere where the cored solution rises above the asymp- totic 𝑟−2 power-law solution at radii comparable to the core radius before aymptoting to constant density at smaller radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Binney & Tremaine 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We note that while there are no simulations which have a phase- space density constraint consistent with the expected free-streaming scale, there have been simulations by Macciò et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (2012) with artificially large initial velocity dispersion and hence a lowered upper limit on phase-space density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Although it is difficult to compare these directly with our profile, it seems that the final cores do saturate the phase-space bound and that they transition relatively quickly to the asymptotic power-law behaviour – both consistent with the profile that we propose here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We find that the annihilation radiation from the cored profile inside radius 𝑟 can be approximated for 𝑟 > 10𝑟core by 𝐽(< 𝑟) = 4𝜋𝐴2(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='531 + log(𝑟/𝑟core)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (14) This is marginally larger than the annihilation radiation one obtains when assuming the power-law profile down to 𝑟core (compare equa- tion (4)) and a uniform density at smaller radii, in which case the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='531 gets replaced by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This is due to the slight enhancement of the profile around 3𝑟core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' When needed, we will use the numerical cored profile throughout this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='3 Impulsive encounters We consider a star with mass 𝑀∗ passing a cusp on a linear orbit at constant relative velocity 𝑣 and minimal distance 𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We can approx- imate the tidal forces acting on cusp particles through a multipole expansion of the potential up to second order, 𝜙(𝒙, 𝑡) = 𝜙𝑠(𝒙 − 𝒙0, 𝑡) − T0(𝑡)(𝒙 − 𝒙0), (15) where 𝜙𝑠 denotes the self-potential of the cusp, 𝒙0 is the location of centre of the cusp and T0(𝑡) is the tidal field of the star evaluated at 𝒙0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Here we have neglected zeroth- and first-order terms, since they do not affect the internal dynamics of the cusp (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This “distant-tide” approximation is valid for particles that are close enough to the centre, ||𝒙 − 𝒙0|| ≪ 𝑏 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We will see that for encounters with stars with masses of order M⊙ the distant tide approximation is excellent for all particles that remain bound to the cusp (which typically have small Δ𝒙) and is only violated for particles that are kicked so strongly that they will leave the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Thus, it is safe to adopt this approximation in all our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Further, we can assume the impulsive approximation which con- siders the limit that particles move very little within the cusp during the time of the encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In this case, the total change in the velocity of a particle due to the encounter can be approximated by Δ𝑣 = �∫ T0(𝑡)d𝑡 � (𝒙 − 𝒙0) = K(𝒙 − 𝒙0), (16) where we have defined the shock tensor 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' It is easy to see that the impulsive approximation is an excellent approximation here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The dynamical time-scale at radius 𝑟 of our cusp is given by 𝑡d(𝑟) = 𝑟 𝑣circ(𝑟) = √︄ 𝑟3 𝐺𝑀(< 𝑟) , (17) where 𝑀(< 𝑟) is the enclosed mass profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The internal dynamical time-scale is shortest at the core-radius, where it is of order 2 × 104 yr for the strongest cusps (which dominate the annihilation distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The impact parameter of the weakest relevant encounters is of order 𝑏 ∼ 104 AU with 𝑣 ∼ 200 km s−1 which gives an encounter time- scale of order 𝑡enc = 𝑏/𝑣 ∼ 200 yr which is two orders of magnitude smaller than the dynamical time scale of the quickest particles in the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Stronger encounters (which are more relevant) will happen on even shorter time-scales and most particles orbit on longer time scales so that the ratio should be even larger in practice and we can safely assume the impulsive limit for all of our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' If we assume that the star is a point mass with tidal field 𝑇𝑖 𝑗 (𝑡) = −𝜕𝑖𝜕𝑗 � − 𝑀∗𝐺 ∥𝒙 − 𝑥∗(𝑡)∥ � , (18) and we assume its trajectory (without loss of generality) to be along the y-direction with the closest encounter at the coordinate (𝑏, 0, 0), then the shock tensor is given by K = 𝐵 �� � 1 0 0 0 0 0 0 0 −1 �� � (19) 𝐵 = 2𝐺𝑀∗ 𝑣𝑏2 (20) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Aguilar & White 1985) where we have defined the shock pa- rameter 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We note that 𝐵 has dimensions of the inverse of time, but to simplify intuitive understanding we will typically state it in units of km s−1 pc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' It is clear that the only relevant parameter for describing the effect of a distant and impulsive stellar tidal shock on a prompt cusp is the tidal shock parameter 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The individual values of 𝑏, 𝑀∗ and 𝑣 matter only insofar that they determine the value of 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We can get a feeling for what range of 𝐵 values will be relevant for typical cusps by considering the values, 𝐵cusp = 𝑣circ(𝑟cusp) 𝑟cusp = � �8𝜋𝐺𝐴 3𝑟3/2 cusp , (21) 𝐵core = 𝑣circ(𝑟core) 𝑟core = √︄ 8𝜋𝐺𝐴 3𝑟3/2 core ∝ 𝐴2/3 𝑓 1/3 max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (22) 𝐵cusp indicates the strength of a tidal shock that is needed to induce a velocity change at the radius 𝑟cusp as large as the circular veloc- ity and 𝐵core is the analogue quantity at the core radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We show the distributions of these two parameters for the fiducial 100 GeV WIMP in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We can expect that tidal shocks with 𝐵 ≳ 𝐵cusp will significantly alter the profile of the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Tidal shocks with 𝐵 ≳ 𝐵core may possibly lead to complete disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Shocks with 𝐵 ≪ 𝐵cusp will leave the cusp largely unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Typical cusps that are relevant for the annihilation rate have values of 𝐵cusp and 𝐵core of order 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='3 km s−1 pc−1 and 30 km s−1 pc−1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The im- pact parameters that are needed to reach such shock parameters for 𝑀∗ = 1 M⊙ and 𝑣 = 200 km s−1 are 1 × 10−2 pc ≈ 2000 AU and 1 × 10−3 pc ≈ 200 AU respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We will see that tidal shocks of this order are not only possible, but also quite likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' It is therefore important to make precise quantitative calculations to evaluate the MNRAS 000, 1–22 (2015) 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 10 2 10 1 Bcusp [km/s / pc] 101 Bcore [km/s / pc] ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' weighted num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' weighted 90% 75% 50% 25% 10% Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The distributions of 𝐵core and 𝐵cusp which indicate the resilience of prompt cusps against tidal shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' A tidal shock with 𝐵 ≳ 𝐵cusp will likely affect the cusp’s profile significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Shocks with 𝐵 ≳ 𝐵core will lead to disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' number of such encounters and the effect they have on the aggregate annihilation luminosity from cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Finally, it is useful to introduce the characteristic spatial scale associated with the action of shocks on a cusp, 𝑟𝐵 = � 8𝜋𝐺𝐴 3𝐵2 �2/3 = � 2𝜋𝐴 3𝐺 �2/3 𝑏8/3𝑣4/3 𝑀4/3 ∗ , (23) which is the radius where the change in velocity induced by the shock is of order the circular velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We will see in Section 4 that the majority of particles beyond 𝑟𝐵 will leave the sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The distant tide approximation is a good approximation if min(𝑟𝐵, 𝑟cusp) ≪ 𝑏 so that it holds for all particles that remain bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We find that typical encounter scenarios with stars for prompt cusps have min(𝑟𝐵, 𝑟cusp) ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1𝑏 for any impact parameter 𝑏, so that the distant tide approximation is always valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' While we do not focus on NFW haloes in this study, we note that many of our sub- sequent derivations are of interest for studies of NFW substructures as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For these, we estimate that 𝑟𝐵,NFW < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1𝑏 (defined through the circular velocity criterion for an NFW profile) holds for typical closest impact parameters with 𝑏 ≲ 1000 AU for haloes with virial masses 𝑚200c ≲ 1𝑀⊙ (and concentration 𝑐 ∼ 30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Many of our results below can therefore also be applied to NFW subhaloes with masses below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0𝑀⊙, but additional care must be taken if larger masses are considered since the distant tide approximation may then fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The goal of the remaining parts of this section is to determine the distribution of shock parameters 𝐵 for a cusp that is moving through an arbitrary stellar distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' a component of the Milky Way).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We note that this calculation is both simpler and more accurate than previous calculations in the literature which have focused on estimating the distribution of the impact parameter 𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='4 The expected number of encounters We want to estimate the expected number of encounters with a tidal shock parameter that is greater than 𝐵, 𝑁(> 𝐵), for a cusp that is orbiting through an arbitrary stellar distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The stellar distri- bution can be described by a mass dependent phase-space (number) density 𝑓∗(𝒙, 𝒗, 𝑀) = d𝑁 d3𝑥 d3𝑣 d𝑀 , (24) which is normalized so that an integral over a phase-space volume and over a stellar mass interval gives the number of stars within that volume and mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' An integral over phase space alone gives the stellar mass function, which is allowed to vary spatially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' If our cusp was passing through a uniform medium without veloc- ity dispersion and with number density 𝑛∗ then the expected number of encounters in a time-interval d𝑡 with impact parameters in the range [𝑏, 𝑏 + d𝑏] is given by d𝑁 = 2𝜋𝑛∗𝑏𝑣 d𝑏 d𝑡, (25) where 𝑣 is the relative velocity with respect to the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Equation (25) arises by considering stars when they get closest to our cusp, which is when they pass through the plane orthogonal to the velocity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The relevant velocity here is the relative velocity between the stars and the cusp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' higher velocities lead to more frequent encounters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For arbitrary phase-space distributions we have to consider each velocity and mass bin individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The number of encounters within time interval d𝑡, impact parameter range (𝑏, 𝑏 + d𝑏) and encounter velocity bin d3𝑣 with stars that have masses in the range (𝑀, 𝑀+ d𝑀) is given by d𝑁 = (2𝜋𝑏 d𝑏)( 𝑓∗(𝒙, 𝒗 − 𝒗rel, 𝑀) d3𝑣 d𝑀)(𝑣 d𝑡), (26) where 𝒗 is the encounter velocity, and 𝑣rel is the relative velocity between our cusp and the zero-point of the stellar phase-space dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Here, we have assumed that the phase space distribution function does not vary significantly over the distance 𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This is an ex- cellent approximation, since typical encounters of interest will have 𝑏 ≪ 1 pc whereas the stellar distribution varies on much larger scales (≫ 100 pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Now, it would be straightforward to estimate, for example, the total number of encounters with impact parameters smaller than some given 𝑏 by integrating (26) over the corresponding variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In general this would turn out to depend both on the phase-space distribution of the stars and the stellar mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, as discussed in the previous subsection, we are actually interested in the distribution of shock parameters 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This can be evaluated by integrating (26) under the constraint that 𝐵 = 𝐺𝑀/𝑣𝑏2 which leads us to d𝑁 d𝐵 = ∫ 6dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 𝛿𝐷 � 2𝑀𝐺 𝑣𝑏2 − 𝐵 � d6𝑁 d3𝑣 d𝑏 d𝑀 d𝑡 d3𝑣 d𝑏 d𝑀 d𝑡, (27) where 𝛿𝐷 is the Dirac delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We evaluate the integral, inte- grating in 𝑏 first: d𝑁 d𝐵 = ∫ ∫ ∫ 𝑓∗(𝒙, 𝒗 − 𝒗rel, 𝑀)𝑣𝑔(𝑀, 𝐵, 𝑣) d3𝑣 d𝑡 d𝑀, (28) 𝑔(𝑀, 𝐵, 𝑣) = ∫ ∞ 0 2𝜋𝑏𝛿𝐷 � 2𝑀𝐺 𝑣𝑏2 − 𝐵 � d𝑏 = ∫ ∞ 0 2𝜋 𝑀𝐺 𝑣𝐵′2 𝛿𝐷 �𝐵′ − 𝐵� d𝐵′ = 2𝜋𝑀𝐺 𝑣𝐵2 , (29) MNRAS 000, 1–22 (2015) Prompt cusps & stellar encounters 7 where we have used the substitution 𝐵′ = 2𝑀𝐺 𝑣𝑏2 to evaluate the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Sorting and evaluating individual terms gives us d𝑁 d𝐵 = 2𝜋𝐺 𝐵2 ∫ ∫ ∫ 𝑀 𝑓∗(𝒙, 𝒗 − 𝒗rel, 𝑀) d3𝑣 d𝑀 d𝑡 = 2𝜋𝐺 𝐵2 ∫ 𝜌∗(𝒙(𝑡)) d𝑡 = 2𝜋𝐺 𝐵2 𝜒∗, (30) where we have used the fact that the mass weighted integral over the phase-space number density gives the stellar mass density 𝜌∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Further we have defined the time integral of the stellar density along the cusp trajectory 𝜒∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' It is further convenient to define the characteristic shock parameter, 𝐵∗ = 2𝜋𝐺𝜒∗, (31) so that d𝑁 d𝐵 = 𝐵∗ 𝐵2 , (32) 𝑁(> 𝐵) = 𝐵∗ 𝐵 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (33) Therefore, we expect on average one encounter with 𝐵 > 𝐵∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' It is worth noting that this result is surprisingly independent of the phase-space distribution function of stars and the stellar mass func- tion, but depends only on the stellar mass density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The mass function is irrelevant, because at a fixed mass density, a reduction in mass leads to an increase in number density, thus making close encounters more likely to the same degree that it decreases the strength of such encounters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' A similar coincidence holds for the velocity dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The number of stars that are encountered increases with the velocity, while at the same time the weight of each encounter decreases with the velocity to such a degree that the two effects cancel exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We call these effects the encounter conspiracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' It is clear that these two simplifications could, in principle, break down at some scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Mass function independence breaks down if the distant tide approximation fails – if we make our perturbers less massive, the encounters get closer for a given value of 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For very small masses e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 10−6 M⊙ the closest approach needed for a significant perturbation would be of order one AU, smaller than core radius of a cusp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' the distant tide approximation would then certainly fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The integral over the mass function in equation (30) should thus have a lower limit, in principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The independence of velocity breaks down for very small encounter velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' If an encounter takes longer than the orbital times within the cusp, then the cusp will react adiabatically, with no long-termchanges inenergy except forparticles that leave the system in the adiabatic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Thus our approximations fail for stars that are moving almost at the same velocity as the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, neither of these problems has a significant effect in practice, since almost all stellar mass is in objects of mass within an order of magnitude or so of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 M⊙ and very few encounter velocities are smaller than, say, 10 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We note that the effects of encounters with other massive objects, such as planets or other prompt cusps would be overestimated if the calculation of this section were applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, the mass density in planets is so much lower than that in stars that planets are quite irrelevant in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The mass density in prompt cusps is non- negligible at large halocentric radii, but when their extended profile is taken into account, we find that the strongest possible shocks are far below 𝐵 ≪ 10−4 km s−1 pc−1 so that cusps cannot significantly shock other cusps (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Therefore, stars pose the only sig- nificant contributor to the distribution of encounter shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 10 1 100 101 102 103 B/B * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 dN / d log B 1st strongest 2nd strongest 3rd strongest Beff [B > 10 1B * ] Beff [B > 10 2B * ] Beff [B > 10 3B * ] Beff [B > 10 4B * ] Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The distribution of shock parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The blue, orange and green lines show the distribution of the strongest, 2nd strongest and 3rd strongest shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The dashed lines show the distribution of the effective shock parameter according to equation (38) when truncating the (divergent) distribution at different values of 𝐵min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 Shock Histories We assume that all aspects of the problem follow Poisson statistics – for example that stars are drawn through a Poisson process from the continuous phase-space distribution and that stellar masses are drawn through a Poisson process from the stellar mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Then also the shock parameter distribution has to follow Poisson statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' That means the probability of having exactly 𝑘 encounters with shock strength bigger than 𝐵 is given by 𝐹(𝑘, 𝐵) = (𝐵∗/𝐵)𝑘 exp(−𝐵∗/𝐵) 𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (34) In particular, the probability of having at least one encounter with shock strength bigger than 𝐵 is given by 𝐹(≥ 1, 𝐵) = 1 − 𝐹(0, 𝐵) = 1 − exp(−𝐵∗/𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (35) The probability density function of the strongest encounter is there- fore 𝑓1(𝐵) = d𝐹(≥ 1, 𝐵) d𝐵 = 𝐵∗ 𝐵2 exp(−𝐵∗/𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (36) It is straightforward to derive the corresponding functions for the 2nd, 3rd etc strongest shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, the distribution of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' the strongest and the second strongest shock are not independent and parameterising the joint distribution is rather cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' When considering a large number of encounters it is more convenient to draw actual realisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This can easily be done by mapping 𝐵 onto another random variable that follows a uniform distribution 𝑥 := 𝐵∗/𝐵, d𝑁 d𝑥 = d𝑁 d𝐵 d𝐵 d𝑥 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (37) We can then create a realization of a history of shocks with 𝐵 > 𝐵min, by sampling 𝑘 uniformly distributed random variables 𝑥𝑖 on the interval [0, 𝐵∗/𝐵min] and then transforming them as 𝐵 = 𝐵∗𝑥−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The number 𝑘 must itself be drawn from a Poisson distribution with mean ⟨𝑘⟩ = 𝐵∗/𝐵min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We note that there will be infinitely many MNRAS 000, 1–22 (2015) 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' encounters as 𝐵min → 0 so that it is in general not possible to sample the full distribution, but only its truncated form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, this is sufficient, since encounters with very small shock parameters become irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In particular, we will show in Section 4 that the joint effect of 𝑘 encounters is more or less equivalent to a single encounter with an effective shock strength, 𝐵eff = � 𝑘 ∑︁ 𝑖=1 𝐵𝑝 𝑖 �1/𝑝 , (38) with 𝑝 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We sample a large number of shock histories and show the strongest three encounters together with the distribution of 𝐵eff in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The distribution of 𝐵eff is already reasonably well approximated when only considering shocks with 𝐵 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1𝐵∗ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' approximately the 10 strongest shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' It is almost fully converged when considering all shocks with 𝐵 > 10−3𝐵∗ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' the 1000 strongest shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Typically the effective shock parameter is not too much larger than the strongest shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Its median lies at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5𝐵∗ whereas the median of the strongest shock lies at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='46𝐵∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, the low-end tail is shifted upwards quite a bit so that there are almost no cases with 𝐵eff < 𝐵∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The high-end tail is virtually identical to the distribution of the strongest shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 3 SHOCK DISTRIBUTION FOR ORBITS IN THE MILKY WAY In this Section we evaluate numerically the quantities that are relevant for describing the full distribution of shock parameters for a cusp that is orbiting in the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We showed in the last section that, for a given orbit, the time integral of the stellar mass density along the cusp’s trajectory, 𝜒∗ = ∫ 𝜌∗(𝒙(𝑡)) d𝑡, (39) is sufficient to parameterise the full distribution of encounter shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Our aim is thus to infer realistic estimates of 𝜒∗ and of the corre- sponding characteristic shock parameter 𝐵∗ = 2𝜋𝐺𝜒∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We assume that the orbital distribution of cusps follows that of dark matter particles within the Milky Way’s halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1 Milky Way Potential For the baryonic components of the Milky Way we assume the pre- scriptions used in the Phat ELVIS simulations (Kelley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2019) at the specific time 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The observational parameters underly- ing these simulations were taken from McMillan (2017) and Bland- Hawthorn & Gerhard (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The stellar disk and the gas disk are each modelled through a superposition of three Miyamoto & Nagai (1975) (hereafter MN) potentials as described by Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The parameters of the MN potentials have been tuned to recreate the mass distribution of an exponential disk up to 1% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For the stellar disk we use a total mass of 𝑀d = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1 × 1010 M⊙ a scale radius 𝑅d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 kpc and a height parameter of 𝑧d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='35 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For the gas disk we use 𝑀d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='9 × 1010 M⊙, 𝑅d = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 kpc and 𝑧d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='08 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We approximate the stellar bulge through a Hernquist (1990) distribution with mass 𝑀B = 9 × 109 M⊙ and with scale-length 𝑟a = 500 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For the Milky Way’s dark matter halo, we assume that in the absence of baryonic components it would correspond to an NFW halo 100 101 102 r [kpc] 109 1010 1011 1012 1013 M( < r) [M ] stellar disk gas disk bulge all baryons halo total NFW (uncontracted) ISS V0 = 220km/s r200c Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Enclosed mass profiles for the different components of our Milky Way model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Solid lines are used for components that contribute to the final potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The brown dashed line shows the uncontracted NFW halo profile whereas the purple curve shows the profile after contraction due to the baryons and is the curve actually used in our modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The dotted grey line indicates the profile of a singular isothermal sphere with 𝑉0 = 220 km s−1 which is sometimes used to approximate the spherically averaged potential of the Milky Way (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Errani & Navarro 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' with concentration 𝑐 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='7 and mass 𝑚200𝑐 = 1012 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, the baryons increase the dark matter density in the inner regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We model this contraction using the semi-analytic approach presented in Cautun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Previously, we used the analytic procedure of Sellwood & McGaugh (2005), but this produces a slightly denser result in the central regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We prefer the Cautun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (2020) approach, since it has been tested in detail both against state-of-the- art simulations and against observations of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We show the result of the contraction of the halo together with profiles of all the different components in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We note that the contracted halo (in purple) is significantly denser in the centre than the uncontracted version (brown dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This contraction is only moderately relevant for getting the correct potential structure of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, it is very relevant for sampling self- consistent orbits of the dark matter distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' To sample orbits we use the (Eddington 1916) inversion method on the density profile of the contracted halo, but inside the joint potential of all components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We have checked that if we sample particles from the contracted halo profile and integrate their orbits in the joint potential, the density profile is stable and evolves very little at later times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 Orbits We create 105 test particles from the contracted halo up to 400 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We use an adaptive sampling method so that their initial number density is proportional to 1/𝑟 times that of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This way we have a good sampling, both at small radii 𝑟 ≲ 1 kpc and close to the virial radius 𝑟 ≳ 200 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Throughout this paper, when showing distributions, we always correct this non-uniform sampling using the appropriate weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We integrate particle orbits in the full 3d Milky Way potential (including bulge, stellar disk, gas disk and contracted halo) using 105 constant timesteps over a period of 10 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Along each orbit we evaluate 𝜒∗ according to equation (39) using the stellar density inferred from the Laplacian of the potential of the stellar components (stellar disk and bulge) 𝜌∗ = Δ𝜙∗/(4𝜋𝐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Note that this can create negative densities in a (very) few locations, since the MN MNRAS 000, 1–22 (2015) Prompt cusps & stellar encounters 9 10 3 10 2 10 1 100 101 102 103 B * [km/s / pc] 100 101 r [kpc] 100 101 102 103 104 105 106 107 200km/s [M /pc2] median Bcusp, Bcore 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='7% region 95% region 68% region Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The distribution of 𝜒∗ as a function of radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' To facilitate inter- pretation, the left y-axis gives 𝜒∗ multiplied by a velocity of 200 km s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' this estimates the total encountered stellar column density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' An alternative y-axis shows the values of 𝐵∗, with red dashed lines showing the critical values, 𝐵core and 𝐵cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' It is clear that shocks will be relevant for almost all cusps that orbit in the central 10 kpc of our Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' potential of the disk can create negative densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We therefore clip 𝜒∗ after the integration at a minimum of 10−8 M⊙/pc2/(km/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This threshold has no impact on any of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The actual minimum should be set by the diffuse stellar halo (and partially by encounters with dark substructures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, none of these aspects matter since, as we will see in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='6, the effect of the smooth tidal field is much larger than this at radii where stellar encounters are infrequent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We find the distributions of 𝜒∗ and 𝐵∗ shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Here and in later plots we assign each particle 1000 times with its final value of 𝜒∗ at different radii chosen uniformly in time over its full orbit history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' To help with an intuitive understanding of the 𝜒∗ distribution, we have multiplied the 𝜒∗ axis by a value of 200 km s−1 so that the left y-axis would correspond to the total stellar column density if the cusp always encountered stars at 200 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Figure 6 shows that a typical cusps orbiting around the solar radius (8 kpc) encounters a stellar column density of about 104 M⊙pc−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' These numbers can easily be understood, as orbits at these radii will have passed through the Galactic disk about 102 times and each disk passage adds a column density of order 102 M⊙pc−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Further, we show as an alternative y-axis in Figure 6 the distri- bution of 𝐵∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Recall that 𝐵∗ indicates the value of 𝐵 such that we expect on average one encounter with 𝐵 > 𝐵∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Therefore we expect typically one encounter with 𝐵 ≳ 1 km s−1 pc−1 for cusps orbiting around the solar radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For each orbit we sample the strongest encounters corresponding to the probability density in equation (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In this way, we find the distribution of strongest shocks shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We also show the corresponding encounter distance for encounters with mass 1 M⊙ and velocity 200 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This Figure is not used in our final results, but is useful to gain intuition for the relevant quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We conclude that inside the central 10 kpc virtually all cusps will have had at least one stellar encounter that might significantly de- crease the annihilation luminosity and possibly even disrupt them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For a reliable evaluation, we have to conduct numerical experiments that evaluate how strongly cusps react to shocks of a given amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 101 102 103 104 105 closest encounter b [AU] 100 101 102 r [kpc] 10 4 10 3 10 2 10 1 100 101 102 103 104 Strongest Shock B [km/s / pc] Bcusp, Bcore median 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='7% region 95% region 68% region Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The distribution of the strongest shock encountered by a cusp as a function of its current Galactocentric radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' An alternative 𝑦-axis shows the corresponding closest approach distance for encounters with a mass of 1 M⊙ and velocity 200 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We see that within the central 10 kpc the strongest shocks encountered will frequently affect the annihilation signal of cusps and in some cases may disrupt them completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 4 THE EFFECT OF IMPULSIVE ENCOUNTERS To evaluate the effects of impulsive encounters on prompt cusps, we ran several sets of N-body simulations addressing scenarios of in- creasing complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The first set considers pure power-law profiles experiencing a single encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The second set considers cored pro- files also with a single encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The third set considers both cored and power-law profiles experiencing multiple stellar encounters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For each case we develop simple descriptions for the annihilation lumi- nosity expected from the remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Finally, at the end of this section we briefly discuss how the effect of the smooth tidal field can be incorporated simply in this formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1 Simulation Setup We consider simulations starting from two different initial profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Power-law simulations start from a Poisson realization of a pure𝑟−3/2 density profile with the phase-space distribution from equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In this case there is no relevant length or mass scale so that the results can be rescaled to any desired tidal shock strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For cored simulations we use the phase-space distribution from equation (8) and the density profile that is self-consistently created from this distribution, as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For these we set 𝑟core = 1 so that length scales are given in units of the core radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' To enhance the dynamic range that can be resolved in these sim- ulations and to minimize the effects of the (numerically required) outer truncation radius, we use a similar non-uniform mass sam- pling strategy to Delos (2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Specifically, we arrange that the same number of particles 𝑁split have pericentres in each of the radial ranges (0, 1), (1, 10), (10, 100), (1000, 10000) which then have par- ticle masses increasing by about a factor 30 between intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We explain this in more detail in Appendix A1, where we also present some numerical stability tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' An implementation of this method is available in the code repository of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We do not find any ar- tifacts due to the non-uniform mass sampling, most likely because we use quite high resolution and because our pericentre criterion min- imizes the intrusion of higher mass particles into higher resolution regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For simulations with a single encounter (presented in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='3) we apply at time 𝑡 = 0 a single kick according to equation MNRAS 000, 1–22 (2015) 10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 10 3 10 2 10 1 100 101 r/rB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 (r)/ 0(r) no kick B (t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5tB) B × 4 (t = 10tB) B × 4 (t = 50tB) fit Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The transfer function for pure 𝑟−3/2 power-law profiles exposed to a tidal shock from a stellar encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' When presented in units of 𝑟𝐵 (𝐵) the transfer functions of simulations with different shock amplitudes (blue versus orange and green) line up perfectly – as is expected from dimensional analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The grey lines show the transfer functions of equivalent simulations without tidal shocks evaluated at the same output times – proving the numerical stability of our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The black line shows the fit to the transfer function given in equation (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (16) using the tensor from equation (19) with shock amplitude 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The shock introduces the characteristic spatial scale, 𝑟𝐵 = � 8𝜋𝐺𝐴 3𝐵2 �2/3 , (40) which is the radius where the tidal shock causes a (maximal) velocity change equal to the circular velocity in a pure power-law profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' By definition, 𝑟𝐵 equals 𝑟cusp and 𝑟core for shocks with strength 𝐵cusp and 𝐵core respectively, so that realistic shocks can easily produce 𝑟𝐵 values that lie at any point of the cusp (compare Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Further, the kick introduces a characteristic time-scale – corresponding to the dynamical time at this radius: 𝑡𝐵 = 𝑟𝐵 𝑣circ(𝑟B) = 1 𝐵 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (41) After the initial shock we evolve the simulation for several dynamical times 𝑡𝐵 until the profile no longer evolves at the radii of interest (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' ∼ 10𝑡𝐵 around 𝑟𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Note that for typical shocks with 𝐵 ∼ 1 km s−1 pc−1, we have 𝑡𝐵 ∼ 1 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 Truncation of Pure Power-law Profiles We ran two high-resolution simulations of shocked power-law pro- files which each have 𝑁split = 222 particles per radial interval, so that they have in total about 2 × 107 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' These two simulations used different amplitudes for the shock parameter, varying by a factor of 4, leading to differing shock radii, 𝑟𝐵 = 20 and 𝑟𝐵 = 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We remind the reader that, in principle, the result of a shocked power-law can be rescaled arbitrarily, so these two simulations differ only in how the physical scale 𝑟𝐵 compares to resolution parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We define the transfer function, 𝑇(𝑟) = 𝜌(𝑟) 𝜌0(𝑟) , (42) where 𝜌(𝑟) and 𝜌0(𝑟) are the final and initial density profiles, respec- tively, and we present transfer functions for our simulations in Figure 8 as functions of 𝑟/𝑟𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We show only the radial range that has reliably converged to a final, stable post-shock profile, and we use different output times to probe different parts of the transfer function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In Ap- pendix A2 we provide convergence tests to determine these scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' At the small end this is the largest radius where two-body relaxation is irrelevant, while at the large end we limit to scales where at least ten dynamical time-scales have passed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' As additional evidence that the simulations are converged, we also show as grey lines in Fig- ure 8 reference simulations evolved to the same times but with no shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Clearly, these are stable and show no discernable numerical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The simulations of different shock strengths line up very well if radii are scaled by 𝑟𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In turn, since 𝐵 ∝ 𝑏−2, 𝑟𝐵 scales with impact parameter as 𝑏8/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This is very different than the scaling of 𝑏8/11 which Ishiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (2010) assumed (without further explanation) to extrapolate their results and to argue that stellar encounters should be irrelevant for the survival of cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Such a scaling is clearly incorrect and is also inconsistent with the profiles of the simulations that Ishiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (2010) presented themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' These simulations, in fact, seem consistent with a 𝑏8/3 scaling and agree qualitatively with what we find here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Inspired by Delos (2019b), we fit the simulated transfer functions jointly using a function of the form, 𝑇(𝑟) = exp(−𝛼(𝑟/𝑟𝐵)𝛽), (43) where we find 𝛼 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='256 and 𝛽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='639 as the best fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Interestingly the value of 𝛽 that we find here for our power-law profile is not too far from the one that Delos (2019b) found (𝛽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='78) for an NFW initial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The small difference presumably reflects the different central slopes, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 and −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We note that our measured profiles disagree slightly with our fit at larger radii 𝑟 ≳ 2𝑟𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, this is quite irrelevant for the calculation of the annihilation luminosity where those contributions are downweighted by a factor 𝑇2(𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For our fitted function we find that the annihilation rate is given by 𝐽pow(𝐵) = 4𝜋 ∫ 𝑟cusp 𝑟min 𝑇(𝑟)2𝜌2 0(𝑟)𝑟2 d𝑟 = 4𝜋𝐴2 𝛽 � Ei � −2𝛼 �𝑟min 𝑟𝐵 �𝛽� − Ei � −2𝛼 �𝑟cusp 𝑟𝐵 �𝛽�� (44) ≈ 4𝜋𝐴2 𝛽 Ei � −2𝛼 �𝑟min 𝑟𝐵 �𝛽� , where Ei is the exponential integral and 𝑟min is a lower limit of inte- gration that is necessary to obtain a finite result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The approximation that is used in the last line, 𝑟cusp → ∞, is already very accurate for 𝑟cusp ≳ 𝑟𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Therefore, if 𝑟𝐵 ≤ 𝑟cusp, it is fine to neglect the ini- tial boundary of the system, since the truncation through the shock sets the actual boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Further it is insightful to consider the limit 𝑟min ≪ 𝑟B in which case 𝐽pow(𝐵) ≈ 4𝜋𝐴2 𝛽 � −𝛾 − log � 2𝛼 �𝑟min 𝑟𝐵 �𝛽�� = 4𝜋𝐴2 � log � 𝑟𝐵 𝑟min � − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='35 � = 4𝜋𝐴2 log � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='096𝑟𝐵 𝑟min � , (45) where 𝛾 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='577 is the Euler–Mascheroni constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This approxi- mation holds (at 5% accuracy or better) for radii 𝑟min < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1𝑟B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' One way to interpret this result is that the annihilation rate of the tidally MNRAS 000, 1–22 (2015) Prompt cusps & stellar encounters 11 𝐵/𝐵core 𝑟core/𝑟𝐵 𝑡/𝑡𝐵 𝐽/4𝜋𝐴2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='008 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='020 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='213 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='050 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='457 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='211 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='125 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='723 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='215 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='338 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='421 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='316 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='527 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='425 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='632 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='543 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='738 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='666 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='000 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Simulation parameters for the shocked cored profiles: The shock parameter in units of the core scale, 𝐵/𝐵core, the ratio of core to shock radius, 𝑟core/𝑟𝐵, the duration of the simulation in units of the dynamical time, 𝑡/𝑡𝐵 and the post-shock annihilation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The last simulation shows complete disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' truncated power-law corresponds to the annihilation rate of a power- law profile that is sharply truncated at 10% of 𝑟𝐵 (compare Equation (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This result is easily understood, since this is approximately the radius where 𝑇2 reaches 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We expect this approximation to be inaccurate if 𝑟core ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1𝑟𝐵, since then shock effects are significant at the core radius where they may be amplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We will consider such cases in the next subsection and derive a general formula for the annihilation rate of shocked cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, we can make a rough estimate of the cored annihilation rate, by assuming the power-law + transfer profile up to the core radius and down-weighting the core contribution by a factor of 𝑇2(𝑟core) 𝐽pow+core(𝐵) = 4𝜋𝐴2 � 𝛽−1 Ei � −2𝛼 �𝑟core 𝑟𝐵 �𝛽� + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='531𝑇2(𝑟core) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (46) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='3 Truncation of cored Profiles We also ran several simulations with cored initial conditions and with different shock parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' These are designed to probe the scales where the shock starts affecting the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Each uses 𝑁split = 220 and therefore in total about 5 × 106 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We evaluate each simulation at a time where the final profile has reached a stable result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This requires a longer integration time for simulations where the central density is reduced significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We give the simulation parameters, durations and final annihilation luminosities in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We show the transfer functions of these simulations in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We can see that as 𝑟core/𝑟𝐵 → 0, the pure power-law behaviour is recovered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' at large radii they have the same transfer function as the power-law case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' At small radii cored profiles are suppressed additionally, and can even disrupt completely for sufficiently strong shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The simulation with 𝑟core/𝑟𝐵 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='666 is the first case which ex- hibits complete disruption – this means that after 50𝑡𝐵 no stable rem- nant is left, and densities keep decreasing everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We checked that this case still disrupts if a four times higher particle number and a smaller softening are used, and that simulations with even larger shocks disrupt also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In addition, we a provide a video of one non- disrupting case and one disrupting case online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' It seems clear that although it is impossible to disrupt centrally divergent power-law profiles, it is indeed possible to disrupt cored profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This agrees with theoretical (Amorisco 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2022) and numerical 4 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='com/jstuecker/cusp-encounters 10 5 10 4 10 3 10 2 10 1 100 101 r/rB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 (r)/ 0(r) rcore/rB 0 rcore/rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='008 rcore/rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='020 rcore/rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='050 rcore/rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='125 rcore/rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='215 rcore/rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='316 rcore/rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='425 rcore/rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='543 rcore/rB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='666 Powerlaw T(r) rrcore Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The transfer functions of cored power-law profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Note that each case has a different value of 𝑟core/𝑟𝐵 so that the initial profiles are not identical as a function of 𝑟/𝑟𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Each simulation is divided by its own initial profile in making this plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Core radii are indicated as vertical dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The transfer functions all show enhanced suppression relative to the power-law transfer function below and slightly above the core radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 10 3 10 2 10 1 100 101 J/(4 A2) Cored Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Powerlaw Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Powerlaw T(r) Fit Disruption 10 3 10 2 10 1 100 B/Bcore 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1 J/J Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Annihilation rates of cored power-law profiles that have been exposed to shocks of varying amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The top panel shows the annihilation rates and the bottom panel the residuals with respect to our fit (blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The black line shows the estimate from equation (46) based on the power- law transfer function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Clearly, the actual annihilation rates are additionally suppressed with respect to the power-law result when shocks get close to the core scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For shocks with 𝐵 ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='65𝐵core we find complete disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' investigations (van den Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Errani & Peñarrubia 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Errani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2022) in the context of NFW haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For the cored profiles we do not attempt to fit a functional form to the transfer functions, but instead directly calculate the annihi- lation radiation from the spherically averaged density profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This works very well numerically, because the part of the profile which is responsible for the bulk of the annihilation radiation (most signifi- cantly around 3𝑟core) is resolved very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 We describe the binning and integration procedures in Appendix A3 and show that all ob- tained annihilation luminosities are accurate at the 3% level – except 5 This is not the case for our power-law profiles, where a major fraction of the radiation always comes from poorly resolved inner radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' MNRAS 000, 1–22 (2015) 12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' for the last non-disrupted case, 𝑟core/𝑟𝐵 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='543, where the relative error is larger, but still less than 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We list the annihilation rates we obtain in Table 1 and plot them in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In this Figure we also show the two power-law simulations with annihilation rates estimated by individually fitting their transfer functions and (arbitrarily) assuming a core radius, 𝑟core = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='05, for the annihilation estimate from equation (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, these two simulations could be arbitrarily rescaled along the black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We are able to fit the post-shock annihilation rates of our cored simulations using the function, 𝐽 = 4𝜋𝐴2 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='531 + 1 𝛽 log(𝑎) − 1 𝛽 Ei � −2𝛼𝑎(𝑥core + 𝑏𝑥3 core)4𝛽/3�� , (47) 𝑥core = 𝐵 𝐵core = �𝑟core 𝑟𝐵 �3/4 , (48) with free parameters 𝑎 and 𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' By construction, this function ap- proaches the behaviour of the power-law truncation fit of equation (46) for 𝐵 ≪ 𝐵core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, it has two degrees of freedom 𝑎 and 𝑏 which can modify the behavior for large values of 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We find that 𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='708, 𝑏 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='98 gives a reasonable fit to all annihilation rates at the 10% level – both in the power-law regime 𝐵 ≪ 𝐵core and when the shock hits the core 𝐵 ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1𝐵core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The function reaches zero at 𝐵dis ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='65𝐵core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We assume that all cusps that had an en- counter with 𝐵 > 𝐵dis are disrupted and have zero contribution to the annihilation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='4 The effect of multiple encounters As a final step, we need to understand what happens to the remnant if it is exposed to multiple shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' To test this, we set up a large number of simulations (for both power-law and cored cases) with a variety of different shock histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For each of these simulations we apply a shock approximately every ten dynamical time-scales 𝑡𝐵 and we use up to 5 shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We always compute the annihilation rate when 10𝑡𝐵 have passed since the last shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We note that the value of 𝑡𝐵 is not very clearly defined in the case with different shock amplitudes, we therefore choose a reference value 𝐵ref to define a 𝑡𝐵 that seems appropriate for each individual history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We have checked that this shock interval is large enough to ensure that the results would not change by slightly decreasing it or by increasing it further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We list all shock histories in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We have created several manually chosen shock histories (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' equal shock strengths, descending shocks, ascending shocks) and we have also sampled a few shock histories from the full distribution of shock histories with 𝐵∗ = 𝐵ref, but only keeping the 5 strongest shocks – see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We sorted some of these histories accidentally, but we also created an additional set of four cored simulations with unsorted shock histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, neither the annihilation rate nor the transfer functions depend much on the order in which shocks happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We present transfer functions for power-law simulations with mul- tiple encounters in Appendix A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Most importantly we note that cases with multiple encounters are still well approximated by the transfer function of equation (43), but with different cut-off radii and we note that for the final transfer function, the order of shocks does not matter, only their amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' These results are both consistent with findings in a previous study of NFW haloes (Delos 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We show the annihilation rates of cusps that have gone through multiple shocks in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The data points in this Figure are ob- tained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For power-law cases, we compute the annihilation rate by first fitting a transfer function to each encounter individually Type 𝐵ref 𝐵1 𝐵2 𝐵3 𝐵4 𝐵5 𝐵eff power-law 1 1 1 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 power-law 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='25 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 power-law 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 1 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 power-law 𝐵∗, S 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='4 power-law 𝐵∗, S 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1 cored 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='00 1 1 1 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='8 cored 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='25 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 cored 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 1 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 cored 𝐵∗, S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='4 cored 𝐵∗, S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1 cored 𝐵∗, S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='11 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='43 10 cored 𝐵∗, S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='97 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='42 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 cored 𝐵∗, U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='34 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='7 cored 𝐵∗, U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='65 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='6 cored 𝐵∗, U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='3 cored 𝐵∗, U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='23 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The different shock histories simulated for the multiple encounters scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 𝐵ref is given in units of 𝐵core and all other shocks parameters are given in units of 𝐵ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 𝐵1 - 𝐵5 indicate the strength of subsequent shocks and 𝐵eff is the final value of the effective shock parameter, defined in equation (49) so that the effect of the whole shock history is equivalent to a single shock with this parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Shock histories that were sampled from the distribution of histories are indicated by a 𝐵∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Those which were accidentally sorted in descending order are marked with an ’S’ whereas unsorted ones are indicated by a ’U’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 10 3 10 2 10 1 100 101 J/(4 A2) Fit Cored Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Powerlaw Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Powerlaw T(r) Disruption 10 2 10 1 100 Beff/Bcore 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='25 J/J 0 1 2 3 4 5 Encounters Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Annihilation rates for power-law and cored profiles after multiple stellar encounters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The encounter histories are summarized through a single effective parameter 𝐵eff so that their annihilation rate is approximately equiv- alent to a single shock with 𝐵eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The blue line shows our previous fit for single encounters, and the bottom panel shows residuals with respect to this fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The effect of multiple encounters is captured to within 20% through a single shock with the effective shock parameter 𝐵eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' using equation (43) and then calculating the annihilation rate accord- ing to equation (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Here we rescale the power-law results to two different core radii 𝑟core = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='05 and 𝑟core = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 so that each power- law simulation appears twice in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We note again that these simulations could be rescaled to be anywhere along the black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For cored profiles we calculate the annihilation rates as explained in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For the 𝐵-axis we calculate an effective shock parameter which summarizes the whole history of encounters of a cusp through a single number, 𝐵eff = 𝑝√︃∑︁ 𝐵𝑝 𝑖 , (49) which is the 𝑝-norm of the shock history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Different values of 𝑝 would MNRAS 000, 1–22 (2015) Prompt cusps & stellar encounters 13 give different importance to stronger versus weaker shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For 𝑝 = 1 the value of 𝐵eff would correspond to the sum of all shock parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For 𝑝 < 1 multiple shocks would have an enhanced effect, and for 𝑝 → ∞ only the strongest shock would matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For reference, we show, how such cases would appear in the Appendix A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, we have found that 𝑝 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 gives excellent predictions, and this is the value of 𝑝 that we use for calculating the 𝐵-values in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We note that some previous studies (of NFW subhaloes) have assumed that multiple shocks can be treated by adding the changes in binding energy (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This would imply 𝑝 = 2 and would give clearly wrong results for the case of prompt cusps and probably also for NFW haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The results of Delos (2019b) indicate that multiple shocks have also a significantly enhanced effect (𝑝 ≪ 2) for NFW haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, it is not clear that our effective description will work equally well for NFW profiles, due to their more complicated form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The blue line in 11 shows the prediction obtained by treating the shock histories as a single shock with effective shock parameter 𝐵eff and inserting this into Equation (47) This predicts the annihilation rates correctly to within 20%6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We show in Appendix A4 that this is much more accurate than results that would be obtained by only considering the strongest shock 𝑝 → ∞ (errors up to factor of a few) or by considering the sum of shocks as the effective shock parameter (errors up to 50%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 Summary of the effect of stellar encounters We conclude that we can estimate the annihilation rate of cored power-law profiles that have gone through complicated shock histo- ries simply by using equation (47) with an effective shock parameter calculated as the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2-norm of all the shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' To account for the ini- tial boundary of cusps when 𝐵 ≪ 𝐵cusp we additionally use the boundary term from equation (44) so that we have 𝐽 = 4𝜋𝐴2𝛽−1 � − Ei � −2𝛼𝑎(𝑥core + 𝑏𝑥3 core)4/3𝛽� + Ei � −2𝛼𝑥4𝛽/3 cusp � + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='531𝛽 − log(𝑎) � , (50) where we have defined 𝑥cusp = 𝐵 𝐵cusp = � 𝑟𝐵 𝑟cusp �3/4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (51) This is the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Note that this function works accurately in all regimes – it recovers the correct suppression for 𝐵 ≫ 𝐵cusp, but it also recovers equation (10) for weak shocks with 𝐵 ≪ 𝐵cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We note that the results we find here suggest that the impact of stellar encounters on the annihilation rate will be quite dramatic for cusps in the inner part of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We show in Figure 12 the distribution of the reduction in annihilation luminosity relative to the initial luminosity as a function of 𝐵∗, the characteristic shock strength of a cusp’s trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Here we assume 𝐵core = 100𝐵cusp, which is a typical ratio between these two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We sample a large number of shock histories, considering all shocks with 𝐵 > 10−3𝐵∗ and evaluate the expected luminosities according to (50) using the effective shock parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For comparison, we show also the lumi- nosities that would be obtained by considering only the strongest 6 Only the last two data points have an error larger than this, but these points also have the largest systematic uncertainty, since they are so close to disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We would have needed to take additional care by choosing later evaluation times to get more precise estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 10 4 10 3 10 2 10 1 100 B * /Bcore 10 3 10 2 10 1 100 J/J0 median only strongest shock Bcusp, Bcore 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='7% region 95% region 68% region Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Reduction in annihilation luminosity due to encounters along a trajectory with a characteristic shock scale 𝐵∗ for a cusp with 𝐵core = 100𝐵cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The shaded regions show our predictions for the distribution of luminosities using the effective shock parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Dotted lines show the results that would have obtained by considering only the strongest shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' These have a slight offset in color so that they can still be seen where they overlap with the shaded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Clearly the effect of shocks with 𝐵 ≳ 𝐵cusp is quite dramatic, and when 𝐵∗ gets close to the core scale, complete disruption is expected in virtually all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Virtually all cusps with 𝐵∗ ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='3𝐵core get completely dis- rupted and cusps with 𝐵∗ ≳ 𝐵cusp are already dramatically affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' As expected (compare also Appendix A4) there is a significant differ- ence between considering only the strongest shock and considering the full history of shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, even when considering only the strongest shock the suppression is quite strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='6 The effect of smooth tides In addition to stellar encounters, the smooth tidal field of the Milky Way can also induce mass-loss, and so a reduction in annihilation luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We do not discuss the effect of smooth tides in great detail here, since this effect was already adequately incorporated by Delos & White (2022b), based on the work of Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, since the effect of smooth tides should be included in addition to that of stellar encounters, we need to make a few modifications to their approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We will discuss these modifications in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In summary, we include the effect of smooth tides by applying the adiabatic-tides model (Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2022) to the cusps that remain after stellar shock truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We find that the joint effect on annihilation radiation of a stellar shock with effective strength 𝐵eff and the smooth tidal field is approximately equivalent to a pure shock with an effective shock parameter, 𝐵eff,𝜆 = √︃ 𝐵2 eff + 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2𝜆, (52) where 𝜆 is the largest eigenvalue of the tidal tensor of the spherically averaged Galactic mass distribution at the pericentre of the cusp’s orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In this way we are able to incorporate the effect of smooth tides simply through a redefinition of the shock parameter used in equa- tion (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In Figure 13 we compare the effective shock parameter from the full history of shocks to the tidal contribution 𝐵𝜆 = √ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2𝜆 and to the total effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' At radii 𝑟 ≤ 20 kpc the effect of encounters dominates, whereas at larger radii the effect of the smooth tide dom- inates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Thus, viewed from the Earth’s position just 8 kpc from the MNRAS 000, 1–22 (2015) 14 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 100 101 102 r [kpc] 10 2 10 1 100 101 102 103 104 Beff [km/s/pc] Bcusp, Bcore Beff [stars] Beff [smooth tide] Beff [total] Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' A comparison between the relative importance of the smooth tidal field and of tidal shocks from stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The shaded regions indicate the 68% regions of the distributions, and the solid lines their medians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For cusps that reach the vicinity of the disk, 𝑟 ≤ 20 kpc, encounters are dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In the outskirts of the Milky Way halo the effect of smooth tides dominates, but is too weak to affect the luminosity of most cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Galactic centre, truncation by stellar encounters has a large effect on the angular distribution of the prompt cusp annihilation signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This must be taken into account when evaluating whether these cusps af- fect interpretation of the Galactic Center Excess measured by Fermi LAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' On the other hand, stellar encounters have much less effect on the radiation seen by a distant observer since most of the mass of the Milky Way’s dark matter halo (and so most of its prompt cusps) are at larger radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 5 RESULTS We combine results from Section 3, on the distribution of stellar shock parameters, and from Section 4, on the effect of shocks and smooth tides on prompt cusps, to estimate the spatial distribution of the annihilation signal of cusps in the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For this we assume the cusp population expected for a WIMP with mass 100 GeV and decoupling temperature 30 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This leads to val- ues 𝐵core and 𝐵cusp through equations (22) and to initial annihilation rates through equation (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We use the 105 orbits and the final 𝐵∗ values inferred in Section 3, but we create a different realization of a cusp and its shock history for each of 1000 different radii sampled uniformly in time along each orbit’s trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For the shock history we consider all shocks stronger than 𝐵 > 10−4𝐵∗ (approximately the 10000 strongest shocks) when evaluating the effective shock param- eter 𝐵eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Additionally we keep track of the smallest radius 𝑟peri that each cusp has reached and we evaluate the tidal field of the spher- ically averaged mass distribution at this point to obtain the value of 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' With this we infer the effective shock parameter 𝐵eff,𝜆 as in equation (52) and evaluate the expected final annihilation luminosity according to equation (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Thus, in total we obtain 108 pairs of radii and annihilation luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We show the distribution of the ratio between initial and final luminosities in the top panel of Figure 14 as a function of radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The percentiles of this distribution give an idea of how dramatic the effect of stellar encounters on cusps is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Typical cusps inside of the central 10 kpc are disrupted by stellar encounters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Within the central 3 kpc less than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5% of cusps survive, and almost all cusps reduce 100 101 102 r [kpc] 10 3 10 2 10 1 100 J/J0 J / J0 (enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' + tide) J / J0 (tide only) J / J0 (only Bmax) Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The distribution of the reduction in annihilation radiation from prompt cusps as a function of their current Galactocentric radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Shaded regions indicate percentiles of the full distribution considering the joint effect of all encounters (colours as in Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The mean (red line) gives the resulting effective reduction of the contribution to the annihilation profile from prompt cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The blue dashed line shows the mean reduction if only the smooth tide is considered, while the orange dashed line shows the reduction caused by the strongest shock alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Clearly, stellar encounters have a dramatic effect on the expected annihilation radiation from cusps in the inner Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' their luminosities by dramatic factors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='7% of cusps reduce their luminosity at least by a factor 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, more relevant than the percentiles of the distribution is the annihilation weighted mean, since this gives the ratio between the initial annihilation profile and the final one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We show this as the red line in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The mean is clearly dominated by the least disrupted cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This is so, since cusps that contribute more annihilation radiation are also more resilient to tides (compare Figure 3) and further, since the bulk of the distribution gets completely disrupted, only the most resilient cusps with the least invasive shock histories contribute to the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Even so, the mean is dramatically suppressed with respect to the case where encounters are neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' At 8 kpc the average luminosity is already suppressed by a factor of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' At smaller radii the effect is even more dramatic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' only about 10−3 of the luminosity expected in the absence of encounters remains near the galactic bulge at about 1 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We show as a blue dashed line the mean annihilation reduction that would be obtained if only the effect of the smooth tide were considered (as in Delos & White 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This greatly over-predicts the luminosity in the central regions, but is accurate at larger radii 𝑟 ≳ 15 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The orange dashed line shows the mean if only the strongest shock is considered instead of the full history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This leads to significantly smaller, although still substantial, suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In Figure 15 we show the annihilation profile as a function of radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Unlike the prediction when only the effect of smooth tides is considered, stellar encounters lead to a non-monotonic profile that decreases towards the centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Its maximum is slightly outside the solar radius at around 10 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' At larger radii the effect of stellar encounters becomes irrelevant so that the original profile is recovered at 𝑟 ≳ 40 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Let us briefly consider, how these effects alter the annihilation luminosity of the Milky Way, as seen by a distant observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For this, we simply have to sum up the luminosity of all cusps out to some truncation radius which we assume to be 𝑟200b = 340 kpc (the radius where the enclosed mean density is 200 times the background MNRAS 000, 1–22 (2015) Prompt cusps & stellar encounters 15 100 101 102 r [kpc] 10 7 10 5 10 3 10 1 101 dJ/dV [M2 /pc3/pc3] Smooth halo cusps (unperturbed) cusps (tides) cusps (enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' + tides) Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The radial distribution of annihilation radiation from cusps in the Milky Way including the effects of both stellar encounters and the mean tide (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For reference, the black line shows the radiation from the smooth halo, the green line the cusp contribution ignoring all disruptive effects, and the blue dashed line including only the effect of smooth tides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' When all effects are included the cusp contribution peaks at around 10 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' initial only tides only enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' tides + enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 𝐽-factor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='32e+11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='90e+11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='02e+11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='79e+11 fraction 100% 82% 87% 77% Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Total dark matter annihilation luminosity of the Milky Way when considering different tidal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 𝐽-factors are in units of M⊙pc−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' These numbers are proportional to the luminosity of the Milky Way as seen by a distant observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Modelling tidal stripping or stellar encounters has only moderate effects on the total luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Additionally we add the contribution of the smooth halo, which is however very small (∼ 2%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We list the resulting total luminosities in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Clearly, the joint effect of tidal stripping and stellar encounters onto the total luminosity is relatively small, only reducing the estimated luminosity by 23%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Stellar encounters do not much affect the total luminosity, since most of the cusps do not get close to the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' As a result, it is not necessary to consider stellar and tidal disruption effects on cusps when inferring the total contribution of extragalactic sources to the IGRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, we observe the dark matter distribution of our own Galaxy from a highly biased view- point which greatly increases the sensitivity to what happens in its inner regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We show in Figure 16 the radiation profile of the Milky Way as it would be observed from the solar radius𝑟 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 kpc if dark matter has a significant self-annihilation cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Here we have inferred for each component the line-of-sight J-factor column-densities, 𝑛𝐽 = ∫ ∞ 0 𝜌2 d𝑙, (53) and then multiplied them by a normalization factor, dΦ𝐸 dΩ = 𝑛𝐽 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='44 × 10−5 MeVcm−2sr−1s−1pc5M−2 ⊙ , (54) that makes the smooth halo profile agree with the Galactic Centre excess (this is the same factor that Delos & White (2022b) have used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Additionally we have overplotted the data points of the observed GCE as given by Di Mauro (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Many error bars are too small to see and 100 101 102 (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=') 10 4 10 3 10 2 10 1 d E d [MeV cm 2 s 1 sr 1] Smooth halo cusps (unperturbed) cusps (tide only) cusps (enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' + tide) total Di Mauro (2021) Ackermann e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (2015) sky average Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Left: The annihilation flux as a function of Galactocentric angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' When including the effect of stellar encounters the angular dependence of the prompt cusp contribution vanishes almost completely, resulting in a total signal which is compatible with the observed Galactic Centre excess (Di Mauro 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Right: The sky average is significantly reduced due to stellar encounters, but cusps still contribute a significant and potentially detectable fraction of the isotropic 𝛾-ray background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' component 𝑛𝐽 Flux Fraction of IGRB smooth halo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='3e+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='9e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='7% cusps (no tides) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='6e+01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='8e-04 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='8% cusps (tides) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5e+01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2e-04 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='3% cusps (tides + enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=') 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='6e+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1e-04 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='8% cusps (extragal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2e+01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='7e-04 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0% total DM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1e+01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0e-04 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5% total DM (previous) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='8e+01 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1e-04 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0% Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Average J-factor column densities (in units of M2 ⊙pc−5) and sky- averaged fluxes of 𝛾-ray radiation (in units of MeV cm−2sr−1s−1) for different components when normalizing fluxes so that the Galactic Centre excess would be explained by the smooth halo signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Fluxes have been averaged over the sky with 𝜃 > 20◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' If the GCE is due to dark matter, annihilation radiation should constitute about 43% of the isotropic 𝛾-ray background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' some data points are only upper limits indicated by arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='7 Stellar encounters affect the profile so strongly that there is only a very small variation with observation angle left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' While the unperturbed profile and the profile including tides alone both seem in tension with the observed shape of the GCE, stellar encounters alleviate this tension and the shape of the GCE is again compatible with dark matter as an explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The signal from cusps in the Galactic halo makes an almost isotropic contribution – only deviating by about 30% from its sky-average at its brightest angle (40◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' When making conclusions from the shape of the signal profile, including the effects of stellar encounters is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, if the GCE is due to dark matter we still expect a signif- icant contribution to the 𝛾-ray background from prompt cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We list the sky-averaged flux that we predict for each component in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' It is most interesting to compare these numbers to the observed isotropic 𝛾-ray background (IGRB) dΦIGRB dΩ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='9+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='8 × 10−4 MeVcm−2sr−1s−1 (55) where the error indicates the systematic uncertainty due to foreground 7 Our halo profile does not fit the GCE perfectly for the innermost angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We did not fit the profile to the GCE, but simply use the one that we have also used for orbital modelling in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' MNRAS 000, 1–22 (2015) 16 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' modelling which we have assumed to accumulate linearly when in- tegrating the spectrum as measured by Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The observed IGRB only considers contributions from galactic latitudes 𝑏 > 20◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' To approximately mimic this, we have only included con- tributions from the Galactocentric angles larger than 𝜃 > 20◦ in the sky-averages listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Additionally we have indicated what fraction of the IGRB each component would comprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Finally, we have also listed the extra-galactic flux here which we estimate in the same manner as Delos & White (2022b) – neglecting the effect from tidal fields, which might reduce the number by around 20% as indicated by Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In comparison to the smooth tide prediction (equivalent to Delos & White 2022b), the predicted background signal due to cusps in our own Milky Way goes down by a factor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 and the signal is by a factor 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 smaller than the unperturbed one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' After taking this correction into account, the total dark matter signal (smooth halo + Milky Way cusps + extra-galactic cusps) is dominated by the extra- galactic signal and goes down by roughly one third.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Delos & White (2022b) argue that the morphology of the signal that is expected from annihilation from prompt cusps and from dark matter decay are essentially the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Therefore, they use constraints on the de- cay of dark matter from Blanco & Hooper (2019) and rescale them to obtain constraints on the dark matter annihilation cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' These constraints are proportional to the predicted background, thus the upper limit on the cross-section has to be increased by about one third.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, we note that these constraints depend quite strongly on how well the astrophysical contributions to the diffuse 𝛾-ray back- ground are understood (Blanco & Hooper 2019) and the assessment of uncertainties has not been very rigorous so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For example, the constraints by Blanco & Hooper (2019) vary by a factor of about 4 simply by considering different assumptions about how correlated the error-bars are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Therefore, to infer reliable constraints it will nec- essary to reanalyse the IGRB with a more sophisticated treatment of the systematic and statistical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Perhaps even more intriguing than the implications for constrain- ing dark matter, the predicted contribution to the IGRB offers an independent test for the dark matter interpretation of the Galactic Centre excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' If the GCE is due to annihilation, then we predict an additional approximately isotropic 𝛾-ray signal that would comprise about 40% of the observed 1 to 10 GeV background for a 100 GeV WIMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' If we additionally consider that different WIMP models can lead to a factor ∼ 2 variation in predicted cusp J-factors (consider Delos & White 2022b, Figure 7 and Equation 2), the total dark mat- ter contribution should range between 20% and 80% of the observed IGRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Given current uncertainties in the (apparently dominant) con- tribution of star-forming galaxies and AGN to the observed signal, it is unclear, whether there is room for such a component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' A careful reevaluation of these other contributions to the 𝛾-ray background is clearly well motivated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Our work here can be used to infer templates for the spatial and spectral shapes of the predicted annihilation signal from prompt cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The detection or exclusion of this additional component would confirm or contradict the dark matter interpretation of the GCE, and so substantially advance our understanding of dark matter itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Firm exclusion could rule out an annihilation interpretation of the GCE, while robust detection would strongly support this interpretation, since it would be a remarkable coincidence to find the GCE and the additional IGRB contribution at the right relative level yet due to different astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 6 DISCUSSION In this article we have modeled the effect of stellar encounters on the expected dark matter annihilation signal from prompt cusps orbiting within the Milky Way’s halo, presenting several advances in the treatment of such encounters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Firstly, we have developed a new method for inferring the full history of the impulsive shocks experienced by a dark substructure as it moves through the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This method is both simpler and more general than previous approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' While we have focused on prompt cusps in this article, our results on the distribution of shock histories (Sections 2 and 3) could be applied to conventional NFW subhaloes also, as long as the dominant shocks are in the distant-encounter regime (masses ≲ 1𝑀⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Secondly, we have performed idealized N-body simulations to in- fer how stellar encounters affect prompt cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Here, we have for the first time considered the phase-space core that any (otherwise) centrally divergent profile must exhibit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Only when this core is con- sidered can a prompt cusp disrupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Our simulations allow us to derive accurate formula to describe the structural effect of encounters with arbitrary combinations of shock history, core radius, truncation ra- dius, characteristic density and smooth tidal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' As a result, we are able to account for the joint effect of smooth tides and of any number of stellar encounters along the entire trajectory of any cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' While our model is relatively complete and comprehensive, a few of its assumptions could still be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We have assumed a static Milky Way potential, and simply integrated cusp orbits for 10 Gyr within it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' A more accurate treatment would consider evolution of the host potential and of the stellar population it contains, and would follow cusps from their initial formation and growth through their accretion onto precursor objects, and finally onto the Milky Way itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' While the early stages of this process remain quite uncertain, we believe that our procedure should be relatively accurate and should be conservative for effects at later times, since the great majority of Milky Way stars formed (approximately) in situ in the disc and the bulge, and most of them are less than 10 Gyr old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Another uncertain point is the precise profile of the phase-space core of prompt cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Here, we used a simple heuristic approach to obtain a stable profile that is consistent with the phase-space density constraint, but a rigorous investigation of the central profiles with N- body simulations would be desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, we expect the results of our study to be quite robust to this uncertainty, since annihilation rates depend relatively weakly on the precise shape of the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' When we apply our modeling to the prompt cusp population in the Milky Way’s halo, we find that stellar encounters have a dra- matic effect on any cusps that enter the star-dominated regions – the vast majority get disrupted within the central 10 kpc, leading to a radial annihilation radiation profile that peaks near 10 kpc, dropping strongly at smaller radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' While this has little effect on the lumi- nosity of the Milky Way as seen by a distant observer (≲ 20%), it strongly affects the annihilation flux observable from Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Stellar encounters destroy cusps so efficiently in the inner Galactic halo, that the surface brightness of cusp annihilation radiation is predicted to vary only slightly between the centre and anticentre directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' As a result, it has no noticeable effect on the Galactic Centre Excess, which therefore remains compatible with production by annihilation of the smooth dark matter distribution in the inner few kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This removes the issue raised by Delos & White (2022b) who included cusp disruption due to the smooth Galactic tide but not that due to stellar encounters, and hence found a total annihilation profile ap- parently incompatible with the GCE profile of Di Mauro (2021, see Figure 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' MNRAS 000, 1–22 (2015) Prompt cusps & stellar encounters 17 This opens up an intriguing new possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' If the GCE is in- deed due to dark matter annihilation, this implies an approximately isotropic annihilation signal from prompt cusps that would have an amplitude in the range 20% − 80% of the observed isotropic 𝛾-ray background, depending on the mass and decoupling temperature of the WIMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The results of Blanco & Hooper (2019) suggest that a signal of this amplitude is inconsistent with the observed IGRB, since the latter appears to be explained entirely by emission from star-forming galaxies and AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We think, however, that our current results warrant a careful reevaluation of those of Blanco & Hooper (2019) since it seems conceivable that some of the fitted templates might absorb a near-isotropic dark matter annihilation signal, or that the statistical treatment of systematic errors is not yet robust enough to exclude such a signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' If such a signal is firmly ruled out, it becomes unlikely that the GCE can be ascribed to annihilation radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' On the other hand, if it were robustly detected, this would confirm the annihilation interpretation of the GCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' ACKNOWLEDGEMENTS JS thanks Sten Delos for answering quickly and clearly all questions regarding the distribution and properties of cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' JS thanks Mattia Di Mauro for providing data related to the GCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' JS and RA thank all members of the cosmology group at Donostia International Physics Center for daily discussions and for the motivating research environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' JS and RA acknowledge the support of the European Research Council through grant number ERC-StG/716151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' GO was supported by the National Key Research and Development Program of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2022FYA1602903) and the Fundamental Research Fund for Chinese Central Universities (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 226-2022-00216).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' DATA AVAILABILITY The code used to generate all results of this study other than the simulation-based analysis of Section 4 is available in an on- line repository under https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='com/jstuecker/cusp-encounters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Some results are based on the adiabatic-tides code which is also publicly available under https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='com/jstuecker/adiabatic-tides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The simulation data from Section 4 will be shared on reasonable re- quest to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' REFERENCES Ackermann M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2015, ApJ, 799, 86 Aguilar L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', White S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 1985, ApJ, 295, 374 Aguirre-Santaella A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Sánchez-Conde M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Ogiya G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Stücker J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Angulo R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='08652 Amorisco N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2021, Cold dark matter subhaloes at arbitrarily low masses (arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='01148) Anderhalden D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Diemand J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2013, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2013, 009 Angulo R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Hahn O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Ludlow A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Bonoli S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2017, MNRAS, 471, 4687 Angus G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Zhao H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2007, MNRAS, 375, 1146 Arcadi G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Dutra M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Ghosh P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Lindner M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Mambrini Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Pierre M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Profumo S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Queiroz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2018, European Physical Journal C, 78, 203 Bardeen J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Bond J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Kaiser N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Szalay A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 1986, ApJ, 304, 15 Bertschinger E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2006, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' D, 74, 063509 Binney J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Tremaine S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2008, Galactic Dynamics: Second Edition Blanco C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Hooper D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2019, Journal of Cosmology and Astroparticle Physics, 2019, 019 Bland-Hawthorn J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Gerhard O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2016, ARA&A, 54, 529 Blas D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Lesgourgues J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Tram T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2011, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2011, 034 Cautun M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2020, Monthly Notices of the Royal Astronomical Society, 494, 4291 Colombi S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2021, A&A, 647, A66 Delos M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2019a, Physical Review D, 100 Delos M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2019b, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' D, 100, 083529 Delos M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', White S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2022a, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='05082 Delos M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', White S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2022b, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='11237 Delos M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Erickcek A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Bailey A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Alvarez M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2018a, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' D, 97, 041303 Delos M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Erickcek A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Bailey A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Alvarez M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2018b, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' D, 98, 063527 Delos M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Bruff M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Erickcek A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2019, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' D, 100, 023523 Di Mauro M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2021, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' D, 103, 063029 Diemand J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Moore B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Stadel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2005, Nature, 433, 389 Eddington A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 1916, MNRAS, 76, 572 Errani R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Navarro J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2021, Monthly Notices of the Royal Astronomical Society, 505, 18–32 Errani R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Peñarrubia J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2020, MNRAS, 491, 4591 Errani R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Navarro J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Peñarrubia J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Famaey B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Ibata R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2022, MNRAS, Goerdt T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Gnedin O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Moore B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Diemand J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Stadel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2007, MNRAS, 375, 191 Grand R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', White S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2022, Monthly Notices of the Royal Astro- nomical Society: Letters, 511, L55 Green A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Goodwin S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2007, MNRAS, 375, 1111 Hernquist L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 1990, ApJ, 356, 359 Hooper D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Goodenough L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2011, Physics Letters B, 697, 412 Hu W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Sugiyama N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 1996, The Astrophysical Journal, 471, 542 Ishiyama T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2014, ApJ, 788, 27 Ishiyama T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Makino J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Ebisuzaki T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2010, ApJ, 723, L195 Kavanagh B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Edwards T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Visinelli L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Weniger C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2021, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' D, 104, 063038 Kelley T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Bullock J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Garrison-Kimmel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Boylan-Kolchin M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Pawlowski M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Graus A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2019, MNRAS, 487, 4409 Macciò A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Paduroiu S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Anderhalden D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Schneider A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Moore B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2012, Monthly Notices of the Royal Astronomical Society, 424, 1105 McMillan P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2017, MNRAS, 465, 76 Miyamoto M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Nagai R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 1975, PASJ, 27, 533 Navarro J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Frenk C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', White S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 1996, ApJ, 462, 563 Ogiya G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Hahn O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2018, MNRAS, 473, 4339 Polisensky E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Ricotti M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2015, MNRAS, 450, 2172 Roszkowski L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Sessolo E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Trojanowski S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2018, Reports on Progress in Physics, 81, 066201 Schneider A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Krauss L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Moore B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2010, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' D, 82, 063525 Sellwood J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', McGaugh S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2005, ApJ, 634, 70 Shen X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Xiao H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Hopkins P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Zurek K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='11276 Sheth R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Mo H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Tormen G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2001, Monthly Notices of the Royal Astronomical Society, 323, 1 Slatyer T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2021, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='02696 Smith R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Flynn C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Candlish G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Fellhauer M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Gibson B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2015, MNRAS, 448, 2934 Stücker J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Angulo R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Busch P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2021, MNRAS, 508, 5196 Stücker J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Ogiya G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Angulo R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Aguirre-Santaella A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Sánchez-Conde M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='00604 Tremaine S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Gunn J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 1979, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 42, 407 van den Bosch F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Ogiya G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Hahn O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', Burkert A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=', 2018, MNRAS, 474, 3043 APPENDIX A: NUMERICAL CONVERGENCE A1 Non-uniform mass initial conditions To simulate the behaviour of our cusps we have to truncate them at some radius 𝑟max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We first tried to run numerical experiments with uniform mass sampling and a sharp truncation radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' With particle MNRAS 000, 1–22 (2015) 18 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' numbers of order 106 one can typically resolve scales down to 10−3- 10−2 𝑟max before two-body relaxation effects become substantial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, at the same time a sharply truncated 𝑟−3/2 power-law is far from equilibrium around the truncation radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We found that trunca- tion still has significant numerical effects below 10−1𝑟max (depend- ing on how many dynamical times are simulated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This behaviour is much worse than for profiles that are steeper in their outer regions, such as Hernquist or NFW profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' As a result, only a small range of radii is reliably resolved and a lot of care has to be taken to get numerically converged results over a substantial radial range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Un- fortunately, the transfer function we wish to measure spans several orders of magnitude in radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We have therefore decided to abolish the uniform mass sampling and instead divide the profile into several populations of particles with differing mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This allows us to get reliably converged results over a dynamical range of 104 in radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We define five populations of particles where the 𝑖th population is defined so that all of its particles have their orbital pericenters in the range, 𝑟peri ∈ [𝑟split,i, 𝑟split,i+1], (A1) and we choose 𝒓split = (0, 1, 10, 100, 1000, 10000)𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In each interval we choose to use the same number of particles 𝑁split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In principle one can create such a realisation by setting up several full profiles at different resolutions and then discarding all particles that do not fulfill the criterion of each population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, this would be very inefficient for generating the innermost popula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Instead, we can directly sample particles from the cumulative en- ergy, pericentre distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The density contribution at radius 𝑟 from particles which have pericentres in a radial range 𝑟1 < 𝑟peri < 𝑟2 and energy smaller than 𝐸 is given by 𝜌(𝑟,𝑟1 < 𝑟peri < 𝑟2, < 𝐸) = 4𝜋 ∫ 𝐸 𝜙∗ ∫ 𝐿2 𝐿1 𝐿 𝑓 (𝐸) 𝑟2 √︃ 2𝐸 − 2𝜙 − 𝐿2 𝑟2 d𝐿 d𝐸, (A2) where the boundaries 𝐿1 and 𝐿2 are chosen so that the integral goes only over particles which fulfill the pericentre criterion: 𝐿1 = max(r1 √︁ 2(E − 𝜙(r1)), 0), (A3) 𝐿2 = min(r2 √︁ 2(E − 𝜙(r2)), r √︁ 2(E − 𝜙(r))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (A4) We find that 𝜌(𝑟,𝑟1 < 𝑟peri < 𝑟2, < 𝐸) = 𝐹(𝐸, 𝜙∗1) √︄ 1 − 𝑟2 1 𝑟2 − 𝐹(𝐸, 𝜙∗2) √︄ 1 − 𝑟2 2 𝑟2 , (A5) 𝜙∗1(𝑟) = 𝜙(𝑟) − 𝜙(𝑟1) 𝑟2 1 𝑟2 1 − 𝑟2 1 𝑟2 , (A6) 𝜙∗2(𝑟) = 𝜙(𝑟) − 𝜙(𝑟2) 𝑟2 2 𝑟2 1 − 𝑟2 2 𝑟2 , (A7) with 𝐹(𝐸, 𝜙) = 8 √ 2𝜋 𝑓0 (𝐸 − 𝜙) 3 2 105 (𝐸 + 𝐸𝑐) 7 2 (𝐸𝑐 + 𝜙)3 � 8𝐸2 + 28𝐸𝐸𝑐 + 12𝐸𝜙 +35𝐸2 𝑐 + 42𝐸𝑐𝜙 + 15𝜙2� , (A8) where the terms containing 𝑟1 have to be set to zero if 𝑟 < 𝑟1 and the terms containing 𝑟2 have to be set to zero for 𝑟 < 𝑟2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Knowledge of this density function is enough to sample radii and energies of particles in each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Radii can be sampled through inverse-distribution function sampling with the cumulative mass pro- file that can be obtained by integrating 𝜌 and using the limit 𝐸 → ∞, but maintaining a finite pericentre range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Energies can be sampled by inverse-distribution function sampling among energies using the normalized version of 𝜌(𝑟, 𝑟1 < 𝑟peri < 𝑟2, < 𝐸).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Finally, angu- lar momenta can be sampled by considering the cumulative angular momentum distribution function, 𝐹(< 𝐿|𝑟, 𝐸) = �√︃ 2𝐸 − 2𝜙 − 𝐿2/𝑟2 � 𝐿2 𝐿1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (A9) An implementation of this scheme for creating initial conditions for cored cusps is also publicly available in the code repository of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We show the radial density profiles of the initial conditions that have been created in this manner for a profile with 𝑟core = 1 and 𝑁split = 218 as solid lines in Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The individual components are shown as coloured lines and the sum is shown as the black lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Clearly this gives an excellent initial representation of the density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The dashed lines in the same Figure show the density profile after the simulation has been evolved for 100 times the dynamical time-scale at the core radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The final profile still agrees excellently with the initial one and we can see that particles of different masses have mixed very little.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Note that the particle number here is still a factor 4 − 16 lower than for the simulations that we actually use in the main text so that we can be fairly confident that all simulations are stable and well converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' A2 Time evolution and convergence of shocked power-law profiles We briefly discuss the convergence aspects of the shocked power-law profiles here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We show the profile of the 𝑟𝐵 = 20 simulation with 𝑁split = 222 at different output times as solid lines in Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The profile is very stable with time over large radial ranges, but there are differences at large radii where at early times the profile has not yet had enough time to relax to its final form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We find that a conservative criterion for the largest radius 𝑟max where the profile has relaxed to its final form is given by 𝑡sim = 10𝑡dyn(𝑟max) = 10 √︄ 𝑟3max 𝑀(< 𝑟max)𝐺 , (A10) where 𝑡sim is the run-time of the simulation and we use the mass profile at the output time (not the initial time) for 𝑀(< 𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We invert this equation numerically to find 𝑟max at each time and mark this point in Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Clearly this is a very conservative criterion and cuts out all parts of the profile that are yet to reach their final form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The innermost radius 𝑟min where we can trust the simulations is MNRAS 000, 1–22 (2015) Prompt cusps & stellar encounters 19 10 6 10 5 10 4 10 3 10 2 10 1 100 101 (r)/ max M = 1Mhr M = 34Mhr M = 962Mhr M = 20934Mhr M = 141945Mhr sum t = 0 t = 100tcore 10 1 100 101 102 103 104 r/rcore 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='4 (r)/ true(r) Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' A cored density profile that has been set up through our non- uniform mass sampling technique (top) and the ratio to the true profile (bot- tom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Black lines show the full profile whereas colored lines show the individ- ual components made of particles with different masses that were separated by their initial pericentre radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The legend indicates the particle mass of each population in units of the mass of the highest resolution particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Dashed lines show the profiles after evolving the simulation for 100 dynamical times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Clearly our non-uniform mass sampling creates very stable profiles over a large dynamical range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' given by the two-body relaxation radius which we can approximate through 𝑡sim = 𝑡relax(𝑟min) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1𝑁(< 𝑟min) log(𝑟0/𝜖) 𝑡dyn(𝑟min) (A11) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Binney & Tremaine 2008) where 𝑟0 is an estimate of the largest two-body encounter radius and 𝜖 is the softening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Since, the depen- dence on 𝑟0 is rather weak we just use 𝑟0 = 𝑟min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This formula only holds strictly for systems that are made of uniform mass particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Therefore, to be extra conservative, we use here not the actual par- ticle number here, but the particle number as if all mass inside of 𝑟min was made up from particles of the second highest resolution level (from the pericentre interval [1, 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Since most particles in the region are actually much lower mass (and none are higher mass), the profiles are actually converged at substantially smaller radii than the thus determined 𝑟min, which we indicate in Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Further, we show two lower resolution simulations with 4 and 16 times fewer particles to demonstrate that the simulations are actually very well 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 (r)/ 0(r) t = 40tB t = 20tB t = 10tB t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5tB t = 10tB, N/4 t = 10tB, N/16 10 1 100 101 102 103 r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 / ref 1 Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Convergence of the transfer function of a shocked power-law profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The top panel shows the density transfer functions, and the bottom panel the residuals with respect to the 𝑡 = 40𝑡B reference case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Different lines indicate different times and/or simulations with different particle numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The dots and triangles indicate the inner and outer convergence radii which are given on the inside by 𝑟min, a conservative estimate of the two-body relaxation radius, and on the outside by 𝑟max, the radius where about 10 dynamical times have passed since the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (The outer orange and red triangles are overlayed by the purple one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=') The simulations are very well converged inside this range and the inferred convergence radii are very conservative estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' converged in the indicated regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In Figure 8 we show only the reliable regions inside the range [𝑟min, 𝑟max].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' A3 Annihilation Rates We describe here how we compute the 𝐽-factors of the cored power- law profiles that we presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In many cases it is numerically problematic to directly infer annihilation rates from N- body simulations of centrally divergent profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This is so since annihilation rates scale as the density squared, and are therefore par- ticularly sensitive to the central regions of the profile which are, however, in general also the regions that suffer the most from numer- ical inaccuracies caused by two-body relaxation, sampling noise and the effects of force softening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, for the simulated cored profiles none of these are prob- lematic, since most of the annihilation radiation comes from radii 𝑟 ≳ 𝑟core which are resolved extremely well in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Therefore, we estimate the annihilation radiation in the following manner: We infer the radial density profile by assigning particles to 50 logarithmically spaced radial bins in the range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1 𝑟core, 104𝑟core] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 10 bins per dex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We then numerically integrate the 𝐽-factor by combining third-order spline interpolation with a large number of integration points which we place between 𝑟min = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2𝑟core and 𝑟max = 6𝑟𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Finally, we add the contribution from 𝑟 ≲ 𝑟min by as- suming that the density is uniform in this range with a value given by the average density within that radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We show the resulting annihi- lation rates as the blue line in Figure A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We then consider different variations of the numerical procedure: using a two times larger value of 𝑟min = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='4𝑟core;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' using a smaller value for 𝑟max = 3𝑟𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' using two MNRAS 000, 1–22 (2015) 20 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 10 2 10 1 100 J/(4 A2) base rrmin × 2 rmax/2 bins ×2 5tB earlier 10 1 100 B/Bcore 10 5 10 4 10 3 10 2 10 1 100 |1 J/Jbase| 3% Figure A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Convergence of the annihilation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The top panel shows the annihilation rates, whereas the bottom shows the difference relative to the reference case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Different lines show variations of numerical parameters with respect to the fiducial ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' All except the last data point have systematic errors well below 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' times the number of bins (20 per dex);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' and by evaluating the profile at a slightly different simulation time (5𝑡𝐵 earlier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Note that the last variation tests that the final profile is both stable and robust to shot noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We show these cases in Figure A3, together with the resid- uals with respect to the reference case in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Clearly none of these deviations has a significant impact on the annihilation radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In all cases the associated relative error is less than 3%, except for the case of the strongest shock considered where the error is significantly larger, but still less than 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' These results are more than accurate enough for the purposes of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' A4 Transfer functions and multiple encounters Here, we present the transfer functions obtained for power-law pro- files that have gone through up to 5 shocks in different sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Each shock is applied from a random direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We use the power- law simulations that are listed in Table 2 and show their transfer functions in Figure A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Additionally, we show the transfer function that is predicted by treating the full history as a single encounter with effective shock parameter 𝐵eff, given by the 𝑝 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 norm of the shock history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Clearly this effective shock parameter predicts the transfer function of multiple encounters reasonably well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Finally, we present alternative versions of Figure 11 that use differ- ent values of 𝑝 for calculating the norm of the shock history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The top panel of Figure A5 assumes 𝑝 = 1, so that 𝐵eff corresponds to the lin- ear sum of 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In this case the prediction (blue line) overestimates the reduction in annihilation radiation by up to 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The bottom panel adopts the value 𝑝 = 100 – effectively selecting only the strongest shock to define 𝐵eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Clearly, considering only the strongest shock can dramatically underpredict the reduction in annihilation luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For example, at 𝐵eff/𝐵core ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1 there are cases where the annihi- lation luminosity based on the full shock history is 10 times smaller than predicted from just the strongest encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 10 3 10 2 10 1 100 101 r/rB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='4 (r)/ 0(r) [1] [1,1] [1,1,1] [1,1,1,1] [1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='25] r × 10 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='25,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='] r × 10 his.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' A, r × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1 his.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' B, r × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1 Figure A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Transfer functions for power-law profiles after experiencing mul- tiple encounters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Dotted lines indicate predictions using an effective shock parameter defined as the 𝑝-norm with 𝑝 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' To avoid clutter, some lines have been offset by a factor 10 up or down in radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For most cases the labels indicate the shock histories and for the last two cases the shock histories can be found in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' APPENDIX B: THE EFFECT OF THE SMOOTH TIDAL FIELD While the main text of this paper discusss the effect of stellar encoun- ters in great detail, the smooth tidal field also affects the annihilation luminosity of prompt cusps orbiting in the Milky Way (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' De- los & White 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' As discussed in Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (2022), the most important parameter determining this effect is the largest eigenvalue 𝜆 of the tidal tensor at the orbital pericentre of a prompt cusp’s or- bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' After a sufficient amount of time (≳ 10 orbits) the orbiting cusp approaches an asymptotic structure which changes little in subse- quent evolution (Errani & Navarro 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (2022) show that the asymptotic remnant can be reasonably approximated by the adiabatic-tides model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This is an analytic description of an object’s reaction when a tidal field is applied very slowly (in the adiabatic limit) and in a spherical approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The corresponding code can be found online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='8 While actual N-body simulations in the Galac- tic context would of course be more accurate, the adiabatic-tides model allows easy exploration of a variety of different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We note that most studies of tidal stripping have focused on NFW subhaloes which are much less resilient to tides than prompt cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' As a result, most previously published results cannot be applied to the case of interest here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' For a pure power-law profile with slope −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 the truncation radius due to smooth tides scales with a characteristic length 𝑟𝜆, which may be defined by equating the cusp’s attractive force and the disruptive force due to the tidal field: 𝜆 · 𝑟𝜆 = ���� 𝜕𝜙 𝜕𝑟 (𝑟𝜆) ���� , (B1) 𝑟𝜆 = � 8𝜋𝐺𝐴 3𝜆 �2/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (B2) For a pure power-law initial profile, the final profile reaches zero density at the tidal radius 𝑟𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='24𝑟𝜆 (Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' If we set up a pure power-law profile and evaluate the structure of the rem- nant using the adiabatic-tides model, we find that its annihilation 8 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='com/jstuecker/adiabatic-tides MNRAS 000, 1–22 (2015) Prompt cusps & stellar encounters 21 10 3 10 2 10 1 100 101 J/(4 A2) p = 1 Fit Cored Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Powerlaw Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Powerlaw T(r) Disruption 10 2 10 1 100 Beff/Bcore 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='25 J/J 0 1 2 3 4 5 Encounters 10 3 10 2 10 1 100 101 J/(4 A2) p = 100 Fit Cored Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Powerlaw Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Powerlaw T(r) Disruption 10 2 10 1 100 Beff/Bcore 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='25 J/J 0 1 2 3 4 5 Encounters Figure A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Multiple encounter annihilation luminosity predictions using dif- ferent values of 𝑝 to calculate the 𝑝-norm of the shock history (for comparison with Figure 11) Top: 𝑝 = 1 corresponding to a linear sum of the shock pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This does not work well, giving annihilation rates off by more than 50% for shocks with 𝐵/𝐵core ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Bottom: 𝑝 = 100 – approximately corresponding to the ∞-norm, thus selecting only the strongest shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This gives very poor predictions (off by factors up to 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' It is clearly inadequate to consider only the effect of the strongest shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' luminosity is given by 𝐽(𝑟 > 𝑟min) = 4𝜋𝐴 log � 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='759 × 10−3𝑟𝜆 𝑟min � , (B3) so that in this case the effect of smooth tide on the annihilation luminosity is equivalent to a sharp truncation of the initial profile at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='78% of 𝑟𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' To model accurately the joint effect of stellar encounters and tidal stripping, the whole encounter and tidal history should, in principle, be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Here, we will simply assume that we can model the joint effect approximately by first considering the effect of all stellar encounters – by truncating the prompt cusp as described in Section 4 – and thereafter applying the pericentre tidal field 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We choose this order because it is technically simpler for us, but we expect that consideration of the effects in reverse order would be just as valid and would give similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We set up a profile following the shocked power-law form, 𝜌(𝑟) = 𝐴𝑟−3/2 exp(−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='256(𝑟/𝑟𝐵)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='639), (B4) and we apply various tidal fields 𝜆 using the adiabatic-tides model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We note that this set-up has one relevant parameter – the ratio between the two truncation scales: 𝑟𝜆 𝑟𝐵 = � 𝐵2 𝜆 �2/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (B5) We note that the annihilation luminosity in equation (B3) is equiva- lent to that in equation (45) for a single stellar encounter with shock parameter 𝐵𝜆 = √ 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='4𝜆 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (B6) Thus, in the limit 𝐵𝜆 ≫ 𝐵 (where the tidal field sets the truncation scale) we expect the annihilation luminosity to be equivalent to that after a single shock with strength 𝐵𝜆, while in the limit 𝐵𝜆 ≪ 𝐵, it should be equivalent to a single shock of strength 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' A reasonable guess for intermediate cases is for the joint effect to be given by a weighted average of the two scales, 𝐵eff,𝜆 = √︁ 𝐵2 + 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' (B7) In principle any 𝑝-norm of 𝐵 and 𝐵𝜆 would be a reasonable guess: we find that 𝑝 = 2 works very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='9 In Figure B1 we compare the transfer functions obtained from the adiabatic-tides calculations to an effective description assuming that the joint effect of encounters and tides is equivalent to a single shock with 𝐵eff,𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This works reasonably well in the regime where 𝜌/𝜌powerlaw > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' In the regime where this ratio is smaller, the approximation is worse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' the tidal truncation is complete at the tidal radius, whereas the encounter truncation has a much longer tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, the tail of the profile is almost irrelevant for the annihilation luminosity, and is unlikely to be correct in the adiabatic-tides description (Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The bottom panel of Figure B1 shows the J-factors obtained by integrating 𝜌2 for the adiabatic remnants over the range [10−6𝑟𝐵, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Note that we chose the lower bound so that it is well in the power-law regime of the profile for all the cases considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Apart from that, it is, of course, arbitrary, and so the absolute offset on the 𝐽-axis is also arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We show additionally the two limiting cases of a pure encounter truncation and a pure tidal truncation, together with the effective description by a single shock with strength 𝐵eff,𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Clearly this recovers the asymptotic limiting cases as well as the intermediate regime very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' To verify that this gives a reasonable description of the joint effect of encounters and of the smooth tidal field in all relevant cases, we must also check that it applies to cored initial profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Here, we only consider the case where 𝐵𝜆 ≫ 𝐵 so that we do not have to deal with two scales at the same time, but only with a single scale 𝐵𝜆/𝐵core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We set up cored profiles as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 and apply tidal fields of varying amplitudes through the adiabatic-tides model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We show the corresponding 𝐽-factors in Figure B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Clearly the effective description works very well for cored pro- files also, as long as 𝐵eff,𝜆 ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2𝐵core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Beyond that scale the cored profiles reach an earlier disruption threshold of 𝐵dis,𝜆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='33𝐵core which is a factor two smaller than the encounter disruption threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' While it would certainly be possible to improve our effective model further to incorporate this reduced disruption threshold, this is unnec- essary, since such large tidal fields are not reached in the Milky Way – especially not at the large radii where smooth tides dominate over stellar encounters and 𝐵𝜆 ≪ 1 km s−1 pc−1 typically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We conclude that treating the joint effect of stellar encounters and smooth tides as that due to a single encounter with a shock parameter 𝐵eff,𝜆 is an excellent approximation within the adiabatic-tides framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' We note that this may slightly overestimate the effects of smooth tides, 9 𝑝 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='8 actually works slightly better, but we stick to 𝑝 = 2 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' MNRAS 000, 1–22 (2015) 22 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 10 5 10 4 10 3 10 2 10 1 100 101 r/rB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 / powerlaw /B2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='001 /B2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='01 /B2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='1 /B2 = 1 /B2 = 10 /B2 = 100 initial profile Beff, approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 10 3 10 2 10 1 100 /B2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='0 J(r > 10 6rB)/(4 A2) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' models Beff, description encounter limit tidal limit Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Top: Transfer functions for shocked power-law profiles that are subsequently adiabatically exposed to a tidal field with amplitude 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The dashed line shows the initial (post-encounter) profile and the dotted lines show the effective description that we propose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Bottom: 𝐽-factors for the corresponding profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The two limiting cases apply when either encounter truncation or tidal truncation dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The intermediate regime is described by an effective shock parameter 𝐵eff,𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 10 2 10 1 100 Beff, /Bcore 0 1 2 3 4 5 J/4 A2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Models Beff, description predicted disruption actual disruption Figure B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 𝐽-factors of cored prompt cusps that were adiabatically exposed to tidal fields of different amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' The 𝐽-factors are very well approximated by the 𝐵eff,𝜆-description, except for the regime 𝐵eff,𝜆 ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content='2𝐵core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, such large tidal fields are anyways not found in the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' since the adiabatic-tides predictions are often a slight overpredic- tion in practice (Stücker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' Aguirre-Santaella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' However, since the effects on cusp annihilation luminosity are in any case rather weak, this seems acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} +page_content=' MNRAS 000, 1–22 (2015)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E3T4oBgHgl3EQfswtu/content/2301.04670v1.pdf'} diff --git a/ANAzT4oBgHgl3EQfhf3F/content/tmp_files/2301.01486v1.pdf.txt b/ANAzT4oBgHgl3EQfhf3F/content/tmp_files/2301.01486v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3420ac388853bce3b1b2128ae6638b9b7dc5f521 --- /dev/null +++ b/ANAzT4oBgHgl3EQfhf3F/content/tmp_files/2301.01486v1.pdf.txt @@ -0,0 +1,2233 @@ +Astronomy & Astrophysics manuscript no. TCrA_Rigliaco +©ESO 2023 +January 5, 2023 +Disk Evolution Study Through Imaging of Nearby Young Stars +(DESTINYS): Characterization of the young star T CrA and its +circumstellar environment ⋆ +E. Rigliaco1, R. Gratton1, S. Ceppi2, C. Ginski.3, 4, M. Hogerheijde3, 4, M. Benisty5, 6, T. Birnstiel7, 8, M. Dima1, S. +Facchini2, A. Garufi9, J. Bae10, M. Langlois11, G. Lodato2, E. Mamajek12, C.F. Manara13, F. Ménard14, Á. Ribas15, and +A. Zurlo16, 17, 18 +1 INAF/Osservatorio Astronomico di Padova, Vicolo dell’osservatorio 5, 35122 Padova e-mail: elisabetta.rigliaco@inaf.it +2 Dipartimento di Fisica, Università Degli Studi di Milano, Via Celoria, 16, Milano, 20133, Italy +3 Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098XH Amsterdam, The Netherlands +4 Leiden Observatory, Leiden University, PO Box 9513, 2300 RA, Leiden, The Netherlands +5 Unidad Mixta Internacional Franco-Chilena de Astronomía, CNRS/INSU UMI 3386, Departamento de Astronomía, Universidad +de Chile, Camino El Observatorio 1515, Las Condes, Santiago, Chile +6 Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France +7 University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität München, Scheinerstr. 1, 81679 Munich, Germany +8 Exzellenzcluster ORIGINS, Boltzmannstr. 2, D-85748 Garching, Germany +9 INAF, Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125, Firenze, Italy +10 Department of Astronomy, University of Florida, Gainesville, FL 32611, United States of America +11 CRAL, UMR 5574, CNRS, Université Lyon 1, 9 avenue Charles André, 69561 Saint-Genis-Laval Cedex, France +12 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA +13 European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching bei München, Germany +14 Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France +15 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK +16 Núcleo de Astronomía, Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Av. Ejercito 441, Santiago, Chile +17 Escuela de Ingeniería Industrial, Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Av. Ejercito 441, Santiago, Chile +18 Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France +Received 12 October 2022; accepted 22 December 2022 +ABSTRACT +Context. In recent years it is emerging a new hot-topic in the star and planet formation field: the interaction between circumstellar +disk and its birth cloud. Birth environments of young stars have strong imprints on the star itself and their surroundings. In this context +we present a detailed analysis of the wealthy circumstellar environment around the young Herbig Ae/Be star T CrA. +Aims. Our aim is to understand the nature of the stellar system and the extended circumstellar structures as seen in scattered light +images. +Methods. We conduct our analysis combining archival data, and new adaptive optics high-contrast and high-resolution images. +Results. The scattered light images reveal the presence of a complex environment around T CrA composed of a bright forward +scattering rim of the disk’s surface that is seen at very high inclination, a dark lane of the disk midplane, bipolar outflows, and streamer +features likely tracing infalling material from the surrounding birth cloud onto the disk. The analysis of the light curve suggests that +the star is a binary with a period of 29.6 years, confirming previous assertions based on spectro-astrometry. The comparison of the +scattered light images with ALMA continuum and 12CO (2–1) line emission shows that the disk is in keplerian rotation, and the +northern side of the outflowing material is receding, while the southern side is approaching to the observer. The overall system lays +on different geometrical planes. The orbit of the binary star is perpendicular to the outflows and is seen edge on. The disk is itself seen +edge-on, with a position angle of ∼7◦. The direction of the outflows seen in scattered light is in agreement with the direction of the +more distant molecular hydrogen emission-line objects (MHOs) associated to the star. Modeling of the spectral energy distribution +(SED) using a radiative transfer scheme well agrees with the proposed configuration, as well as the hydrodynamical simulation +performed using a Smoothed Particle Hydrodynamics (SPH) code. +Conclusions. We find evidence of streamers of accreting material around T CrA. These streamers connect the filament along which +T CrA is forming with the outer parts of the disk, suggesting that the strong misalignment between the inner and outer disk is due to +a change in the direction of the angular momentum of the material accreting on the disk during the late phase of star formation. This +impacts the accretion on the components of the binary, favoring the growth of the primary with respect the secondary, as opposite to +the case of aligned disks. +Key words. stars: pre-main sequence, circumstellar matter – protoplanetary disks – ISM: individual object: T CrA – ISM: jets and +outflows +Article number, page 1 of 17 +arXiv:2301.01486v1 [astro-ph.SR] 4 Jan 2023 + +A&A proofs: manuscript no. TCrA_Rigliaco +1. Introduction +Herbig Ae/Be stars (Herbig 1960) are pre-main sequence stars +with intermediate mass covering the range between low-mass T +Tauri stars (TTSs) and the embedded massive young stellar ob- +jects. The formation of stars in the low and intermediate-mass +regimes involves accreting disks formed during the collapse of +the protostar, and fast collimated outflows and jets. The circum- +stellar environment of these objects is highly dynamic and multi- +wavelengths observations show large photometric and spectro- +scopic variability (e.g., Pikhartova et al. 2021; Mendigutía et al. +2011) that can be used as a tool to understand the physics of ac- +cretion and ejection related to the interaction between the star +and its circumstellar environment. +T CrA (RA=19:01:58.79 DEC=-36:57:50.33) is an Herbig +Ae/Be star member of the Coronet Cluster, belonging to the +Corona Australis star-forming region, which is one of the near- +est (149.4±0.4 pc, Galli et al. 2020) and most active regions of +ongoing star formation. The Coronet Cluster is centered on the +Herbig Ae/Be stars R CrA and T CrA. It is very active in star +formation (e.g. Lindberg & Jørgensen 2012), harboring many +Herbig-Haro (HHs) objects and Molecular Hydrogen emission- +line Objects (MHOs). It has been target of many surveys, and +all studies agree in assigning the Coronet an age <3 Myr (e.g. +Meyer & Wilking 2009; Sicilia-Aguilar et al. 2011). In this pa- +per we investigate the variable star T CrA. T CrA is classified +as F0 by Joy (1945) with effective temperature Teff=7200 K, +and according to Cazzoletti et al. (2019) and Herczeg & Hil- +lenbrand (2014) this corresponds to L∗ ∼29 L⊙, and stellar mass +∼2.25 M⊙ using the evolutionary tracks by Siess et al. 2000, and +adopting the average distance of 154 pc calculated by Dzib et al. +(2018). The Gaia-DR2 and DR3 catalogs (Gaia Collaboration +et al. 2016, 2021) do not provide proper motion or parallax for +T CrA. This star was not observed by the Hipparcos satellite and +it is also not listed in the UCAC5 catalog. The former UCAC4 +catalog (Zacharias et al. 2012) provides a proper motion result +(µα cos δ = 2.0 ± 3.8 mas yr−1, µδ=-22.6±3.8 mas yr−1), which +is consistent with membership in Corona-Australis (within the +large uncertainties of that solution). Galli et al. (2020) provided +an updated census of the stellar population in the Corona Aus- +tralis deriving an average distance of 149.4±0.4 pc. This is the +distance we will use throughout the paper. A deep H2 v=1–0 +S(1) 2.12 µm narrow-band imaging survey of the northern part +of the Corona Australis cloud conducted by Kumar et al. (2011) +identified many new MHOs (Davis et al. 2010). Among these +objects, two are considered unambiguously associated to T CrA: +MHO2013 and MHO2015, see Figure 3 in Kumar et al. (2011). +MHO 2015 is a clear bow-shock feature, lying to the south of +T CrA, and it marks the southern lobe of the bipolar outflow +originating from T CrA. MHO 2013 marks the northern lobe. +The hypothetical line connecting the two MHOs crosses the po- +sition of T CrA. This is the only unambiguously detected bipolar +outflow traced by two complementing bow-shock features in the +entire Coronet region (Kumar et al. 2011). We reproduce the im- +age shown in Kumar et al. (2011) in the left panel of Fig. 1. +T CrA was suggested to be a binary system by Bailey (1998) +and Takami et al. (2003) who adopted spectro-astrometry in the +Hα line suggesting that the system is a binary with a compan- +ion at >0.14′′. However, no companion has been detected us- +ing spectro-astrometry in the fundamental rovibrational band of +CO at 4.6µm (Pontoppidan et al. 2011) nor with K-band speckle +⋆ Based on observations collected at the European Organisation for +Astronomical Research in the Southern Hemisphere under ESO pro- +gramme 1104.C-0415(H). +imaging (Ghez et al. 1997; Köhler et al. 2008). In the same +years, infrared speckle observations performed by Ghez et al. +(1997) did not show the presence of a stellar companion. The +non-detection of the companion by Ghez et al. (1997) implies +that the possible companion has a contrast in the K-band larger +than 3 mag (that is a K-magnitude fainter than 10.5) or a sep- +aration smaller than 0.1 arcsec at the epoch of the observation +(April 26, 1994; see also Takami et al. 2003). +Recently, the circumstellar environment of T CrA has been +investigated. SOFIA/FORCAST (Faint Object infraRed CAm- +era for the SOFIA Telescope, Herter et al. 2018) observations +show very strong excess in the far-IR. T CrA was also de- +tected in all Herschel/PACS (Photodetector Array Camera and +Spectrometer) bands (Sandell et al. 2021), highlighting the pres- +ence of warm or hot dust. Mid-infrared interferometric data +obtained with VLT/MIDI (MID-infrared Interferometric instru- +ment) show the presence of disk emission from the inner regions, +where the temperature is sufficiently high (Varga et al. 2018). +The presence of the inner disk is also given by the spectral en- +ergy distribution (SED) which shows near-IR excess emission +(Sicilia-Aguilar et al. 2013). Optical and IR spectra covering +the [OI] λ6300 and [NeII] 12.81 µm lines (Pascucci et al. 2020) +show emission attributed to a jet nearly in the plane of the sky. +Moreover, continuum ALMA observations of T CrA at 1.3 mm +(230 GHz) were conducted as part of the survey of protoplan- +etary disks in Corona Australis (Cazzoletti et al. 2019) and the +data show a ∼22σ detection at 1.34′′ from the nominal Spitzer +position that is considered as detection. The 1.3 mm continuum +flux is then converted into a dust mass (Mdust) under the assump- +tion of optically thin and isothermal sub-millimeter emission, +yielding Mdust=3.64±0.27 M⊕. No information on the 12CO(2- +1) gas content in the disk are provided. The average disk mass +in CrA is 6±3 M⊕, and it is significantly lower than that of disks +in other young (1–3 Myr) star forming regions (Lupus, Taurus, +Chamaeleon I, and Ophiuchus) and appears to be consistent with +the average disk mass of the 5–10 Myr-old Upper Sco (Cazzo- +letti et al. 2019). +In this paper we analyze images of T CrA acquired with the +Very Large Telescope at ESO’s Paranal Observatory in Chile. +We employ polarimetric differential imaging (PDI) observations +obtained with SPHERE (Spectro-Polarimetric High-contrast Ex- +oplanet REsearch, Beuzit et al. 2019) in the H band to explore +the circumstellar environment by tracing light scattered by the +small (µm-sized) dust grains. Moreover, we use archival pho- +tometric and imaging data to investigate the multiplicity of the +system. The paper is organized as follows. In Sect. 2 we describe +the data collected from the archive and newly acquired. In Sect. 3 +we describe the data analysis. First we discuss the multiplicity of +the system as suggested by the photometric data, the analysis of +the proper motion and the analysis of the PSF subtracted images. +Second we analyze the geometry of the system with the analy- +sis of the disk and the extended emission seen in scattered light. +In Sect. 4 we propose a scenario that reconciles all the findings, +showing a model of the system, and discussing a modeling of +the spectral energy distribution and hydrodynamical simulation. +In Sect. 5 we summarize and conclude. +2. Observations +2.1. SPHERE data +T CrA was observed on 2021 June 6th with SPHERE/IRDIS +(InfraRed Dual-band Imager and Spectrograph (IRDIS; Dohlen +et al. 2008) in dual-beam polarimetric imaging mode (DPI; de +Article number, page 2 of 17 + +Rigliaco et al.: DESTINYS–TCrA +Boer et al. 2020; van Holstein et al. 2020) in the broadband H +filter with pupil tracking setting, as part of the DESTINYS pro- +gram (Disk Evolution Study Through Imaging of Nearby Young +Stars, Ginski et al. (2020, 2021)). An apodized Lyot coronagraph +with an inner working angle of 92.5 mas was used to mask the +central star. The individual frame exposure time was set to 32 s, +and a total of 136 frames were taken separately in 34 polari- +metric cycles of the half-wave plate. The total integration time +was 72.5 minutes. Observing conditions were excellent with an +average seeing of 0.8′′ and an atmosphere coherence time of +6.4 ms. In addition to the science images, flux calibration images +were obtained by offsetting the star position by about 0.5 arcsec +with respect to the coronagraph using the SPHERE tip/tilt mir- +ror, and inserting a suitable neutral density filter to avoid image +saturation. Two flux calibration sequences were acquired, before +and after the science observation. We used the public IRDAP +pipeline (IRDIS Data reduction for Accurate Polarimetry; van +Holstein et al. 2020) to reduce the data. The images were astro- +metrically calibrated using the pixel scale and true north offset +given in Maire et al. (2016). Because the data were taken in pupil +tracking mode, we were able to perform an angular differential +imaging (ADI; Marois et al. 2006) reduction in addition to the +polarimetric reduction, resulting in a total intensity image and +a polarized intensity image. We show the initial combined and +flux calibrated Stokes Q and U images as well as the QΦ and UΦ +images in Appendix A. +Additional SPHERE observations of T CrA were acquired +in 2016 and 2018 with the ESO programs 097.C-0591(A) and +0101.C-0686(A) (P.I. Schmidt) in classical imaging mode, using +a classical Lyot coronagraph and the broadband H filter (BB_H). +The data were reduced through the SPHERE Data Center (De- +lorme et al. 2017). The 2016 data have very low S/N ratio and +they are not usable for this work. The 2018 IRDIS data are in- +stead of good quality and are used to confirm the features de- +tected in the 2021 images. +2.2. NACO data +To perform our analysis we also employed archival NACO data. +Adaptive optics corrected near-infrared imaging of T CrA was +obtained with NAOS-CONICA (NACO; Lenzen et al. 2003; +Rousset et al. 2003) at the VLT in July 12th 2007 (program ID +079.C-0103(A)), March 29th 2016 (program ID 097.C-0085(A)) +and May 21st 2017 (program ID 099.C-0563(A)). In all cases +images were obtained in Ks band (λc=2.18 µm) using the S13 +camera, with a 13.72 mas/pixel scale. In 2007, 3000 frames of +0.6 seconds were taken with an average seeing of 0.8. In 2016, +540 frames of 0.5 seconds each were taken with average see- +ing of 1.5. In 2017, 756 frames of 0.35 seconds each were taken +with average seeing of 1.4. The final images are obtained as the +median of all the exposures for each year, after re-centering and +rotating the single-exposure images. +2.3. Photometric data +We collected long-term optical photometry of T CrA from the +AAVSO Database1 (American Association of Variable Star Ob- +servers: Kafka 2020) in order to investigate its secular evolution. +We also considered data acquired within the ASAS (Pojman- +1 https://www.aavso.org/data-access +ski 1997)2 and ASAS-SN surveys (Shappee et al. 2014)3. While +more accurate than the AAVSO data, they have a much more +limited temporal coverage. Results are fully consistent with the +long-term light curve obtained from the AAVSO data, but no fur- +ther insight could be obtained. So we will not discuss the ASAS +data further. +2.4. ALMA data +T CrA was observed by ALMA on 2016 August 1–2 (project +2015.1.01058.S). Details of the observations and calibration are +described in Cazzoletti et al. (2019). These authors also present +an analysis of the continuum emission. For the current paper, +the continuum emission was imaged using Hogböm CLEANing +with Brigss weighting, a robust parameter of 0.5, and a manu- +ally drawn CLEAN mask. The resulting beam size is 0.36×0.27 +arcsec (PA +78◦). The noise level is 0.12 mJy, and a continuum +flux of 3.1 mJy is detected. These values are not corrected for +the primary beam response, which can be expected to affect the +results since the observations was not centered on the target. A +2D Gaussian fit to the continuum emission shows that the con- +tinuum emission is slightly resolved, with a size of 0.54 × 0.37 +and a PA of +23◦. +The 12CO line emission was imaged using natural weight- +ing and 0.5 km s−1 channels, from VLSR = −5 to +15 km s−1; +no emission was detected outside this range. We used hand +drawn masks for each individual channel and applied multi-scale +CLEAN with scales of 0,5,15,25 pixels. A pixel scale of 12.251 +mas was used, coincident with the SPHERE pixel scale. Because +the CrA region contains extended CO emission around the sys- +temic velocity of T CrA (e.g., Cazzoletti et al. 2019), we re- +moved all baselines shorter than 55 kλ. This removed most, but +not all, of the extended line flux but also limits the recovered +spatial scales to ∼ 3.75 arcsec. +3. Data Analysis +The new and archival data described in the previous section al- +low us to investigate the nature of T CrA as young stellar object. +In this section we will analyze the observational evidences we +have for the stellar system, its environment, and the geometry of +the extended structures visible in scattered light. In Sect. 3.1 we +analyze the clues related to the binarity of the system. In Sect. 3.2 +we show the newly acquired polarized light image in H-band of +T CrA, describing all the features that we see in the image. +3.1. T CrA as binary system +The light curve (Fig. 2) shows alternate and periodic maxima +and minima. The photometric time series analyzed in this study +consists of more than 5100 V-band data points collected from the +AAVSO Database and taken in a period of over 100 years, be- +tween 1910 and 2010. Each point in Figure 2 is the mean value +over each year. The secular evolution of the light curve is well +reproduced by a sinusoidal function with a period of 29.6 years. +Sinusoidal light curves, like the one observed in T CrA, can be +due to different reasons such as rotation, pulsation, the presence +of eclipsing binaries, or occulting binaries. In the case of oc- +culting binaries, the period is generally longer than in the other +cases, and the occultation is not only due to the stars, but also +2 http://www.astrouw.edu.pl/asas/?page=aasc&catsrc= +asas3 +3 https://asas-sn.osu.edu/variables +Article number, page 3 of 17 + +A&A proofs: manuscript no. TCrA_Rigliaco +Fig. 1: SPHERE/IRDIS polarized light image in H-band of T CrA. Left panel: H2 image of the Coronet sub-region. The image is +adapted from Kumar et al. 2011. The red line shows the line connecting the two MHOs associated to T CrA. The orange box shows +the IRDIS field of view. Middle panel: Field of view (∼12.5′′) of the SPHERE/IRDIS polarized light image in H-band of T CrA. +The extended emission features analyzed in the manuscript are labeled. The orange box shows the innermost region of the system. +Right panel: Zoom-in of the innermost 2′′ around the central system. The disk and the shielded disk mid-plane seen as dark lane are +labeled. +Fig. 2: Secular light curve of T CrA with the photometry col- +lected from the AAVSO archive. Each point is the mean value +for each year; error bar is the standard deviation of the mean.The +horizontal dashed lines show the ∆V-mag variation. The period +of the light curve, measured as the mean between the difference +of the first and third maxima and minima, is labeled. +to the circumstellar disks surrounding one or both the stars. The +light curve of T CrA is suggestive of the motion of an occulting +binary star. The variation (∆V) in V-magnitude is of the order of +∼1.4±0.2 mag (see Fig. 2). +Evidence of the presence of a binary star is also provided by +the peculiar proper motion of T CrA. Indeed T CrA shows a rela- +tive average motion of 7.5±3.8 mas yr−1 with respect to the clus- +ter in the direction (PAPM)=156±30◦ over the period 1998 (mean +epoch of UCAC4 and PPMXL observation) and 2016 (epoch of +Gaia DR3). These values are given by the difference between +the proper motion of T CrA, µα cos δ = 4.2 ± 2.5 mas yr−1 in +RA and µδ=-6.2±2.9 mas yr−1 in DEC (see Appendix B), and +the average proper motion of the on-cloud Coronet cluster mem- +bers (µα cos δ = 4.3 mas yr−1 and µδ=-27.3 mas yr−1, Galli et al. +2020). This result might indicate a peculiar (large) motion of +T CrA with respect to the Coronet cluster. However the position +of T CrA is also constrained and defined by the position of the +two associated MHOs (Kumar et al. 2011). We measured the po- +sition angle of the straight line connecting MHO 2013 and 2015, +that are thought to be connected to the star (Kumar et al. 2011), +and crossing T CrA, finding the position angle of the bipolar out- +flow (PAMHO) to be PAMHO ≃33◦. This represents the direction +of the large scale bipolar outflows. We notice that the minimum +distance between T CrA and the line connecting the two MHOs +is only 0.44′′. While this small offset is within the errors in the +MHO positions, it can be used to set an upper limit to the relative +proper motion of T CrA with the Coronet cloud in the direction +perpendicular to this straight line, that is roughly along the di- +rection where we found an offset between the proper motion of +T CrA measured above and that of the Coronet cluster. The exact +value depends on the time elapsed between the expulsion of the +material responsible for the MHO and the observation by Ku- +mar et al. (2011). Given the projected distances from the star +of the MHO’s are 217′′ (MHO 2013) and 64′′ (MHO 2015), +considering the distance of the Coronet cluster and assuming +the collimated fast outflowing gas has a speed of approximately +200 km/s as typical for jets from young stars (e.g., Frank et al. +2014), we obtain that the material was expelled 765 year ago (for +MHO2013) and 224 years ago (for MHO 2015). The upper limit +on the proper motion of T CrA with respect to the cloud is then +obtained by dividing the measured offset between the barycenter +of the system that includes T Cra and the line connecting the two +MHOs: the result is about 1 mas/yr, an order of magnitude less +than the offset in proper motions considered above and consis- +tent with the typical scatter of stars in the Coronet cluster. We +conclude that this offset is not due to a real peculiar motion of +Article number, page 4 of 17 + +36:54:00.0 +F103 +N +MHO2013 +PAdisk +E +dark lane + 102 +36:56:00.0 +Extended +Extended emission +DEC (J2000) +emission +(feature 1) +(feature 2) +101 +tail +36:58:00.0 +disk +TCrA +H2image of +the Coronet +subregion +MHO2015 +10° +2" +0.5"-75AU +PA +-37:00:00.0 +02:10.019:02:00.001:50.0 +RA (J2000)Rigliaco et al.: DESTINYS–TCrA +T CrA, that moves as the Coronet cluster, and should then be +an apparent or transient effect, that might be due to the orbital +motion of the central binary star. +Additional evidence of T CrA as a binary system can also be +found in the images acquired with IRDIS in 2018 and 2021 and +NACO in 2007, 2016 and 2017. We subtracted a median PSF, +obtained by rotating and averaging the PSF image in steps of 1 +degree, to the raw NACO images taken in 2007 and 2016, 2017. +For IRDIS, we used the flux calibration images that are acquired +before and after the science sequence. The technique, described +by Bonavita et al. (2021), allows to make a differential image +that cancels static aberrations. The output of the procedure is +a contrast map that allows to spot stellar companions. Due to +the contrast limit and to the limits imposed by the diffraction +patterns, none of the images obtained allows us to clearly and +uniquely detect the presence of a companion star. However, The +PSF of the NACO 2016 and 2017 data set clearly show an exten- +sion in the same direction (see Fig. 3), namely NW–SE, but in +the NACO 2007 data set we do not see this extension. A slight +extension can be seen in the SPHERE 2018 data set, while no ex- +tension in the SPHERE 2021 data set. The observed extensions, +all in the same direction, are very unlikely to be caused by adap- +tive optic effect, but might indicate a distortion of the PSF due to +an unresolved companion. +3.2. The geometry of the system +Figure 1 shows the polarized light image in H-band of T CrA. +The image shows several structures, as annotated. In the right +panel the brightly illuminated top-side of the outer disk is clearly +visible, as well as the shielded disk mid-plane, seen as a stark +dark lane in approximately the N-S direction. On larger scale, +in the middle panel, we can identify two different extended +emissions. The extended emission labeled as "feature 1" is two- +lobed and extends in the NE–SW direction, up to 2′′ from the +central source. The extended emission labeled as "feature 2" +appears two lobed as well, it is approximately oriented along +the N-S direction. The South lobe extends out to the edge of +SPHERE/IRDIS field of view, while the North lobe extends up +to ∼5′′ from the central source. In the following section we will +analyze these different structures. +3.2.1. Outer Disk +Figure 1 in the right panel shows a very prominent morpholog- +ical feature composed by a dark lane and a bright region that +represents the disk surface. This outer disk appears highly in- +clined, and oriented almost edge-on with respect to the observer, +and extends almost to the edge of the coronagraph. The dark +lane has a maximum width of ∼0.2′′ along the E–W direction, +corresponding to ∼30 au if it were seen exactly edge-on. More- +over, the disk seen as a dark lane shows an offset with respect +to the center of the image that corresponds to ∼10 pixels in the +West direction (∼122 mas) that is about four times the FWHM +of the point spread function. The disk surface is instead shown +by the bright regions that extend further out. The PA of the disk +measures PAdisk=7±2◦, shown as green line in Fig. 1. The disk +appears highly inclined and seen as a dark lane, as in the case +for DoAr25 (Garufi et al. 2020), MY Lup and IM Lup (Aven- +haus et al. 2018). From the images we cannot provide a precise +estimate of the disk inclination, but we can make a few con- +siderations. The brightness asymmetry between the bright disk +top-side, and the diffuse disk bottom-side, indicates that the disk +is not exactly seen edge-on, indeed in that case we should expect +top- and bottom-side of the disk to be equally bright. Moreover, +the offset between the dark-lane and the center of the image pro- +vides another hint of a non-exactly edge-on disk. From simple +trigonometric consideration, we can measure the inclination of +the disk from the angle between the center of the image and +the center of the dark lane and dividing for half the lengths of +the dark lane, finding an inclination of ∼87◦. We can conserva- +tively assume that the T CrA disk, identified as a dark lane in +the SPHERE image has an inclination between 85-90◦. Another +possible interpretation for the dark lane could be that it is due to +a shadow cast by a highly inclined inner disk close to the cen- +ter, as in the case of SU Aur (Ginski et al. 2021). However, in +this scenario, we can not reconcile the brightness asymmetry be- +tween the bright top-side and the diffuse bottom-side of the disk. +Moreover, we should expect the shadow to cross the center of +the image, while it appears shifted to the west by ∼10 pixels. +In order to investigate the innermost region of the outer disk, +we have plotted the radial profile of the flux seen in QΦ scat- +tered light along a slice oriented as the disk, seven pixels wide +and 2.5′′ long. The radial profile, normalized to the brightness +peak of the disk, is shown in Fig. 4 as a black line. The grey +area shows the coronagraph. The disk has a gap that extends up +to ∼25 au and is quite symmetric in the innermost region. As +far as 60 au the disk start to look asymmetric, and extends up to +∼100 au. The observed asymmetry might be due to the outflow- +ing material that overlaps with the disk itself in the north side (as +discussed in the next section). From this analysis we consider for +the outer disk an inner rim with radius rin=0.17′′ (∼25 au) and +an outer rim rout=0.67′′ (∼100 au). We performed the same anal- +ysis of the radial profile in the direction orthogonal to the disk, +and shown as blue-dotted line in Fig. 4. In the East side there is +emission from the scattered light down to the border of the coro- +nagraph (rin−east ≲14 au), and inside the disk rim measured along +the disk direction. As expected, in the West-side the emission +starts further out, due to the presence of the disk’s dark silhou- +ette (rin−west ∼30 au). We notice that in the West direction at ra- +dial distances >50 au there is contamination with the outflowing +material. We will discuss the presence of scattered light emission +inside the outer disk gap in the following section, showing that it +may suggest the presence of an intermediate circumbinary disk +surrounding the central binary system. +3.2.2. Extended emission +The structure seen in scattered light in the NE–SW direction, +identified as feature 1, is consistent with an outflow in the di- +rection of the line connecting the two MHOs (MHO2013 and +MHO2015) that are unambiguously associated to T CrA (show +in the left panel of Fig. 1), which are however at a projected sepa- +ration of ∼35,000 au and ∼10,000 au, respectively. The presence +of the MHOs is a clear sign that the source has in the past al- +ready experienced outflowing phenomena, hence it is consistent +to consider the emission seen in scattered light in the same direc- +tions as associated to outflowing material close to the star. From +a geometrical point of view, the dust seen in scattered light in +the direction of the outflow has a position angle PAoutflow ∼35◦ +with semi-aperture of ∼25◦, consistent with the PAMHO previ- +ously discussed. +The extended emission that elongates approximately in the +N-S direction, and identified as feature 2, is two lobed as well. +In the North it extends up to 4.5′′ from the center, and appears +bent toward the West direction. The Southern feature 2 extends +up to the edge of the field of view and appears brighter than +Article number, page 5 of 17 + +A&A proofs: manuscript no. TCrA_Rigliaco +0.2 +0.1 +0.0 +0.1 +0.2 +0.2 +0.1 +0.0 +0.1 +0.2 +∆Dec (arcsec) +NACO/2007 +0.2 +0.1 +0.0 +0.1 +0.2 +NACO/2016 +0.2 +0.1 +0.0 +0.1 +0.2 +∆RA (arcsec) +NACO/2017 +0.2 +0.1 +0.0 +0.1 +0.2 +SPHERE/2018 +0.2 +0.1 +0.0 +0.1 +0.2 +0.2 +0.1 +0.0 +0.1 +0.2 +SPHERE/2021 +Fig. 3: PSF for all the epochs T CrA was observed. The size of the PSF for every single epochs is shown in the bottom-right corner. +For NACO 2016, 2017 data sets we can notice an elongation of the PSF in the NW–SW direction. +Fig. 4: Radial profile of the Qφ image. The black profile shows +the radial profile obtained along a 2.5′′ long slice centered on +the star in the N-S direction, with PA=7◦ and extending along +the disk (black-dashed box in the insert). The blue-dotted profile +shows the radial profile obtained in the orthogonal direction (E- +W, blue-dashed box in the insert). All profiles are normalized to +the brightness peak of the disk. The gray area shows the radius +of the coronagraph. +the North feature. We can also detect a faint dust tail extend- +ing from the main disk toward SE. As it happens in the case of +SU Aur, where several tails are detected (Ginski et al. 2021), +we can trace the tail structure until it merges with the disk. Fea- +ture 2 is most likely showing the presence of accretion streamers +that bring material from the forming cloud filament to the outer +disk. From the polarized (Fig. 1) and total intensity images of +T CrA we can see that in both cases the northern streamer is +fainter than the southern streamer, indicating that we overall re- +ceive more photons from the South than from the North side of +the extended structure. Moreover, the ratio between the polarized +and total intensity image shows that the overall degree of polar- +ization is similar on both sides. This indicates that light from the +South streamer is scattered with angles smaller than 90◦, favor- +ing the forward scattering. Because the Northern streamer shows +a similar degree of polarization, but overall fainter signal, we +conclude that the light is scattered with angles larger than 90◦. +Hence, the South streamer is angled toward the observed and the +North streamer is angled away from the observer. +4. Discussion +The environment around T CrA is very complex and the analysis +of new and archival data shows several features. In the following +we will discuss each of the evidences presented in the previous +sections, and we will provide a global picture of its circumstel- +lar environment. A cartoon of the proposed model, showing all +the observational evidences analyzed in the previous section, is +shown in Fig. 5. +Fig. 5: Not-to-scale cartoon of the proposed model for the T CrA +system. All the features seen in the scattered light images are +labeled. Moreover, the central binary system, and the size of the +coronagraph is shown. +4.1. Modeling of the light curve +Motivated by the light curve, the peculiar proper motion and the +PSF distortion, we conducted a detailed analysis of the pho- +tometric and proper motion data, to be compared to the new +information on the system’s geometry gathered thanks to the +Article number, page 6 of 17 + +[argsec] +-0.5 +0.5 +1 +N +Rin(N-s) = 0.17" +Rout(N-s) = 0.70" +1 +E +W +intensity +Arbitrary +0.5 +South-side +North-side +0 +East-side +West-side +-100 +0 +100 +Radial distance (AU)Accretion +streamer +Outflow +Flows from +outer to inner +disk +Intermediate +(circumbinary) +Coronagraph +disk +edge +Outer disk +dark lane +Outer disk +surface +tailRigliaco et al.: DESTINYS–TCrA +Parameters +Value +log(q) +-0.27±0.17 M⊙ +T0 +2006.06±0.4 years +AV0 +6.7±1.1 mag +Disk Thickness +54.7±20.2 mas +Disk Offset +90.7±19.2 mas +Table 1: +Stellar parameters obtained from the modeling of +T CrA as a binary star. The primary mass star is assumed to be +1.7M⊙, the orbit to be circular, and period 29.6 years. +SPHERE’s images. In the attempt to reproduce the observed +light curve and the H-band magnitude collected from 2MASS, +we develop a Monte Carlo (MC) model that accounts for the +light emitted from a binary system and partially absorbed by a +disk seen edge-on, modeled as a slab with an exponential pro- +file, and inclined with respect to the binary’s orbit by 35◦, corre- +sponding to an orbit perpendicular to the outflow’s direction. For +this simplified model we assume for the binary system a circular +orbit seen itself edge-on. While the circular orbit is an assump- +tion made to reduce the number of free parameters, and hence +avoid degeneracy in the models, the high-inclination of the bi- +nary orbit is supported by the observation. Indeed, as discussed +in Pascucci et al. (2020), evidence from the small blueshift of +the [OI] and [NeII] forbidden lines of T CrA suggests that the +inner disk is itself close to edge-on, with the microjets close to +the plane of the sky. We assume for the F0 star a mass of 1.7M⊙ +for the primary star, corresponding to 2 Myrs from the BHAC +evolutionary tracks (Baraffe et al. 2015), circular orbit, and a +period of 29.6 years as found from the light curve. The model +provides the mass ratio (q) between the primary and secondary +component of the binary system, the epoch of the minimum dis- +tance between the two components (T0, in years), the offset of +the center of mass with respect to the absorbing slab (disk offset, +in mas), the disk thickness (in mas) and the maximum absorption +at the disk center (AV0, in mag). The proper motion between the +1998 and 2016 is also measured to be compared to the apparent +proper motion of T CrA.A corner plot of the derived quantities +is shown in Appendix C. The MC model computes one million +random sampling of the priors, and provides solutions with re- +duced χ2 <2.3. Figure 6 shows the comparison between the ob- +served secular evolution of T CrA and the light curve obtained +from the model. There is a very good agreement between the +observed and modeled light curve. The best fit parameters for +each of the computed values, obtained as the median value of all +the solutions with χ2 <2.3, are reported in Table 1. According +to this model T CrA is a binary system, whose primary star is +a 1.7M⊙ star, and the secondary is a ∼0.9M⊙, and it is orbiting +with a 29.6 years period. The corresponding semi-major axis of +the orbit is ∼12 au, seen edge-on, and with the line of nodes of +the orbit almost perpendicular to the position angle determined +for the outflow. Moreover, we check the consistency between +the apparent motion as measured from Gaia and ground-based +facilities, and the one measured by assuming the motion of the +modeled binary system. We find that the offset between the two +epochs (1998 and 2016) corresponds to 72±26 mas, which is +consistent with the value of 130±66 mas measured via Gaia and +UCAC4/PPMXL observations, hence justifying the large proper +motion of T CrA with respect to the Coronet motion as due to the +motion of the binary system. We will further discuss the results +from the model in the next Section. +Fig. 6: Light curve of T CrA (red points) compared to the light +curves computed with the MC model (black lines) assuming a +period of 29.6 years. +4.2. Disk and extended emission +Thanks to the new images acquired with SPHERE/IRDIS, and +to the wealth of literature data on this target, we have now a bet- +ter knowledge of the disk and extended structure around T CrA, +and it appears very composite. The disk itself is composed by in- +ner (circumstellar) disk(s) surrounding the primary (secondary) +star of the binary system, an intermediate (circumbinary) disk, +slightly visible in scattered light, and an outer (circumbinary) +disk that is the most prominent in scattered light. Together with +the extended emission features, we will discuss all these features +in the following subsections. +Disks. The outer disk around T CrA is not continuous. The +scattered light images and the radial profile analysis of the QΦ +image show that the bright top-side of the outer disk extends up +to ∼100 au in the N-S direction, and show a gap in the same +direction that extends down to ∼25 au. +Evidence of an inner (circumstellar) disk(s) surrounding the +primary (secondary) star of the binary system comes from the +several tracers of gas and dust well beyond the dust gap. Pas- +cucci et al. (2020) analyze the [OI] λ6300 and [NeII] 12.81 µm +emission lines observed in high-resolution optical and infrared +spectra, and conclude that they are associated to fast and col- +limated microjets. In addition, the presence of gas can also be +inferred from the non-negligible level of mass accretion rate +( ˙Macc ∼8.1×10−9 M⊙/yr, Dong et al. 2018; Takami et al. 2003). +This gas is most likely distributed into an inner circumstellar +disk, that allows accretion onto the system. The presence of the +inner disk is also highlighted by mid-infrared interferometric +data of the thermal emission of disk (Varga et al. 2018), and by +the SED (Sicilia-Aguilar et al. 2013; Sandell et al. 2021). +The images acquired with SPHERE show the presence of +scattered light down to the edge of the coronagraph in the E- +W direction. The origin of such emission, highly inclined with +respect to the outer disk, is not clear. However, as we will see +in section 4.4, it might be due to an intermediate circumbinary +disk, that is a natural transient consequence of the breaking of +the innermost circumstellar disks due to the different inclination +of inner and outer disks. Evidence of emission very close to the +coronagraph edges are also found by Cugno et al. 2022 using +the NaCo imager with the L′ filter (λ=3.6 µm) within the NaCo- +ISPY large program. +Feature 1. The PA of the extended emission identified as +feature 1 is consistent with the large scale MHOs and coinci- +Article number, page 7 of 17 + +12 +(mag) +1.3 +14 +15 +1900 +1920 +1940 +1960 +1980 +2000 +2020 +Time +(yr)A&A proofs: manuscript no. TCrA_Rigliaco +dent with the small scale microjets detected through forbidden +lines (Pascucci et al. 2020). Hence, we reasonably assume that +it is representing outflows detected in scattered light, and that +this feature is orthogonal to the inner and intermediate disk. The +innermost disks (inner and intermediate) are misaligned with re- +spect to the outer disk, with a PA for the inner disk of ∼125◦, +measured as PAoutflow+90◦. Considering the outer disk is seen +with PAdisk=7◦, the resulting misalignment between innermost +and outer disk is of the order of 62◦ with an uncertainty of ±10◦. +This feature is illuminated by the central system. The shape of +the outflow is due to higher density regions of dust, generated by +instabilities created by two or more layers of material with dif- +ferent densities and velocities resulting in a wind-blown cavity +(Liang et al. 2020). The regions with different physical prop- +erties are the highly collimated microjet (as seen from the de- +tection of forbidden lines, e.g., Pascucci et al. 2020), and the +surrounding wider-angle disk wind, or parent cloud. The impact +between these two regions, besides carving out a large and slow +massive outflow cavity into the parent cloud (Frank et al. 2014), +creates regions of high density where dust grains accumulate, be- +coming brighter in scattered light. We also notice that there is a +good agreement between the small scale outflow seen in the po- +larimetric images, and the large scale outflows determined by the +MHOs, supporting the scenario of highly collimated jets carving +a cavity and creating high density regions. We have also tested +the emission seen in scattered light versus the continuum thermal +emission at 1.3 mm, and the 12CO emission seen with ALMA. +In Figure 7, we show the continuum emission and the red- and +blue-shifted line emission overlayed on the SPHERE scattered +light image. The continuum emission, shown as white contours, +is slightly resolved, compact, and it is distinctly different from +the orientation of the beam. The comparison with the SPHERE +image is not quite conclusive in the direction of the emission, if +along the disk or the extended emission identified as feature 1. +12CO line emission was clearly detected in the channels, consis- +tent with a structure of ∼ 2.5 arcsec in diameter. The emission is +most likely due to the combination from emission aligned with +the disk orientation inferred from SPHERE, and emission from +the outflowing material in the same direction as the MHOs. The +gas emission close to the N-S direction might trace the gas in the +outer disk, and the velocity structure of the line emission is con- +sistent with Keplerian rotation. The emission from the outflow- +ing material is in the same direction as the MHOs. The velocities +of the extended emission span from -3 km s−1 to 11 km s−1. The +low velocities for the outflowing material confirm that the emis- +sion must happen close to the plane of the sky, as also found +by Pascucci et al. (2020). In both cases, either when tracing the +outer disk or the outflowing material, the N-E side is receding +and the S-W side is approaching the observer. +Misalignment between the inner and the outer disks are not +rare. As an example, Bohn et al. (2021) have recently inves- +tigated misalignment between inner and outer disks in transi- +tional disks, finding that out of a sample of 20 objects ana- +lyzed, six clearly show evidence of misalignment, five do not +show evidence of misalignment and the others can not be eval- +uated with the current data. Misaligned disks, and disks whose +orientations vary with time can be due to their formation in a +turbulent, chaotic environment (Bate 2018). Moreover, the evo- +lution of the stellar and disk spin axes during the formation of +a star which is accreting in a variable fashion from an inher- +ently chaotic environment might affect the disk orientation as +well (Bate et al. 2010). Also late infalling events, which carry +along a specific angular momentum with respect to the star, may +tilt the pre-existing disk to another rotation axis depending on the +Fig. 7: Overlay of SPHERE (color scale) and ALMA (contours) +data. On the left all the extended structure as seen with SPHERE, +on the right a zoom-in of the innermost 2′′. White contours are +ALMA 1.3 mm continuum, plotted at contours starting at, and +increasing with, 3σ=0.37 mJy beam−1. Red and blue contours +are integrated 12CO 2–1 emission over 10 km s−1 blue- and red- +shifted relative to the source velocity, taken as VLSR=4.5 km s−1. +Red and blue contours are also drawn starting at, and increasing +with, 3σ = 0.12 Jy beam−1 km s−1. The ALMA data are aligned +with the SPHERE data to have the stellar position at the center +of the image; the continuum emission peaks ∼ 0.06′′ North of +that position. +mass ratio of the mass accreted and the disk (Dullemond et al. +2019; Kuffmeier et al. 2021). This was indeed recently observed +within the DESTINYS program for the SU Aur system (Ginski +et al. 2021), which shows large scale streamers in scattered light, +similar to those observed in our new observations of T CrA and +which were shown to trace infalling material. Stellar properties, +such as strong stellar magnetic dipole, can cause a warp or mis- +alignment in the innermost region of the disk (e.g., Matsumoto +& Tomisaka 2004; Machida et al. 2006; Matsumoto et al. 2006; +Hennebelle & Ciardi 2009; Joos et al. 2012; Krumholz et al. +2013; Li et al. 2013; Lewis et al. 2015; Lewis & Bate 2017; +Wurster & Li 2018). Additionally, the presence of a compan- +ion, either stellar or substellar, can also cause inner and outer +disks misalignment (e.g., Facchini et al. 2013, 2018; Zhu 2019; +Nealon et al. 2020), as in the case of HD142527 (Owen & Lai +2017; Price et al. 2018a). +Indeed, T CrA and HD142527 show several similarities even +if the inclinations at which the outer disks are seen are very dif- +ferent (almost edge-on in the case of T CrA and almost face- +on for HD142527). HD142527 is a binary system characterized +by a primary 2.0 M⊙ star surrounded by an inner disk signifi- +cantly misaligned (59◦) with respect to the outer disk (Balmer +et al. 2022). For T CrA the outer disk is seen almost edge-on +and the misalignment between outer and inner disk is coinci- +dent with the inclination of the inner disk orbit, namely ∼55◦. +The primary star in both cases is an F-type Herbig. In the case +of HD142527 all the main observational features (spirals, shad- +ows seen in scattered light, horseshoe dust structure, radial flows +and streamers) can be explained by the interaction between the +disk and the observed binary companion (Price et al. 2018a). The +analysis done on HD142527 led the authors (Price et al. 2018a) +to conclude that the disk around this Herbig star is a circumbi- +nary rather than transitional disk, with an inclined inner disk, and +Article number, page 8 of 17 + +103 +102 +1.0"Rigliaco et al.: DESTINYS–TCrA +with streamers of material connecting the inner and outer disk. +In the case of T CrA, if we assume that the inner disk is aligned +perpendicular to the outflowing material, and hence misaligned +with respect to the outer disk, the configuration is similar. Hints +of dusty material inside and misaligned with respect to the outer +disk come from the radial profile of the scattered light signal +seen from SPHERE/IRDIS and shown in Fig. 4, where in the +East-side of the disk in the direction orthogonal to the disk there +is material down to the coronagraph edge. However, we cannot +say from these images if this material is organized into a disk- +structure itself, or if it represents a streamer of material accreting +from the outer disk onto the inner regions of the system. How- +ever, as opposite to HD142527, we must mention the absence +of obvious shadowing features in scattered light in T CrA, that +can nevertheless be due to the different viewing geometry. In the +following section we will present a 3D hydrodynamical model +as the one developed for HD142527 to explain the observed fea- +tures as disk–binary interaction. +Feature 2. The extended emission identified as feature 2 ap- +pears very extended and resemble material falling onto the disk +as in the case of SU Aur (Ginski et al. 2021). Unfortunately, +the strong foreground contamination due to the overall cloud +does not allow to clearly detect the 12CO (2–1), 13CO (2–1), +and C18O (2–1) transitions at distances larger than ∼2.5′′, thus +we cannot perform a detailed analysis of the kinematics of the +material, as it was done, for example, in the case of SU Aur +(Ginski et al. 2021). Indeed, some parts of the CO disk may +be missing from from Fig. 7 because the cloud contaminates +the signal. Moreover, the large scale streamers do not show any +emission due to the removing of any sensitivity to large scale +emission in the data reduction process. They may exist, but they +are very hard to image. The disk may also be more extended +than seen here. Hence we cannot be conclusive on the nature +of the extended emission in feature 2. It is highly unlikely that +this emission is itself indicating outflowing material, as feature +1, but it can be most likely due to streamers of material that is +falling onto the disk connecting the disk itself to the surrounding +cloud material, as for SU Aur. To some extent we might con- +sider the scattered light morphology of T CrA as an edge-on +view of SU Aur, where we can see the streamers of infalling +material and at least one tail of accretion. The same stream- +ers of accretion were already seen, but not interpreted as such, +by Ward-Thompson et al. (1985); Clark et al. (2000). Ward- +Thompson et al. (1985) used linear polarization mapping of the +region in R-band and identified a jet-like structure with a pro- +jected lengths of 20′′ emerging from T CrA, in the direction of, +but pointing away from R CrA. Clark et al. (2000) performed +near-infrared linear imaging polarimetry in J, H and Kn bands, +and circular imaging polarimetry in the H band and interpreted +the images as bipolar cavities, where the SE emission is visi- +ble as far as ∼15′′ from T CrA. They stress the presence of a +pronounced asymmetry in the polarized intensity images, sug- +gestive of fairly sudden depolarization of the dust grains caused +by foreground material in the reflection nebula. The identifica- +tion of the MHOs, and the analysis of the images acquired with +NACO and SPHERE is now showing that the features observed +in the past were not associated with jets but more likely the same +streamer of accretion seen in scattered light. A possible test to +ascertain the origin of feature 2 can be done using the SO2 tran- +sition from ALMA. Garufi et al. (2022) have indeed shown that +for the source IRAS 04302+2247, the SO2 emission does not +probe the disk region, but rather originates at the intersection be- +tween extended streamers and disks. We notice that the presence +of streamers of material feeding the disk of T CrA would also +go in the direction of mitigating the issue of the low disk masses +found in CrA. Indeed, it was found that the average disk mass +in CrA is significantly lower than that of disks in other young +(1-3 Myr) star forming regions (Lupus, Taurus, Chamaeleon I, +and Ophiuchus, Cazzoletti et al. 2019). If there is accretion of +fresh material onto the disk, one could have lower measured disk +masses at the beginning, and mitigate the issue (Manara et al. +2018). The observed increase in disk masses with time (e.g., +Testi et al. 2022; Cazzoletti et al. 2019) should otherwise be ex- +plained with other mechanisms such as planetesimal collisions +(Bernabò et al. 2022). +Moreover, the presence of streamers of accretion is also in +agreement with the orientation of T CrA with respect to R CrA, +both belonging to the Coronet Cluster. These two stars formed +within the same filament, which is oriented at PA=124◦ pro- +jected on sky (this is also the PA of T CrA relative to R CrA). +This orientation is indeed similar to that of the orbit proposed +for the central binary of T CrA and very close to perpendicular +to the PA of the MHO objects (PA=33◦); these values are well +consistent with the direction of the same structures seen in the +neighbor star R CrA (Rigliaco et al. 2019; Mesa et al. 2019)). +This suggests that the bulk of the inflow of material that formed +the T CrA system was coplanar with this filament and that the +original disk of T CrA was likely oriented at the PA of the fil- +ament; this is actually the case also for the disk around R CrA. +However, the current outer disk of T CrA has a very different +orientation (PA=7 degree), though it seems to be still fed by the +same filament. This is because T CrA appears to be presently +offset by a few hundreds au (a few arcsec on sky) with respect +to the filament. Considering the age of T CrA (likely 1-3 Myr), +this offset is indeed very small, corresponding to a minuscule ve- +locity of only ∼ 1m/s. This suggests that the generation of mis- +aligned structure is very likely whenever accretion on the disk is +prolonged over such long intervals of time. +4.3. Spectral Energy Distribution +We model the SED of T CrA using the dust radiative transfer +model developed by Whitney et al. (2003b,a). The code uses a +Monte Carlo radiative transfer scheme that follows photon pack- +ets emitted by the central star as they are scattered, absorbed, +and re-emitted throughout the disk. For the modeling we have +assumed that the geometry of the star+disk system is comprised +by a central 2.0 M⊙ source emitting photons and a gapped and +misaligned circumstellar disk as described above. The total mass +of the disk Mdisk=10−3M⊙, which is in agreement with Mdust re- +trieved by Cazzoletti et al. (2019) using the 1.3 mm continuum +flux, assuming an ISM gas-to-dust ratio of 100. The outcome of +the model, shown in orange in Fig. 8, well reproduces the ob- +served photometric points collected in Table 2, suggesting that +the interpretation of inner and outer disks misaligned with re- +spect to each other is in very good agreement with the collected +photometry 4. For comparison, we also show the SED obtained +with the same parameters, in the case where no misalignment +between inner and outer disk is assumed (red profile). In this +case the curve does not well reproduce the observed photome- +try at wavelengths longer than ∼10–15 µm. We must notice that +the radiative transfer model does not account for the binary star, +hence it may cause deviation in the illumination of the disk. In +particular, in their orbit the two stars spend time above the disk +midplane, hence illuminating the circumbinary disk from above. +4 The apparent oscillations of the model at wavelengths longer than +300 µm is due to low number statistics and has no physical meaning. +Article number, page 9 of 17 + +A&A proofs: manuscript no. TCrA_Rigliaco +λc +Flux +Facility +Reference +(µm) +(Jy) +0.349 +0.00531 +SkyMapper +Wolf et al. (2018) +0.444 +0.00792 +CTIO +Henden et al. (2016) +0.444 +0.00988 +UCAC4-RPM +Nascimbeni et al. (2016) +0.482 +0.0138 +CTIO +Henden et al. (2016) +0.497 +0.0126 +SkyMapper +Wolf et al. (2018) +0.504 +0.0142 +GAIA +Gaia Collaboration (2020) +0.554 +0.0163 +Hamilton +Herbig & Bell (1988) +0.554 +0.0171 +CTIO +Henden et al. (2016) +0.554 +0.0204 +UCAC4-RPM +Nascimbeni et al. (2016) +0.604 +0.0181 +SkyMapper +Wolf et al. (2018) +0.762 +0.045 +GAIA +Gaia Collaboration (2020) +0.763 +0.0539 +CTIO +Henden et al. (2016) +1.24 +0.425 +2MASS-J +Cutri et al. (2003) +1.65 +0.871 +2MASS-H +Cutri et al. (2003) +2.16 +1.55 +2MASS-K +Cutri et al. (2003) +3.55 +1.93 +Spitzer/IRAC +Gutermuth et al. (2009) +4.49 +2.07 +Spitzer/IRAC +Gutermuth et al. (2009) +5.73 +2.38 +Spitzer/IRAC +Gutermuth et al. (2009) +11.6 +3.48 +WISE/W3 +Cutri & et al. (2012) +19.7 +23.4 +SOFIA +Sandell et al. (2021) +22.1 +23.8 +WISE/W4 +Cutri & et al. (2012) +25.3 +30.7 +SOFIA +Sandell et al. (2021) +31.5 +29.0 +SOFIA +Sandell et al. (2021) +37.1 +29.3 +SOFIA +Sandell et al. (2021) +70.0 +19.3 +Herschel +Herschel Group et al. (2020) +100.0 +14.2 +Herschel +Herschel Group et al. (2020) +160.0 +5.0 +Herschel +Herschel Group et al. (2020) +1300 +0.00499 +ALMA +Cazzoletti et al. (2019) +Table 2: List of the fluxes at different wavelengths collected +from the literature used for the SED. +In the two SEDs shown in Fig. 8 we do not account for this ef- +fect. +4.4. Hydrodynamical Simulation +We perform a 3D hydrodynamical simulation of the T CrA con- +figuration considered in this work using the Smoothed Particle +Hydrodynamics (SPH) code Phantom (Price et al. 2018b; Mon- +aghan 2005; Price 2012). The initial conditions of the system +are set following the observational constraints acquired so far. +T CrA is modeled as a binary system with masses 1.7 M⊙, and +1.0 M⊙ for the primary and secondary component, respectively. +Each star is simulated as a sink particle (Price et al. 2018b; Bate +et al. 1995) with an accretion radius of 0.5 au. The orbit is eccen- +tric, and the period of the binary star is 29.6 years, correspond- +ing to a semi-major axis of 13.3 au. The orbit is seen edge-on +with an inclination of 90◦, and PAorbit is perpendicular to the out- +flowing material (PAorbit=145◦). The outer disk, extending from +Rin = 25 au to Rout = 100 au is simulated with 8 × 105 SPH par- +ticles, resulting in a smoothing length ≈ 0.2 times the disk scale +height. The inner disk, extending from rin = 1 au to rout = 5 au, +and co-planar to the orbit of the binary star, is simulated with +2 × 105 SPH particles, resulting in a smoothing length of about +the disk scale height. Outflows and inflows are not considered in +this model. Viscosity is implemented with the artificial viscosity +method (Lucy 1977; Gingold & Monaghan 1977) that results in +an Shakura & Sunyaev (1973) α-viscosity as shown by Lodato +& Price (2010). We use α ≈ 5 × 10−3. We run the full hydrody- +namical model (with both the outer and the inner disk) for 100 +binary orbits in order to relax the initial condition and to produce +a synthetic image of the system to compare with the observation. +To perform a direct comparison with observations of T CrA we +Fig. 8: SED of TCrA. The black asterisks show the published +photometry as reported in Table 2. The orange curve shows the +total emission. The magenta line shows the SED component due +to stellar origin, in blue the component due to the disk, and in +green the component due to the envelope. The red curve shows +the emission if no misalignment between the intermediate and +outer disk is assumed. The oscillations in the model curves at +the longest wavelengths are artifacts related to the finite number +of photon packets considered in the Monte Carlo scheme. +post-processed our simulation using the Monte Carlo radiative +transfer code MCFOST (Pinte et al. 2016) in order to produce +synthetic images of the hydrodynamical model. MCFOST maps +the physical quantities in the SPH simulation (e.g. dust and gas +density, temperature) onto a Voronoi mesh directly built around +the SPH particles, without interpolation. We adopt a gas-to-dust +mass ratio equals to 100 and we assume micrometer grains to be +well coupled with the gas. These grains scatter the stellar light +collected by SPHERE and are assumed to be spherical and ho- +mogeneous (as in the Mie theory). Their chemical composition +is 60% astronomical silicates and 15% amorphous carbons (as +DIANA standard dust composition, Woitke et al. 2016) and they +have a porosity of 10%. The gas mass is directly taken from the +SPH simulation. We use the same distance from the source used +in this paper (149.4 pc) and ≈ 106 photon packets to compute +the temperature profile of the model and ≈ 1010 photon packets +to compute the source function of the model in order to produce +the scattered light image at 2 µm wavelength. +The total intensity polarized light image obtained with the +hydrodynamical simulation is show in the left panel of Fig. 9. +The middle panel is the synthetic image convolved to the +SPHERE/IRDIS resolution and in the right panel we show the +observed image. There are a few features that are clearly repro- +duced in the simulation: the dark lane, the offset of the dark lane +with respect to the center of the image, the top-surface of the disk +brighter than the bottom-side of the disk. There are two bright +spots in the East-West direction on the convolved synthetic im- +age, that are also observed in the real image. These points are +due to the intermediate circumbinary disk that breaks from the +outer regions, precessing as a rigid body, and leading to its evo- +lution. The breaking of the inner disk generates an intermediate +disk, that is visible as bright spots at the East and West side of +the coronagraph. We must notice that the simulation does not +take into consideration the outflowing material, and does not ac- +Article number, page 10 of 17 + +1000 +collectedphotometry +Itotal +stellarorigin +100 +- disk origin +LL +:envelopeorigin +Itotal-nodisksmisalignment +10 +TTT +(Jy) +Flux +1 +LLL +0.1 +TT +0.01 +L +* +0.001 +0.1 +1 +10 +100 +1000 +104 +Wavelength (μm)Rigliaco et al.: DESTINYS–TCrA +count for the replenishment of the outer disk due to the accretion +streamers (hence slowing down its expansion). A more detailed +simulation is needed for T CrA, but it is beyond the scope of this +observational paper and will be discussed in a separate publica- +tion. +In order to measure how the circumbinary disk mass dis- +tributes among the binary stars, we run a second hydrodynamical +model as the one described above but without the circumprimary +disk. Indeed, accretion into a binary system happens via the for- +mation of up to three disks (two circumstellar disks, one around +each component, and a circumbinary disk, Monin et al. (2007)). +The two circumstellar disks are periodically replenished by ac- +cretion streamers pulled from the inner edge of the circumbi- +nary disks by the stars (Artymowicz & Lubow 1994; Tofflemire +et al. 2017). In a quasi-steady state regime, the mass flux en- +tering the Roche lobe of a star via the gas streamers equals the +star accretion rate. Thus, we can reliably measure the fraction of +mass accreted onto a star by simulating only the circumbinary +disk, provided that the stellar Roche lobes are resolved by the +simulation and the central part of the disk has relaxed (as done +with SPH simulations e.g. in Young & Clarke 2015 and recently +tested in Ceppi et al. 2022). In general, simulations of accretion +into binary systems find that the primordial mass ratio is pushed +towards unity (that is, closer to equal masses in the binary com- +ponents) by accretion from a circumbinary disk (Clarke 2012). +This is due to the ease with which the secondary component ac- +cretes the infalling gas, as it lies farther from the binary barycen- +ter and closer to the disk edge. Its differential velocity with re- +spect to the gas is also low, allowing it to accrete efficiently. In +the case of T CrA the primary star is still accreting more than the +secondary (see Fig. 10). This is due to the misalignment between +inner and outer disk that makes the secondary to be at consider- +able height over/below the disk for a large fraction of its orbit. +5. Summary and Conclusions +We investigate new and archival data of the Herbig Ae/Be star +T CrA collected with different instruments. The analysis of the +data shows that T CrA is a very interesting and complex system, +belonging to one of the nearest and most active region of star +formation. Combining archival NACO imaging data with pho- +tometric data, and new and archival SPHERE adaptive optics +images we study the complex stellar environment around T CrA +and the stellar properties: +– the outer disk is seen edge-on as a dark lane elongated ap- +proximately in the N-S. The dark lane is shifted by 122 mas +with respect to the center of the image, and it is seen with +a PA of 7◦. This value is in very good agreement with the +value recently found by Cugno et al. 2022 using a different +instrument and set of data; +– the bright illuminated top-side of the disk surface is clearly +visible in scattered light; +– extended emission in the NE–SW direction, identified as fea- +ture 1, is consistent in direction with the line connecting the +two-lobed MHOs seen on larger scale. It is most likely out- +flowing material, with PA=33◦, consistent with the PA of the +two MHOs. +– extended emission in the N-S direction, identified as feature +2, is interpreted as large scale streamers of material likely in- +falling onto the disk. In the North the streamer extends up to +∼4.5′′ from the central system, while in the South it extends +up to the edge of the field of view, and probably beyond, as +suggested by previous stellar polarization images in the op- +tical and near-IR; +– the periodic behavior of the light curve suggests a cen- +tral binary with a period of 29.6 years. Even if the non- +coronagraphic images acquired with NACO and SPHERE do +not show direct evidence of the presence of a stellar compan- +ion, a detailed comparison of the position of the secondary +along the proposed orbit at the epochs of the observations +acquired so far with NACO and SPHERE shows that in all +of them it was too close to the primary star for detection as +a separate object. According to our modeling results the two +components will be at their maximum separation in 2027: ap- +propriate high-contrast images at that epoch should provide +direct evidence of the binary system. +Overall, we find that the binary system and intermediate +circumbinary disk lay on different geometrical planes, placing +T CrA among the objects with a misaligned inner disk. Inner +and outer disk misalignment is not rare, and in very recent years, +thanks to high-contrast imaging, it is becoming clear that the +misalignment can also be due to the accretion history of the star- +forming cloud onto the disk. Indeed in the case of T CrA (as well +as SU Aur) we found evidences of the presence of streamers of +accreting material that connect the filament along which the star +has formed with the outer part of the disk. These streamers have +an angular momentum with respect to the star whose direction is +very different from that of the system (in the case of T CrA, this +is dominated by the binary) causing a misalignment between an +inner and outer disk. +Besides characterizing the disk/outflow structures around +T CrA, we have also modeled its spectral energy distribution, +showing that the disk geometry obtained is well consistent with +the observed SED, and such consistency is not reached if we +do not consider the misalignment between inner and outer disk. +Moreover, we have performed hydrodynamical simulation of the +configuration for 100 orbits of the binary star. The model is +consistent with the observations and the analysis of the accre- +tion rates of the individual stars shows that the accretion hap- +pens mainly onto the primary star, rather than on the secondary, +as a consequence of the inclination between inner/intermediate +and outer disk. Also the light curve is easily explained assum- +ing the configuration of two misaligned disks. Comparison of +the ALMA continuum and 12CO emission have also been per- +formed. While for the continuum emission we cannot clearly +point out the region where the dust is located, if along the disk +or the outflowing material, the gas emission is most likely due +to the combination from emission aligned with the disk orienta- +tion inferred from SPHERE, and emission from the outflowing +material in the same direction as the MHOs. +The analysis conducted on T CrA has confirmed its ex- +tremely interesting and complex nature. As in the case of +HD142527, the misalignment between inner and outer disk can +be due to the interaction between the disk and the central binary +system. On the other hand, the large scale streamers observed in +the N–S direction are very similar to the disk-cloud interaction +observed for SU Aur, that represents material infalling onto the +disk, and inner and outer disk misalignment might be caused by +this interaction. It comes clear the need for high resolution obser- +vations to disentangle the different effects that shape early plan- +etary system formation. T CrA is an excellent target/laboratory +to better understand the impact of binarity and the environment +in the evolution of protoplanetary disks. +Acknowledgements. We would like to thank the referee Roubing Dong, whose +careful and constructive comments improved the quality of this manuscript. E.R. +was supported by the European Union’s Horizon 2020 research and innovation +programme under the Marie Skłodowska-Curie grant agreement No 664931. +This work has been supported by the project PRIN INAF 2016 The Cradle of Life +Article number, page 11 of 17 + +A&A proofs: manuscript no. TCrA_Rigliaco +Fig. 9: Snapshot of the SPH simulation compared to the observed image. The left panel shows the result in total intensity of the +SPH simulation, with a resolution of 4.0 mas/pixel. In the middle panel the same image convolved to the SPHERE/IRDIS resolution +(12.25 mas/pixel). On the right the observed total intensity image. All images have a 2′′ field of view. +Fig. 10: Mass accretion rate ratio of secondary and primary star +as a function of the number of orbits. +- GENESIS-SKA (General Conditions in Early Planetary Systems for the rise of +life with SKA) and by the "Progetti Premiali" funding scheme of the Italian Min- +istry of Education, University, and Research. C.F.M acknowledges funding from +the European Union under the European Union’s Horizon Europe Research & +Innovation Programme 101039452 (WANDA). Views and opinions expressed +are however those of the author(s) only and do not necessarily reflect those of +the European Union or the European Research Council. Neither the European +Union nor the granting authority can be held responsible for them. T.B. acknowl- +edges funding from the European Research Council (ERC) under the European +Union’s Horizon 2020 research and innovation programme under grant agree- +ment No 714769 and funding by the Deutsche Forschungsgemeinschaft (DFG, +German Research Foundation) under grants 361140270, 325594231, and Ger- +many’s Excellence Strategy - EXC-2094 - 390783311. A.R. has been supported +by the UK Science and Technology research Council (STFC) via the consoli- +dated grant ST/S000623/1 and by the European Union’s Horizon 2020 research +and innovation programme under the Marie Sklodowska-Curie grant agreement +No. 823823 (RISE DUSTBUSTERS project). This paper makes use of the fol- +lowing ALMA data: ADS/JAO.ALMA#2016.0.01058.S. ALMA is a partnership +of ESO (representing its member states), NSF (USA) and NINS (Japan), together +with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Ko- +rea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is +operated by ESO, AUI/NRAO and NAOJ. MRH acknowledges the assistance of +Allegro, the ARC node in the Netherlands, who assisted with the calibration of +this data set. This work is partly based on data products produced at the SPHERE +Data Centre hosted at OSUG/IPAG, Grenoble. We thank P. Delorme and E. La- +gadec (SPHERE Data Centre) for their efficient help during the data reduction +process. SPHERE is an instrument designed and built by a consortium consist- +ing of IPAG (Grenoble, France), MPIA (Heidelberg, Germany), LAM (Marseille, +France), LESIA (Paris, France), Laboratoire Lagrange (Nice, France), INAF Os- +servatorio Astronomico di Padova (Italy), Observatoire de Genève (Switzerland), +ETH Zurich (Switzerland), NOVA (Netherlands), ONERA (France) and AS- +TRON (Netherlands) in collaboration with ESO. SPHERE was funded by ESO, +with additional contributions from CNRS (France), MPIA (Germany), INAF +(Italy), FINES (Switzerland) and NOVA (Netherlands). SPHERE also received +funding from the European Commission Sixth and Seventh Framework Pro- +grammes as part of the Optical Infrared Coordination Network for Astronomy +(OPTICON) under grant number RII3-Ct-2004-001566 for FP6 (2004-2008), +grant number 226604 for FP7 (2009-2012), and grant number 312430 for FP7 +(2013-2016). This work has made use of data from the European Space Agency +(ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by +the Gaia Data Processing and Analysis Consortium (DPAC, https://www. +cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has +been provided by national institutions, in particular the institutions participating +in the Gaia Multilateral Agreement. We acknowledge with thanks the variable +star observations from the AAVSO International Database contributed by ob- +servers worldwide and used in this research. +References +Artymowicz, P. & Lubow, S. H. 1994, ApJ, 421, 651 +Avenhaus, H., Quanz, S. P., Garufi, A., et al. 2018, ApJ, 863, 44 +Bailey, J. 1998, MNRAS, 301, 161 +Balmer, W. O., Follette, K. B., Close, L. M., et al. 2022, arXiv e-prints, +arXiv:2206.00687 +Baraffe, I., Homeier, D., Allard, F., & Chabrier, G. 2015, A&A, 577, A42 +Bate, M. R. 2018, MNRAS, 475, 5618 +Bate, M. R., Bonnell, I. A., & Price, N. M. 1995, MNRAS, 277, 362 +Bate, M. R., Lodato, G., & Pringle, J. E. 2010, MNRAS, 401, 1505 +Bernabò, L. M., Turrini, D., Testi, L., Marzari, F., & Polychroni, D. 2022, ApJ, +927, L22 +Beuzit, J. L., Vigan, A., Mouillet, D., et al. 2019, A&A, 631, A155 +Bohn, +A. +J., +Benisty, +M., +Perraut, +K., +et +al. +2021, +arXiv +e-prints, +arXiv:2112.00123 +Bonavita, M., Gratton, R., Desidera, S., et al. 2021, arXiv e-prints, +arXiv:2103.13706 +Cazzoletti, P., Manara, C. F., Liu, H. B., et al. 2019, A&A, 626, A11 +Ceppi, S., Cuello, N., Lodato, G., et al. 2022, MNRAS[arXiv:2205.08784] +Clark, S., McCall, A., Chrysostomou, A., et al. 2000, MNRAS, 319, 337 +Clarke, C. J. 2012, in From Interacting Binaries to Exoplanets: Essential Model- +ing Tools, ed. M. T. Richards & I. Hubeny, Vol. 282, 409–416 +Cugno, G., Pearce, T. D., Launhardt, R., et al. 2022, arXiv e-prints, +arXiv:2211.15434 +Cutri, R. M. & et al. 2012, VizieR Online Data Catalog, II/311 +Cutri, R. M., Skrutskie, M. F., van Dyk, S., et al. 2003, 2MASS All Sky Catalog +of point sources. +Davis, C. J., Gell, R., Khanzadyan, T., Smith, M. D., & Jenness, T. 2010, A&A, +511, A24 +de Boer, J., Langlois, M., van Holstein, R. G., et al. 2020, A&A, 633, A63 +Article number, page 12 of 17 + +1.0 +0.8 +M +0.6 +M2/ +0.4 +0.2 +0.0 +0 +50 +100 +150 +200 +250 +300 +Orbits10-1 +6× 10-1 +10-1 +10-3 +4× 10-1 +3× 10-1 +10-5 +10-2 +2×10-1 +10-7 +1.0" +1.0" +1.0"Rigliaco et al.: DESTINYS–TCrA +Delorme, P., Meunier, N., Albert, D., et al. 2017, in SF2A-2017: Proceedings of +the Annual meeting of the French Society of Astronomy and Astrophysics, +ed. C. Reylé, P. Di Matteo, F. Herpin, E. Lagadec, A. Lançon, Z. Meliani, & +F. Royer, Di +Dohlen, K., Langlois, M., Saisse, M., et al. 2008, in Society of Photo-Optical +Instrumentation Engineers (SPIE) Conference Series, Vol. 7014, Proc. SPIE, +70143L +Dong, R., Najita, J. R., & Brittain, S. 2018, ApJ, 862, 103 +Dullemond, C. P., Küffmeier, M., Goicovic, F., et al. 2019, A&A, 628, A20 +Dzib, S. A., Loinard, L., Ortiz-León, G. N., Rodríguez, L. F., & Galli, P. A. B. +2018, ApJ, 867, 151 +Facchini, S., Juhász, A., & Lodato, G. 2018, MNRAS, 473, 4459 +Facchini, S., Lodato, G., & Price, D. J. 2013, MNRAS, 433, 2142 +Frank, A., Ray, T. P., Cabrit, S., et al. 2014, in Protostars and Planets VI, ed. +H. Beuther, R. S. Klessen, C. P. Dullemond, & T. Henning, 451 +Gaia Collaboration. 2020, VizieR Online Data Catalog, I/350 +Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2021, A&A, 649, A1 +Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1 +Galli, P. A. B., Bouy, H., Olivares, J., et al. 2020, A&A, 634, A98 +Garufi, A., Avenhaus, H., Pérez, S., et al. 2020, A&A, 633, A82 +Garufi, A., Podio, L., Codella, C., et al. 2022, A&A, 658, A104 +Ghez, A. M., McCarthy, D. W., Patience, J. L., & Beck, T. L. 1997, ApJ, 481, +378 +Gingold, R. A. & Monaghan, J. J. 1977, MNRAS, 181, 375 +Ginski, C., Facchini, S., Huang, J., et al. 2021, ApJ, 908, L25 +Ginski, C., Ménard, F., Rab, C., et al. 2020, A&A, 642, A119 +Gutermuth, R. A., Megeath, S. T., Myers, P. C., et al. 2009, ApJS, 184, 18 +Henden, A. A., Templeton, M., Terrell, D., et al. 2016, VizieR Online Data Cat- +alog, II/336 +Hennebelle, P. & Ciardi, A. 2009, A&A, 506, L29 +Herbig, G. H. 1960, ApJS, 4, 337 +Herbig, G. H. & Bell, K. R. 1988, Third Catalog of Emission-Line Stars of the +Orion Population : 3 : 1988 +Herczeg, G. J. & Hillenbrand, L. A. 2014, ApJ, 786, 97 +Herschel Group, Marton, G., Calzoletti, L., et al. 2020, VizieR Online Data Cat- +alog, VIII/106 +Herter, T. L., Adams, J. D., Gull, G. E., et al. 2018, Journal of Astronomical +Instrumentation, 7, 1840005 +Joos, M., Hennebelle, P., & Ciardi, A. 2012, A&A, 543, A128 +Joy, A. H. 1945, ApJ, 102, 168 +Kafka, S. 2020, in European Planetary Science Congress, EPSC2020–314 +Köhler, R., Neuhäuser, R., Krämer, S., et al. 2008, A&A, 488, 997 +Krumholz, M. R., Crutcher, R. M., & Hull, C. L. H. 2013, ApJ, 767, L11 +Kuffmeier, M., Dullemond, C. P., Reissl, S., & Goicovic, F. G. 2021, arXiv e- +prints, arXiv:2110.04309 +Kumar, M. S. N., Sharma, S., Davis, C. J., Borissova, J., & Grave, J. M. C. 2011, +A&A, 533, A137 +Lenzen, R., Hartung, M., Brandner, W., et al. 2003, in Society of Photo-Optical +Instrumentation Engineers (SPIE) Conference Series, Vol. 4841, Instrument +Design and Performance for Optical/Infrared Ground-based Telescopes, ed. +M. Iye & A. F. M. Moorwood, 944–952 +Lewis, B. T. & Bate, M. R. 2017, MNRAS, 467, 3324 +Lewis, B. T., Bate, M. R., & Price, D. J. 2015, MNRAS, 451, 288 +Li, H.-b., Fang, M., Henning, T., & Kainulainen, J. 2013, MNRAS, 436, 3707 +Liang, L., Johnstone, D., Cabrit, S., & Kristensen, L. E. 2020, ApJ, 900, 15 +Lindberg, J. E. & Jørgensen, J. K. 2012, A&A, 548, A24 +Lodato, G. & Price, D. J. 2010, MNRAS, 405, 1212 +Lucy, L. B. 1977, AJ, 82, 1013 +Machida, M. N., Matsumoto, T., Hanawa, T., & Tomisaka, K. 2006, ApJ, 645, +1227 +Maire, A.-L., Langlois, M., Dohlen, K., et al. 2016, in Society of Photo-Optical +Instrumentation Engineers (SPIE) Conference Series, Vol. 9908, Ground- +based and Airborne Instrumentation for Astronomy VI, ed. C. J. Evans, +L. Simard, & H. Takami, 990834 +Manara, C. F., Morbidelli, A., & Guillot, T. 2018, A&A, 618, L3 +Marois, C., Lafrenière, D., Doyon, R., Macintosh, B., & Nadeau, D. 2006, ApJ, +641, 556 +Matsumoto, T., Nakazato, T., & Tomisaka, K. 2006, ApJ, 637, L105 +Matsumoto, T. & Tomisaka, K. 2004, ApJ, 616, 266 +Mendigutía, I., Eiroa, C., Montesinos, B., et al. 2011, A&A, 529, A34 +Mesa, D., Bonnefoy, M., Gratton, R., et al. 2019, A&A, 624, A4 +Meyer, M. R. & Wilking, B. A. 2009, PASP, 121, 350 +Monaghan, J. J. 2005, Reports on Progress in Physics, 68, 1703 +Monin, J. L., Clarke, C. J., Prato, L., & McCabe, C. 2007, in Protostars and +Planets V, ed. B. Reipurth, D. Jewitt, & K. Keil, 395 +Nascimbeni, V., Piotto, G., Ortolani, S., et al. 2016, MNRAS, 463, 4210 +Nealon, R., Cuello, N., & Alexander, R. 2020, MNRAS, 491, 4108 +Owen, J. E. & Lai, D. 2017, MNRAS, 469, 2834 +Pascucci, I., Banzatti, A., Gorti, U., et al. 2020, ApJ, 903, 78 +Pikhartova, M., Long, Z. C., Assani, K. D., et al. 2021, ApJ, 919, 64 +Pinte, C., Dent, W. R. F., Ménard, F., et al. 2016, ApJ, 816, 25 +Pojmanski, G. 1997, Acta Astron., 47, 467 +Pontoppidan, K. M., Blake, G. A., & Smette, A. 2011, ApJ, 733, 84 +Price, D. J. 2012, Journal of Computational Physics, 231, 759 +Price, D. J., Cuello, N., Pinte, C., et al. 2018a, MNRAS, 477, 1270 +Price, D. J., Wurster, J., Tricco, T. S., et al. 2018b, PASA, 35, e031 +Rigliaco, E., Gratton, R., Mesa, D., et al. 2019, A&A, 632, A18 +Roeser, S., Demleitner, M., & Schilbach, E. 2010, AJ, 139, 2440 +Rousset, G., Lacombe, F., Puget, P., et al. 2003, in Society of Photo-Optical In- +strumentation Engineers (SPIE) Conference Series, Vol. 4839, Adaptive Op- +tical System Technologies II, ed. P. L. Wizinowich & D. Bonaccini, 140–149 +Sandell, G., Reipurth, B., Vacca, W. D., & Bajaj, N. S. 2021, ApJ, 920, 7 +Shakura, N. I. & Sunyaev, R. A. 1973, in IAU Symposium, Vol. 55, X- and +Gamma-Ray Astronomy, ed. H. Bradt & R. Giacconi, 155 +Shappee, B. J., Prieto, J. L., Grupe, D., et al. 2014, ApJ, 788, 48 +Sicilia-Aguilar, A., Henning, T., Kainulainen, J., & Roccatagliata, V. 2011, ApJ, +736, 137 +Sicilia-Aguilar, A., Henning, T., Linz, H., et al. 2013, A&A, 551, A34 +Siess, L., Dufour, E., & Forestini, M. 2000, A&A, 358, 593 +Takami, M., Bailey, J., & Chrysostomou, A. 2003, A&A, 397, 675 +Testi, L., Natta, A., Manara, C. F., et al. 2022, A&A, 663, A98 +Tofflemire, B. M., Mathieu, R. D., Ardila, D. R., et al. 2017, ApJ, 835, 8 +van Holstein, R. G., Girard, J. H., de Boer, J., et al. 2020, A&A, 633, A64 +Varga, J., Ábrahám, P., Chen, L., et al. 2018, A&A, 617, A83 +Ward-Thompson, D., Warren-Smith, R. F., Scarrott, S. M., & Wolstencroft, R. D. +1985, MNRAS, 215, 537 +Whitney, B. A., Wood, K., Bjorkman, J. E., & Cohen, M. 2003a, ApJ, 598, 1079 +Whitney, B. A., Wood, K., Bjorkman, J. E., & Wolff, M. J. 2003b, ApJ, 591, +1049 +Woitke, P., Min, M., Pinte, C., et al. 2016, A&A, 586, A103 +Wolf, C., Onken, C. A., Luvaul, L. C., et al. 2018, PASA, 35, e010 +Wurster, J. & Li, Z.-Y. 2018, Frontiers in Astronomy and Space Sciences, 5, 39 +Young, M. D. & Clarke, C. J. 2015, MNRAS, 452, 3085 +Zacharias, N., Finch, C. T., Girard, T. M., et al. 2012, VizieR Online Data Cata- +log, I/322A +Zhu, Z. 2019, MNRAS, 483, 4221 +Article number, page 13 of 17 + +A&A proofs: manuscript no. TCrA_Rigliaco +Appendix A: SPHERE polarimetric images +In Figure A.1 we present the Stokes Q and U, as well as the +derived QΦ and UΦ images of T CrA. The flux calibration was +carried out by measuring the flux of the central star in the non- +coronagraphic flux calibration images, taken at the beginning +and end of the observation sequence. To convert pixel counts +to physical units we used the 2MASS H-band magnitude of the +star. +4 +3 +2 +1 +0 +1 +2 +3 +4 +4 +3 +2 +1 +0 +1 +2 +3 +4 +Q +4 +3 +2 +1 +0 +1 +2 +3 +4 +4 +3 +2 +1 +0 +1 +2 +3 +4 +U +4 +3 +2 +1 +0 +1 +2 +3 +4 +4 +3 +2 +1 +0 +1 +2 +3 +4 +Qφ +4 +3 +2 +1 +0 +1 +2 +3 +4 +4 +3 +2 +1 +0 +1 +2 +3 +4 +Uφ +∆RA (arcsec) +∆Dec (arcsec) +60 +45 +30 +15 +0 +15 +30 +45 +60 +(mJy/arcsec2) +Fig. A.1: Flux calibrated image of the Q, U, QΦ and UΦ frames. +Appendix B: Proper motion analysis +The average proper motion for the on-cloud Coronet cluster +members obtained using Gaia DR2 data from Galli et al. 2020, +is µα cos δ = 4.3 mas yr−1 and µδ=-27.3 mas yr−1 with a small +dispersion of less than 1 mas/yr for the individual objects. As we +mentioned in the introduction, Gaia does not provide astrometric +solutions and proper motion for the star T CrA. However, we can +check for peculiar/transient motion of the star using the follow- +ing procedure. We collect from UCAC4 (Zacharias et al. 2012), +PPMXL (Roeser et al. 2010) and Gaia DR3 (Gaia Collaboration +et al. 2016, 2021) database a list of 10 bright stars in the T CrA +surroundings for which proper motion are available in all cata- +logs. These stars, listed in Table B.1, are selected such that they +have magG ≤14 and lay within 10′ from T CrA. For these stars +we measure a long term proper motion given by the difference in +position between the UCAC4, PPMXL, and Gaia DR3 epochs. +The long term proper motion, defined as the motion of the star +between different epochs of observations is measured as: +µα cos δ = (RAEpoch1 − RAEpoch2) ∗ cos(DECEpoch1) +(Epoch1 − Epoch2) +(B.1) +µδ = (DECEpoch1 − DECEpoch2) +(Epoch1 − Epoch2) +(B.2) +The analysis of the proper motion between the various epochs of +the selected stars allows us to find a systematic offset between the +average of the coordinate systems of UCAC4 and PPMXL with +respect to Gaia DR3, that averages to 0.41±2.46 mas yr−1 in RA +and 9.67 mas yr−1±2.93 mas yr−1 in DEC for this specific region +of the sky. For T CrA we obtain an estimate of the proper motion +of the star by correcting the long term proper motion with the +systematic offset, finding as values µα cos δ = 8.5 ± 2.5 mas yr−1 +and µδ=-33.5±2.9 mas yr−1. Considering the average proper +motion of the on-cloud members we find for T CrA an ap- +parent motion µα cos δ = 4.2 ± 2.5 mas yr−1 in RA and µδ=- +6.2±2.9 mas yr−1 in DEC. +Fig. B.1: Proper motion of the ten stars reported in Table B.1. +The cyan square represents the average proper motion for the on- +cloud Coronet cluster obtained using Gaia DR2 data (Galli et al. +2020). The blue star is the calculated apparent proper motion +of T CrA after correcting the long term proper motion for the +systematic offset. In orange the direction of the systematic offset +of the proper motion due to the different coordinate system. +Article number, page 14 of 17 + +-10 +PMfromGaiafortheselectedstars +average PM of the on-cloud members +apparent PM of TCrA +-20 +μ(mas/yr) +-30 +40 +average systematic offset +betweenGaiaandUCAC4/PPMXL +50 +40 +-20 +0 +20 +μαcosd(mas/yr)Rigliaco et al.: DESTINYS–TCrA +T CrA +V709 CrA +HD176269 +HD176270 +TY CrA +HD176423 +V702 CrA +HD176386 +HD176497 +HD176018 +CD-36 13202 +(UCAC4) +RA +285.494904 +285.395229 +285.263532 +285.267908 +285.420107 +285.460076 +285.508229 +285.412217 +285.528297 +284.930902 +284.920385 +Dec +-36.963871 +-37.015723 +-37.060898 +-37.061555 +-36.876063 +-36.664651 +-37.128761 +-36.890712 +-36.361622 +-36.788004 +-36.588797 +Ep. RA +1997.40 +1985.45 +1991.25 +1991.25 +1991.09 +1990.50 +1985.88 +1991.25 +1990.57 +1988.91 +1995.62 +Ep. Dec +1997.77 +1985.43 +1991.25 +1991.25 +1990.52 +1989.91 +1984.44 +1991.25 +1990.28 +1988.04 +1995.74 +(PPMXL) +RA +285.494908 +285.395224 +285.263532 +285.267908 +285.420102 +285.460076 +285.508229 +285.412219 +285.528303 +284.930902 +284.920394 +Dec +-36.963869 +-37.015722 +-37.060898 +-37.061555 +-36.876064 +-36.664651 +-37.128761 +-36.890703 +-36.361625 +-36.788007 +-36.588800 +Ep. RA +1999.95 +1988.00 +1991.73 +1991.18 +1991.53 +1991.41 +1997.44 +1991.14 +1991.32 +1991.23 +1997.69 +Ep. Dec +1999.95 +1986.70 +1991.64 +1991.19 +1991.76 +1991.62 +1998.14 +1991.09 +1991.38 +1991.64 +1998.44 +(Gaia DR3) +RA +285.494959 +285.395278 +285.263578 +285.267966 +285.420142 +285.460103 +285.508268 +285.412238 +285.528324 +284.930856 +284.920226 +Dec +-36.963983 +-37.015845 +-37.061040 +-37.061682 +-36.876201 +-36.664770 +-37.128870 +-36.890838 +-36.361752 +-36.788188 +-36.589008 +Ep. RA +2016.0 +2016.0 +2016.0 +2016.0 +2016.0 +2016.0 +2016.0 +2016.0 +2016.0 +2016.0 +2016.0 +Ep. Dec +2016.0 +2016.0 +2016.0 +2016.0 +2016.0 +2016.0 +2016.0 +2016.0 +2016.0 +2016.0 +2016.0 +Table B.1: List of the stars used to measure the proper motion offset. +Article number, page 15 of 17 + +A&A proofs: manuscript no. TCrA_Rigliaco +Epoch +Offset B-A (mas) +dH (mag) +2007.54 (NACO) +-26.0±7.0 +1.0±0.6 +2016.25 (NACO) +-72.0±5.0 +0.0±0.7 +2016.60 (SPHERE) +-69.0±5.0 +0.2±0.7 +2018.36 (SPHERE ) +-44.0±7.0 +0.3±0.6 +2021.50 (SPHERE) +11.0±7.0 +1.0±0.5 +Table C.1: Relative position of the secondary star (B) with re- +spect to the primary star (A), and relative contrast (dH) in H- +band of the secondary star with respect to the primary star, for a +period of 29.6 years. The offset is defined in the direction of the +semi-major axis of the stellar orbit. +Appendix C: Binarity and light curve +The light curve of T CrA appears to be periodic. The period is +found to be 29.6 years and it can be due to the presence of a +binary star at the center of the T CrA system with a mass ratio +q∼0.5±0.2, that is partially obscured by a disk seen edge-on, that +has an offset with respect to the photocenter of the binary star of +∼90 mas. The model of this binary system, described in Sect 3.1, +is also able to account for the large apparent proper motion mea- +sured in the period between 1998 and 2016. None of the images +acquired in recent years with NACO (in 2007, 2016 and 2017) +and SPHERE (in 2016, 2018 and 2021) shows clear evidence of +a binary system for T CrA. Hence, we have checked what was +the relative position of the secondary star with respect to the pri- +mary for every single epoch for which we have an image, and +the H-band contrast that should be observed. These quantities +are shown in Table C.1. These value are all consistent with the +fact that the binary system is not clearly resolved. Indeed, in the +2007, 2018 and 2021 epochs the separation between the two stars +is too small to see the two sources separately. On the contrary, +the two 2016 epochs have a larger separation, though still within +2×λ/D, that is so close that the secondary cannot be clearly sep- +arated from the primary. We notice however, that in both images +acquired around this epoch with NACO and SPHERE, the PSF +appears elongated in the NW-SE direction, that corresponds to +the direction of the major axis of the orbital motion of the bi- +nary system. The average position angle of the elongated PSFs +acquired in 2016 is 130±15◦, in very good agreement with the +direction of the peculiar proper motion (PAPM) measured, and +with the hypothesis that the orbit of the binary system is seen +edge-on, and perpendicular to the outflow. This elongation in +different epochs supports then the scenario of a binary star. In +a few years, namely in 2027, when the system is at its highest +separation, the secondary component should be detectable with +high-contrast images. +We have also considered that the period of the system +might be double than the period measured in Sect. 2.3, namely +59.2 years. While the light curve can be, also in this case, eas- +ily reproduced, there are several observational shortcomings in +this interpretation. First of all we must notice that in this case +the model predicts a mass ratio q as high as 0.9, and an off- +set of the disk of ∼10 mas. This last quantity is in disagree- +ment with the observations, that instead show that the disk +dark lane has an offset ten times larger. Moreover, the position +of the center of the binary system as retrieved by assuming a +59.2 years period is not consistent with the motion of the system +obtained from UCAC4/PPMXL and Gaia DR3 data. Addition- +ally, in the SPHERE image acquired in 2021, the predicted sep- +aration between the primary and secondary component should +be 108±6 mas, with a contrast dH=0.5±0.1 mag, making it visi- +ble as a separate point source in the image. The relative position +Epoch +Offset B-A (mas) +dH (mag) +2007.54 (NACO) +91.0±7.0 +1.0±0.2 +2016.25 (NACO) +-39.0±8.0 +0.0±0.3 +2016.60 (SPHERE) +-44.0±8.0 +0.0±0.2 +2018.36 (SPHERE) +-70.0±7.0 +0.0±0.2 +2021.50 (SPHERE) +-108.0±6..0 +0.0 ± 0.1 +Table C.2: Relative position of the secondary star (B) with re- +spect to the primary star (A), and relative contrast (dH) in H- +band of the secondary star with respect to the primary star, for a +period of 59.2 years. The offset is defined in the direction of the +semi-major axis of the stellar orbit. +of the secondary star with respect to the primary for every sin- +gle epoch for which we have an image, and the H-band contrast +that should be observed are reported in Table C.2. The image +does not reveal the presence of the secondary star. Given these +shortcomings between observations and the output of the model, +we exclude that the period of the binary star is 59.2 years. Fig- +ure C.1 and C.2 show the corner plot of the derived quantities +of the model used in Sect. 4.1 to model the light curve assuming +a period of 29.6 years or the double (59.2 years). The light curve +for the period of 59.2 years is shown in Fig. C.3. +Fig. C.1: Corner plot showing the results of the MC parameters +estimation for the model described in the paper when a period of +29.2 years is considered. The plots show the 2D joints posterior +densities of all couple of parameters. +Article number, page 16 of 17 + +2009 +2008 +2007 + 2006 +2005 +2004 +2003 +-1.0-0.8-0.6-0.4-0.2 0.0 +log q +(mag) +12 +Max absorption +10 +8 +6 +4 +1.0-0.8-0.6-0.4-0.2 0.0 +log q +2003 +2004 +2005 +2006 +2007 +2008 +2009 +TO +Disk Thickness (mas) +Disk Thickness (mas) +3688885 +40 +140 +120 +8688867 +100E +Thickness +80 E +60 E +40 +20 E +1.0-0.8-0.6-0.4-0.2 0.0 +6 +8 +10 +12 +log q +20032004 +2005 +2006 +2007 +20082009 +Max absorption (mag) +TO +148 +160E +(mas) +140E +offset (mas) +160 +(mas) +160 +148 +(sDw) +120 E +2888898 +offset +Disk offset +100日 +offset +80 E +Disk +60 E +Disk +40 E +1.0-0.8-0.6-0.4-0.2 0.0 +20日 +4 +6 +8 +10 +12 +20 40 60 80100120140 +b 6ol +2003 +20042005 +2006 +200720082009 +Max absorption (mag) +Disk thickness (mas) +TORigliaco et al.: DESTINYS–TCrA +Fig. C.2: Corner plot showing the results of the MC parameters +estimation for the model described in the paper when a period of +59.6 years is considered. The plots show the 2D joints posterior +densities of all couple of parameters. +Fig. C.3: Light curve of T CrA (red points) compared to the light +curves computed with the MC model (black lines) assuming a +period of 59.2 years. +Article number, page 17 of 17 + +2017 +2016 +2015 + 2014 +2013 +2012 +2011 +1.0-0.8-0.6-0.4-0.2 0.0 +log q +absorption +XDW +.5 +1.0-0.8-0.6-0.4-0.2 0.0 +log q +2011 +2012 +2013 +2014 +2015 +20162017 +TO +(mas) +(mas) +140 +40 +140 +2888898 +120 +Disk Thickness +Thickness +00 +100 +80 +80 +60 +60 +40 +MSI +40 +Disk +20 +1.0-0.8-0.6-0.4-0.2 0.0 +4.0 4.5 5.0 5.5 6.0 6.5 7.0 +log q +2011 +2012 +2013 +2014 +2015 +2016 +2017 +Max absorption (mag) +TO +< offset (mas) +(mas) +(mas) +40 +40 +40 +40 +SDU +20 +20 +20 +offset +< offset +0 +0 +0 +Disk +Disk +20 +Disk +Disk +20 +-1.0-0.8-0.6-0.4-0.2 0.0 +4.0 4.5 5.0 5.5 6.0 6.5 7.0 +20 40 60 80100120140 +log q +201120122013 +2014 +201520162017 +Max absorption (mag) +Disk thickness (mas) +TO12 +mag +13 +15 +1900 +1920 +1940 +1960 +1980 +2000 +2020 +Time +(yr \ No newline at end of file diff --git a/ANAzT4oBgHgl3EQfhf3F/content/tmp_files/load_file.txt b/ANAzT4oBgHgl3EQfhf3F/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b4fbe5b994af486923245c7e043e7b19521028f3 --- /dev/null +++ b/ANAzT4oBgHgl3EQfhf3F/content/tmp_files/load_file.txt @@ -0,0 +1,1964 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf,len=1963 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' TCrA_Rigliaco ©ESO 2023 January 5, 2023 Disk Evolution Study Through Imaging of Nearby Young Stars (DESTINYS): Characterization of the young star T CrA and its circumstellar environment ⋆ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Rigliaco1, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Gratton1, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Ceppi2, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Ginski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3, 4, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Hogerheijde3, 4, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Benisty5, 6, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Birnstiel7, 8, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Dima1, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Facchini2, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Garufi9, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Bae10, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Langlois11, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Lodato2, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Mamajek12, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Manara13, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Ménard14, Á.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Ribas15, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Zurlo16, 17, 18 1 INAF/Osservatorio Astronomico di Padova, Vicolo dell’osservatorio 5, 35122 Padova e-mail: elisabetta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='rigliaco@inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='it 2 Dipartimento di Fisica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Università Degli Studi di Milano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Via Celoria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Milano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 20133,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Italy 3 Anton Pannekoek Institute for Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' University of Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Science Park 904,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1098XH Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The Netherlands 4 Leiden Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Leiden University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' PO Box 9513,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2300 RA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Leiden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The Netherlands 5 Unidad Mixta Internacional Franco-Chilena de Astronomía,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' CNRS/INSU UMI 3386,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Departamento de Astronomía,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Universidad de Chile,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Camino El Observatorio 1515,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Las Condes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Santiago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Chile 6 Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France 7 University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität München, Scheinerstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1, 81679 Munich, Germany 8 Exzellenzcluster ORIGINS, Boltzmannstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' D-85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Germany 9 INAF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Osservatorio Astrofisico di Arcetri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Largo Enrico Fermi 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 50125,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Firenze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Italy 10 Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' University of Florida,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Gainesville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' FL 32611,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' United States of America 11 CRAL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' UMR 5574,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Université Lyon 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 9 avenue Charles André,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 69561 Saint-Genis-Laval Cedex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' France 12 Jet Propulsion Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' California Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 4800 Oak Grove Drive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Pasadena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' CA 91109,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' USA 13 European Southern Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Karl-Schwarzschild-Strasse 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 85748 Garching bei München,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Germany 14 Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France 15 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK 16 Núcleo de Astronomía, Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Ejercito 441, Santiago, Chile 17 Escuela de Ingeniería Industrial, Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Ejercito 441, Santiago, Chile 18 Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France Received 12 October 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' accepted 22 December 2022 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In recent years it is emerging a new hot-topic in the star and planet formation field: the interaction between circumstellar disk and its birth cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Birth environments of young stars have strong imprints on the star itself and their surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In this context we present a detailed analysis of the wealthy circumstellar environment around the young Herbig Ae/Be star T CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Our aim is to understand the nature of the stellar system and the extended circumstellar structures as seen in scattered light images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We conduct our analysis combining archival data, and new adaptive optics high-contrast and high-resolution images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The scattered light images reveal the presence of a complex environment around T CrA composed of a bright forward scattering rim of the disk’s surface that is seen at very high inclination, a dark lane of the disk midplane, bipolar outflows, and streamer features likely tracing infalling material from the surrounding birth cloud onto the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The analysis of the light curve suggests that the star is a binary with a period of 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6 years, confirming previous assertions based on spectro-astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The comparison of the scattered light images with ALMA continuum and 12CO (2–1) line emission shows that the disk is in keplerian rotation, and the northern side of the outflowing material is receding, while the southern side is approaching to the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The overall system lays on different geometrical planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The orbit of the binary star is perpendicular to the outflows and is seen edge on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The disk is itself seen edge-on, with a position angle of ∼7◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The direction of the outflows seen in scattered light is in agreement with the direction of the more distant molecular hydrogen emission-line objects (MHOs) associated to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Modeling of the spectral energy distribution (SED) using a radiative transfer scheme well agrees with the proposed configuration, as well as the hydrodynamical simulation performed using a Smoothed Particle Hydrodynamics (SPH) code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We find evidence of streamers of accreting material around T CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' These streamers connect the filament along which T CrA is forming with the outer parts of the disk, suggesting that the strong misalignment between the inner and outer disk is due to a change in the direction of the angular momentum of the material accreting on the disk during the late phase of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This impacts the accretion on the components of the binary, favoring the growth of the primary with respect the secondary, as opposite to the case of aligned disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' stars: pre-main sequence, circumstellar matter – protoplanetary disks – ISM: individual object: T CrA – ISM: jets and outflows Article number, page 1 of 17 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='01486v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='SR] 4 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' TCrA_Rigliaco 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Introduction Herbig Ae/Be stars (Herbig 1960) are pre-main sequence stars with intermediate mass covering the range between low-mass T Tauri stars (TTSs) and the embedded massive young stellar ob- jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The formation of stars in the low and intermediate-mass regimes involves accreting disks formed during the collapse of the protostar, and fast collimated outflows and jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The circum- stellar environment of these objects is highly dynamic and multi- wavelengths observations show large photometric and spectro- scopic variability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Pikhartova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Mendigutía et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2011) that can be used as a tool to understand the physics of ac- cretion and ejection related to the interaction between the star and its circumstellar environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' T CrA (RA=19:01:58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='79 DEC=-36:57:50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='33) is an Herbig Ae/Be star member of the Coronet Cluster, belonging to the Corona Australis star-forming region, which is one of the near- est (149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4 pc, Galli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020) and most active regions of ongoing star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The Coronet Cluster is centered on the Herbig Ae/Be stars R CrA and T CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' It is very active in star formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Lindberg & Jørgensen 2012), harboring many Herbig-Haro (HHs) objects and Molecular Hydrogen emission- line Objects (MHOs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' It has been target of many surveys, and all studies agree in assigning the Coronet an age <3 Myr (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Meyer & Wilking 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Sicilia-Aguilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In this pa- per we investigate the variable star T CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' T CrA is classified as F0 by Joy (1945) with effective temperature Teff=7200 K, and according to Cazzoletti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2019) and Herczeg & Hil- lenbrand (2014) this corresponds to L∗ ∼29 L⊙, and stellar mass ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='25 M⊙ using the evolutionary tracks by Siess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2000, and adopting the average distance of 154 pc calculated by Dzib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The Gaia-DR2 and DR3 catalogs (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2016, 2021) do not provide proper motion or parallax for T CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This star was not observed by the Hipparcos satellite and it is also not listed in the UCAC5 catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The former UCAC4 catalog (Zacharias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2012) provides a proper motion result (µα cos δ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='8 mas yr−1, µδ=-22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='8 mas yr−1), which is consistent with membership in Corona-Australis (within the large uncertainties of that solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Galli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2020) provided an updated census of the stellar population in the Corona Aus- tralis deriving an average distance of 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This is the distance we will use throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A deep H2 v=1–0 S(1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='12 µm narrow-band imaging survey of the northern part of the Corona Australis cloud conducted by Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2011) identified many new MHOs (Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Among these objects, two are considered unambiguously associated to T CrA: MHO2013 and MHO2015, see Figure 3 in Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' MHO 2015 is a clear bow-shock feature, lying to the south of T CrA, and it marks the southern lobe of the bipolar outflow originating from T CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' MHO 2013 marks the northern lobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The hypothetical line connecting the two MHOs crosses the po- sition of T CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This is the only unambiguously detected bipolar outflow traced by two complementing bow-shock features in the entire Coronet region (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We reproduce the im- age shown in Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2011) in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' T CrA was suggested to be a binary system by Bailey (1998) and Takami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2003) who adopted spectro-astrometry in the Hα line suggesting that the system is a binary with a compan- ion at >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='14′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' However, no companion has been detected us- ing spectro-astrometry in the fundamental rovibrational band of CO at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6µm (Pontoppidan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2011) nor with K-band speckle ⋆ Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO pro- gramme 1104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='C-0415(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' imaging (Ghez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Köhler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In the same years, infrared speckle observations performed by Ghez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (1997) did not show the presence of a stellar companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The non-detection of the companion by Ghez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (1997) implies that the possible companion has a contrast in the K-band larger than 3 mag (that is a K-magnitude fainter than 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5) or a sep- aration smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 arcsec at the epoch of the observation (April 26, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' see also Takami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Recently, the circumstellar environment of T CrA has been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' SOFIA/FORCAST (Faint Object infraRed CAm- era for the SOFIA Telescope, Herter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018) observations show very strong excess in the far-IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' T CrA was also de- tected in all Herschel/PACS (Photodetector Array Camera and Spectrometer) bands (Sandell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2021), highlighting the pres- ence of warm or hot dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Mid-infrared interferometric data obtained with VLT/MIDI (MID-infrared Interferometric instru- ment) show the presence of disk emission from the inner regions, where the temperature is sufficiently high (Varga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The presence of the inner disk is also given by the spectral en- ergy distribution (SED) which shows near-IR excess emission (Sicilia-Aguilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Optical and IR spectra covering the [OI] λ6300 and [NeII] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='81 µm lines (Pascucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020) show emission attributed to a jet nearly in the plane of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Moreover, continuum ALMA observations of T CrA at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3 mm (230 GHz) were conducted as part of the survey of protoplan- etary disks in Corona Australis (Cazzoletti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2019) and the data show a ∼22σ detection at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='34′′ from the nominal Spitzer position that is considered as detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3 mm continuum flux is then converted into a dust mass (Mdust) under the assump- tion of optically thin and isothermal sub-millimeter emission, yielding Mdust=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='64±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='27 M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' No information on the 12CO(2- 1) gas content in the disk are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The average disk mass in CrA is 6±3 M⊕, and it is significantly lower than that of disks in other young (1–3 Myr) star forming regions (Lupus, Taurus, Chamaeleon I, and Ophiuchus) and appears to be consistent with the average disk mass of the 5–10 Myr-old Upper Sco (Cazzo- letti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In this paper we analyze images of T CrA acquired with the Very Large Telescope at ESO’s Paranal Observatory in Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We employ polarimetric differential imaging (PDI) observations obtained with SPHERE (Spectro-Polarimetric High-contrast Ex- oplanet REsearch, Beuzit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2019) in the H band to explore the circumstellar environment by tracing light scattered by the small (µm-sized) dust grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Moreover, we use archival pho- tometric and imaging data to investigate the multiplicity of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2 we describe the data collected from the archive and newly acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 3 we describe the data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' First we discuss the multiplicity of the system as suggested by the photometric data, the analysis of the proper motion and the analysis of the PSF subtracted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Second we analyze the geometry of the system with the analy- sis of the disk and the extended emission seen in scattered light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 4 we propose a scenario that reconciles all the findings, showing a model of the system, and discussing a modeling of the spectral energy distribution and hydrodynamical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 5 we summarize and conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Observations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' SPHERE data T CrA was observed on 2021 June 6th with SPHERE/IRDIS (InfraRed Dual-band Imager and Spectrograph (IRDIS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Dohlen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2008) in dual-beam polarimetric imaging mode (DPI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' de Article number, page 2 of 17 Rigliaco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' : DESTINYS–TCrA Boer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' van Holstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020) in the broadband H filter with pupil tracking setting, as part of the DESTINYS pro- gram (Disk Evolution Study Through Imaging of Nearby Young Stars, Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2020, 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' An apodized Lyot coronagraph with an inner working angle of 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 mas was used to mask the central star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The individual frame exposure time was set to 32 s, and a total of 136 frames were taken separately in 34 polari- metric cycles of the half-wave plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The total integration time was 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Observing conditions were excellent with an average seeing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='8′′ and an atmosphere coherence time of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In addition to the science images, flux calibration images were obtained by offsetting the star position by about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 arcsec with respect to the coronagraph using the SPHERE tip/tilt mir- ror, and inserting a suitable neutral density filter to avoid image saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Two flux calibration sequences were acquired, before and after the science observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We used the public IRDAP pipeline (IRDIS Data reduction for Accurate Polarimetry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' van Holstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020) to reduce the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The images were astro- metrically calibrated using the pixel scale and true north offset given in Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Because the data were taken in pupil tracking mode, we were able to perform an angular differential imaging (ADI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Marois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2006) reduction in addition to the polarimetric reduction, resulting in a total intensity image and a polarized intensity image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We show the initial combined and flux calibrated Stokes Q and U images as well as the QΦ and UΦ images in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Additional SPHERE observations of T CrA were acquired in 2016 and 2018 with the ESO programs 097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='C-0591(A) and 0101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='C-0686(A) (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Schmidt) in classical imaging mode, using a classical Lyot coronagraph and the broadband H filter (BB_H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The data were reduced through the SPHERE Data Center (De- lorme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The 2016 data have very low S/N ratio and they are not usable for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The 2018 IRDIS data are in- stead of good quality and are used to confirm the features de- tected in the 2021 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' NACO data To perform our analysis we also employed archival NACO data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Adaptive optics corrected near-infrared imaging of T CrA was obtained with NAOS-CONICA (NACO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Lenzen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Rousset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2003) at the VLT in July 12th 2007 (program ID 079.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='C-0103(A)), March 29th 2016 (program ID 097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='C-0085(A)) and May 21st 2017 (program ID 099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='C-0563(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In all cases images were obtained in Ks band (λc=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='18 µm) using the S13 camera, with a 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='72 mas/pixel scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In 2007, 3000 frames of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6 seconds were taken with an average seeing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In 2016, 540 frames of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 seconds each were taken with average see- ing of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In 2017, 756 frames of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='35 seconds each were taken with average seeing of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The final images are obtained as the median of all the exposures for each year, after re-centering and rotating the single-exposure images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Photometric data We collected long-term optical photometry of T CrA from the AAVSO Database1 (American Association of Variable Star Ob- servers: Kafka 2020) in order to investigate its secular evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We also considered data acquired within the ASAS (Pojman- 1 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='aavso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='org/data-access ski 1997)2 and ASAS-SN surveys (Shappee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2014)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' While more accurate than the AAVSO data, they have a much more limited temporal coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Results are fully consistent with the long-term light curve obtained from the AAVSO data, but no fur- ther insight could be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' So we will not discuss the ASAS data further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' ALMA data T CrA was observed by ALMA on 2016 August 1–2 (project 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='01058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Details of the observations and calibration are described in Cazzoletti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' These authors also present an analysis of the continuum emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' For the current paper, the continuum emission was imaged using Hogböm CLEANing with Brigss weighting, a robust parameter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5, and a manu- ally drawn CLEAN mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The resulting beam size is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='36×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='27 arcsec (PA +78◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The noise level is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='12 mJy, and a continuum flux of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 mJy is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' These values are not corrected for the primary beam response, which can be expected to affect the results since the observations was not centered on the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A 2D Gaussian fit to the continuum emission shows that the con- tinuum emission is slightly resolved, with a size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='54 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='37 and a PA of +23◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The 12CO line emission was imaged using natural weight- ing and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 km s−1 channels, from VLSR = −5 to +15 km s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' no emission was detected outside this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We used hand drawn masks for each individual channel and applied multi-scale CLEAN with scales of 0,5,15,25 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A pixel scale of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='251 mas was used, coincident with the SPHERE pixel scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Because the CrA region contains extended CO emission around the sys- temic velocity of T CrA (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Cazzoletti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2019), we re- moved all baselines shorter than 55 kλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This removed most, but not all, of the extended line flux but also limits the recovered spatial scales to ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='75 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Data Analysis The new and archival data described in the previous section al- low us to investigate the nature of T CrA as young stellar object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In this section we will analyze the observational evidences we have for the stellar system, its environment, and the geometry of the extended structures visible in scattered light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 we analyze the clues related to the binarity of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 we show the newly acquired polarized light image in H-band of T CrA, describing all the features that we see in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' T CrA as binary system The light curve (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2) shows alternate and periodic maxima and minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The photometric time series analyzed in this study consists of more than 5100 V-band data points collected from the AAVSO Database and taken in a period of over 100 years, be- tween 1910 and 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Each point in Figure 2 is the mean value over each year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The secular evolution of the light curve is well reproduced by a sinusoidal function with a period of 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Sinusoidal light curves, like the one observed in T CrA, can be due to different reasons such as rotation, pulsation, the presence of eclipsing binaries, or occulting binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In the case of oc- culting binaries, the period is generally longer than in the other cases, and the occultation is not only due to the stars, but also 2 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='astrouw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='pl/asas/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='page=aasc&catsrc= asas3 3 https://asas-sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='osu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='edu/variables Article number, page 3 of 17 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' TCrA_Rigliaco Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1: SPHERE/IRDIS polarized light image in H-band of T CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Left panel: H2 image of the Coronet sub-region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The image is adapted from Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The red line shows the line connecting the two MHOs associated to T CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The orange box shows the IRDIS field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Middle panel: Field of view (∼12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5′′) of the SPHERE/IRDIS polarized light image in H-band of T CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The extended emission features analyzed in the manuscript are labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The orange box shows the innermost region of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Right panel: Zoom-in of the innermost 2′′ around the central system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The disk and the shielded disk mid-plane seen as dark lane are labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2: Secular light curve of T CrA with the photometry col- lected from the AAVSO archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Each point is the mean value for each year;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' error bar is the standard deviation of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='The horizontal dashed lines show the ∆V-mag variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The period of the light curve, measured as the mean between the difference of the first and third maxima and minima, is labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' to the circumstellar disks surrounding one or both the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The light curve of T CrA is suggestive of the motion of an occulting binary star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The variation (∆V) in V-magnitude is of the order of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 mag (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Evidence of the presence of a binary star is also provided by the peculiar proper motion of T CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Indeed T CrA shows a rela- tive average motion of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='8 mas yr−1 with respect to the clus- ter in the direction (PAPM)=156±30◦ over the period 1998 (mean epoch of UCAC4 and PPMXL observation) and 2016 (epoch of Gaia DR3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' These values are given by the difference between the proper motion of T CrA, µα cos δ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 mas yr−1 in RA and µδ=-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='9 mas yr−1 in DEC (see Appendix B), and the average proper motion of the on-cloud Coronet cluster mem- bers (µα cos δ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3 mas yr−1 and µδ=-27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3 mas yr−1, Galli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This result might indicate a peculiar (large) motion of T CrA with respect to the Coronet cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' However the position of T CrA is also constrained and defined by the position of the two associated MHOs (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We measured the po- sition angle of the straight line connecting MHO 2013 and 2015, that are thought to be connected to the star (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2011), and crossing T CrA, finding the position angle of the bipolar out- flow (PAMHO) to be PAMHO ≃33◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This represents the direction of the large scale bipolar outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We notice that the minimum distance between T CrA and the line connecting the two MHOs is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='44′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' While this small offset is within the errors in the MHO positions, it can be used to set an upper limit to the relative proper motion of T CrA with the Coronet cloud in the direction perpendicular to this straight line, that is roughly along the di- rection where we found an offset between the proper motion of T CrA measured above and that of the Coronet cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The exact value depends on the time elapsed between the expulsion of the material responsible for the MHO and the observation by Ku- mar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Given the projected distances from the star of the MHO’s are 217′′ (MHO 2013) and 64′′ (MHO 2015), considering the distance of the Coronet cluster and assuming the collimated fast outflowing gas has a speed of approximately 200 km/s as typical for jets from young stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Frank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2014), we obtain that the material was expelled 765 year ago (for MHO2013) and 224 years ago (for MHO 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The upper limit on the proper motion of T CrA with respect to the cloud is then obtained by dividing the measured offset between the barycenter of the system that includes T Cra and the line connecting the two MHOs: the result is about 1 mas/yr, an order of magnitude less than the offset in proper motions considered above and consis- tent with the typical scatter of stars in the Coronet cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We conclude that this offset is not due to a real peculiar motion of Article number, page 4 of 17 36:54:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 F103 N MHO2013 PAdisk E dark lane 102 36:56:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 Extended Extended emission DEC (J2000) emission (feature 1) (feature 2) 101 tail 36:58:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 disk TCrA H2image of the Coronet subregion MHO2015 10° 2" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5"-75AU PA 37:00:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 02:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='019:02:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='001:50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 RA (J2000)Rigliaco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' : DESTINYS–TCrA T CrA, that moves as the Coronet cluster, and should then be an apparent or transient effect, that might be due to the orbital motion of the central binary star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Additional evidence of T CrA as a binary system can also be found in the images acquired with IRDIS in 2018 and 2021 and NACO in 2007, 2016 and 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We subtracted a median PSF, obtained by rotating and averaging the PSF image in steps of 1 degree, to the raw NACO images taken in 2007 and 2016, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' For IRDIS, we used the flux calibration images that are acquired before and after the science sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The technique, described by Bonavita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2021), allows to make a differential image that cancels static aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The output of the procedure is a contrast map that allows to spot stellar companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Due to the contrast limit and to the limits imposed by the diffraction patterns, none of the images obtained allows us to clearly and uniquely detect the presence of a companion star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' However, The PSF of the NACO 2016 and 2017 data set clearly show an exten- sion in the same direction (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 3), namely NW–SE, but in the NACO 2007 data set we do not see this extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A slight extension can be seen in the SPHERE 2018 data set, while no ex- tension in the SPHERE 2021 data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The observed extensions, all in the same direction, are very unlikely to be caused by adap- tive optic effect, but might indicate a distortion of the PSF due to an unresolved companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The geometry of the system Figure 1 shows the polarized light image in H-band of T CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The image shows several structures, as annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In the right panel the brightly illuminated top-side of the outer disk is clearly visible, as well as the shielded disk mid-plane, seen as a stark dark lane in approximately the N-S direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' On larger scale, in the middle panel, we can identify two different extended emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The extended emission labeled as "feature 1" is two- lobed and extends in the NE–SW direction, up to 2′′ from the central source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The extended emission labeled as "feature 2" appears two lobed as well, it is approximately oriented along the N-S direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The South lobe extends out to the edge of SPHERE/IRDIS field of view, while the North lobe extends up to ∼5′′ from the central source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In the following section we will analyze these different structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Outer Disk Figure 1 in the right panel shows a very prominent morpholog- ical feature composed by a dark lane and a bright region that represents the disk surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This outer disk appears highly in- clined, and oriented almost edge-on with respect to the observer, and extends almost to the edge of the coronagraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The dark lane has a maximum width of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2′′ along the E–W direction, corresponding to ∼30 au if it were seen exactly edge-on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' More- over, the disk seen as a dark lane shows an offset with respect to the center of the image that corresponds to ∼10 pixels in the West direction (∼122 mas) that is about four times the FWHM of the point spread function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The disk surface is instead shown by the bright regions that extend further out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The PA of the disk measures PAdisk=7±2◦, shown as green line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The disk appears highly inclined and seen as a dark lane, as in the case for DoAr25 (Garufi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020), MY Lup and IM Lup (Aven- haus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' From the images we cannot provide a precise estimate of the disk inclination, but we can make a few con- siderations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The brightness asymmetry between the bright disk top-side, and the diffuse disk bottom-side, indicates that the disk is not exactly seen edge-on, indeed in that case we should expect top- and bottom-side of the disk to be equally bright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Moreover, the offset between the dark-lane and the center of the image pro- vides another hint of a non-exactly edge-on disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' From simple trigonometric consideration, we can measure the inclination of the disk from the angle between the center of the image and the center of the dark lane and dividing for half the lengths of the dark lane, finding an inclination of ∼87◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We can conserva- tively assume that the T CrA disk, identified as a dark lane in the SPHERE image has an inclination between 85-90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Another possible interpretation for the dark lane could be that it is due to a shadow cast by a highly inclined inner disk close to the cen- ter, as in the case of SU Aur (Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' However, in this scenario, we can not reconcile the brightness asymmetry be- tween the bright top-side and the diffuse bottom-side of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Moreover, we should expect the shadow to cross the center of the image, while it appears shifted to the west by ∼10 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In order to investigate the innermost region of the outer disk, we have plotted the radial profile of the flux seen in QΦ scat- tered light along a slice oriented as the disk, seven pixels wide and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5′′ long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The radial profile, normalized to the brightness peak of the disk, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 4 as a black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The grey area shows the coronagraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The disk has a gap that extends up to ∼25 au and is quite symmetric in the innermost region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' As far as 60 au the disk start to look asymmetric, and extends up to ∼100 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The observed asymmetry might be due to the outflow- ing material that overlaps with the disk itself in the north side (as discussed in the next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' From this analysis we consider for the outer disk an inner rim with radius rin=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='17′′ (∼25 au) and an outer rim rout=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='67′′ (∼100 au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We performed the same anal- ysis of the radial profile in the direction orthogonal to the disk, and shown as blue-dotted line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In the East side there is emission from the scattered light down to the border of the coro- nagraph (rin−east ≲14 au), and inside the disk rim measured along the disk direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' As expected, in the West-side the emission starts further out, due to the presence of the disk’s dark silhou- ette (rin−west ∼30 au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We notice that in the West direction at ra- dial distances >50 au there is contamination with the outflowing material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We will discuss the presence of scattered light emission inside the outer disk gap in the following section, showing that it may suggest the presence of an intermediate circumbinary disk surrounding the central binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Extended emission The structure seen in scattered light in the NE–SW direction, identified as feature 1, is consistent with an outflow in the di- rection of the line connecting the two MHOs (MHO2013 and MHO2015) that are unambiguously associated to T CrA (show in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1), which are however at a projected sepa- ration of ∼35,000 au and ∼10,000 au, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The presence of the MHOs is a clear sign that the source has in the past al- ready experienced outflowing phenomena, hence it is consistent to consider the emission seen in scattered light in the same direc- tions as associated to outflowing material close to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' From a geometrical point of view, the dust seen in scattered light in the direction of the outflow has a position angle PAoutflow ∼35◦ with semi-aperture of ∼25◦, consistent with the PAMHO previ- ously discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The extended emission that elongates approximately in the N-S direction, and identified as feature 2, is two lobed as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In the North it extends up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5′′ from the center, and appears bent toward the West direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The Southern feature 2 extends up to the edge of the field of view and appears brighter than Article number, page 5 of 17 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' TCrA_Rigliaco 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 ∆Dec (arcsec) NACO/2007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 NACO/2016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 ∆RA (arcsec) NACO/2017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 SPHERE/2018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 SPHERE/2021 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 3: PSF for all the epochs T CrA was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The size of the PSF for every single epochs is shown in the bottom-right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' For NACO 2016, 2017 data sets we can notice an elongation of the PSF in the NW–SW direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 4: Radial profile of the Qφ image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The black profile shows the radial profile obtained along a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5′′ long slice centered on the star in the N-S direction, with PA=7◦ and extending along the disk (black-dashed box in the insert).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The blue-dotted profile shows the radial profile obtained in the orthogonal direction (E- W, blue-dashed box in the insert).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' All profiles are normalized to the brightness peak of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The gray area shows the radius of the coronagraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' the North feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We can also detect a faint dust tail extend- ing from the main disk toward SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' As it happens in the case of SU Aur, where several tails are detected (Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2021), we can trace the tail structure until it merges with the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Fea- ture 2 is most likely showing the presence of accretion streamers that bring material from the forming cloud filament to the outer disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' From the polarized (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1) and total intensity images of T CrA we can see that in both cases the northern streamer is fainter than the southern streamer, indicating that we overall re- ceive more photons from the South than from the North side of the extended structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Moreover, the ratio between the polarized and total intensity image shows that the overall degree of polar- ization is similar on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This indicates that light from the South streamer is scattered with angles smaller than 90◦, favor- ing the forward scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Because the Northern streamer shows a similar degree of polarization, but overall fainter signal, we conclude that the light is scattered with angles larger than 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Hence, the South streamer is angled toward the observed and the North streamer is angled away from the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Discussion The environment around T CrA is very complex and the analysis of new and archival data shows several features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In the following we will discuss each of the evidences presented in the previous sections, and we will provide a global picture of its circumstel- lar environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A cartoon of the proposed model, showing all the observational evidences analyzed in the previous section, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 5: Not-to-scale cartoon of the proposed model for the T CrA system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' All the features seen in the scattered light images are labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Moreover, the central binary system, and the size of the coronagraph is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Modeling of the light curve Motivated by the light curve, the peculiar proper motion and the PSF distortion, we conducted a detailed analysis of the pho- tometric and proper motion data, to be compared to the new information on the system’s geometry gathered thanks to the Article number, page 6 of 17 [argsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 1 N Rin(N-s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='17" Rout(N-s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='70" 1 E W intensity Arbitrary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 South-side North-side 0 East-side West-side 100 0 100 Radial distance (AU)Accretion streamer Outflow Flows from outer to inner disk Intermediate (circumbinary) Coronagraph disk edge Outer disk dark lane Outer disk surface tailRigliaco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' : DESTINYS–TCrA Parameters Value log(q) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='27±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='17 M⊙ T0 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='06±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4 years AV0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 mag Disk Thickness 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='7±20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 mas Disk Offset 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='7±19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 mas Table 1: Stellar parameters obtained from the modeling of T CrA as a binary star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The primary mass star is assumed to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='7M⊙, the orbit to be circular, and period 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' SPHERE’s images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In the attempt to reproduce the observed light curve and the H-band magnitude collected from 2MASS, we develop a Monte Carlo (MC) model that accounts for the light emitted from a binary system and partially absorbed by a disk seen edge-on, modeled as a slab with an exponential pro- file, and inclined with respect to the binary’s orbit by 35◦, corre- sponding to an orbit perpendicular to the outflow’s direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' For this simplified model we assume for the binary system a circular orbit seen itself edge-on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' While the circular orbit is an assump- tion made to reduce the number of free parameters, and hence avoid degeneracy in the models, the high-inclination of the bi- nary orbit is supported by the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Indeed, as discussed in Pascucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2020), evidence from the small blueshift of the [OI] and [NeII] forbidden lines of T CrA suggests that the inner disk is itself close to edge-on, with the microjets close to the plane of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We assume for the F0 star a mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='7M⊙ for the primary star, corresponding to 2 Myrs from the BHAC evolutionary tracks (Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2015), circular orbit, and a period of 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6 years as found from the light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The model provides the mass ratio (q) between the primary and secondary component of the binary system, the epoch of the minimum dis- tance between the two components (T0, in years), the offset of the center of mass with respect to the absorbing slab (disk offset, in mas), the disk thickness (in mas) and the maximum absorption at the disk center (AV0, in mag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The proper motion between the 1998 and 2016 is also measured to be compared to the apparent proper motion of T CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='A corner plot of the derived quantities is shown in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The MC model computes one million random sampling of the priors, and provides solutions with re- duced χ2 <2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Figure 6 shows the comparison between the ob- served secular evolution of T CrA and the light curve obtained from the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' There is a very good agreement between the observed and modeled light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The best fit parameters for each of the computed values, obtained as the median value of all the solutions with χ2 <2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3, are reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' According to this model T CrA is a binary system, whose primary star is a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='7M⊙ star, and the secondary is a ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='9M⊙, and it is orbiting with a 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6 years period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The corresponding semi-major axis of the orbit is ∼12 au, seen edge-on, and with the line of nodes of the orbit almost perpendicular to the position angle determined for the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Moreover, we check the consistency between the apparent motion as measured from Gaia and ground-based facilities, and the one measured by assuming the motion of the modeled binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We find that the offset between the two epochs (1998 and 2016) corresponds to 72±26 mas, which is consistent with the value of 130±66 mas measured via Gaia and UCAC4/PPMXL observations, hence justifying the large proper motion of T CrA with respect to the Coronet motion as due to the motion of the binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We will further discuss the results from the model in the next Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 6: Light curve of T CrA (red points) compared to the light curves computed with the MC model (black lines) assuming a period of 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Disk and extended emission Thanks to the new images acquired with SPHERE/IRDIS, and to the wealth of literature data on this target, we have now a bet- ter knowledge of the disk and extended structure around T CrA, and it appears very composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The disk itself is composed by in- ner (circumstellar) disk(s) surrounding the primary (secondary) star of the binary system, an intermediate (circumbinary) disk, slightly visible in scattered light, and an outer (circumbinary) disk that is the most prominent in scattered light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Together with the extended emission features, we will discuss all these features in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The outer disk around T CrA is not continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The scattered light images and the radial profile analysis of the QΦ image show that the bright top-side of the outer disk extends up to ∼100 au in the N-S direction, and show a gap in the same direction that extends down to ∼25 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Evidence of an inner (circumstellar) disk(s) surrounding the primary (secondary) star of the binary system comes from the several tracers of gas and dust well beyond the dust gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Pas- cucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2020) analyze the [OI] λ6300 and [NeII] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='81 µm emission lines observed in high-resolution optical and infrared spectra, and conclude that they are associated to fast and col- limated microjets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In addition, the presence of gas can also be inferred from the non-negligible level of mass accretion rate ( ˙Macc ∼8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1×10−9 M⊙/yr, Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Takami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This gas is most likely distributed into an inner circumstellar disk, that allows accretion onto the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The presence of the inner disk is also highlighted by mid-infrared interferometric data of the thermal emission of disk (Varga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018), and by the SED (Sicilia-Aguilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Sandell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The images acquired with SPHERE show the presence of scattered light down to the edge of the coronagraph in the E- W direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The origin of such emission, highly inclined with respect to the outer disk, is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' However, as we will see in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4, it might be due to an intermediate circumbinary disk, that is a natural transient consequence of the breaking of the innermost circumstellar disks due to the different inclination of inner and outer disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Evidence of emission very close to the coronagraph edges are also found by Cugno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2022 using the NaCo imager with the L′ filter (λ=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6 µm) within the NaCo- ISPY large program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Feature 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The PA of the extended emission identified as feature 1 is consistent with the large scale MHOs and coinci- Article number, page 7 of 17 12 (mag) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3 14 15 1900 1920 1940 1960 1980 2000 2020 Time (yr)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' TCrA_Rigliaco dent with the small scale microjets detected through forbidden lines (Pascucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Hence, we reasonably assume that it is representing outflows detected in scattered light, and that this feature is orthogonal to the inner and intermediate disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The innermost disks (inner and intermediate) are misaligned with re- spect to the outer disk, with a PA for the inner disk of ∼125◦, measured as PAoutflow+90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Considering the outer disk is seen with PAdisk=7◦, the resulting misalignment between innermost and outer disk is of the order of 62◦ with an uncertainty of ±10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This feature is illuminated by the central system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The shape of the outflow is due to higher density regions of dust, generated by instabilities created by two or more layers of material with dif- ferent densities and velocities resulting in a wind-blown cavity (Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The regions with different physical prop- erties are the highly collimated microjet (as seen from the de- tection of forbidden lines, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Pascucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020), and the surrounding wider-angle disk wind, or parent cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The impact between these two regions, besides carving out a large and slow massive outflow cavity into the parent cloud (Frank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2014), creates regions of high density where dust grains accumulate, be- coming brighter in scattered light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We also notice that there is a good agreement between the small scale outflow seen in the po- larimetric images, and the large scale outflows determined by the MHOs, supporting the scenario of highly collimated jets carving a cavity and creating high density regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We have also tested the emission seen in scattered light versus the continuum thermal emission at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3 mm, and the 12CO emission seen with ALMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In Figure 7, we show the continuum emission and the red- and blue-shifted line emission overlayed on the SPHERE scattered light image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The continuum emission, shown as white contours, is slightly resolved, compact, and it is distinctly different from the orientation of the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The comparison with the SPHERE image is not quite conclusive in the direction of the emission, if along the disk or the extended emission identified as feature 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 12CO line emission was clearly detected in the channels, consis- tent with a structure of ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 arcsec in diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The emission is most likely due to the combination from emission aligned with the disk orientation inferred from SPHERE, and emission from the outflowing material in the same direction as the MHOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The gas emission close to the N-S direction might trace the gas in the outer disk, and the velocity structure of the line emission is con- sistent with Keplerian rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The emission from the outflow- ing material is in the same direction as the MHOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The velocities of the extended emission span from -3 km s−1 to 11 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The low velocities for the outflowing material confirm that the emis- sion must happen close to the plane of the sky, as also found by Pascucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In both cases, either when tracing the outer disk or the outflowing material, the N-E side is receding and the S-W side is approaching the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Misalignment between the inner and the outer disks are not rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' As an example, Bohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2021) have recently inves- tigated misalignment between inner and outer disks in transi- tional disks, finding that out of a sample of 20 objects ana- lyzed, six clearly show evidence of misalignment, five do not show evidence of misalignment and the others can not be eval- uated with the current data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Misaligned disks, and disks whose orientations vary with time can be due to their formation in a turbulent, chaotic environment (Bate 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Moreover, the evo- lution of the stellar and disk spin axes during the formation of a star which is accreting in a variable fashion from an inher- ently chaotic environment might affect the disk orientation as well (Bate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Also late infalling events, which carry along a specific angular momentum with respect to the star, may tilt the pre-existing disk to another rotation axis depending on the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 7: Overlay of SPHERE (color scale) and ALMA (contours) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' On the left all the extended structure as seen with SPHERE, on the right a zoom-in of the innermost 2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' White contours are ALMA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3 mm continuum, plotted at contours starting at, and increasing with, 3σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='37 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Red and blue contours are integrated 12CO 2–1 emission over 10 km s−1 blue- and red- shifted relative to the source velocity, taken as VLSR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Red and blue contours are also drawn starting at, and increasing with, 3σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='12 Jy beam−1 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The ALMA data are aligned with the SPHERE data to have the stellar position at the center of the image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' the continuum emission peaks ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='06′′ North of that position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' mass ratio of the mass accreted and the disk (Dullemond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Kuffmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This was indeed recently observed within the DESTINYS program for the SU Aur system (Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2021), which shows large scale streamers in scattered light, similar to those observed in our new observations of T CrA and which were shown to trace infalling material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Stellar properties, such as strong stellar magnetic dipole, can cause a warp or mis- alignment in the innermost region of the disk (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Matsumoto & Tomisaka 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Machida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Matsumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Hennebelle & Ciardi 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Joos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Krumholz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Lewis & Bate 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Wurster & Li 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Additionally, the presence of a compan- ion, either stellar or substellar, can also cause inner and outer disks misalignment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Facchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2013, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Zhu 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Nealon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020), as in the case of HD142527 (Owen & Lai 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Price et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Indeed, T CrA and HD142527 show several similarities even if the inclinations at which the outer disks are seen are very dif- ferent (almost edge-on in the case of T CrA and almost face- on for HD142527).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' HD142527 is a binary system characterized by a primary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 M⊙ star surrounded by an inner disk signifi- cantly misaligned (59◦) with respect to the outer disk (Balmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' For T CrA the outer disk is seen almost edge-on and the misalignment between outer and inner disk is coinci- dent with the inclination of the inner disk orbit, namely ∼55◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The primary star in both cases is an F-type Herbig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In the case of HD142527 all the main observational features (spirals, shad- ows seen in scattered light, horseshoe dust structure, radial flows and streamers) can be explained by the interaction between the disk and the observed binary companion (Price et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The analysis done on HD142527 led the authors (Price et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018a) to conclude that the disk around this Herbig star is a circumbi- nary rather than transitional disk, with an inclined inner disk, and Article number, page 8 of 17 103 102 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0"Rigliaco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' : DESTINYS–TCrA with streamers of material connecting the inner and outer disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In the case of T CrA, if we assume that the inner disk is aligned perpendicular to the outflowing material, and hence misaligned with respect to the outer disk, the configuration is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Hints of dusty material inside and misaligned with respect to the outer disk come from the radial profile of the scattered light signal seen from SPHERE/IRDIS and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 4, where in the East-side of the disk in the direction orthogonal to the disk there is material down to the coronagraph edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' However, we cannot say from these images if this material is organized into a disk- structure itself, or if it represents a streamer of material accreting from the outer disk onto the inner regions of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' How- ever, as opposite to HD142527, we must mention the absence of obvious shadowing features in scattered light in T CrA, that can nevertheless be due to the different viewing geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In the following section we will present a 3D hydrodynamical model as the one developed for HD142527 to explain the observed fea- tures as disk–binary interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Feature 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The extended emission identified as feature 2 ap- pears very extended and resemble material falling onto the disk as in the case of SU Aur (Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Unfortunately, the strong foreground contamination due to the overall cloud does not allow to clearly detect the 12CO (2–1), 13CO (2–1), and C18O (2–1) transitions at distances larger than ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5′′, thus we cannot perform a detailed analysis of the kinematics of the material, as it was done, for example, in the case of SU Aur (Ginski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Indeed, some parts of the CO disk may be missing from from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 7 because the cloud contaminates the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Moreover, the large scale streamers do not show any emission due to the removing of any sensitivity to large scale emission in the data reduction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' They may exist, but they are very hard to image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The disk may also be more extended than seen here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Hence we cannot be conclusive on the nature of the extended emission in feature 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' It is highly unlikely that this emission is itself indicating outflowing material, as feature 1, but it can be most likely due to streamers of material that is falling onto the disk connecting the disk itself to the surrounding cloud material, as for SU Aur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' To some extent we might con- sider the scattered light morphology of T CrA as an edge-on view of SU Aur, where we can see the streamers of infalling material and at least one tail of accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The same stream- ers of accretion were already seen, but not interpreted as such, by Ward-Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (1985);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Ward- Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (1985) used linear polarization mapping of the region in R-band and identified a jet-like structure with a pro- jected lengths of 20′′ emerging from T CrA, in the direction of, but pointing away from R CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2000) performed near-infrared linear imaging polarimetry in J, H and Kn bands, and circular imaging polarimetry in the H band and interpreted the images as bipolar cavities, where the SE emission is visi- ble as far as ∼15′′ from T CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' They stress the presence of a pronounced asymmetry in the polarized intensity images, sug- gestive of fairly sudden depolarization of the dust grains caused by foreground material in the reflection nebula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The identifica- tion of the MHOs, and the analysis of the images acquired with NACO and SPHERE is now showing that the features observed in the past were not associated with jets but more likely the same streamer of accretion seen in scattered light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A possible test to ascertain the origin of feature 2 can be done using the SO2 tran- sition from ALMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Garufi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2022) have indeed shown that for the source IRAS 04302+2247, the SO2 emission does not probe the disk region, but rather originates at the intersection be- tween extended streamers and disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We notice that the presence of streamers of material feeding the disk of T CrA would also go in the direction of mitigating the issue of the low disk masses found in CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Indeed, it was found that the average disk mass in CrA is significantly lower than that of disks in other young (1-3 Myr) star forming regions (Lupus, Taurus, Chamaeleon I, and Ophiuchus, Cazzoletti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' If there is accretion of fresh material onto the disk, one could have lower measured disk masses at the beginning, and mitigate the issue (Manara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The observed increase in disk masses with time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Testi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Cazzoletti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2019) should otherwise be ex- plained with other mechanisms such as planetesimal collisions (Bernabò et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Moreover, the presence of streamers of accretion is also in agreement with the orientation of T CrA with respect to R CrA, both belonging to the Coronet Cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' These two stars formed within the same filament, which is oriented at PA=124◦ pro- jected on sky (this is also the PA of T CrA relative to R CrA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This orientation is indeed similar to that of the orbit proposed for the central binary of T CrA and very close to perpendicular to the PA of the MHO objects (PA=33◦);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' these values are well consistent with the direction of the same structures seen in the neighbor star R CrA (Rigliaco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This suggests that the bulk of the inflow of material that formed the T CrA system was coplanar with this filament and that the original disk of T CrA was likely oriented at the PA of the fil- ament;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' this is actually the case also for the disk around R CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' However, the current outer disk of T CrA has a very different orientation (PA=7 degree), though it seems to be still fed by the same filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This is because T CrA appears to be presently offset by a few hundreds au (a few arcsec on sky) with respect to the filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Considering the age of T CrA (likely 1-3 Myr), this offset is indeed very small, corresponding to a minuscule ve- locity of only ∼ 1m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This suggests that the generation of mis- aligned structure is very likely whenever accretion on the disk is prolonged over such long intervals of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Spectral Energy Distribution We model the SED of T CrA using the dust radiative transfer model developed by Whitney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2003b,a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The code uses a Monte Carlo radiative transfer scheme that follows photon pack- ets emitted by the central star as they are scattered, absorbed, and re-emitted throughout the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' For the modeling we have assumed that the geometry of the star+disk system is comprised by a central 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 M⊙ source emitting photons and a gapped and misaligned circumstellar disk as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The total mass of the disk Mdisk=10−3M⊙, which is in agreement with Mdust re- trieved by Cazzoletti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2019) using the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3 mm continuum flux, assuming an ISM gas-to-dust ratio of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The outcome of the model, shown in orange in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 8, well reproduces the ob- served photometric points collected in Table 2, suggesting that the interpretation of inner and outer disks misaligned with re- spect to each other is in very good agreement with the collected photometry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' For comparison, we also show the SED obtained with the same parameters, in the case where no misalignment between inner and outer disk is assumed (red profile).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In this case the curve does not well reproduce the observed photome- try at wavelengths longer than ∼10–15 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We must notice that the radiative transfer model does not account for the binary star, hence it may cause deviation in the illumination of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In particular, in their orbit the two stars spend time above the disk midplane, hence illuminating the circumbinary disk from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 4 The apparent oscillations of the model at wavelengths longer than 300 µm is due to low number statistics and has no physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Article number, page 9 of 17 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' TCrA_Rigliaco λc Flux Facility Reference (µm) (Jy) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='349 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='00531 SkyMapper Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2018) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='00792 CTIO Henden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2016) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='00988 UCAC4-RPM Nascimbeni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2016) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='482 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0138 CTIO Henden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2016) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='497 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0126 SkyMapper Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2018) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='504 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0142 GAIA Gaia Collaboration (2020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0163 Hamilton Herbig & Bell (1988) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0171 CTIO Henden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2016) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0204 UCAC4-RPM Nascimbeni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2016) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='604 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0181 SkyMapper Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2018) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='762 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='045 GAIA Gaia Collaboration (2020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='763 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0539 CTIO Henden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2016) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='425 2MASS-J Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2003) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='871 2MASS-H Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2003) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='55 2MASS-K Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2003) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='93 Spitzer/IRAC Gutermuth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2009) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='49 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='07 Spitzer/IRAC Gutermuth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2009) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='73 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='38 Spitzer/IRAC Gutermuth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2009) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='48 WISE/W3 Cutri & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2012) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='7 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4 SOFIA Sandell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2021) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='8 WISE/W4 Cutri & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2012) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='7 SOFIA Sandell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2021) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 SOFIA Sandell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2021) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3 SOFIA Sandell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2021) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3 Herschel Herschel Group et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2020) 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 Herschel Herschel Group et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2020) 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 Herschel Herschel Group et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2020) 1300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='00499 ALMA Cazzoletti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2019) Table 2: List of the fluxes at different wavelengths collected from the literature used for the SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In the two SEDs shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 8 we do not account for this ef- fect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Hydrodynamical Simulation We perform a 3D hydrodynamical simulation of the T CrA con- figuration considered in this work using the Smoothed Particle Hydrodynamics (SPH) code Phantom (Price et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Mon- aghan 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Price 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The initial conditions of the system are set following the observational constraints acquired so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' T CrA is modeled as a binary system with masses 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='7 M⊙, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 M⊙ for the primary and secondary component, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Each star is simulated as a sink particle (Price et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Bate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1995) with an accretion radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The orbit is eccen- tric, and the period of the binary star is 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6 years, correspond- ing to a semi-major axis of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The orbit is seen edge-on with an inclination of 90◦, and PAorbit is perpendicular to the out- flowing material (PAorbit=145◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The outer disk, extending from Rin = 25 au to Rout = 100 au is simulated with 8 × 105 SPH par- ticles, resulting in a smoothing length ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 times the disk scale height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The inner disk, extending from rin = 1 au to rout = 5 au, and co-planar to the orbit of the binary star, is simulated with 2 × 105 SPH particles, resulting in a smoothing length of about the disk scale height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Outflows and inflows are not considered in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Viscosity is implemented with the artificial viscosity method (Lucy 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Gingold & Monaghan 1977) that results in an Shakura & Sunyaev (1973) α-viscosity as shown by Lodato & Price (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We use α ≈ 5 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We run the full hydrody- namical model (with both the outer and the inner disk) for 100 binary orbits in order to relax the initial condition and to produce a synthetic image of the system to compare with the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' To perform a direct comparison with observations of T CrA we Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 8: SED of TCrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The black asterisks show the published photometry as reported in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The orange curve shows the total emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The magenta line shows the SED component due to stellar origin, in blue the component due to the disk, and in green the component due to the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The red curve shows the emission if no misalignment between the intermediate and outer disk is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The oscillations in the model curves at the longest wavelengths are artifacts related to the finite number of photon packets considered in the Monte Carlo scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' post-processed our simulation using the Monte Carlo radiative transfer code MCFOST (Pinte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2016) in order to produce synthetic images of the hydrodynamical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' MCFOST maps the physical quantities in the SPH simulation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' dust and gas density, temperature) onto a Voronoi mesh directly built around the SPH particles, without interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We adopt a gas-to-dust mass ratio equals to 100 and we assume micrometer grains to be well coupled with the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' These grains scatter the stellar light collected by SPHERE and are assumed to be spherical and ho- mogeneous (as in the Mie theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Their chemical composition is 60% astronomical silicates and 15% amorphous carbons (as DIANA standard dust composition, Woitke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2016) and they have a porosity of 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The gas mass is directly taken from the SPH simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We use the same distance from the source used in this paper (149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4 pc) and ≈ 106 photon packets to compute the temperature profile of the model and ≈ 1010 photon packets to compute the source function of the model in order to produce the scattered light image at 2 µm wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The total intensity polarized light image obtained with the hydrodynamical simulation is show in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The middle panel is the synthetic image convolved to the SPHERE/IRDIS resolution and in the right panel we show the observed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' There are a few features that are clearly repro- duced in the simulation: the dark lane, the offset of the dark lane with respect to the center of the image, the top-surface of the disk brighter than the bottom-side of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' There are two bright spots in the East-West direction on the convolved synthetic im- age, that are also observed in the real image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' These points are due to the intermediate circumbinary disk that breaks from the outer regions, precessing as a rigid body, and leading to its evo- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The breaking of the inner disk generates an intermediate disk, that is visible as bright spots at the East and West side of the coronagraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We must notice that the simulation does not take into consideration the outflowing material, and does not ac- Article number, page 10 of 17 1000 collectedphotometry Itotal stellarorigin 100 disk origin LL :envelopeorigin Itotal-nodisksmisalignment 10 TTT (Jy) Flux 1 LLL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 TT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='01 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 1 10 100 1000 104 Wavelength (μm)Rigliaco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' : DESTINYS–TCrA count for the replenishment of the outer disk due to the accretion streamers (hence slowing down its expansion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A more detailed simulation is needed for T CrA, but it is beyond the scope of this observational paper and will be discussed in a separate publica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In order to measure how the circumbinary disk mass dis- tributes among the binary stars, we run a second hydrodynamical model as the one described above but without the circumprimary disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Indeed, accretion into a binary system happens via the for- mation of up to three disks (two circumstellar disks, one around each component, and a circumbinary disk, Monin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' (2007)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The two circumstellar disks are periodically replenished by ac- cretion streamers pulled from the inner edge of the circumbi- nary disks by the stars (Artymowicz & Lubow 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Tofflemire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In a quasi-steady state regime, the mass flux en- tering the Roche lobe of a star via the gas streamers equals the star accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Thus, we can reliably measure the fraction of mass accreted onto a star by simulating only the circumbinary disk, provided that the stellar Roche lobes are resolved by the simulation and the central part of the disk has relaxed (as done with SPH simulations e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' in Young & Clarke 2015 and recently tested in Ceppi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In general, simulations of accretion into binary systems find that the primordial mass ratio is pushed towards unity (that is, closer to equal masses in the binary com- ponents) by accretion from a circumbinary disk (Clarke 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This is due to the ease with which the secondary component ac- cretes the infalling gas, as it lies farther from the binary barycen- ter and closer to the disk edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Its differential velocity with re- spect to the gas is also low, allowing it to accrete efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In the case of T CrA the primary star is still accreting more than the secondary (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This is due to the misalignment between inner and outer disk that makes the secondary to be at consider- able height over/below the disk for a large fraction of its orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Summary and Conclusions We investigate new and archival data of the Herbig Ae/Be star T CrA collected with different instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The analysis of the data shows that T CrA is a very interesting and complex system, belonging to one of the nearest and most active region of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Combining archival NACO imaging data with pho- tometric data, and new and archival SPHERE adaptive optics images we study the complex stellar environment around T CrA and the stellar properties: – the outer disk is seen edge-on as a dark lane elongated ap- proximately in the N-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The dark lane is shifted by 122 mas with respect to the center of the image, and it is seen with a PA of 7◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This value is in very good agreement with the value recently found by Cugno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2022 using a different instrument and set of data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' – the bright illuminated top-side of the disk surface is clearly visible in scattered light;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' – extended emission in the NE–SW direction, identified as fea- ture 1, is consistent in direction with the line connecting the two-lobed MHOs seen on larger scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' It is most likely out- flowing material, with PA=33◦, consistent with the PA of the two MHOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' – extended emission in the N-S direction, identified as feature 2, is interpreted as large scale streamers of material likely in- falling onto the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In the North the streamer extends up to ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5′′ from the central system, while in the South it extends up to the edge of the field of view, and probably beyond, as suggested by previous stellar polarization images in the op- tical and near-IR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' – the periodic behavior of the light curve suggests a cen- tral binary with a period of 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Even if the non- coronagraphic images acquired with NACO and SPHERE do not show direct evidence of the presence of a stellar compan- ion, a detailed comparison of the position of the secondary along the proposed orbit at the epochs of the observations acquired so far with NACO and SPHERE shows that in all of them it was too close to the primary star for detection as a separate object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' According to our modeling results the two components will be at their maximum separation in 2027: ap- propriate high-contrast images at that epoch should provide direct evidence of the binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Overall, we find that the binary system and intermediate circumbinary disk lay on different geometrical planes, placing T CrA among the objects with a misaligned inner disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Inner and outer disk misalignment is not rare, and in very recent years, thanks to high-contrast imaging, it is becoming clear that the misalignment can also be due to the accretion history of the star- forming cloud onto the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Indeed in the case of T CrA (as well as SU Aur) we found evidences of the presence of streamers of accreting material that connect the filament along which the star has formed with the outer part of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' These streamers have an angular momentum with respect to the star whose direction is very different from that of the system (in the case of T CrA, this is dominated by the binary) causing a misalignment between an inner and outer disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Besides characterizing the disk/outflow structures around T CrA, we have also modeled its spectral energy distribution, showing that the disk geometry obtained is well consistent with the observed SED, and such consistency is not reached if we do not consider the misalignment between inner and outer disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Moreover, we have performed hydrodynamical simulation of the configuration for 100 orbits of the binary star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The model is consistent with the observations and the analysis of the accre- tion rates of the individual stars shows that the accretion hap- pens mainly onto the primary star, rather than on the secondary, as a consequence of the inclination between inner/intermediate and outer disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Also the light curve is easily explained assum- ing the configuration of two misaligned disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Comparison of the ALMA continuum and 12CO emission have also been per- formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' While for the continuum emission we cannot clearly point out the region where the dust is located, if along the disk or the outflowing material, the gas emission is most likely due to the combination from emission aligned with the disk orienta- tion inferred from SPHERE, and emission from the outflowing material in the same direction as the MHOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The analysis conducted on T CrA has confirmed its ex- tremely interesting and complex nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' As in the case of HD142527, the misalignment between inner and outer disk can be due to the interaction between the disk and the central binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' On the other hand, the large scale streamers observed in the N–S direction are very similar to the disk-cloud interaction observed for SU Aur, that represents material infalling onto the disk, and inner and outer disk misalignment might be caused by this interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' It comes clear the need for high resolution obser- vations to disentangle the different effects that shape early plan- etary system formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' T CrA is an excellent target/laboratory to better understand the impact of binarity and the environment in the evolution of protoplanetary disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We would like to thank the referee Roubing Dong, whose careful and constructive comments improved the quality of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 664931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This work has been supported by the project PRIN INAF 2016 The Cradle of Life Article number, page 11 of 17 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' TCrA_Rigliaco Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 9: Snapshot of the SPH simulation compared to the observed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The left panel shows the result in total intensity of the SPH simulation, with a resolution of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 mas/pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In the middle panel the same image convolved to the SPHERE/IRDIS resolution (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='25 mas/pixel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' On the right the observed total intensity image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' All images have a 2′′ field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 10: Mass accretion rate ratio of secondary and primary star as a function of the number of orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' GENESIS-SKA (General Conditions in Early Planetary Systems for the rise of life with SKA) and by the "Progetti Premiali" funding scheme of the Italian Min- istry of Education, University, and Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='M acknowledges funding from the European Union under the European Union’s Horizon Europe Research & Innovation Programme 101039452 (WANDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Neither the European Union nor the granting authority can be held responsible for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' acknowl- edges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agree- ment No 714769 and funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under grants 361140270, 325594231, and Ger- many’s Excellence Strategy - EXC-2094 - 390783311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' has been supported by the UK Science and Technology research Council (STFC) via the consoli- dated grant ST/S000623/1 and by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 823823 (RISE DUSTBUSTERS project).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This paper makes use of the fol- lowing ALMA data: ADS/JAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='ALMA#2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='01058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Ko- rea), in cooperation with the Republic of Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' MRH acknowledges the assistance of Allegro, the ARC node in the Netherlands, who assisted with the calibration of this data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This work is partly based on data products produced at the SPHERE Data Centre hosted at OSUG/IPAG, Grenoble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We thank P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Delorme and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' La- gadec (SPHERE Data Centre) for their efficient help during the data reduction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' SPHERE is an instrument designed and built by a consortium consist- ing of IPAG (Grenoble, France), MPIA (Heidelberg, Germany), LAM (Marseille, France), LESIA (Paris, France), Laboratoire Lagrange (Nice, France), INAF Os- servatorio Astronomico di Padova (Italy), Observatoire de Genève (Switzerland), ETH Zurich (Switzerland), NOVA (Netherlands), ONERA (France) and AS- TRON (Netherlands) in collaboration with ESO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' SPHERE was funded by ESO, with additional contributions from CNRS (France), MPIA (Germany), INAF (Italy), FINES (Switzerland) and NOVA (Netherlands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' SPHERE also received funding from the European Commission Sixth and Seventh Framework Pro- grammes as part of the Optical Infrared Coordination Network for Astronomy (OPTICON) under grant number RII3-Ct-2004-001566 for FP6 (2004-2008), grant number 226604 for FP7 (2009-2012), and grant number 312430 for FP7 (2013-2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='int/web/gaia/dpac/consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We acknowledge with thanks the variable star observations from the AAVSO International Database contributed by ob- servers worldwide and used in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' References Artymowicz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' & Lubow, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1994, ApJ, 421, 651 Avenhaus, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Quanz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Garufi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018, ApJ, 863, 44 Bailey, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1998, MNRAS, 301, 161 Balmer, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Follette, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Close, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='00687 Baraffe, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Homeier, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Allard, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Chabrier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2015, A&A, 577, A42 Bate, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018, MNRAS, 475, 5618 Bate, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Bonnell, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Price, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1995, MNRAS, 277, 362 Bate, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Lodato, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Pringle, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2010, MNRAS, 401, 1505 Bernabò, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Turrini, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Testi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Marzari, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Polychroni, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2022, ApJ, 927, L22 Beuzit, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Vigan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Mouillet, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2019, A&A, 631, A155 Bohn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Benisty, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Perraut, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2021, arXiv e-prints, arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='00123 Bonavita, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Gratton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Desidera, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2021, arXiv e-prints, arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='13706 Cazzoletti, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Manara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2019, A&A, 626, A11 Ceppi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Cuello, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Lodato, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2022, MNRAS[arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='08784] Clark, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', McCall, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Chrysostomou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2000, MNRAS, 319, 337 Clarke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2012, in From Interacting Binaries to Exoplanets: Essential Model- ing Tools, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Richards & I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Hubeny, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 282, 409–416 Cugno, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Pearce, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Launhardt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='15434 Cutri, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2012, VizieR Online Data Catalog, II/311 Cutri, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Skrutskie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', van Dyk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2003, 2MASS All Sky Catalog of point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Davis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Gell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Khanzadyan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Smith, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Jenness, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2010, A&A, 511, A24 de Boer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Langlois, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', van Holstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020, A&A, 633, A63 Article number, page 12 of 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='8 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6 M2/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 0 50 100 150 200 250 300 Orbits10-1 6× 10-1 10-1 10-3 4× 10-1 3× 10-1 10-5 10-2 2×10-1 10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0" 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0" 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0"Rigliaco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' : DESTINYS–TCrA Delorme, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Meunier, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Albert, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2017, in SF2A-2017: Proceedings of the Annual meeting of the French Society of Astronomy and Astrophysics, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Reylé, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Di Matteo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Herpin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Lagadec, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Lançon, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Meliani, & F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Royer, Di Dohlen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Langlois, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Saisse, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2008, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 7014, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' SPIE, 70143L Dong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Najita, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Brittain, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018, ApJ, 862, 103 Dullemond, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Küffmeier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Goicovic, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2019, A&A, 628, A20 Dzib, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Loinard, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Ortiz-León, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Rodríguez, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Galli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018, ApJ, 867, 151 Facchini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Juhász, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Lodato, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018, MNRAS, 473, 4459 Facchini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Lodato, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Price, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2013, MNRAS, 433, 2142 Frank, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Ray, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Cabrit, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2014, in Protostars and Planets VI, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Beuther, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Klessen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Dullemond, & T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Henning, 451 Gaia Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020, VizieR Online Data Catalog, I/350 Gaia Collaboration, Brown, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Vallenari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2021, A&A, 649, A1 Gaia Collaboration, Prusti, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', de Bruijne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2016, A&A, 595, A1 Galli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Bouy, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Olivares, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020, A&A, 634, A98 Garufi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Avenhaus, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Pérez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020, A&A, 633, A82 Garufi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Podio, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Codella, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2022, A&A, 658, A104 Ghez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', McCarthy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Patience, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Beck, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1997, ApJ, 481, 378 Gingold, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' & Monaghan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1977, MNRAS, 181, 375 Ginski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Facchini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2021, ApJ, 908, L25 Ginski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Ménard, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Rab, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020, A&A, 642, A119 Gutermuth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Megeath, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Myers, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2009, ApJS, 184, 18 Henden, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Templeton, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Terrell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2016, VizieR Online Data Cat- alog, II/336 Hennebelle, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' & Ciardi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2009, A&A, 506, L29 Herbig, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1960, ApJS, 4, 337 Herbig, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' & Bell, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1988, Third Catalog of Emission-Line Stars of the Orion Population : 3 : 1988 Herczeg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' & Hillenbrand, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2014, ApJ, 786, 97 Herschel Group, Marton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Calzoletti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020, VizieR Online Data Cat- alog, VIII/106 Herter, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Adams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Gull, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018, Journal of Astronomical Instrumentation, 7, 1840005 Joos, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Hennebelle, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Ciardi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2012, A&A, 543, A128 Joy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1945, ApJ, 102, 168 Kafka, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020, in European Planetary Science Congress, EPSC2020–314 Köhler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Neuhäuser, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Krämer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2008, A&A, 488, 997 Krumholz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Crutcher, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Hull, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2013, ApJ, 767, L11 Kuffmeier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Dullemond, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Reissl, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Goicovic, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2021, arXiv e- prints, arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='04309 Kumar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Sharma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Davis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Borissova, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Grave, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2011, A&A, 533, A137 Lenzen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Hartung, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Brandner, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2003, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 4841, Instrument Design and Performance for Optical/Infrared Ground-based Telescopes, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Iye & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Moorwood, 944–952 Lewis, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' & Bate, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2017, MNRAS, 467, 3324 Lewis, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Bate, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Price, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2015, MNRAS, 451, 288 Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='-b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Fang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Henning, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Kainulainen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2013, MNRAS, 436, 3707 Liang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Johnstone, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Cabrit, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Kristensen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020, ApJ, 900, 15 Lindberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' & Jørgensen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2012, A&A, 548, A24 Lodato, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' & Price, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2010, MNRAS, 405, 1212 Lucy, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1977, AJ, 82, 1013 Machida, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Matsumoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Hanawa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Tomisaka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2006, ApJ, 645, 1227 Maire, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Langlois, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Dohlen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 9908, Ground- based and Airborne Instrumentation for Astronomy VI, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Evans, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Simard, & H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Takami, 990834 Manara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Morbidelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Guillot, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018, A&A, 618, L3 Marois, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Lafrenière, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Doyon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Macintosh, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Nadeau, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2006, ApJ, 641, 556 Matsumoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Nakazato, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Tomisaka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2006, ApJ, 637, L105 Matsumoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' & Tomisaka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2004, ApJ, 616, 266 Mendigutía, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Eiroa, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Montesinos, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2011, A&A, 529, A34 Mesa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Bonnefoy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Gratton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2019, A&A, 624, A4 Meyer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' & Wilking, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2009, PASP, 121, 350 Monaghan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2005, Reports on Progress in Physics, 68, 1703 Monin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Clarke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Prato, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & McCabe, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2007, in Protostars and Planets V, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Reipurth, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Jewitt, & K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Keil, 395 Nascimbeni, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Piotto, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Ortolani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2016, MNRAS, 463, 4210 Nealon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Cuello, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Alexander, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020, MNRAS, 491, 4108 Owen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' & Lai, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2017, MNRAS, 469, 2834 Pascucci, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Banzatti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Gorti, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020, ApJ, 903, 78 Pikhartova, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Long, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Assani, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2021, ApJ, 919, 64 Pinte, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Dent, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Ménard, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2016, ApJ, 816, 25 Pojmanski, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1997, Acta Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', 47, 467 Pontoppidan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Blake, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Smette, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2011, ApJ, 733, 84 Price, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2012, Journal of Computational Physics, 231, 759 Price, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Cuello, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Pinte, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018a, MNRAS, 477, 1270 Price, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Wurster, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Tricco, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018b, PASA, 35, e031 Rigliaco, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Gratton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Mesa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2019, A&A, 632, A18 Roeser, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Demleitner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Schilbach, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2010, AJ, 139, 2440 Rousset, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Lacombe, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Puget, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2003, in Society of Photo-Optical In- strumentation Engineers (SPIE) Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 4839, Adaptive Op- tical System Technologies II, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Wizinowich & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Bonaccini, 140–149 Sandell, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Reipurth, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Vacca, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Bajaj, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2021, ApJ, 920, 7 Shakura, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' & Sunyaev, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1973, in IAU Symposium, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 55, X- and Gamma-Ray Astronomy, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Bradt & R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Giacconi, 155 Shappee, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Prieto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Grupe, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2014, ApJ, 788, 48 Sicilia-Aguilar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Henning, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Kainulainen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Roccatagliata, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2011, ApJ, 736, 137 Sicilia-Aguilar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Henning, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Linz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2013, A&A, 551, A34 Siess, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Dufour, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Forestini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2000, A&A, 358, 593 Takami, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Bailey, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Chrysostomou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2003, A&A, 397, 675 Testi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Natta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Manara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2022, A&A, 663, A98 Tofflemire, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Mathieu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Ardila, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2017, ApJ, 835, 8 van Holstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Girard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', de Boer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020, A&A, 633, A64 Varga, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Ábrahám, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018, A&A, 617, A83 Ward-Thompson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Warren-Smith, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Scarrott, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Wolstencroft, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 1985, MNRAS, 215, 537 Whitney, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Wood, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Bjorkman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Cohen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2003a, ApJ, 598, 1079 Whitney, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Wood, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Bjorkman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', & Wolff, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2003b, ApJ, 591, 1049 Woitke, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Min, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Pinte, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2016, A&A, 586, A103 Wolf, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Onken, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Luvaul, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018, PASA, 35, e010 Wurster, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' & Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2018, Frontiers in Astronomy and Space Sciences, 5, 39 Young, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' & Clarke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2015, MNRAS, 452, 3085 Zacharias, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Finch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', Girard, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2012, VizieR Online Data Cata- log, I/322A Zhu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2019, MNRAS, 483, 4221 Article number, page 13 of 17 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' TCrA_Rigliaco Appendix A: SPHERE polarimetric images In Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 we present the Stokes Q and U, as well as the derived QΦ and UΦ images of T CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The flux calibration was carried out by measuring the flux of the central star in the non- coronagraphic flux calibration images, taken at the beginning and end of the observation sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' To convert pixel counts to physical units we used the 2MASS H-band magnitude of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 4 3 2 1 0 1 2 3 4 4 3 2 1 0 1 2 3 4 Q 4 3 2 1 0 1 2 3 4 4 3 2 1 0 1 2 3 4 U 4 3 2 1 0 1 2 3 4 4 3 2 1 0 1 2 3 4 Qφ 4 3 2 1 0 1 2 3 4 4 3 2 1 0 1 2 3 4 Uφ ∆RA (arcsec) ∆Dec (arcsec) 60 45 30 15 0 15 30 45 60 (mJy/arcsec2) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1: Flux calibrated image of the Q, U, QΦ and UΦ frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Appendix B: Proper motion analysis The average proper motion for the on-cloud Coronet cluster members obtained using Gaia DR2 data from Galli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020, is µα cos δ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3 mas yr−1 and µδ=-27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3 mas yr−1 with a small dispersion of less than 1 mas/yr for the individual objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' As we mentioned in the introduction, Gaia does not provide astrometric solutions and proper motion for the star T CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' However, we can check for peculiar/transient motion of the star using the follow- ing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We collect from UCAC4 (Zacharias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2012), PPMXL (Roeser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2010) and Gaia DR3 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2016, 2021) database a list of 10 bright stars in the T CrA surroundings for which proper motion are available in all cata- logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' These stars, listed in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1, are selected such that they have magG ≤14 and lay within 10′ from T CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' For these stars we measure a long term proper motion given by the difference in position between the UCAC4, PPMXL, and Gaia DR3 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The long term proper motion, defined as the motion of the star between different epochs of observations is measured as: µα cos δ = (RAEpoch1 − RAEpoch2) ∗ cos(DECEpoch1) (Epoch1 − Epoch2) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1) µδ = (DECEpoch1 − DECEpoch2) (Epoch1 − Epoch2) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2) The analysis of the proper motion between the various epochs of the selected stars allows us to find a systematic offset between the average of the coordinate systems of UCAC4 and PPMXL with respect to Gaia DR3, that averages to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='41±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='46 mas yr−1 in RA and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='67 mas yr−1±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='93 mas yr−1 in DEC for this specific region of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' For T CrA we obtain an estimate of the proper motion of the star by correcting the long term proper motion with the systematic offset, finding as values µα cos δ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 mas yr−1 and µδ=-33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='9 mas yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Considering the average proper motion of the on-cloud members we find for T CrA an ap- parent motion µα cos δ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 mas yr−1 in RA and µδ=- 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='9 mas yr−1 in DEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1: Proper motion of the ten stars reported in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The cyan square represents the average proper motion for the on- cloud Coronet cluster obtained using Gaia DR2 data (Galli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The blue star is the calculated apparent proper motion of T CrA after correcting the long term proper motion for the systematic offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In orange the direction of the systematic offset of the proper motion due to the different coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Article number, page 14 of 17 10 PMfromGaiafortheselectedstars average PM of the on-cloud members apparent PM of TCrA 20 μ(mas/yr) 30 40 average systematic offset betweenGaiaandUCAC4/PPMXL 50 40 20 0 20 μαcosd(mas/yr)Rigliaco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' : DESTINYS–TCrA T CrA V709 CrA HD176269 HD176270 TY CrA HD176423 V702 CrA HD176386 HD176497 HD176018 CD-36 13202 (UCAC4) RA 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='494904 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='395229 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='263532 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='267908 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='420107 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='460076 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='508229 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='412217 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='528297 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='930902 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='920385 Dec 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='963871 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='015723 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='060898 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='061555 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='876063 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='664651 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='128761 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='890712 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='361622 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='788004 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='588797 Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' RA 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='40 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='45 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='25 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='25 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='09 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='50 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='88 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='25 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='57 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='91 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='62 Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Dec 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='77 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='43 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='25 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='25 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='52 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='91 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='44 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='25 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='28 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='04 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='74 (PPMXL) RA 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='494908 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='395224 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='263532 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='267908 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='420102 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='460076 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='508229 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='412219 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='528303 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='930902 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='920394 Dec 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='963869 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='015722 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='060898 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='061555 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='876064 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='664651 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='128761 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='890703 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='361625 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='788007 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='588800 Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' RA 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='95 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='00 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='73 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='18 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='53 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='41 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='44 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='14 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='32 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='23 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='69 Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Dec 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='95 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='70 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='64 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='19 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='76 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='62 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='14 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='09 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='38 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='64 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='44 (Gaia DR3) RA 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='494959 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='395278 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='263578 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='267966 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='420142 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='460103 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='508268 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='412238 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='528324 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='930856 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='920226 Dec 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='963983 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='015845 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='061040 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='061682 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='876201 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='664770 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='128870 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='890838 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='361752 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='788188 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='589008 Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' RA 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Dec 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1: List of the stars used to measure the proper motion offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Article number, page 15 of 17 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' TCrA_Rigliaco Epoch Offset B-A (mas) dH (mag) 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='54 (NACO) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='25 (NACO) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='7 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='60 (SPHERE) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='7 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='36 (SPHERE ) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='50 (SPHERE) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1: Relative position of the secondary star (B) with re- spect to the primary star (A), and relative contrast (dH) in H- band of the secondary star with respect to the primary star, for a period of 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The offset is defined in the direction of the semi-major axis of the stellar orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Appendix C: Binarity and light curve The light curve of T CrA appears to be periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The period is found to be 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6 years and it can be due to the presence of a binary star at the center of the T CrA system with a mass ratio q∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2, that is partially obscured by a disk seen edge-on, that has an offset with respect to the photocenter of the binary star of ∼90 mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The model of this binary system, described in Sect 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1, is also able to account for the large apparent proper motion mea- sured in the period between 1998 and 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' None of the images acquired in recent years with NACO (in 2007, 2016 and 2017) and SPHERE (in 2016, 2018 and 2021) shows clear evidence of a binary system for T CrA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Hence, we have checked what was the relative position of the secondary star with respect to the pri- mary for every single epoch for which we have an image, and the H-band contrast that should be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' These quantities are shown in Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' These value are all consistent with the fact that the binary system is not clearly resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Indeed, in the 2007, 2018 and 2021 epochs the separation between the two stars is too small to see the two sources separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' On the contrary, the two 2016 epochs have a larger separation, though still within 2×λ/D, that is so close that the secondary cannot be clearly sep- arated from the primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We notice however, that in both images acquired around this epoch with NACO and SPHERE, the PSF appears elongated in the NW-SE direction, that corresponds to the direction of the major axis of the orbital motion of the bi- nary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The average position angle of the elongated PSFs acquired in 2016 is 130±15◦, in very good agreement with the direction of the peculiar proper motion (PAPM) measured, and with the hypothesis that the orbit of the binary system is seen edge-on, and perpendicular to the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This elongation in different epochs supports then the scenario of a binary star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' In a few years, namely in 2027, when the system is at its highest separation, the secondary component should be detectable with high-contrast images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' We have also considered that the period of the system might be double than the period measured in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3, namely 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' While the light curve can be, also in this case, eas- ily reproduced, there are several observational shortcomings in this interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' First of all we must notice that in this case the model predicts a mass ratio q as high as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='9, and an off- set of the disk of ∼10 mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' This last quantity is in disagree- ment with the observations, that instead show that the disk dark lane has an offset ten times larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Moreover, the position of the center of the binary system as retrieved by assuming a 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 years period is not consistent with the motion of the system obtained from UCAC4/PPMXL and Gaia DR3 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Addition- ally, in the SPHERE image acquired in 2021, the predicted sep- aration between the primary and secondary component should be 108±6 mas, with a contrast dH=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 mag, making it visi- ble as a separate point source in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The relative position Epoch Offset B-A (mas) dH (mag) 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='54 (NACO) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='25 (NACO) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='60 (SPHERE) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='36 (SPHERE) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='50 (SPHERE) 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='.0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2: Relative position of the secondary star (B) with re- spect to the primary star (A), and relative contrast (dH) in H- band of the secondary star with respect to the primary star, for a period of 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The offset is defined in the direction of the semi-major axis of the stellar orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' of the secondary star with respect to the primary for every sin- gle epoch for which we have an image, and the H-band contrast that should be observed are reported in Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The image does not reveal the presence of the secondary star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Given these shortcomings between observations and the output of the model, we exclude that the period of the binary star is 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Fig- ure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 show the corner plot of the derived quantities of the model used in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1 to model the light curve assuming a period of 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6 years or the double (59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The light curve for the period of 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 years is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='1: Corner plot showing the results of the MC parameters estimation for the model described in the paper when a period of 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 years is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The plots show the 2D joints posterior densities of all couple of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Article number, page 16 of 17 2009 2008 2007 2006 2005 2004 2003 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 log q (mag) 12 Max absorption 10 8 6 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 log q 2003 2004 2005 2006 2007 2008 2009 TO Disk Thickness (mas) Disk Thickness (mas) 3688885 40 140 120 8688867 100E Thickness 80 E 60 E 40 20 E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 6 8 10 12 log q 20032004 2005 2006 2007 20082009 Max absorption (mag) TO 148 160E (mas) 140E offset (mas) 160 (mas) 160 148 (sDw) 120 E 2888898 offset Disk offset 100日 offset 80 E Disk 60 E Disk 40 E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 20日 4 6 8 10 12 20 40 60 80100120140 b 6ol 2003 20042005 2006 200720082009 Max absorption (mag) Disk thickness (mas) TORigliaco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' : DESTINYS–TCrA Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2: Corner plot showing the results of the MC parameters estimation for the model described in the paper when a period of 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6 years is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' The plots show the 2D joints posterior densities of all couple of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='3: Light curve of T CrA (red points) compared to the light curves computed with the MC model (black lines) assuming a period of 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content=' Article number, page 17 of 17 2017 2016 2015 2014 2013 2012 2011 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 log q absorption XDW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 log q 2011 2012 2013 2014 2015 20162017 TO (mas) (mas) 140 40 140 2888898 120 Disk Thickness Thickness 00 100 80 80 60 60 40 MSI 40 Disk 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 log q 2011 2012 2013 2014 2015 2016 2017 Max absorption (mag) TO < offset (mas) (mas) (mas) 40 40 40 40 SDU 20 20 20 offset < offset 0 0 0 Disk Disk 20 Disk Disk 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='6-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} +page_content='0 20 40 60 80100120140 log q 201120122013 2014 201520162017 Max absorption (mag) Disk thickness (mas) TO12 mag 13 15 1900 1920 1940 1960 1980 2000 2020 Time (yr' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfhf3F/content/2301.01486v1.pdf'} diff --git a/AtAzT4oBgHgl3EQfhv1C/content/2301.01488v1.pdf b/AtAzT4oBgHgl3EQfhv1C/content/2301.01488v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0b9c3a12f21183aa4977c5d074968742f452fe6d --- /dev/null +++ b/AtAzT4oBgHgl3EQfhv1C/content/2301.01488v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f135afe587d47d94f4b1fc93bdb9231dac3b2a822bf9e7085a09db8e79b19a38 +size 7087447 diff --git a/AtAzT4oBgHgl3EQfhv1C/vector_store/index.faiss b/AtAzT4oBgHgl3EQfhv1C/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..64f86529ac48e00007f9c10f3bcad85e77d66aac --- /dev/null +++ b/AtAzT4oBgHgl3EQfhv1C/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8a24fba9464fb72034a52193092609b9f485b4e68c63d74a129770f8f309434e +size 5373997 diff --git a/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf b/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..081341aae9f5bcb1b95779ab86631fa39db5dc62 --- /dev/null +++ b/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:405bf38c3f46093548e8aed4951156b388a763d676ed6f2fda9623e31b5f2f7f +size 697519 diff --git a/AtAzT4oBgHgl3EQfv_4W/vector_store/index.pkl b/AtAzT4oBgHgl3EQfv_4W/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..88ad64cffd136c4fa06f73c8a46ce966a1298737 --- /dev/null +++ b/AtAzT4oBgHgl3EQfv_4W/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8dafb05b2578290c9e409c18b6f96a74e06d6d3207f5f6a0b6b2377f8248f5e9 +size 86679 diff --git a/BdFKT4oBgHgl3EQfXC6p/content/tmp_files/2301.11793v1.pdf.txt b/BdFKT4oBgHgl3EQfXC6p/content/tmp_files/2301.11793v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ee5bdea77c8598c2ed56c5c33f6ea2f57a940717 --- /dev/null +++ b/BdFKT4oBgHgl3EQfXC6p/content/tmp_files/2301.11793v1.pdf.txt @@ -0,0 +1,792 @@ +arXiv:2301.11793v1 [hep-th] 27 Jan 2023 +1 +Schwinger-Dyson equation in complex plane +− The (1 + 1)-dimensional Gross-Neveu model − +Hidekazu Tanaka ∗) and Shuji Sasagawa +Rikkyo University, Tokyo 171-8501, Japan +ABSTRACT +Effective mass and energy of fermions are investigated using the Schwinger- +Dyson equation (SDE) in the complex plane. As a simple example, we solve the +SDE for the (1+1)-dimensional Gross-Neveu model and study some properties of +the effective mass and energy of fermions in the complex plane. +∗) E-mail:tanakah@rikkyo.ac.jp + +2 +§1. +Introduction +Behavior of effective mass and energy in non-perturbative region is one of in- +teresting problems to be studied, because they are related to the properties of the +propagator in non-perturbative region. +Particularly, interesting phenomena are expected in Minkowski space. In some +studies, it has been pointed out that the positivity of the gluon spectral function in +quantum chromodynamics (QCD) appears to be violated in strong coupling region. +[1,2] This indicates that gluons do not have asymptotic states, suggesting that gluons +are confined to hadrons. +Unfortunately, lattice simulations for studying non-perturbative region do not +allow direct evaluation of the imaginary part of the effective mass in Minkowski space. +One useful tool for studying non-perturbative phenomena is the Schwinger-Dyson +equation (SDE) [3,4]. The structure of the gluon propagator has been evaluated by +the SDE, in which the squared momentum for the gluon is extended to the complex +value. [5,6]. They found that the gluon propagator has poles not on the real axis +in the squared momentum plane at zero temperature. In their framework, they also +showed that the spectral function of the gluon violates positive value condition. +In evaluations using the SDE, one of difficulties in Minkowski space is the exis- +tence of poles in propagator. This requires knowledge of the precise pole positions of +the propagator in the self-energy calculation. To avoid this, it is computed by Wick- +rotating the axis of integration from the real axis to the imaginary axis. However, +the Wick rotation requires the location of the poles to be known in advance, but the +value of the mass in the non-perturbative region is non-trivial. +In this paper, as a starting point for thinking about these problems, we examine +the (1 + 1)-dimensional (one dimension of time and one dimension of space) Gross- +Neveu (GN) model at zero temperature. [7] We extend the SDE to the complex +plane, and integrate the loop momentum around poles of the propagator in the self- +energy with two different integration paths in the complex energy plane. Then we +examine the properties of the solutions obtained by the SDE in the complex plane. +In Section 2, we formulate the SDE for the (1 + 1)-dimensional GN model in +terms of complex mass and energy. +In Section 3, we discuss analytical solutions +for effective mass and energy in the complex plane with finite cutoff values of the +momentum. In Section 4, we numerically calculate the effective mass and energy +using the SDE. Section 5 is devoted to the summary and some comments. Explicit +expressions of the complex mass and energy implemented in calculations are given +in Appendix. + +3 +§2. +The SDE for effective mass of fermion in complex plane +The Lagrangian density of the GN model is given by +L = i ¯ψ∂/ψ + g2 +2 ( ¯ψψ)2, +(2.1) +where ψ and g2 are the 2-component fermion field in (1 + 1) dimensions and the +coupling constant of 4-fermion interaction, respectively. +In this paper, we evaluate the fermion effective mass M using the SDE. In order +to obtain the effective mass, we calculate the one-loop self-energy Σ of the fermion +in (1 + 1) dimensions, which is given by +Σ = i +g2 +(2π)2 +� +d2QTr[S(Q)]. +(2.2) +In Eq.(2 · 2), S(Q) is an effective propagator of the fermion with momentum Q = +(q0, q), which is given by +iS(Q) = +i +Q/ − Σ + iε +(2.3) +Here, we define Σ ≡ M, because the wave-function renormalization constant of the +fermion is √Z2 = 1 in one-loop order of perturbation. +Therefore, the SDE for the effective mass M is given by +M = i 2g2 +(2π)2 +� +d2Q +M +Q2 − M2 + iε = iλ +� +dq0dq +M +q2 +0 − q2 − M2 + iε, +(2.4) +where we define λ ≡ 2g2/(2π)2 for simplicity. The propagator S(Q) has poles, which +satisfies q2 +0 − q2 − M2 + iε = 0. +In this paper, we extend q0 as a complex value z and the effective mass M is also +extended as a complex value. Explicitly, they are written as q0 = (q0)R + i(q0)I ≡ +zR + izI = z and M ≡ MR + iMI, respectively. Here, we write the denominator of +the fermion propagator S(Q) as +z2 − q2 − M2 + iε ≡ z2 − E2(q) = (z − E(q))(z + E(q)) +(2.5) +with +E(q) ≡ +� +E2(q) = +� +q2 + M2 − iε ≡ ER(q) + iEI(q). +(2.6) +Therefore, the poles are located at z = ±E(q) in the complex z plane. Here, we define +ER(q) > 0. Explicit relations among the complex values are given in Appendix. +The SDE for the effective fermion mass in terms of the complex values is written +as +M = iλ +� +dq +� +C +dz +M +z2 − q2 − M2 + iε = iλ +� +dq +� +C +dz +M +(z − E(q))(z + E(q)).(2.7) + +4 +Here, we write above equation as +M = 1 +2M(+) + 1 +2M(−), +(2.8) +where +M(±) = iλ +� +dq +� +C +dz +1 +z − z± +� +M +z + z± +� +≡ iλ +� +dq +� +C +dz +1 +z − z± +f (±)(z, q) (2.9) +with z± = ±E(q) and +f (±)(z, q) = +M +z + z± +. +(2.10) +In our calculation, we integrate Eq.(2·9) around z = z± with following two +integral paths. +(1) Integral path including the imaginary axis +In this case, we separate the integral path around the poles z± = ±E(q) to C1 +and C2 as follows: +For the integral path around z+ = E(q), we take −iΛ0 − η < z < iΛ0 − η as +the path C1, and the path C2 is defined as clockwise rotation in right-half on the +complex energy plane with z = Λ0eiθ, where we take the integration from θ = π/2 +to θ = −π/2. +On the other hand, for the integral path around z− = −E(q), we take −iΛ0+η < +z < iΛ0 + η as the path C1, and the path C2 is defined as anticlockwise rotation in +left-half on the complex energy plane with z = Λ0eiθ, where we take the integration +from θ = π/2 to θ = 3π/2. ∗) +Integrating over the integral path C around the pole z± = ±E(q) in the right- +hand side of Eq. (2·9), we have +M(±) = iλ +� +dq(∓2πi)f (±)(z±, q) = πλ +� +dq M +E(q) +(2.11) +for Λ0 → ∞. Therefore, the SDE for the effective mass is given by +M = 1 +2M(+) + 1 +2M(−) = πλ +� +dq M +E(q). +(2.12) +For η → 0, this case corresponds to the SDE for Euclidian momentum integration +with the complex mass M, which is given as +M = λ +� +dq +� ∞ +−∞ +dq4 +M +q2 +4 + q2 + M2 − iε +(2.13) +with z = iq4 in Eq. (2·7). +∗) In order to evaluate the contributions from the singular poles on the imaginary axis, we sift +the integral path by ∓η from the imaginary axis. + +5 +(2) Integral path including the real axis +In this case, we separate the integral path around the poles z± = ±E(q) to C1 +and C2 as follows: +For the integral path around z+ = E(q), we take −Λ0 − iη < z < Λ0 − iη as the +path C1 if EI > 0, and the path C2 is defined as anticlockwise rotation in upper-half +on the complex energy plane with z = Λ0eiθ, where we take the integration from +θ = 0 to θ = π. If EI < 0, we take −Λ0 + iη < z < Λ0 + iη as the path C1, and the +path C2 is defined as clockwise rotation in lower-half on the complex energy plane +with z = Λ0eiθ, where we take the integration from θ = 0 to θ = −π. +For the integral path around z− = −E(q), we take −Λ0 −iη < z < Λ0 −iη as the +path C1 if EI < 0, and the path C2 is defined as anticlockwise rotation in upper-half +on the complex energy plane with z = Λ0eiθ, where we take the integration from +θ = 0 to θ = π. If EI > 0, we take −Λ0 + iη < z < Λ0 + iη as the path C1, and the +path C2 is defined as clockwise rotation in lower-half on the complex energy plane +with z = Λ0eiθ, where we take the integration from θ = 0 to θ = −π. ∗) +Integrating over the integral path C around the pole z± = ±E(q) in the right- +hand side of Eq. (2·9), we have +M(±) = iλ +� +dq(±2πi) +� EI(q) +|EI(q)| +� +f (±)(z±, q) = −πλ +� +dq +� EI(q) +|EI(q)| +� M +E(q)(2.14) +for Λ0 → ∞. +Therefore, the effective mass is given as +M = 1 +2M(+) + 1 +2M(−) = −πλ +� +dq +� EI(q) +|EI(q)| +� M +E(q). +(2.15) +For η → 0, this case corresponds to the SDE for Minkowski momentum integra- +tion with the complex mass M, which is given as +M = iλ +� +dq +� ∞ +−∞ +dq0 +M +q2 +0 − q2 − M2 + iε +(2.16) +with z = q0 in Eq. (2·7). +§3. +Analytical solutions +We can find analytical solutions for the SDE obtained in the previous section. +Here, we write the SDE for two different integral paths as +M = πsλ +� +dq M +E(q) +(3.1) +with s = 1 for the case (1) and s = −EI(q)/|EI(q)| = −[(M2)I − ε]/|(M2)I − ε| for +the case (2), respectively. +∗) In order to evaluate the contributions from the singular poles on the real axis, we sift the +integral path by ∓iη from the real axis. + +6 +From Eq.(3·1), nontrivial solutions with M ̸= 0 are given by solving the equation +1 − πsλ +� +dq +1 +E(q) = 0. +(3.2) +Thus, the real and imaginary parts of Eq.(3·2) satisfy +1 − πsλ +� +dq ER(q) +|E(q)|2 = 0 +(3.3) +and +πsλ +� +dq EI(q) +|E(q)|2 = 0, +(3.4) +respectively. +Here, Eq. (3·4) is written as +πsλ +� +dq EI(q) +|E(q)|2 = πsλ((M2)I − ε) +� +dq +1 +2ER(q)|E(q)|2 = 0. +(3.5) +For ER(q) > 0, we obtain (M2)I − ε = 0 for sλ ̸= 0, which gives EI(q) = 0.∗) +Moreover, from +(M2)I − ε = 2MRMI − ε = 0, +(3.6) +the imaginary part of the effective mass MI is given by +MI = +ε +2MR +. +(3.7) +In the calculation below, we neglect the imaginary part of the effective mass for small +ε. Thus, we approximate as (M2)R ≃ M2 +R for simplicity. +Using EI(q) = 0, and introducing a ultraviolet cutoff Λ and an infrared cutoff δ +for the momentum q, we write Eq. (3·3) as +1 = πsλ +� Λ +−Λ +dq 1 +ER +θ(|q| − δ) = 2πsλ +� Λ +δ +dq +1 +� +q2 + M2 +R +. +(3.8) +Here, θ(|q| − δ) denotes the step function for restriction of the momentum q. +Eq. (3·8) gives +1 = 2πsλ log +������ +Λ + +� +Λ2 + M2 +R +δ + +� +δ2 + M2 +R +������ +, +(3.9) +which is satisfied if sλ > 0. Therefore, for λ > 0, s = −EI(q)/|EI(q)| = 1 should be +satisfied for the case (2). +∗) As shown in the next section, EI(q) is determined by an asymptotic value, which is numerically +calculated by the SDE with a given initial value of the mass M. + +7 +Defining mR = MR/Λ, ¯δ = δ/Λ and +1 + +� +1 + m2 +R +¯δ + +� +¯δ2 + m2 +R += e1/(2πsλ) ≡ ζ, +(3.10) +Eq. (3·10) is written as +m2 +R(Am2 +R − B) = 0 +(3.11) +with A = (1 − ζ2)2 and B = 4ζ(1 − ¯δζ)(ζ − ¯δ). +The solution for m2 +R ̸= 0 is given as +m2 +R = B +A = 4ζ(1 − ¯δζ)(ζ − ¯δ) +(1 − ζ2)2 +. +(3.12) +For sλ > 0, ζ − ¯δ > 0 is satisfied. Moreover, m2 +R > 0 demands 1 − ¯δζ > 0, which +gives +ζ = e1/(2πsλ) < 1 +¯δ = Λ +δ . +(3.13) +Eq. (3·13) restricts the coupling constant λ as +λ > +1 +2π log Λ +δ +≡ λc +(3.14) +with s = 1. +For above restriction of λ, the real part of the effective mass is given as +mR = MR +Λ += ± +� +4ζ(1 − ¯δζ)(ζ − ¯δ) +(1 − ζ2)2 +. +(3.15) +§4. +Numerical solutions +In this section, we calculate the SDE for two different integral paths. The SDE +is given in Eq. (3·1). In numerical calculation, we write the SDE for the real and +imaginary parts of the mass as +MR = 2πsλ +� Λ +δ +dq[M(E(q))∗]R +|E(q)|2 += 2πsλ +� Λ +δ +dqMRER(q) + MIEI(q) +|E(q)|2 +(4.1) +and +MI = 2πsλ +� Λ +δ +dq[M(E(q))∗]I +|E(q)|2 += 2πsλ +� Λ +δ +dqMIER(q) − MREI(q) +|E(q)|2 +, +(4.2) +,respectively with |E(q)|2 = E2 +R(q) + E2 +I (q). + +8 +We solve the SDE by iteration method from some initial input values for the +real and imaginary parts of the effective mass denoted by MR(0) and MI(0). +For the case (1), we can start from any values of the mass to solve the SDE, since +s is independent on the mass. However, for the case (2), the SDE has non-trivial +solutions only for s = −EI(q)/|EI(q)| = −[(M2)I−ε]/|(M2)I−ε| = 1. Since (M2)I = +2MRMI, we set initial input values of the real and imaginary parts of the mass, which +satisfy (M2)I(0) = 2MR(0)MI(0) < 0. +1e-010 +1e-008 +1e-006 +0.0001 +0.01 +1 +100 +0 +20 +40 +60 +80 +100 + |M|/Λ + I +λ=0.020 +λ=0.025 +λ=0.030 +Fig. 1. +The convergence behaviors of |M|/Λ for λ = 0.020, 0.025, 0.030 with MR(0) = −MI(0) = +0.01Λ. The horizontal axis denotes the number of iterations. +In Fig.1, we present the convergence behaviors of |M|/Λ = +� +M2 +R + M2 +I /Λ near +the critical coupling constant λc denoted in Eq. (3·14) with δ/Λ = 10−3, which gives +λ > 0.023. Here, we set the input values of the mass as MR(0) = −MI(0) = 0.01Λ.∗) +From Fig.1, we can conclude that λc locates between λ = 0.020 and λ = 0.025. +0.001 +0.01 +0.1 +1 +10 +100 +0 +0.5 +1 +1.5 +2 +2.5 +3 + |M|/Λ + λ +Solution by SDE +Analytical solution +Fig. 2. +The λ dependence of |M|/Λ for 0.03 ≤ λ ≤ 3 with MR(0) = −MI(0) = 0.01Λ. The dotted +curve denotes the calculated result by the analytical solution divided by Λ. +∗) We set ε = 10−5Λ2. + +9 +In Fig.2, we present the λ dependence of the absolute value of the effective mass +|M|/Λ for 0.03 ≤ λ ≤ 3 with MR(0) = −MI(0) = 0.01Λ. The dotted curve denotes the +calculated result using Eqs. (3·7) and (3·15). +In the following calculations, we set four initial values for the mass as MR(0) = +±0.01Λ and MI(0) = ±0.01Λ, respectively. +0 +0.5 +1 +1.5 +2 +2.5 +3 +0 +5 +10 +15 +20 + ER/Λ + I +MR(0)>0 MI(0)>0 +MR(0)>0 MI(0)<0 +Analytical soltion for MR>0 +MR(0)<0 MI(0)>0 +MR(0)<0 MI(0)<0 +Analytical soltion for MR<0 +Fig. 3. +The convergence behavior of ER(q)/Λ with q = 0. The straight lines denote the energy +divided by Λ calculated using the analytical solutions of the mass. The horizontal axis denotes +the number of iterations. +-2e-005 +-1.5e-005 +-1e-005 +-5e-006 +0 +5e-006 +1e-005 +1.5e-005 +2e-005 +0 +5 +10 +15 +20 + EI/Λ + I +MR(0)>0 MI(0)>0 +MR(0)>0 MI(0)<0 +Analytical soltion for MR>0 +MR(0)<0 MI(0)>0 +MR(0)<0 MI(0)<0 +Analytical soltion for MR<0 +Fig. 4. +The convergence behavior of EI(q)/Λ with q = 0. The straight lines denote the analytical +solutions of energy, which is EI(q)/Λ = 0. The horizontal axis denotes the number of iterations. +In Figs.3 and 4, we show the convergence behaviors of the real and imaginary +parts of the energy with the momentum q = 0, respectively. The straight lines denote +the energy calculated using the analytical solutions of the mass given in Eqs.(3·7) +and (3·15).(See Appendix.) +Since the real part of the energy is defined to be positive, the numerical results +do not depend on the sign of initial values of the mass. The imaginary part of the +energy converges EI → 0, in which the convergence behavior depends on the sign of + +10 +(M2)I(0). The calculated results shown in Figs. 1-4 are common for the two integral +paths (1) and (2). +On the other hand, the convergence behaviors for the effective mass are different +for the two integral paths. +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +0 +5 +10 +15 +20 + MR/Λ + I +MR(0)>0 MI(0)>0 +MR(0)>0 MI(0)<0 +Analytical soltion for MR>0 +MR(0)<0 MI(0)>0 +MR(0)<0 MI(0)<0 +Analytical soltion for MR<0 +Fig. 5. +The convergence behavior of MR/Λ for the case (1). The straight lines denote the analytical +solutions of the real part of the effective mass divided by Λ. The horizontal axis denotes the +number of iterations. +-2e-005 +-1.5e-005 +-1e-005 +-5e-006 +0 +5e-006 +1e-005 +1.5e-005 +2e-005 +0 +5 +10 +15 +20 + MI/Λ + I +MR(0)>0 MI(0)>0 +MR(0)>0 MI(0)<0 +Analytical soltion for MI>0 +MR(0)<0 MI(0)>0 +MR(0)<0 MI(0)<0 +Analytical soltion for MI<0 +Fig. 6. +The convergence behavior of MI/Λ for the case (1). The straight lines denote the analytical +solutions of the imaginary part of the effective mass divided by Λ. The horizontal axis denotes +the number of iterations. +For the case (1), the real and imaginary parts of the effective mass calculated +by the SDE are shown in Figs.5 and 6, respectively. +As shown in Fig. +5, the +convergent solution splits into two values depending on the sign of MR(0)/Λ. As +shown in Fig.6, the imaginary part of the effective mass is small and it depends +on ε. Moreover, MI/Λ initially behaves according to the sign of the initial value +of MI(0)/Λ, but the convergent solution depends on the sign of the initial value +MR(0)/Λ. + +11 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +0 +5 +10 +15 +20 + MR/Λ + I +MR(0)>0 MI(0)>0 +MR(0)>0 MI(0)<0 +Analytical soltion for MR>0 +MR(0)<0 MI(0)>0 +MR(0)<0 MI(0)<0 +Analytical soltion for MR<0 +Fig. 7. +The convergence behavior of MR/Λ for the case (2). The straight lines denote the analytical +solutions of the real part of the effective mass divided by Λ. The horizontal axis denotes the +number of iterations. +-2e-005 +-1.5e-005 +-1e-005 +-5e-006 +0 +5e-006 +1e-005 +1.5e-005 +2e-005 +0 +5 +10 +15 +20 + MI/Λ + I +MR(0)>0 MI(0)>0 +MR(0)>0 MI(0)<0 +Analytical soltion for MI>0 +MR(0)<0 MI(0)>0 +MR(0)<0 MI(0)<0 +Analytical soltion for MI<0 +Fig. 8. +The convergence behavior of MI/Λ for the case (2). The straight lines denote the analytical +solutions of the imaginary part of the effective mass divided by Λ. The horizontal axis denotes +the number of iterations. +For the case (2), the convergence behaviors of the effective mass calculated by +the SDE are shown in Figs.7 and 8, respectively. As shown in Figs. 7 and 8, the +convergent solution splits into two values depending on the sign of the initial value +of MR(0)/Λ for (M2)I(0) < 0. However, for (M2)I(0) > 0, the iterated values are +oscillated. In this case, since s < 0, the SDE has no non-trivial solution. +§5. +Summary and Comments +In this paper, we examined the (1 + 1)-dimensional Gross-Neveu (GN) model at +zero temperature and solved the Schwinger-Dyson equation (SDE) in the complex +plane. We compered the effective mass and energy calculated in two different integral +paths in the complex energy plane. Then we examined the properties of the solutions + +12 +obtained by the SDE. +First, we investigated the effect of the momentum cutoff on chiral symmetry +breaking. Though the cutoff on the momentum is an artificial parameter for numer- +ical calculations, this example suggests a possibility of changing the critical point in +a physical system with restricted momentum. +In the model treated in this paper, the imaginary part of the energy is zero and +the poles of the effective propagator are on the real axis, which is different situation +in QCD pointed out in Ref.[5,6], in which the poles are not on the real axis. +We also investigated the dependence of the solutions obtained by the SDE on +the initial input parameters. The effective mass obtained by the SDE depends on +the sign of the input initial input values. Our calculations suggest that the SDE +may lead to multiple solutions depending on the initial input values. Moreover, it +can be seen that, for the integral path including the real axis, which corresponds to +the SDE in Minkowski space, the input values leading to chiral symmetry broken +phases are limited than the case with the integral path including the imaginary axis, +which corresponds to the SDE in Euclidean space. This result suggests that the +calculation of SDE requires careful selection of input values. On the other hand, in +our example, when an oscillating solution exists, there exists a solution with broken +chiral symmetry for input values of appropriate sign. +The SDE extended to complex plane may be useful for investigating a wider +class of non-perturbative solutions. Although further computational techniques will +be required, it is expected that the method presented in this paper can be applied +to other models such as non-perturbative QCD in Minkowski space. +Appendix. Complex mass and energy +In order to solve the SDE in complex energy plane, we need the explicit forms +ofcomplex mass and energy. +We define the complex mass as M = MR + iMI and the squared of the mass as +M2 = (M2)R + i(M2)I. Here, (M2)R and (M2)I are given by +(M2)R = M2 +R − M2 +I , +(M2)I = 2MRMI. +The squared energy E2 is defined by +E2 = q2 + M2 − iε ≡ (E2)R + i(E2)I. +with +(E2)R = q2 + (M2)R, +(E2)I = (M2)I − ε. +On the other hand, using the complex energy E = ER + iEI, (E2)R and (E2)I are +also written as +(E2)R = E2 +R − E2 +I , +(E2)I = 2EREI. +Therefore the imaginary part of the energy is written as +EI = (E2)I +2ER +. + +13 +Substituting above equation to (E2)R = E2 +R − E2 +I , we have a quadratic equation for +E2 +R as +(E2 +R)2 − E2 +R(E2)R − (E2)2 +I /4 = 0. +The solution of the equation for E2 +R > 0 is given by +E2 +R = (E2)R + |E2| +2 +with |E2| = +� +[(E2)R]2 + [(E2)I]2. +Therefore, we have the solution +ER = +� +(E2)R + |E2| +2 +, +EI = (E2)I +2ER +for ER > 0. +- +References +1) D.Dudal,O.Oliveira and P.J.Silva, Phys.Rev.D89,014010(2014) [arXiv:1310:4069 [hep- +lat]]. +2) F.Siringo, Phys.Rev.D94,0114036(2016) [arXiv:1605:07357 [hep-ph]]. +3) F.J.Dyson, Phys.Rev.75 (1949) ,1736. +4) J.S.Schwinger,Proc.Nat.Acad.Sci.37 (1951),452. +5) S.Strauss, +C.S.Fischer +and +C.Kellermann, +Phys. +Rev. +Lett. +109 +(2012),252001 +[arXiv:1208:6239 [hep-ph]]. +6) C.S.Fischer and M.Q.Huber, Phys. Rev. D102 (2020),094005 [arXiv:2007.11505]. +7) D.J.Gross and A.Neveu, Phys. Rev. D10 (1974),3235. + diff --git a/BdFKT4oBgHgl3EQfXC6p/content/tmp_files/load_file.txt b/BdFKT4oBgHgl3EQfXC6p/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3c6c16bc84736a4ef232f58a61ee9593f32ac41d --- /dev/null +++ b/BdFKT4oBgHgl3EQfXC6p/content/tmp_files/load_file.txt @@ -0,0 +1,314 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf,len=313 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='11793v1 [hep-th] 27 Jan 2023 1 Schwinger-Dyson equation in complex plane − The (1 + 1)-dimensional Gross-Neveu model − Hidekazu Tanaka ∗) and Shuji Sasagawa Rikkyo University, Tokyo 171-8501, Japan ABSTRACT Effective mass and energy of fermions are investigated using the Schwinger- Dyson equation (SDE) in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' As a simple example, we solve the SDE for the (1+1)-dimensional Gross-Neveu model and study some properties of the effective mass and energy of fermions in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' ∗) E-mail:tanakah@rikkyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='jp 2 §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Introduction Behavior of effective mass and energy in non-perturbative region is one of in- teresting problems to be studied, because they are related to the properties of the propagator in non-perturbative region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Particularly, interesting phenomena are expected in Minkowski space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' In some studies, it has been pointed out that the positivity of the gluon spectral function in quantum chromodynamics (QCD) appears to be violated in strong coupling region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' [1,2] This indicates that gluons do not have asymptotic states, suggesting that gluons are confined to hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Unfortunately, lattice simulations for studying non-perturbative region do not allow direct evaluation of the imaginary part of the effective mass in Minkowski space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' One useful tool for studying non-perturbative phenomena is the Schwinger-Dyson equation (SDE) [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The structure of the gluon propagator has been evaluated by the SDE, in which the squared momentum for the gluon is extended to the complex value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' They found that the gluon propagator has poles not on the real axis in the squared momentum plane at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' In their framework, they also showed that the spectral function of the gluon violates positive value condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' In evaluations using the SDE, one of difficulties in Minkowski space is the exis- tence of poles in propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' This requires knowledge of the precise pole positions of the propagator in the self-energy calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' To avoid this, it is computed by Wick- rotating the axis of integration from the real axis to the imaginary axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' However, the Wick rotation requires the location of the poles to be known in advance, but the value of the mass in the non-perturbative region is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' In this paper, as a starting point for thinking about these problems, we examine the (1 + 1)-dimensional (one dimension of time and one dimension of space) Gross- Neveu (GN) model at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' [7] We extend the SDE to the complex plane, and integrate the loop momentum around poles of the propagator in the self- energy with two different integration paths in the complex energy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Then we examine the properties of the solutions obtained by the SDE in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' In Section 2, we formulate the SDE for the (1 + 1)-dimensional GN model in terms of complex mass and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' In Section 3, we discuss analytical solutions for effective mass and energy in the complex plane with finite cutoff values of the momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' In Section 4, we numerically calculate the effective mass and energy using the SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Section 5 is devoted to the summary and some comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Explicit expressions of the complex mass and energy implemented in calculations are given in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 3 §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The SDE for effective mass of fermion in complex plane The Lagrangian density of the GN model is given by L = i ¯ψ∂/ψ + g2 2 ( ¯ψψ)2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='1) where ψ and g2 are the 2-component fermion field in (1 + 1) dimensions and the coupling constant of 4-fermion interaction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' In this paper, we evaluate the fermion effective mass M using the SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' In order to obtain the effective mass, we calculate the one-loop self-energy Σ of the fermion in (1 + 1) dimensions, which is given by Σ = i g2 (2π)2 � d2QTr[S(Q)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='2) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (2 · 2), S(Q) is an effective propagator of the fermion with momentum Q = (q0, q), which is given by iS(Q) = i Q/ − Σ + iε (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='3) Here, we define Σ ≡ M, because the wave-function renormalization constant of the fermion is √Z2 = 1 in one-loop order of perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Therefore, the SDE for the effective mass M is given by M = i 2g2 (2π)2 � d2Q M Q2 − M2 + iε = iλ � dq0dq M q2 0 − q2 − M2 + iε, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='4) where we define λ ≡ 2g2/(2π)2 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The propagator S(Q) has poles, which satisfies q2 0 − q2 − M2 + iε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' In this paper, we extend q0 as a complex value z and the effective mass M is also extended as a complex value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Explicitly, they are written as q0 = (q0)R + i(q0)I ≡ zR + izI = z and M ≡ MR + iMI, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Here, we write the denominator of the fermion propagator S(Q) as z2 − q2 − M2 + iε ≡ z2 − E2(q) = (z − E(q))(z + E(q)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='5) with E(q) ≡ � E2(q) = � q2 + M2 − iε ≡ ER(q) + iEI(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='6) Therefore, the poles are located at z = ±E(q) in the complex z plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Here, we define ER(q) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Explicit relations among the complex values are given in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The SDE for the effective fermion mass in terms of the complex values is written as M = iλ � dq � C dz M z2 − q2 − M2 + iε = iλ � dq � C dz M (z − E(q))(z + E(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='7) 4 Here, we write above equation as M = 1 2M(+) + 1 2M(−), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='8) where M(±) = iλ � dq � C dz 1 z − z± � M z + z± � ≡ iλ � dq � C dz 1 z − z± f (±)(z, q) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='9) with z± = ±E(q) and f (±)(z, q) = M z + z± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='10) In our calculation, we integrate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (2·9) around z = z± with following two integral paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (1) Integral path including the imaginary axis In this case, we separate the integral path around the poles z± = ±E(q) to C1 and C2 as follows: For the integral path around z+ = E(q), we take −iΛ0 − η < z < iΛ0 − η as the path C1, and the path C2 is defined as clockwise rotation in right-half on the complex energy plane with z = Λ0eiθ, where we take the integration from θ = π/2 to θ = −π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' On the other hand, for the integral path around z− = −E(q), we take −iΛ0+η < z < iΛ0 + η as the path C1, and the path C2 is defined as anticlockwise rotation in left-half on the complex energy plane with z = Λ0eiθ, where we take the integration from θ = π/2 to θ = 3π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' ∗) Integrating over the integral path C around the pole z± = ±E(q) in the right- hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (2·9), we have M(±) = iλ � dq(∓2πi)f (±)(z±, q) = πλ � dq M E(q) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='11) for Λ0 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Therefore, the SDE for the effective mass is given by M = 1 2M(+) + 1 2M(−) = πλ � dq M E(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='12) For η → 0, this case corresponds to the SDE for Euclidian momentum integration with the complex mass M, which is given as M = λ � dq � ∞ −∞ dq4 M q2 4 + q2 + M2 − iε (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='13) with z = iq4 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (2·7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' ∗) In order to evaluate the contributions from the singular poles on the imaginary axis, we sift the integral path by ∓η from the imaginary axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 5 (2) Integral path including the real axis In this case, we separate the integral path around the poles z± = ±E(q) to C1 and C2 as follows: For the integral path around z+ = E(q), we take −Λ0 − iη < z < Λ0 − iη as the path C1 if EI > 0, and the path C2 is defined as anticlockwise rotation in upper-half on the complex energy plane with z = Λ0eiθ, where we take the integration from θ = 0 to θ = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' If EI < 0, we take −Λ0 + iη < z < Λ0 + iη as the path C1, and the path C2 is defined as clockwise rotation in lower-half on the complex energy plane with z = Λ0eiθ, where we take the integration from θ = 0 to θ = −π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' For the integral path around z− = −E(q), we take −Λ0 −iη < z < Λ0 −iη as the path C1 if EI < 0, and the path C2 is defined as anticlockwise rotation in upper-half on the complex energy plane with z = Λ0eiθ, where we take the integration from θ = 0 to θ = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' If EI > 0, we take −Λ0 + iη < z < Λ0 + iη as the path C1, and the path C2 is defined as clockwise rotation in lower-half on the complex energy plane with z = Λ0eiθ, where we take the integration from θ = 0 to θ = −π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' ∗) Integrating over the integral path C around the pole z± = ±E(q) in the right- hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (2·9), we have M(±) = iλ � dq(±2πi) � EI(q) |EI(q)| � f (±)(z±, q) = −πλ � dq � EI(q) |EI(q)| � M E(q)(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='14) for Λ0 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Therefore, the effective mass is given as M = 1 2M(+) + 1 2M(−) = −πλ � dq � EI(q) |EI(q)| � M E(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='15) For η → 0, this case corresponds to the SDE for Minkowski momentum integra- tion with the complex mass M, which is given as M = iλ � dq � ∞ −∞ dq0 M q2 0 − q2 − M2 + iε (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='16) with z = q0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (2·7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Analytical solutions We can find analytical solutions for the SDE obtained in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Here, we write the SDE for two different integral paths as M = πsλ � dq M E(q) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='1) with s = 1 for the case (1) and s = −EI(q)/|EI(q)| = −[(M2)I − ε]/|(M2)I − ε| for the case (2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' ∗) In order to evaluate the contributions from the singular poles on the real axis, we sift the integral path by ∓iη from the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 6 From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (3·1), nontrivial solutions with M ̸= 0 are given by solving the equation 1 − πsλ � dq 1 E(q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='2) Thus, the real and imaginary parts of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (3·2) satisfy 1 − πsλ � dq ER(q) |E(q)|2 = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='3) and πsλ � dq EI(q) |E(q)|2 = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='4) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Here, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (3·4) is written as πsλ � dq EI(q) |E(q)|2 = πsλ((M2)I − ε) � dq 1 2ER(q)|E(q)|2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='5) For ER(q) > 0, we obtain (M2)I − ε = 0 for sλ ̸= 0, which gives EI(q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='∗) Moreover, from (M2)I − ε = 2MRMI − ε = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='6) the imaginary part of the effective mass MI is given by MI = ε 2MR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='7) In the calculation below, we neglect the imaginary part of the effective mass for small ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Thus, we approximate as (M2)R ≃ M2 R for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Using EI(q) = 0, and introducing a ultraviolet cutoff Λ and an infrared cutoff δ for the momentum q, we write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (3·3) as 1 = πsλ � Λ −Λ dq 1 ER θ(|q| − δ) = 2πsλ � Λ δ dq 1 � q2 + M2 R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='8) Here, θ(|q| − δ) denotes the step function for restriction of the momentum q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (3·8) gives 1 = 2πsλ log ������ Λ + � Λ2 + M2 R δ + � δ2 + M2 R ������ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='9) which is satisfied if sλ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Therefore, for λ > 0, s = −EI(q)/|EI(q)| = 1 should be satisfied for the case (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' ∗) As shown in the next section, EI(q) is determined by an asymptotic value, which is numerically calculated by the SDE with a given initial value of the mass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 7 Defining mR = MR/Λ, ¯δ = δ/Λ and 1 + � 1 + m2 R ¯δ + � ¯δ2 + m2 R = e1/(2πsλ) ≡ ζ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='10) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (3·10) is written as m2 R(Am2 R − B) = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='11) with A = (1 − ζ2)2 and B = 4ζ(1 − ¯δζ)(ζ − ¯δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The solution for m2 R ̸= 0 is given as m2 R = B A = 4ζ(1 − ¯δζ)(ζ − ¯δ) (1 − ζ2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='12) For sλ > 0, ζ − ¯δ > 0 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Moreover, m2 R > 0 demands 1 − ¯δζ > 0, which gives ζ = e1/(2πsλ) < 1 ¯δ = Λ δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='13) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (3·13) restricts the coupling constant λ as λ > 1 2π log Λ δ ≡ λc (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='14) with s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' For above restriction of λ, the real part of the effective mass is given as mR = MR Λ = ± � 4ζ(1 − ¯δζ)(ζ − ¯δ) (1 − ζ2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='15) §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Numerical solutions In this section, we calculate the SDE for two different integral paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The SDE is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (3·1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' In numerical calculation, we write the SDE for the real and imaginary parts of the mass as MR = 2πsλ � Λ δ dq[M(E(q))∗]R |E(q)|2 = 2πsλ � Λ δ dqMRER(q) + MIEI(q) |E(q)|2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='1) and MI = 2πsλ � Λ δ dq[M(E(q))∗]I |E(q)|2 = 2πsλ � Λ δ dqMIER(q) − MREI(q) |E(q)|2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='2) ,respectively with |E(q)|2 = E2 R(q) + E2 I (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 8 We solve the SDE by iteration method from some initial input values for the real and imaginary parts of the effective mass denoted by MR(0) and MI(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' For the case (1), we can start from any values of the mass to solve the SDE, since s is independent on the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' However, for the case (2), the SDE has non-trivial solutions only for s = −EI(q)/|EI(q)| = −[(M2)I−ε]/|(M2)I−ε| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Since (M2)I = 2MRMI, we set initial input values of the real and imaginary parts of the mass, which satisfy (M2)I(0) = 2MR(0)MI(0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 1e-010 1e-008 1e-006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='01 1 100 0 20 40 60 80 100 |M|/Λ I λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='020 λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='025 λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='030 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The convergence behaviors of |M|/Λ for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='020, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='025, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='030 with MR(0) = −MI(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='01Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The horizontal axis denotes the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='1, we present the convergence behaviors of |M|/Λ = � M2 R + M2 I /Λ near the critical coupling constant λc denoted in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (3·14) with δ/Λ = 10−3, which gives λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Here, we set the input values of the mass as MR(0) = −MI(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='01Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='∗) From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='1, we can conclude that λc locates between λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='020 and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='1 1 10 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='5 3 |M|/Λ λ Solution by SDE Analytical solution Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The λ dependence of |M|/Λ for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='03 ≤ λ ≤ 3 with MR(0) = −MI(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='01Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The dotted curve denotes the calculated result by the analytical solution divided by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' ∗) We set ε = 10−5Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 9 In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='2, we present the λ dependence of the absolute value of the effective mass |M|/Λ for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='03 ≤ λ ≤ 3 with MR(0) = −MI(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='01Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The dotted curve denotes the calculated result using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (3·7) and (3·15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' In the following calculations, we set four initial values for the mass as MR(0) = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='01Λ and MI(0) = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='01Λ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='5 3 0 5 10 15 20 ER/Λ I MR(0)>0 MI(0)>0 MR(0)>0 MI(0)<0 Analytical soltion for MR>0 MR(0)<0 MI(0)>0 MR(0)<0 MI(0)<0 Analytical soltion for MR<0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The convergence behavior of ER(q)/Λ with q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The straight lines denote the energy divided by Λ calculated using the analytical solutions of the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The horizontal axis denotes the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 2e-005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='5e-005 1e-005 5e-006 0 5e-006 1e-005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='5e-005 2e-005 0 5 10 15 20 EI/Λ I MR(0)>0 MI(0)>0 MR(0)>0 MI(0)<0 Analytical soltion for MR>0 MR(0)<0 MI(0)>0 MR(0)<0 MI(0)<0 Analytical soltion for MR<0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The convergence behavior of EI(q)/Λ with q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The straight lines denote the analytical solutions of energy, which is EI(q)/Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The horizontal axis denotes the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='3 and 4, we show the convergence behaviors of the real and imaginary parts of the energy with the momentum q = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The straight lines denote the energy calculated using the analytical solutions of the mass given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (3·7) and (3·15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' (See Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=') Since the real part of the energy is defined to be positive, the numerical results do not depend on the sign of initial values of the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The imaginary part of the energy converges EI → 0, in which the convergence behavior depends on the sign of 10 (M2)I(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The calculated results shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 1-4 are common for the two integral paths (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' On the other hand, the convergence behaviors for the effective mass are different for the two integral paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 3 2 1 0 1 2 3 4 5 0 5 10 15 20 MR/Λ I MR(0)>0 MI(0)>0 MR(0)>0 MI(0)<0 Analytical soltion for MR>0 MR(0)<0 MI(0)>0 MR(0)<0 MI(0)<0 Analytical soltion for MR<0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The convergence behavior of MR/Λ for the case (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The straight lines denote the analytical solutions of the real part of the effective mass divided by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The horizontal axis denotes the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 2e-005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='5e-005 1e-005 5e-006 0 5e-006 1e-005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='5e-005 2e-005 0 5 10 15 20 MI/Λ I MR(0)>0 MI(0)>0 MR(0)>0 MI(0)<0 Analytical soltion for MI>0 MR(0)<0 MI(0)>0 MR(0)<0 MI(0)<0 Analytical soltion for MI<0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The convergence behavior of MI/Λ for the case (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The straight lines denote the analytical solutions of the imaginary part of the effective mass divided by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The horizontal axis denotes the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' For the case (1), the real and imaginary parts of the effective mass calculated by the SDE are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='5 and 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 5, the convergent solution splits into two values depending on the sign of MR(0)/Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='6, the imaginary part of the effective mass is small and it depends on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Moreover, MI/Λ initially behaves according to the sign of the initial value of MI(0)/Λ, but the convergent solution depends on the sign of the initial value MR(0)/Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 11 3 2 1 0 1 2 3 4 5 0 5 10 15 20 MR/Λ I MR(0)>0 MI(0)>0 MR(0)>0 MI(0)<0 Analytical soltion for MR>0 MR(0)<0 MI(0)>0 MR(0)<0 MI(0)<0 Analytical soltion for MR<0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The convergence behavior of MR/Λ for the case (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The straight lines denote the analytical solutions of the real part of the effective mass divided by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The horizontal axis denotes the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 2e-005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='5e-005 1e-005 5e-006 0 5e-006 1e-005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='5e-005 2e-005 0 5 10 15 20 MI/Λ I MR(0)>0 MI(0)>0 MR(0)>0 MI(0)<0 Analytical soltion for MI>0 MR(0)<0 MI(0)>0 MR(0)<0 MI(0)<0 Analytical soltion for MI<0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The convergence behavior of MI/Λ for the case (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The straight lines denote the analytical solutions of the imaginary part of the effective mass divided by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The horizontal axis denotes the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' For the case (2), the convergence behaviors of the effective mass calculated by the SDE are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='7 and 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' As shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 7 and 8, the convergent solution splits into two values depending on the sign of the initial value of MR(0)/Λ for (M2)I(0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' However, for (M2)I(0) > 0, the iterated values are oscillated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' In this case, since s < 0, the SDE has no non-trivial solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Summary and Comments In this paper, we examined the (1 + 1)-dimensional Gross-Neveu (GN) model at zero temperature and solved the Schwinger-Dyson equation (SDE) in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' We compered the effective mass and energy calculated in two different integral paths in the complex energy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Then we examined the properties of the solutions 12 obtained by the SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' First, we investigated the effect of the momentum cutoff on chiral symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Though the cutoff on the momentum is an artificial parameter for numer- ical calculations, this example suggests a possibility of changing the critical point in a physical system with restricted momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' In the model treated in this paper, the imaginary part of the energy is zero and the poles of the effective propagator are on the real axis, which is different situation in QCD pointed out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' [5,6], in which the poles are not on the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' We also investigated the dependence of the solutions obtained by the SDE on the initial input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The effective mass obtained by the SDE depends on the sign of the input initial input values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Our calculations suggest that the SDE may lead to multiple solutions depending on the initial input values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Moreover, it can be seen that, for the integral path including the real axis, which corresponds to the SDE in Minkowski space, the input values leading to chiral symmetry broken phases are limited than the case with the integral path including the imaginary axis, which corresponds to the SDE in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' This result suggests that the calculation of SDE requires careful selection of input values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' On the other hand, in our example, when an oscillating solution exists, there exists a solution with broken chiral symmetry for input values of appropriate sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The SDE extended to complex plane may be useful for investigating a wider class of non-perturbative solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Although further computational techniques will be required, it is expected that the method presented in this paper can be applied to other models such as non-perturbative QCD in Minkowski space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Complex mass and energy In order to solve the SDE in complex energy plane, we need the explicit forms ofcomplex mass and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' We define the complex mass as M = MR + iMI and the squared of the mass as M2 = (M2)R + i(M2)I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Here, (M2)R and (M2)I are given by (M2)R = M2 R − M2 I , (M2)I = 2MRMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The squared energy E2 is defined by E2 = q2 + M2 − iε ≡ (E2)R + i(E2)I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' with (E2)R = q2 + (M2)R, (E2)I = (M2)I − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' On the other hand, using the complex energy E = ER + iEI, (E2)R and (E2)I are also written as (E2)R = E2 R − E2 I , (E2)I = 2EREI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Therefore the imaginary part of the energy is written as EI = (E2)I 2ER .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 13 Substituting above equation to (E2)R = E2 R − E2 I , we have a quadratic equation for E2 R as (E2 R)2 − E2 R(E2)R − (E2)2 I /4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' The solution of the equation for E2 R > 0 is given by E2 R = (E2)R + |E2| 2 with |E2| = � [(E2)R]2 + [(E2)I]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Therefore, we have the solution ER = � (E2)R + |E2| 2 , EI = (E2)I 2ER for ER > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' References 1) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Dudal,O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Oliveira and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Silva, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='D89,014010(2014) [arXiv:1310:4069 [hep- lat]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 2) F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Siringo, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='D94,0114036(2016) [arXiv:1605:07357 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 3) F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Dyson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='75 (1949) ,1736.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 4) J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Schwinger,Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='37 (1951),452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 5) S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Strauss, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Fischer and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Kellermann, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 109 (2012),252001 [arXiv:1208:6239 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 6) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Fischer and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Huber, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' D102 (2020),094005 [arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='11505].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' 7) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Gross and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content='Neveu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} +page_content=' D10 (1974),3235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFKT4oBgHgl3EQfXC6p/content/2301.11793v1.pdf'} diff --git a/BtAyT4oBgHgl3EQf4PqZ/vector_store/index.faiss b/BtAyT4oBgHgl3EQf4PqZ/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..5a4eaf701427d296db208fe54dc220792802f4d9 --- /dev/null +++ b/BtAyT4oBgHgl3EQf4PqZ/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:87bf19ba8cfba3cdfc781a5cd6ce955f27f5396cb210b75d496ff4923cd70318 +size 4194349 diff --git a/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf b/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6a31bc68c499188d8e6434ac80db469803e90585 --- /dev/null +++ b/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5e4378bd4e534d78a8ae8900f6fdefc9fead1b3d5d4fbe156285be348db35ba5 +size 18686188 diff --git a/E9E1T4oBgHgl3EQfqgWQ/vector_store/index.faiss b/E9E1T4oBgHgl3EQfqgWQ/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..a1722c45ab2258d94b644a4dacd1d967783b9b53 --- /dev/null +++ b/E9E1T4oBgHgl3EQfqgWQ/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c6fe55497b5c562283ecf96af9cfb3d4ddcba1c661fe28ac2fdb25ad6550ab00 +size 7340077 diff --git a/ENE2T4oBgHgl3EQfogjY/content/tmp_files/load_file.txt b/ENE2T4oBgHgl3EQfogjY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ab49093270a48444d899309cbd94ce42b34cd035 --- /dev/null +++ b/ENE2T4oBgHgl3EQfogjY/content/tmp_files/load_file.txt @@ -0,0 +1,7619 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf,len=7618 +page_content='Jian Guo 1, 2 IDEA Research & The Hong Kong University of Science and Technology (Guangzhou) Saizhuo Wang 1 The Hong Kong University of Science and Technology & IDEA Research Lionel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Ni The Hong Kong University of Science and Technology (Guangzhou) & The Hong Kong University of Science and Technology Heung-Yeung Shum 2 IDEA Research & The Hong Kong University of Science and Technology Keywords: AGI, Artificial Intelligence, AutoML, Causality Engineering, Deep Learning, Feature Engineering, Investment Engineering, Knowledge Graph, Knowledge Reasoning , Knowledge Representation, Model Compression, NAS, Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0, Quantitative Investment, Risk Graph, XAI idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='cn IDEA Research Report Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0: Engineering Quantitative Investment with Automated, Explainable and Knowledge-driven Artificial Intelligence Future Trend and Perspective 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Equal Contribution 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Corresponding Author arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='04020v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='CP] 13 Dec 2022 idea INTERNATIONAL DIGITAL ECONOMY ACADEMYTable of Contents 1 Introduction 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Wealth Management and Quant .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Quant Strategies .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Components of Quant Strategy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Examples of Popular Strategies .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3 Fundamental Principles of Asset Management .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Fundamental Law of Active Management 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Impossible Trinity of Investment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 History of Quantitative Investment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Q-Quant and P-Quant .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Landmarks in Q-Quant .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3 Landmarks in P-Quant .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 Development of Quant in Industry .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='5 Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0: Why and What .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Limitations of Quant 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 What is Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 8 2 Automated AI for Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Automating Quant Research Pipeline .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Traditional Quant Pipeline .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Automated AI Quant Pipeline .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Automating Factor Mining .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Symbolic Factors .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Machine Learning Factors .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3 Automated Modeling .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Search Space .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Search Algorithm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3 Accelerating Evaluation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 Automated One-click Deployment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Acceleration by Model Compilation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Acceleration by Model Compression .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 16 3 Explainable AI for Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Overview of Explainable AI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Model-intrinsic Explanation in XAI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Model-agnostic Explanation in XAI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Explainable AI for Quant .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Explanation on Stock .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Explanation on Time .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3 Explanation on Factors .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 21 4 Knowledge-driven AI for Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Knowledge Representation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Knowledge Base Techniques .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Knowledge Graph Techniques .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Knowledge Reasoning .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Symbolic Reasoning .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Neural Reasoning .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3 Neurosymbolic Reasoning .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3 Application in Quant .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 26 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Building a Financial Knowledge Graph 26 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Knowledge Reasoning for Quant .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 26 5 Building Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0: Engineering & Architecture 27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 System for Offline Research .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Hardware Platform Architecture .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Design of Data System .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3 Factor Mining System .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 Knowledge-based System .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='5 Modeling System .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 System for Online Trading .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Model Deployment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Trading Execution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 31 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3 Trading Analysis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 31 6 Discussion on 10 Challenges in Quant Technology 31 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Exponentially Growing Demand of Computing Power .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 31 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 and Supercomputers .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 31 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Solving Computing Power Dilemma .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 33 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Alternative Data Technology .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 33 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Examples of Alternative Data .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 34 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Problems in Data Acquisition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 34 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3 Problems in Data Aggregation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 34 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3 Financial Knowledge Engineering .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 35 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Difficulties in Knowledge Engineering .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 35 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Knowledge Engineering vs Large Model 35 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 Financial Metaverse & World Model Simulator 35 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Financial Metaverse Market Simulator .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 35 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 World Model for Simulation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 36 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='5 Cognitive AI & Causality Engineering .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 36 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Cognitive AI for Investment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 36 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Causality Engineering .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 37 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='6 AI Risk Graph & Systematic Modeling .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 37 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Risk Graph for Systematic Modeling .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 37 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Complex Risk Measure for Investment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 37 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='7 Spatiotemporal Modeling .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 37 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Unifying Cross-section & Time-series .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 38 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Spatiotemporal Graph for Quant .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 38 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='8 Universal Modeling .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 38 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 Pretraining-Funetuning Paradigm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 38 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Challenge in Quant Pretraining .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 38 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='9 Robust Modeling .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 38 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='10 End-to-end Modeling .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 39 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 End-to-end Consistent Optimization .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 40 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 Learning Unstructured Data .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 40 7 Conclusion and Perspective 40 Acknowledgement 40 References 40 Author Biographies 53 2 Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0: Engineering Quantitative Investment with Automated, Explainable and Knowledge-driven Artificial Intelligence Jian Guoa,c,1,∗, Saizhuo Wanga,b,1,2, Lionel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Nib,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Heung-Yeung Shuma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='∗ aIDEA Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' International Digital Economy Academy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 5 Shihua Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Futian District,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Shenzhen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 518045,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Guangdong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' China bThe Hong Kong University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Clear Water Bay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Kowloon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Hong Kong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 999077,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' China cThe Hong Kong University of Science and Technology (Guangzhou),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 1st Duxue Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Nansha District,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Guangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 518055,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Guangdong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' China Abstract Quantitative investment (“quant”) is an interdisciplinary field combining financial engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' computer science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Quant has become one of the mainstream investment methodologies over the past decades, and has experienced three generations: Quant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0, trading by mathematical modeling to discover mis-priced assets in markets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Quant 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0, shifting quant research pipeline from small “strategy workshops” to large “alpha factories”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Quant 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0, applying deep learning techniques to dis- cover complex nonlinear pricing rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Despite its advantage in prediction, deep learning relies on extremely large data volume and labor-intensive tuning of “black-box” neural network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' To address these limitations, in this paper, we introduce Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 and provide an engineering perspective for next-generation quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 has three key differentiating components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' First, Automated AI changes quant pipeline from traditional hand-craft modeling to the state-of-the-art automated modeling, practicing the philosophy of “algorithm produces algorithm, model builds model, and eventually AI creates AI”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Second, Explainable AI de- velops new techniques to better understand and interpret investment decisions made by machine learning black-boxes, and explains complicated and hidden risk exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Third, Knowledge-driven AI is a supplement to data-driven AI such as deep learning and it incorporates prior knowledge into modeling to improve investment decision, in particular for quantitative value investing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' More- over, we discuss how to build a system that practices the Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Finally, we propose ten challenging research problems for quant technology, and discuss potential solutions, research directions, and future trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Introduction Quantitative investment is an important part of wealth man- agement (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' asset management) industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' This section con- tains introductory knowledge about quant, including market sit- uation, classification and principles of strategy development, historical landmarks, and concepts of Quant1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0–Quant4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Wealth Management and Quant The wealth management industry is one of the largest sec- tors of the world’s economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' According to a global wealth re- port from Boston Consulting Group (BCG) [1] and the illus- tration in Figure 1, the volume of global financial wealth has grown from 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='6 trillion USD in 2016 to 274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 trillion USD in 2021, almost three times as the global nominal GDP in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Moreover, the company predicts this number will increase to 355 trillion USD in 2026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' It is not surprising that North Amer- ica, Asia, and Europe are the three biggest regional markets of wealth management in the world, with approximately 46%, 26%, and 21% of the global market size in 2021, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Email addresses: guojian@idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='cn (Jian Guo), swangeh@connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='hk (Saizhuo Wang), ni@ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='hk (Lionel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Ni), hshum@idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='cn (Heung-Yeung Shum) 1Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2This work was done during the internship at IDEA Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' We also see the stable and sustainable growth of the wealth management market, both globally and regionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Figure 2 shows an ecosystem of the wealth management industry, where investment funds as well as fund managers (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' investment managers) play core roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' They raise money from various cap- ital providers, such as endowment foundations, fund of funds (FOF), family offices, billionaires, insurance companies, pen- sion/sovereign funds and retail clients, and invest this money into financial markets to bet return and profit for their cus- tomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Many types of investment instruments are liked by fund managers, such as stocks, exchange-traded funds (ETFs), bonds, futures, options, and foreign exchange [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Some in- vestment funds even borrow money from depository institutions such as banks or peer-to-peer lending companies for investment and profit from the difference between investment return and loan interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' With the rapid development of digital economy, big data, and artificial intelligence, more and more new tech- nologies are applied in the wealth management industry, lead- ing to a branch of financial technology/engineering, called “in- vestment engineering” [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Consequently, the pipeline of in- vestment research, trading execution, and risk management is becoming a systematic, automated, and intelligent process, and this philosophy has been practiced in the recent evolution of quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' As an important family of players in financial markets and the wealth management industry, contemporary quant applies 1 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='6 274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 355 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='8 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='5 328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3 100 50 0 50 100 150 200 250 300 350 400 2016 2021 2026 (est) Global Wealth Market Size and Growth Financial Wealth Real Assets Liabili�es (a) Global market sizes of wealth management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 71 2016 2021 2026 Europe 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='8 2016 2021 2026 Africa 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='7 2016 2021 2026 Latin America 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='5 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='8 2016 2021 2026 Asia 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='8 5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='5 2016 2021 2026 Middle East 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='8 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='6 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 2016 2021 2026 North America 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='9 2016 2021 2026 Oceania (b) Regional market sizes of wealth management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Figure 1: Global and regional market sizes of wealth management industry (unit: trillion USD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Panel (a) illustrates the volume of financial wealth, real assets and liabilities in 2016, 2021 and 2026 (estimated) in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Panel (b) shows the distribution of financial wealth in seven regional markets around the world in 2016, 2021 and 20226 (estimated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Data come from the report of BCG [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Figure 2: The asset management ecosystem [4] rigorous mathematical and statistical modeling techniques, ma- chine learning techniques, and algorithmic trading techniques to discover asset pricing abnormalities in financial markets and make money from the following arbitrage or investment oppor- tunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Compared with traditional fundamental and techni- cal investment, quantitative investment has a number of advan- tages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Firstly, the performance of quant strategies can be ex- amined and evaluated beforehand using back-test experiments based on historical data before the beginning of real trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Secondly, quant trading has speed superiority in bidding orders with the best price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Thirdly, it eliminates the negative effect of human emotion in decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Finally, quant research has significant advantages in data analysis with much deeper, broader and diversified coverage of information about financial markets and sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In the past 30 years, information infras- tructure and computer technology are widely applied by finan- cial exchange markets around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Nowadays, massive financial data are generated and millions of orders are executed every second, leading to the rapid growth of the quant industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Taking the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' stock market as an example, over 60% of over- all trading volumes comes from the orders placed by computer trading algorithms rather than human traders [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Quant Strategies A quant strategy is a systematic function or trading method- ology used for trading securities in financial markets based on predefined rules or trained models for making trading decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Strategies are usually the core intelligent property of a quanti- tative fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Components of Quant Strategy A standard quant strategy contains a series of components, such as investment instrument, trading frequency, trading mode, strategy type and data type, and we introduce them one by one (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Investment instrument specifies which financial instruments are put in the universe by the strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Popular candidate instruments include stocks, ETFs, bonds, foreign exchanges, convertible bonds, and cryptocurrencies, as well as more com- plicated financial derivatives such as futures, options, swaps, and forwards [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' An investment strategy could trade either a single type of instrument (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=', a strategy for trading ETFs) or multiple types of instruments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=', an alpha hedging strategy that longs stocks and shorts index futures to eliminate market risks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Trading frequency specifies how to hold your asset in port- folio and how frequently to trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Usually, high-frequency trading holds a position in several minutes or seconds, while low-frequency trading may hold an asset over several months or years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Comparing high-frequency trading and low-frequency trading, the dramatic discrepancy of holding periods result in very different consideration in strategy design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' For exam- ple, asset capacity limitations and trading costs are big issues for high-frequency trading, while how to control the risk of drawdown [7] is what we should carefully think about for low-frequency trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Model type characterizes how to formally model the trading problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Examples include cross-sectional trading, time- series trading, and event-driven trading [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Cross-sectional trading is used commonly in stock selection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' where all stocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='CapitalProviders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Principal&Interest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Depository ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Retail Clients ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Pension/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='DebtCapital ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Institutions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='$100K) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Sovereign Funds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='MassAffluent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='($100K-$1MM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Endowments/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Principal & ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Debt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Foundations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Investment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Interest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Capital ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='HNWIS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Funds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='($1MM-$50MM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Insurance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Investment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Investment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Companies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='UHNWIS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='InvestmentTargets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='($50MM-$100MM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='HedgeFunds/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Mutual Funds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='NetReturn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='GrossReturn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Fund Managers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='FamilyOffice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='($100MM-$1B) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Development ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Financial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Fees ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Billionaires ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Finance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Services ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='(>$1B) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Institutions (DFls) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Financial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Intermediaries ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Financial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Fees ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Services ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Fees ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='OtherIntermediaries ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='(Consultants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Lawyers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='TechnicalAssistance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Wealth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='ServiceProviders) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Advisors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='DirectInvestment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='InvestmentReturn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Key ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Tracksflowofcapital ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Tracks relationshipMomentum & ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Mean-reversion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Hedging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Arbitrage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Market Marking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Trade Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Lead-lag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Stocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='ETFs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Bonds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Forex ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Convertible Bond ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Crypto ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Futures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Options ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Swaps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Forwards ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Basics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Derivatives ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Instrument ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Model Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Time-series ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Cross-sectional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Event-driven ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='High Frequency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Low Frequency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Frequency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Medium Frequency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Cross- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='sectional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Time- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='series ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Event- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='driven ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Long-only / Short-only ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Hedging / Long-Short ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Arbitrage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� Stock Multifactor Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='with Long-only Trading ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� Future Time-series CTA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� Stock Time-series Intraday ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Trading ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� High-frequency Future ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Time-series Trading ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� Activist Stock Investment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� Distressed Cooperation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Bond Investment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� Stock Multifactor Long- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Short Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� Future Cross-sectional CTA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� Event-driven Global Macro ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� Even-driven High- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='frequency Alpha ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� Multi-asset Statistical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Arbitrage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� Triangular Arbitrage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� High-frequency Cross- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='market Arbitrage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� High-frequency Calendar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Spread Arbitrage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� Stock Time-series Trading ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Hedged by Index Future or ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Index Option ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� Merger Arbitrage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� Even-driven Convertible ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Arbitrage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Examples of Investment Strategies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Data Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Limit Order Book ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Price / Volume ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Financial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Statement / Report ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='News & ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Social Media ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Alternative Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Figure 3: Classification of common strategies and investment instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' in a universe are ranked according to their scores of expected future returns predicted by a model, and portfolio managers could long stocks with the highest scores and short those with the lowest scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Time series trading is relatively simple, where long/short trading operates only on a single instrument such as a certain stock or a certain future contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Event- driven trading differs from time-series trading because the time intervals between events are not evenly distributed over time, while investment decisions and trading executions are triggered by the occurrence of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Trade type is a series of thinking templates for us to design a strategy quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Examples include momentum trading [8], mean-reversion trading [9], arbitrage trading [10], hedging [11], market making [12], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' By leveraging these strategy types, traders can explore profit chances from different as- pects of financial markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Specifically, momentum trading assumes the price trend is sustainable in the following time window and it follows this trend direction to trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Mean- reversion trading, on the contrary, bets the price trend will move towards the opposite direction in recent future and buy opposite positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Hedging is the purchase of one asset with the intention of reducing the risk of loss from another asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Arbitrage is simultaneously longing and shorting the same asset in different markets or a pair of highly correlated assets in order to profit from the convergence of price discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Market making is a liquidity-providing trade that quotes both a buy and a sell price in a tradable asset held in inventory, hoping to make a profit on the bid–ask spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Data type means what type of data is used in a strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Typ- ical data types include quote data, limit order books [13] b, news data, financial statements, analysts’ reports, and alter- native data such as sentimental data, location data, satellite images, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' A strategy researcher must consider what kind of data he has and what kind of data he needs in a strategy development process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' For example, limit order book streams are usually used in building high-frequency trading strate- gies, while news data are used more commonly in event- driven strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Examples of Popular Strategies Figure 3 also list a number of popular strategies as exam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' For example, stock hedging strategy based on multifactor model [14] is very popular in many main markets around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' This strategy hedges market risk by longing the most fa- vorable stocks and shorting the other end (in some markets pro- hibiting shorting, short the corresponding index future or index option instead).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' If we trade stocks with multifactor models in a long-only way without shorting and constrain the risk expo- sure between selected portfolios and certain stock indices, it is an enhanced indexing strategy, which is almost the most popu- lar quantitative strategy in China’s stock market if measured by assets under management (AUM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Fundamental Principles of Asset Management Similar to the situation that learning law of energy conser- vation could help avoid the trap of perpetual motion machine, it is beneficial to learn some fundamental principles of asset management so as to get rid of some common traps in strategy development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Fundamental Law of Active Management The first principle is the fundamental law of active man- agement developed by Richard Grinold and Ronald Kahn [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' This principle states that the performance of an active invest- ment manager (or equivalently quant model) depends on the quality of investment skills and, consequently, the frequency of investment opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' This law can be expressed mathemat- ically as follows: IR = IC × √ Breadth (1) where IC is the information coefficient (correlation between the predicted return and true return in a future time window) 3 evaluating investment quality, Breadth means the number of independent investment decisions in a year, and IR is the ra- tio of portfolio returns above the returns of a benchmark to the volatility of returns, measuring the performance of asset man- agement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Mathematically, the fundamental law of active man- agement can be regarded as an application of the central limit theorem in mathematical statistics [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' When applying this law in practice, we have to notice that IC and Breadth are usually not independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' For example, given a strategy, we may in- crease its Breadth by relaxing the threshold of trading signals, but in this way, IC may decrease because more false-positive noise is introduced to our decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Therefore, a good strat- egy should find an optimal trade-off between these two coupled variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Figure 4a illustrates the distribution of various pop- ular strategies on IC and Breadth, and their corresponding IR performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Impossible Trinity of Investment The second principle is the impossible trinity of asset man- agement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Specifically, any investment strategy can not meet the following three conditions simultaneously, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=', high return, low risk (or equivalently high stability), and high capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Fig- ure 4b illustrates the impossible trinity using a radar chart with three variables return, stability and capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' For example, high- frequency market making and calendar arbitrage strategy could reach high return and stability (low portfolio volatility), but the capacity of its AUM is usually small, typically hard to exceed several billions of USD even in global trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' On the contrary, stock fundamental strategy has high capacity up to trillions of USD, but its return and stability are not as good as those of high-frequency trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' History of Quantitative Investment The origin of quant can trace back to over a century ago when French mathematician Louis Bachelier published his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' thesis “The Theory of Speculation” in 1900 [17] and he exhib- ited how to use probability law and mathematical tools to study the movement of stock prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' As a pioneer exploring the ap- plication of advanced mathematics in financial markets, Bache- lier’s work inspired academic research of quantitative finance despite the lack of industry application due to data scarcity at his age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Quantitative investment was first practiced by Amer- ican mathematics professor Edward Thorp, who used proba- bility theory and statistical analysis to win blackjack games, and his research was subsequently used to seek systematic and consistent returns in stock markets [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In this subsection, we introduce the history and landmarks in the development of quantitative finance through two routes: research landmarks in academia and evolution of quant in industry practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Q-Quant and P-Quant People in academia and investment industry classify quan- titative finance into two branches, which are usually referred to as “Q-quant” and “P-quant”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' These two branches are named af- ter their differentiation in modeling based on risk-neural mea- sure and probability measure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Generally speak- ing, Q-quant studies the problem of derivative pricing and ex- trapolate the present, using a model-driven research framework where data is usually used to adjust the parameters of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' On the other hand, P-quant studies quantitative risk and port- folio management to model the future, using a data-driven re- search framework where different models are built to improve the fitting of historical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Usually, Q-quant research is con- ducted in sell-side institutes such as investment banks and secu- rity companies, while P-quant is popular in buy-side institutes such as mutual funds and hedge funds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Table 1 compares the characteristics of these two types of quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Table 1: Comparisons of P-quant and Q-quant [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Q-quant P-quant Goal Extrapolate the present Model the future Scenario Derivatives pricing Portfolio management Measure Risk-neural measure Probability measure Modeling Continuous stochastic process Discrete time series Example Black-Scholes model Multifactor model Algorithm Ito calculus, PDEs Statistics, Machine Learning Challenge Calibration Estimation/Prediction Business Sell-side Buy-side 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Landmarks in Q-Quant In 1965, Paul Samuelson, American economist and the win- ner of 1970 Nobel Memorial Prize in Economic Sciences, intro- duced stochastic process and stochastic calculus tools in analyz- ing financial markets and modeling the stochastic movement of stock prices [20], and in 1965, he published a paper studying the lifetime portfolio selection problem using a stochastic pro- gramming method [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In the same year, another American economist Robert Merton published his work about lifetime portfolio selection as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Different from Samuelson’s work using discrete-time stochastic process, Merton’s work modeled the random uncertainty of portfolio using continuous-time stochas- tic calculus [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Almost in the same year, economists Fischer Black and Myron Scholes demonstrated that the expected re- turn and risk of assets under management could be removed by dynamically revising a portfolio, and thus inventing the risk- neutral strategy for derivative investment [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' They applied the theory to real market trading and published it in 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The risk- neutral formula was later named in honor of them and called Black-Scholes Model [24], a partial differential equation (PDE) tool for pricing a financial market containing derivative invest- ment instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Specifically, the Black–Scholes model estab- lishes a partial differential equation governing the price evolu- tion of a European option call or European option put, as fol- lows: ∂V ∂t + 1 2σ2S 2 ∂2V ∂S 2 + rS ∂V ∂S − rV = 0 (2) where V is the price of the option as a function of stock price S and time t, r is the risk-free interest rate, and σ is the volatility of the stock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' This PDE has a closed-form solution called Black- Scholes Formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Since Robert Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model, he was usually credited with the contribution of this theory as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Merton and Scholes received the 1997 4 High-freq Statistical Arbitrage Breadth |IC| Stock Cross-sectional High-freq Alpha Foundamental Value Investment Stock Cross-sectional Foundamental Alpha Cross-Sectional CTA Low High |IR| Event-Driven Arbitrage High-freq Market Making Global Macro Risk-free Arbitrage Quant CTA (a) Common investment strategies under the fundamental law of active management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The figure illustrates the relationship between the magnitude of IR with breadth and the magnitude of IC for different strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Return Stability Capacity Stock Fundamental Investment Future High-frequency Arbitrage Fix Income Investment (b) Illustration of the impossible trinity of return, capacity, and stability for ac- tive management using three typical strategies: stock fundamental investment, future high-frequency arbitrage and fixed income investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Figure 4: Illustration of the principles for active investment management with specific strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Théorie de la Spéculation Bachelier 1970 Nobel Prize Paul Samuelson 🏅 1900 Stochastic Calculus for Asset Pricing Paul Samuelson 1965 Capital Asset Pricing Model (1961-1966) Jack Treynor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' William Sharpe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' John Linter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Jan Mossi 1966 Continuous Option Pricing Robert Merton 1969 Black-Scholes Model Black,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Scholes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Merton 1973 Arbitrage Pricing Theory Stephen Ross 1976 Vector Autoregression Christopher Sims 1980 Fundamental Theorem of Asset Pricing Harrison and Pliska 1981 2012 Nobel Prize Lloyd Shapley Contribution: Shapley Value 🏅 ARCH/GARCH Robert Engle 1982 Co-integration Robert Engle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Clive Granger 1987 Fama-French Three Factor Model Eugene Fama,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Kenneth French 1992 Local Average Treatment Effect Guido Imbens,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Joshua Angrist 1994 1990 Nobel Prize Markowitz & Sharpe 🏅 2013 Nobel Prize Eugene Fama 🏅 2003 Nobel Prize Robert Engle & Clive Granger 🏅 2011 Nobel Prize Christopher Sims 🏅 2021 Nobel Prize Guido Imbens,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Joshua Angrist 🏅 2011 Turing Award Judea Pearl Contribution: Calculus for Probabilistic and Causal Reasoning 🏅 2018 Turing Award G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Hinton, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Bengio, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Lecun Contribution: Deep Learning 🏅 1997 Nobel Prize Scholes & Merton 🏅 Heston Model Steven Heston 1993 Gaussian Copula David X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Li 2000 SABR Model Hagan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2002 P-Quant Q-Quant Deep Hedging Buehler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2018 Modern Portfolio Theory Harry Markowitz 1952 Figure 5: Main academic contributors and their works that deeply influence the development of quantitative investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Photo credit: Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Nobel Memorial Prize in Economic Sciences for their discov- ery of the risk-neutral dynamic revision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The original Black- Scholes model was extended later for deterministically variable rates and volatilities, and was extended to characterize the price of European options on instruments paying dividends, as well as American options and binary options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' As a pioneering work in risk-neutral theory, Black-Scholes model has many limitations, one of which is the assumption that the underlying volatility is constant over the life of the deriva- tive, and is unaffected by the changes in the price level of the underlying security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' This assumption usually contradicts the phenomenon of the smile and skew shapes of implied volatility surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' A possible solution is to relax the constant volatility assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' By characterizing the volatility of the underlying price using stochastic process, it is possible to model derivatives more accurately in practice, and this idea leads to a series of works about stochastic volatility, such as the Heston model [25] and the SABR model [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' As a commonly used stochastic volatility model, the Heston model assumes the variation of the volatility process varies as a square root of the variance itself, and it exhibits a reversion trend towards the long-term mean of variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Another popular stochastic volatility model is the SABR model, commonly used in interest rate derivative mar- kets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' This model uses stochastic differential equations to de- scribe a single forward (such as a LIBOR forward rate, a for- ward swap rate, or a forward stock price) as well as its volatil- ity, and has the ability to reproduce the effect of volatility smile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In recent years,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' deep learning and reinforcement learning tech- niques are applied to integrate with risk neural Q-quant mod- eling and a concept learning to trade was introduced by Hans Buehler [27],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' who proposed the deep hedging model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' a frame- work for hedging a portfolio of derivatives in the presence of market frictions such as transaction costs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' market impact,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' liq- uidity constraints or risk limits and for modeling the volatility stochastic process using deep reinforcement learning and mar- ket simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' It does not use the Greeks anymore and natu- rally captures co-movements of relevant market parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In addition to derivative pricing models, market efficiency theory and risk modeling theory are also very important in Q- quant, both in academia and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In 1980s, Harrison and Pliska established the fundamental theorem of asset pricing [28], which provides a series of necessary and sufficient conditions for an efficient market to be arbitrage free as well as complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In 2000, David X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Li introduced the statistical model Gaus- sian copula [29] to evaluate the value-at-risk (VaR) of derivative pricing and portfolio optimization, especially the collateralized debt obligations (CDO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Gaussian copula quickly became a tool for financial institutions to correlate associations between mul- 5 tiple financial securities since it is relatively simple in modeling even for those assets too complex to price previously, such as mortgages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' PFE GM GNC MSFT Modern Portfolio Theory ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='09 𝑤!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 𝑤" 𝑤# 𝑤#$!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 𝑤% Optimized Portfolio Positions Portfolio Cumulative Return Trading and hold the positions in next N days Shift time window Extract returns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='8 1 (a) Illustration of Markowitz’ portfolio optimization theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' (b) Illustration of efficient frontier first formulated by Harry Markowitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Figure cited from [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Figure 6: Portfolio optimization 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Landmarks in P-Quant Q-quant plays an extremely important role in quantitative fi- nance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In this article, however, we stand on a buy-side point of view and focus on asset prediction and portfolio optimization problems, and thus all discussions about quantitative invest- ment in the following content assume a P-quant statement un- less otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The origin of P-quant started from the establishment of modern portfolio theory introduced by Harry Markowitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The theory was initialized in his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' thesis “Port- folio Selection” and later published in Journal of Finance in 1952 [31], and an extension published in his book Portfolio Se- lection: Efficient Diversification of Investments [32] in 1959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' According to the old adage “Don’t put all your eggs in one bas- ket”, Markowitz came up with the concept of efficient frontier of asset investment in financial market and formalized it math- ematically as a quadratic optimization problem by maximizing the expected return of the portfolio given its risk (usually mea- sured by the variance of the assets in a portfolio) at a certain level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Figure 6 illustrates the application of Markowitz’s the- ory for allocating the best positions for assets in portfolio and illustrates the concept of efficient frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Based on the modern portfolio theory, the Capital Asset Pricing Model (CAPM) was later introduced by Jack Treynor (1961, 1962) [33], William F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Sharpe (1964) [34], John Lintner (1965) [35] and Jan Mossin (1966) [36] independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' CAPM aims to describe the relationship between systematic risk from the market and expected return for assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' E(Rp) − Rf = α + β · (E(Rm) − Rf ) (3) where E(Rp) is the expect return of portfolio, Rf is the risk- free return, E(Rm) is the expected return of market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Specifi- cally, CAPM decomposes asset return and risk into two sepa- rate parts, alpha and beta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Alpha measures the performance of a portfolio compared to a benchmark index (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=', S&P500 index), while beta measures the variance of the portfolio in relation to a benchmark index, characterizing the risk from market volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' One of the main contributors to CAPM, William Sharpe, shared the 1990 Nobel Prize with Harry Markowitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' A following im- portant step in quantitative finance is the establishment of ar- bitrage pricing theory (APT) by MIT economist Stephen Ross in 1976 [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' APT improved its predecessor CAPM by fur- ther introducing the multifactor model framework to build the relationship between asset price and various macroeconomic risk variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Under the multifactor model framework, Nobel Prize-winning economist Eugene Fama proposed the famous Fama–French Three-Factor Model with his colleague Kenneth French at the University of Chicago in 1992 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' E(Rp)−Rf = β0 +β1 ·(E(Rm)−Rf )+β2 ·S MB+β3 · HML (4) This model establishes the relationship between the expected portfolio return (up to subtracting a risk-free return) E(rp) − r f with respect to three systematic risk factors: expected mar- ket return E(Rm) − Rf , size S MB (the spread between small capitalization stocks and large capitalization stocks), book-to- market values HML (the spread between high book-to-market companies and low book-to-market companies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The three- factor model was then extended to Fama and French Five Factor Model in 2015 [38], by adding two more factors: profitability (return spread of the most profitable firms minus the least prof- itable) and investment aggressiveness (the return spread of firms that invest conservatively minus aggressively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Parallel with the progress of multifactor models, a num- ber of significant research about time-series analysis appears in 1980s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In 1980, Nobel Prize winner Christopher Sims intro- duced the Vector Autoregression (VAR) model into economics and finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' As an extension of single sequence autoregres- sive (AR) model and autoregressive-moving-average (ARMA) model commonly used in time-series analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=" VAR charac- terizes the autoregressive properties over time across multiple 6 15% Efficient Frontier of All Risky Securities Market Portfolio = Efficient Portfolio IBM OGE O GM Capital 10% Disney Expected Return Market McDonald's Line Merck O Exxon Mobil Campbell Soup 5% Anheuser-Busch Edison International Risk-Free Investment 0% 0% 5% 10% 15% 20% 25% 30% 35% 40% Volatility (standard deviation)45." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='5 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='5 28 Sept 29 Sept 30 Sept 3 Oct8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='6 29 Sept 30 Sept 3 Oct 4 Oct242 240 238 236 234 232 28 Sept 29 Sept 30 Sept 3 Oct36 35 34 33 32- 28 Sept 29 Sept 30 Sept 3 Octm maxw1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='.Wm Wiri) i=1 subject to Wi ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' i= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='.m m <)pis Wiri) ≤ θ i=1times series and it assumes constant variance of error terms in the regression formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In 1982, Robert Engle introduced the Autoregressive Conditional Heteroskedasticity (ARCH) model and extend it to Generalized Autoregressive Conditional Het- eroskedasticity (GARCH) model to characterize the pattern of financial volatility in the market by specifying stochastic vari- ance in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In 1987, he introduced co-integration method with Clive Granger (inventor of Granger Causality for mod- eling lead-lag patterns among multiple time series) for testing the significance of mean-reversing patterns in financial time se- ries, and co-integration test has been widely used in discovering promising asset pairs for statistical arbitrage strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Both Engle and Granger received the 2003 Nobel Prize for their con- tribution to time series analysis which has been widely applied in quantitative finance for market forecasting and investment research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In 2018, three pioneers in deep learning techniques, Yoshua Bengio, Geoffrey Hinton and Yann LeCun, are granted the Turing Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Nowadays, deep learning has been widely used by academic researchers in finance and quant researchers in financial institutions to build complex nonlinear models in order to learn the relationship between financial signals and ex- pected returns and to predict asset prices, and its powerful abil- ity in fitting big data significantly improves the performance of market prediction and portfolio management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Although accurately predicting the future trend of asset price is a very important task in P-Quant, how to explain the effect of model prediction and interpret how a model is really working seems more important for quant researchers since “know how” is more crucial than “know what” in risk management for port- folio managers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Causal effect analysis [39] and factor impor- tance analysis are two core tasks in quant model interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Clive Granger invented the Granger Causality Test in 1969 [40] for determining whether one time series is useful in forecasting another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The original Granger causality test does not account for latent confounding effects and does not capture instanta- neous and non-linear causal relationships, though several ex- tensions have been proposed to address these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Although there is an argument about whether Granger causality test can evaluate “real” causality in terms of statistics, this method has been widely applied in quant research such as searching and evaluating pairs of stocks with significant lead-lag effect and trading with corresponding strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In 1994, Guido Imbens and Joshua Angrist introduced the local average treatment ef- fect (LATE) model to characterize the statistical causal effect in economics, finance and social sciences, and they shared the 2021 Nobel Prize in economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Another important contribu- tor in causal inference is the Turing Award winner Judea Pearl, who invented the causal diagram (Bayesian network) and it can be used to mine the causal effect among factors and returns in multifactor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' On the other hand, in the area of factor importance analysis, Shapley Value has become an important criterion for measuring the contribution of single feature in a complex nonlinear machine learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In fact, it is inter- esting that this criterion was invented originally to measure the contribution of individual player/agent in a cooperative gaming process when it was first proposed by Lloyd Shapely, a Nobel Prize winner and a pioneer in game theory research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Development of Quant in Industry The blooming era of quantitative investment funds started from 1990s, along with the emergence of the Internet and the development of electronic trading in exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Here we briefly introduce the evolution of quant operating models and clas- sify them into three generations, denoted as Quant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0, and summarize their characteristics in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Quant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 appeared in the early age of quantitative invest- ment but it is still the most popular quant operating model in contemporary market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The features of Quant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 includes: 1) Small but elite team, typically led by an experienced portfo- lio manager and composed of a few genius researchers and traders with strong mathematics, physics or computer sci- ence background;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2) applying or even inventing mathemat- ical and statistical tools to analyze financial market analysis and discover mispriced assets for trading;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 3) trading signals and trading strategies are usually simple, understandable and interpretable to reduce the risk of in-sample over-fitting in modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' This operating model has high efficiency in quant trading but low robustness in management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Especially, the success of a Quant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 team relies too much on particular ge- nius researchers or traders, and such a team may decline or even bankrupt rapidly with the departure of genius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In ad- dition, such a small “strategy workshop” limits the research efficiency on complex investment strategies such as quanti- tative stock alpha strategy which depends on diversified fi- nancial data types, extremely large data volume, and com- plex modeling techniques such as super large deep learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Quant 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 changes quant operating model from small genius’ workshop to an industrialized and standardized alpha fac- tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In this model, hundreds or even thousands of invest- ment researchers work on the same pipeline to mine effec- tive alpha factors [41] out of the plethora of financial data, using standardized evaluation criteria, standardized back-test processes and standardized parameter configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' These alpha mining researchers are rewarded by submitting quali- fied alpha factors which usually have high back-test returns, high Sharpe ratio, reasonable turnover rate and low correla- tion with existing factors in the alpha database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Tradition- ally, each alpha factor is a mathematical expression char- acterizing some pattern or profile of stocks, or some rela- tionship between stocks, although more and more compli- cated machine learning factors are mined as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Typical al- pha factors include momentum factors, mean-reversion fac- tors, event-driven factors, volume-price dispersion factors, growth factors, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Many alpha factors submitted by alpha researchers are combined into statistical models or machine learning models by portfolio managers to find the optimal as- set positions after appropriate risk neutralization, expecting to obtain a stable and promising excess return in the mar- ket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' However, large-scale team work results in huge costs for human resources, and the situation gets more and more seri- ous with the team growing larger and larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Specifically, we could expect the number of discovered effective alphas fol- lows an approximately linear trend with the team size (actu- 7 Mathematical modeling Interpretable robust signal Small genius team Quant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 Intensive factor mining Scalable modeling pipeline Reward mechanism Quant 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 Deep learning blackbox Intensive model tuning Very large data volume Quant 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 End-to-end automated AI Explainable AI system Knowledge graph reasoning Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 Human-centric Trading System-centric Trading Figure 7: The development history of of quantitative investment in industry, from Quant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 to Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' ally in practice, discovering new effective alphas is more and more difficult when the size of accumulated factors is already large), but the portfolio return grows significantly lower than the expand of alpha volume and team size, and this results in the profit margin getting smaller and smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' This phe- nomenon is caused by a number of reasons such as the lim- itation of strategy market capacity, the growing difficulty in discovering new effective alphas, and even the limitation of human intelligence in searching all possibilities in strategy space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Quant 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 emerges with the rapid development of deep learn- ing techniques which have exhibited success in many domain areas such as computer vision and natural language process- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Different from Quant 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 which puts more research ef- forts and human labor into mining sophisticated alpha fac- tors, Quant 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 pays more attention to deep learning model- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' With relatively simpler factors, deep learning still has the potential to learn a prediction model performing as well as a Quant 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 model, by leveraging its powerful end-to-end learning ability and its flexible model fitting ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In Quant 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0, the cost of human labor of alpha mining is at least par- tially replaced by the cost of computing power, especially for the expensive GPU servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' But generally speaking, it is a more efficient way for quant research in the long run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0: Why and What 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Limitations of Quant 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 Although Quant 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 has demonstrated its success in some strategy scenarios such as high-frequency stock and future trad- ing, it has three primary limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Traditionally, building a “good” deep neural network is time- consuming and labor-intensive, because of the heavy work in network architecture design and model hyperparameter tuning, as well as the tedious work in model deployment and maintenance in trading ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' It is a challenge to read understandable messages from a model encoded by deep learning black box, making it very unfriendly to investors and researchers who care much about the mechanism of financial markets and expect to know the source of profit and loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The good performance of deep learning relies heavily on ex- tremely large volumes of data, and thus only high-frequency trading (or at least medium cross-sectional alpha trading with large breadth) belongs to the strategy pool that deep learning favorites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' This phenomenon prevents deep learning tech- niques from application in low-frequency investment sce- narios such as value investing, fundamental CTA and global macro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' New research and new techniques are needed to address these limitations, and this leads to our proposal for Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Automated AI ➢ AIs create AIs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' models build models ➢ Make AI economic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' efficient & scalable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Explainable AI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='➢ Make AI transparent in investment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='➢ Interpretability for return and risk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Knowledge-driven AI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='➢ Data-driven to knowledge-driven ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='➢ Financial knowledge logic reasoning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Manual Deep Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='➢ Labor intensive in hand-craft modeling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='➢ Heavy workload in model deployment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Blackbox Machine Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='➢ Lack of transparency in trading risk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='➢ Hard to control model over-fitting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Data-driven Modeling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='➢ Performance depends on data volume ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='➢ Difficult in low-frequency investment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Limitations of Quant 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 Advantage of Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 Figure 8: The three key components of Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' What is Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' We believe the limitations of Quant 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 are very likely to be solved or at least partially solved in the future with the quick development of the artificial intelligence (AI) technology fron- tier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0, the next-generation quant technology, is prac- ticing the philosophy of “end-to-end going all-in on AI” and “AI creates AI” by incorporating the state-of-the-art automated AI, explainable AI and knowledge-driven AI and plotting a new picture for quant industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Automated AI aims to build end-to-end automation for quant research and trading, in order to significantly reduce the cost of labor and time for quant research including data prepro- cessing, feature engineering, model construction and model deployment, and to dramatically improve R&D’s efficiency and sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In particular, we introduce state-of-the- art AutoML [42] techniques to automate every module in the whole strategy development pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In this way, we pro- pose to change traditional hand-craft modeling to an auto- mated modeling workflow in an “algorithm produces algo- rithm, model builds model” manner, and eventually move to- wards a technical philosophy of “AI creates AI”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Besides AI 8 All-inon AlFactorDatabase 000Deep Learning Model Base 000automation, another important task is to make AI more trans- parent, which is essentially important for investment risk man- agement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Explainable AI, usually abbreviated as XAI in machine learn- ing area, attempts to open the black box encapsulating deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Pure black-box modeling is unsafe for quant research because people can not calibrate the risk accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' It is difficult to know, for example, where returns come from and whether they rely on certain market styles, and what the reason for a specific drawdown is, under black-box model- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' More and more new techniques in the field of XAI could be applied in quant to enhance the transparency of machine learning modeling, and thus we recommend quant researchers to pay more attention to XAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' We have to no- tice that improving model explainability has costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Figure 9 shows an impossible trinity of versatility, accuracy and ex- plainability, and tells us that we have to sacrifice at least one apex in the triangle to obtain the benefit from the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' For example, physical law E = mc2 establishes an explain- able and accurate relationship among energy, mass and speed of light, but this formula can be only applied in specific do- mains of physics and sacrifices versatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Imagining that we provide more prior knowledge or domain experience in a model, it is equivalent to reducing the versatility to pro- tect the performance of accuracy and explainability simulta- neously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Examples of Versatile & Accurate Modeling Deep Learning Kernel Support Vector Machine Examples of Versatile & Explainable Modeling Linear Model (Feature Explainability) Decision Tree (Rule Explainability) Examples of Accurate & Explainable Modeling Black-Scholes Model for Option Pricing Physical Laws Such as 𝐸 = 𝑚𝑐2 Knowledge: money oversupply → inflation Versatility Accuracy Explainability Figure 9: Impossible trinity of versatility, accuracy and explainability in mod- eling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Knowledge-driven AI differs from the data-driven AI which heavily depends on large volumes of data samples and thus is appropriate for investment strategies with large breadth such as high-frequency trading or stock cross-sectional trad- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' It is an important complement to data-driven AI tech- niques such as deep learning (illustrated in Figure 10 us- ing Bayes’ theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In this paper, we introduce knowledge graph which represents knowledge with a network structure composed of entities and relations, and stores knowledge with semantic triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' A knowledge graph of financial behaviors and events could be analyzed and inferred for investment de- cisions using symbolic reasoning and neural reasoning tech- niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' This implies potential applications to those invest- ment scenarios with low trading frequency but intensive fun- damental information in collection and analysis, including value investing and global macro investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 𝑃 ȁ 𝜃 𝑥 ∝ 𝑃 𝜃 × 𝑃 ȁ 𝑥 𝜃 Posterior Distribution for Decision Making Data-driven Likelihood (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=', Deep Learning) Knowledge-driven Prior (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=', Domain Knowledge Graph Reasoning) Figure 10: Complementary function of data-driven AI and knowledge-driven AI in decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Automated AI for Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 Automated AI for Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 covers the automation of the full quant pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In this section, we will first give an overview of the pipeline and then introduce how to upgrade it to an auto- mated AI pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Automating Quant Research Pipeline 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Traditional Quant Pipeline Over decades of development, quant research has formed a standard workflow as shown in Figure 11 (blue part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' This workflow consists of a number of modules, including data pre- processing, factor mining, modeling, portfolio optimization, or- der execution, and risk analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Data preprocessing is usually the first step in quant research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Original raw data may have many issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Firstly, financial data usually have missing records, more or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' For exam- ple, in technical analysis, you may not receive price data at some time points due to packet loss during communication, or you may miss the price data on some trading days because of stock suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Similarly, in fundamental analysis, you may miss part of financial statement data since they are not reported on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Although conventional statistical data im- putation methods could be used to estimate and fill in miss- ing records, we must avoid using future information in the imputation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Secondly, financial data contain extreme values and outliers which may come from misrecording, data storage issues, data transfer issues, or extreme markets, and these outliers may lead to risky biases in investment deci- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Outliers could be eliminated by data winsorization methods [43] which limit extreme values in a certain per- centile range, but we have to notice that some outliers are ac- tually strong signals for quant trading rather than noise, and must differentiate the two during data preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Thirdly, many financial data, such as news event data, have low data coverage and irregular updating frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' We must align these types of data with high coverage and regular frequency such as quotes data for the convenience of downstream fac- tor mining and modeling tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Fourthly, different data fea- tures have quite different scales in value range and thus some “large” features may dominate “small” features in modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Therefore, data standardization methods are used to normal- ize the range of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' We have to take care of the way to standardize the data in order to reduce information loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Raw Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Exchange ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Traditional Quant Research Pipeline ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Investment Research ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Execution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Adjustments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Pre-processing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Cleaning / ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Imputation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Aggregation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Standardization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Factors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Factor Mining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Data Analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Factor Design ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Evaluation & ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Selection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Predictions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Modelling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Construction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Backtest Analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Stress Testing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Portfolio Optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Positions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Mean-Variance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Risk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Neuralization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Turnover Control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Orders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Order Execution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Real-time Risk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Algorithmic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Trading ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Trading ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Acceleration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Monitor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Returns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Risk Analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Risk Factor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Exposure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Decomposition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Loss Analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Quant Research Pipeline with Automated AI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Investment Research ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Execution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Adjustments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Meta Factors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Pre-processing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Cleaning / ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Imputation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Aggregation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Standardization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Factors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Feature Engineering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Symbolic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Regression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Representation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Hybrid Models ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Predictions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='AutoML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Architecture ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Search ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Hyperparameter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Training Objective ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Selection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Automated Positioning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Positions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Reinforcement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Position ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Risk Calculation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Orders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Automated Trading ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Order Execution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Reinforcement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Trading Latency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Monitor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Returns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Risk Tracking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Real-time Monitor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Risk Analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Risk-Control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Intervention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='One-click ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Deployment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Figure 11: A prototypical workflow of quantitative investment with comparisons between the current quantitative investment system (manual,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' upper blue part) and AI investment engineering (automated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' lower orange part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Factor mining is a task of feature engineering [44], which uses financial and economic domain knowledge to design, search, or extract factors (features for downstream modeling) from raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Usually, a larger factor value indicates a more significant trading signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The motivation of factor mining is to find those signals from raw data for market prediction and improve the quality of downstream modeling tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Tradi- tionally, financial factors could be represented as either alge- braic formulas or rule-based expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Let’s take a simple stock alpha factor as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' factor = −ts corr(rank(close), rank(volume), 50) (5) where the ts corr() function computes the correlation of daily close price and volume along time using the data from the previous 50 trading days, representing how similar the trend of close time series and volume time series are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The rank() function maps the values in a cross-section to their orders and normalizes them to the range of [−1, +1] evenly according to their descending order, in order to remove the effect of ex- treme values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' This factor prefers to select those stocks when their price and volume move in opposite directions, and the idea behind it is based on the assumption that a price trend can not sustain without the support of volume growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Tra- ditionally, factor mining is a labor-intensive job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Most quant researchers can only discover a limited number of “good” factors in a year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Different financial institutions have different definitions or criteria for a “good” factor, but most of them consider a few common aspects, such as return, Sharpe ra- tio, maximum drawdown, turnover rate, and correlation with other factors [41], and moreover, some institutions require the factors must be meaningful, understandable and explain- able in economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Modeling is a task to build statistical or machine learning models using factors and to predict market trends, asset price movements, best trading times, or most/least valuable assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Usually, prediction models are evaluated through back-test experiments which simulate the prediction and trading pro- cess using historical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Choices of models must consider a number of factors, such as prediction accuracy, model ex- plainability, model robustness, and computational complex- ity, and find the best tradeoff according to the ultimate goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In particular, we must notice that most statistical or machine learning models are not specifically developed for financial time series, and we have to adjust the application of these models in quant modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Firstly, financial time series pre- diction must avoid using future information, and thus we prefer forward-validation [45] (splitting the time series into training, validation, and test blocks over time) rather than cross-validation in model hyperparameter optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Sec- ondly, financial time series are usually significantly nonsta- tionary, far from the independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=') assumption required by many machine learning mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Therefore, data transformation is needed to make the data distribution closer to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' and if possible, look more like a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Thirdly, market style moves over time and it results in the shift of financial time-series distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Therefore, periodic model retraining is necessary for keeping the model adapted to market style variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Portfolio optimization aims to find the optimal asset alloca- tion to expect high return and low risk simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' While prediction models tell us what or when to buy/sell, portfo- lio optimization specifies how much to buy/sell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' A typi- cal portfolio optimizer attempts to solve a constrained con- vex quadratic programming problem which is extended from 10 Markowitz’s efficient frontier theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' max wt wT t rt subject to wT t Σwt ≤ C1 |wt − wt−1| ≤ C2 0 ≤ wi,t ≤ C3 ≤ 1, for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' , n where rt = (r1,t, r2,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' , rn,t)T is the returns of n assets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=', stocks) at time t, and wt = (w1,t, w2,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' , wn,t)T is the cor- responding position weights (percentages of capital alloca- tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' C1,C2,C3 are positive constraint bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Σ is the volatility matrix of the n asset returns at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The target function tries to maximize the portfolio return and control the upper bound of risk and turnover rate (to reduce transac- tion cost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The key in this optimization problem is how to estimate the volatility matrix Σ whose estimation is usually unstable if historical data is not long enough, and in this case dimension reduction tricks such as regularization and factor- ization can be helpful to improve estimation robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Order execution is a task that buys or sells orders with opti- mal prices and minimal market impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Usually buying (or selling) a big order at one time will push the price of the tar- get asset in a harmful direction (market impact by this big or- der), and therefore increase the trading cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' A widely used solution is order splitting, which divides a big order into a number of small orders to reduce market impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Algorith- mic trading provides a series of mathematical tools for or- der splitting, from the simplest time-weighted average price (TWAP) and volume-weighted average price (VWAP) to the complicated reinforcement learning methods [46] in which optimal order flow is modeled as a (partially observable) Markov decision process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Risk analysis is an indispensable task for quant research and quant trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' We must discover and understand every pos- sible risk exposure in order to better control unnecessary and harmful risks in quant research and trading [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In the mon- itor module, risks are measured in real-time and these mes- sages and analysis are sent back to help quant researcher im- prove their strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The most popular risk model in stock trading is the BARRA model [48] which decomposes portfo- lio volatility into the exposures of a number of predefined risk factors, including style factors (size, growth, liquidity, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=') and industry factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' However, the BARRA model could ex- plain only about 30% of total volatility, leaving the risk hid- den in the remaining 70% part still unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Automated AI Quant Pipeline The automated pipeline of Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 is shown in Figure 11 (orange part), where modules in the pipeline are automated by applying state-of-the-art AI technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In the following part of this section, we will concentrate on three core modules in the automated pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Automated factor mining (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2) applies automated feature engineering techniques to search and evaluate significant fi- nancial factors generated from meta factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' We will intro- duce popular search algorithms and demonstrate how to de- sign the algorithmic workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Automated modeling (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3) applies AutoML techniques to discover optimal deep learning models, automatically se- lecting the most appropriate models and the optimal model structures, and tuning the best hyperparameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' One-click deployment (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4) builds an automated workflow to deploy trained large models on trading servers with lim- ited computing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' It executes model compression, task scheduling, and model parallelization automatically, saving a lot of labor and time for tedious “dirty” work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Automating Factor Mining Feature engineering for quant refers to the process of ex- tracting financial factors from original data, on which effective pattern recognition is difficult due to their intrinsic noisiness [49, 50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Traditionally, financial factors with significant “alpha” are explored and developed by quant researchers manu- ally, they rely on professional domain expertise and comprehen- sive knowledge of financial markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Although some financial institutions started using random search or generic program- ming algorithms, these techniques are mainly used as small- scale auxiliary tools to help improve the productivity of quant researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0, We propose to automate the factor mining process by formulating feature engineering as a search problem and utilize corresponding algorithms to generate fac- tors with satisfactory backtest performance at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In partic- ular, according to their expression form, we classify factors as 1) symbolic [52] factors which are symbolic equations or sym- bolic rules, and 2) machine learning factors which are expressed by neural networks, and we will elaborate on the details in the following part of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Symbolic Factors Symbolic factor mining can be regarded as a special case of symbolic regression [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Traditional symbolic regression algo- rithms usually generate a large number of symbolic expressions from given operands and operators and select the symbolic ex- pressions that maximize the predefined objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Fig- ure 12 shows a framework for automated symbolic factor min- ing, which consists of four core parts: operand space, operator space, search algorithm, and evaluation criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Operand Space defines which meta factors could be used for factor mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Meta factors are fundamental components for factor construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Typical meta factors include basic price and volume information, sector categorizations, basic features extracted from limit/order books, common techni- cal indices, basic statistics from financial analysts, impor- tant signals from financial reports, announcements and other research reports from public companies, sentiment signals from investor emotions [57, 58], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Operator space defines which operators could be used in the factor mining process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' in cross-sectional stock selection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' the operators could be classified as main opera- tors for constructing symbolic factors and post-processing 11 Pre- process Search Space Data Operand Space Volume-price Fundamental Sentiment Grouping Operator Space Main Search Algorithm Random Search Equation Neural Network Evaluation Criteria Alpha Factor Noise Style Factor Risk Factor IC stdev IC magnitude Generate Feedback Group group_mean(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' group),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' group_sum(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' group),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' group_rank(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' group),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Time-series ts_corr(x, y, interval), ts_mean(x, y, interval), ts_rank(x, interval), ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Cross-sectional rank(x), quantile(x, quarter), ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Element-wise add(x,y), cos(x), mul(x, y) ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Post-processing Standardization Grouping Neutralization Decay ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Quote Data Limit Order Book Financial Statement Relational Data Text Data ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Genetic Programming Equation Generation Embedding Prediction Symbolic Factor Machine Learning Factor Figure 12: An example factor mining pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The search space is defined by operators and meta-factors, where meta-factors are extracted from raw data in various forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The search space is explored by search algorithms that are discrete or continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The evaluation module provides feedback to search algorithms based on certain criteria that serve as guidance for the next search iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Part of this figure is cited from [53, 54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' operators for standardizing the factors for different trading environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Main operators could be classified further as element-wise operators such as √() and log(), time-series op- erators such as ts rank() and ts mean() which compute the rank order and mean along each stock respectively, cross- sectional operators such as rank() and quantile() which com- pute the rank and quantile along the cross-section at a spe- cific trading time, and group operators such as group rank() which compute rank order in each group (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=', industry or sector) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Post-processing operators are used to “fine-tune” the generated factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Typical post-processing operators are standardization operators such as winsorization for outlier clipping [43] and normalization for unifying data range, neutralization operators for risk balancing, grouping operators for restricting the universe of stock selection, and decay operators for controlling turnover rate so as to reduce transaction cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Search algorithms aim to search and find effective or quali- fied factors as efficiently as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' A simple way to gener- ate new factors is the Monte Carlo (MC) algorithm which randomly picks the elements in the operand and operator spaces and generates a symbolic expression tree recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Unfortunately, the search time may grow exponentially with the length and complexity of the generated formula, and push us to consider more efficient alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The first option is Markov-chain Monte Carlo (MCMC) algorithm [59], which generates factors in sampling with importance way from a posterior distribution [60], and thus it is more efficient than MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The second option is genetic programming [61], which is a special evolutionary algorithm for sampling and optimiz- ing tree-type data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The third option is about gradient-based methods such as neural networks, which approximate the dis- crete symbolic formulas with continuous nonlinear functions and search along the gradient direction, significantly more efficient than random search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Evaluation criteria measure the quality of factors found by search algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The performance of the generated fac- tors is evaluated using backtest experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Typical eval- uation criteria include information coefficient (IC), informa- tion ratio based on information coefficient (ICIR), as well as annualized return, maximum drawdown, Sharpe ratio, and turnover rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In addition, it is very important to keep in- formation diverse among factors by filtering out redundant factors which highly correlated with other factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Due to their importance in factor mining, we introduce about two types of search algorithms in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Genetic programming (GP) [64] (Figure 13), an extension of genetic algorithm [65], is a metaheuristic algorithm for search- ing tree-structured symbolic factor expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In an algo- rithmic loop, GP starts from a number of initial factors, and it uses an evolutionary mechanism to produce the next gener- ation of factors, aiming to improve factor performance mea- sured by the fitness function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' There are two types of evo- lutionary mechanism in GP: mutation which replace a node randomly with another operand of an operator, and crossover where two trees swap their subtrees randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In each iter- ation, all factors are evaluated using IC or alternatives and only the best-performing factors are kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' This process is repeated until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Neural symbolic regression utilizes gradient information to accelerate the search process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Neural networks are used to learn a continuous and nonlinear function to approximate those discrete symbolic expressions and use this function to 12 IndustryNeutralization FactorValue0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2sell limit order ordercancel il.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='i LOBbid-sidelevels Volume i Price LOBask-sidelevels buy limit order mid-priceRETURNONASSETS RETURNONEQUITY DEBT-EQUITYRATIO 21,5% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='7% 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='6% SHAREPRICE P/ERatio WORKINGCAPITALRATIO 207E 31,5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='8Parent Google company Alphabet Boeing 747 LLC Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Item Produce Owned operated Citigroup by BlackRock Boeing United Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Airlines, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='ComputerSoftware Transportation Pharmaceuticals (Technology) (Travel) (Healthcare) Groupings Norwegian Synopsys Vertex Cruise Line Pharmaceuticals Expedia Adobe Group Industry Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Autodesk Regeneron Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' AmericanAirlines Pharmaceuticals Group Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=" =BUSINESS NEWS Businessnewstoday aboutgrowingstock monev Businessnews today Business newstoday Businessnewstoday aboutgrowing stock about growing stock aboutgrowingstock moneyto'0 EO'O 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='01 2 0 2 valueTechnology Healthcare Energy Telcom Materials Cgtaumer Industrials Real Estate FinancialsMutation Crossover rank rank open volume corr sign mul high low sqrt sub rank open rank rank open volume corr neg −𝑐𝑜𝑟𝑟(𝑟𝑎𝑛𝑘 𝑜𝑝𝑒𝑛 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 𝑟𝑎𝑛𝑘 𝑣𝑜𝑙𝑢𝑚𝑒 ) s𝑖𝑔𝑛(𝑐𝑜𝑟𝑟 𝑟𝑎𝑛𝑘 𝑜𝑝𝑒𝑛 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 𝑟𝑎𝑛𝑘 𝑣𝑜𝑙𝑢𝑚𝑒 ) rank rank open volume corr neg −𝑐𝑜𝑟𝑟(𝑟𝑎𝑛𝑘 𝑜𝑝𝑒𝑛 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 𝑟𝑎𝑛𝑘 𝑣𝑜𝑙𝑢𝑚𝑒 ) mul high low sqrt sub 𝑠𝑞𝑟𝑡 ℎ𝑖𝑔ℎ ∗ 𝑙𝑜𝑤 − 𝑠𝑞𝑟𝑡(ℎ𝑖𝑔ℎ) 𝑠𝑞𝑟𝑡 ℎ𝑖𝑔ℎ ∗ 𝑙𝑜𝑤 − 𝑐𝑜𝑟𝑟(𝑟𝑎𝑛𝑘 𝑜𝑝𝑒𝑛 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 𝑟𝑎𝑛𝑘 𝑣𝑜𝑙𝑢𝑚𝑒 ) sqrt high rank volume corr neg −𝑐𝑜𝑟𝑟(𝑠𝑞𝑟𝑡(ℎ𝑖𝑔ℎ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 𝑟𝑎𝑛𝑘 𝑣𝑜𝑙𝑢𝑚𝑒 ) sqrt high Figure 13: Illustrations of two evolutionary mechanism used in genetic pro- gramming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' generate new formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' We introduce two works about neu- ral symbolic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The first paper [62] (Figure 14a) builds a transformer generative model from a number of ex- isting symbolic expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In the training stage, a special transformer model (called set transformer [66]) encodes the formulas in the training set into embedded vectors, and they are delivered to a transformer decoder to update symbolic expressions in an autoregressive way using beam search, and this process is repeated until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The generative model is trained by minimizing the cross-entropy loss be- tween the generated expression and the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In the test stage, the trained model is used to generate new symbolic expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The second paper [63] (Figure 14b), designed a new neural network specifically for expressing symbolic for- mulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In particular, the activation functions in this network are replaced with symbolic operators such as sin(·) and √(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' This special neural network has the flexibility to generate al- most all formula expressions need to use in factor mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Machine Learning Factors Symbolic factors have their advantages in simplicity and understandability, and are thus widely used in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' How- ever, their representation ability is limited by the richness of operands and operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Machine learning factors, on the other hand, have more flexibility in representation to fit more com- plicated nonlinear relationships [67], and thus they have the chance to perform better in market prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In particular, mining machine learning factors [68, 69, 70, 71] is a process to fit neural networks, where gradients provide the optimal direc- tion for fast search of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' As shown in Figure 15, most deep neural networks for stock prediction follows the encoder- (a) Neural symbolic regression in a sequence generation manner [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' (b) Neural symbolic regression that directly uses the neural network as symbolic expressions [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Figure 14: Illustrations of neural symbolic regression algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' … Encoder RSI ADX MACD MA Raw Data MSFT AAPL … AAPL MSFT … GOOGL Stock Embeddings Decoder AAPL MSFT … GOOGL 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='52 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='63 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='78 Prediction … … GOOGL Figure 15: Illustration of the encoder-decoder architecture used in stock predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Both embeddings and predictions can be used as factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' decoder architecture [72], where the encoder maps meta factors to a latent vector representation and the decoder transforms this embedding to some outcome such as future return [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In fact, not only the final outcome, but also the embedding itself could be used as a (high-dimensional) machine learning factor [74], and further applied to various downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Machine learning factors have some limitations as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Firstly, they are usually hard to interpret and understand because of the black-box nature of machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Secondly, gradient search used by neural networks may be stuck at some local op- tima and result in model instability problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Finally, neural networks may suffer more serious overfitting due to their flex- ibility, and this situation gets worse in quant because data are extremely noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Automated Modeling The automation of statistical machine learning such as SVM, decision tree and boosting has been extensively researched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' A simple and direct automation method is the brute-force enumer- ation of all possible configurations, each including the choice of machine learning algorithms and the corresponding hyper- parameters (Figure 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 13 Pre-Training Test Prediction Data Generator Cross Entropy RNG Skeleton y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='5 exp(-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2c²) y = sin(Cia) + C2 y = sin(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2α) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3 BFGS Transformer fitting constants RNG Constants Decoder C1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2,C2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3 Beam Search Set Transformer [(ci, yi)li RNG Support Encoder = [-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2, -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='7, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=',3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='9, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='8] Set Transformer Transformer [(ai, yi) Encoder Decoder(1) y(1) id id α Sin (sin) W(1) (T)M (all-to-all) (all-to-all) (all-to-all)<>>><>>><>>>Search Space Cell-based Hierachical Entire-structured Search Strategy Recursive Differentiable Performance Estimation Strategy Early-stopping Lower-fidelity Weight Inheritance Weight Sharing One-shot Define Evaluate Feedback Figure 16: The automated modeling pipeline for architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The structure of this figure is adapted from [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Illustrations of search spaces and search strategies are cited from [42, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Figure 17: Illustration of early-stage automated machine learning based on brute-force search of algorithms and hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Figure is cited from [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In this article, we focus on the state-of-the-art deep learning automation problem, which is more complex due to the end-to- end property and network architecture issue in modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The configuration of a deep learning model consists of three parts: architecture, hyperparameters, and objectives, and they jointly determine the final performance of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Traditionally, these configurations are tuned manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0, they are searched and optimized using various AutoML algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' A standard AutoML system needs to answer the following three questions: what to search (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=', search space §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1), how to search (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=', search algorithm §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2), and why to search (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=', performance evaluation §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Search Space Search space is designed for the three configuration settings that need to be optimized in an automatic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Architecture configures a network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' For example, the architecture of a multi-layer perceptron is specified by the number of hidden layers and the number of neurons at each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The architecture of a convolution neural net- work needs to consider more configurations such as the num- ber of convolution kernels as well as their strides and recep- tive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The architecture of large-scale models such as Transformer is composed of a number of predefined blocks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' self-attention blocks, residual blocks) linked together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' As discussed above, architecture is complex and may have a hierarchical structure at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Accordingly, the search space can be defined at various granularities, ranging from low-level operators such as convolutions and attentions to high-level modules such as LSTM cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Early search al- gorithms run on the finest granularity and optimize the low- level structure of the neural network [78, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Such a search process is flexible in network structure but inefficient in in- corporating prior knowledge and abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' One solution is to assume a hierarchical structure in network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Specifically, at a high level, the network is designed to be a graph of cells (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' blocks/motifs [42, 80]), each of which is a subnetwork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Many cells share the same internal struc- ture at a low level in order to reduce the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Cell-based search algorithms [81, 82] need to find both high- level structures between cells as well as low-level structures within the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Hyperparameter controls the overall training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' For example, learning rates determines the step size moving to- wards a minimum of a loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' A smaller learning rate is more accurate in solution but slower in convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The batch size determines the number of samples involved in a batch for gradient estimation, which also has an influence on training efficiency and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The search space for hyper- parameters is simpler than that for architecture since most hyperparameters are continuous (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=', learning rate) or ap- proximately continuous values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=', batch size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Objective specifies the loss functions and labels used for train- ing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The loss function is the key component of ma- chine learning models since it provides a goal towards which a model should be trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Besides classic loss functions such as mean square loss and cross-entropy loss, new loss functions specifically designed for quant tasks can be also selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Labels define the “ground-truth” target the model aims to fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' For example, either price raise/fall or future re- turns in different holding time windows can be considered in the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='SVM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='LDA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Kernel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Gamma ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Solver ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='(n_components) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='polynomial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='rbf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='sigmoid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Isgr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='eigen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='svd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='coeffo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='coeffo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='shrinkage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='degree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='shrinkageoutput ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Cell k+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='redation cel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Cell k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='concat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='normal cell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Block11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='eduction cell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Block o ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='mal cell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='reductikon cell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='rormal cell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Cell k-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='xn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Cellk-2output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='国国 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Celln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Block3-B3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Cell3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Block 3-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Cell2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Bloxk 1-B1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Cell1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Block 1-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='inputoutput ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='L4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='conv 3x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='conv 3x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='L3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='conv 5xs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='conv 5x5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='L2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='conv 3x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='conv 3x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='LI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='max pool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='max pool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='inputop A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='index A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='op B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='index B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='op A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='index A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='op B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='index B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='skip ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='max- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='5x5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='pool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='hidden ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='state ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Empty ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='skip ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='max- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='5x5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='pool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Block 1 of cell k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Block 2 of cell k0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3 3 3 (a) (b) (c) (p)0 Operation Operation2 Operation3 2 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Search Algorithm Given the search space, we could use search algorithms to find the best model configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Table 2 lists various types of search algorithms and their corresponding tasks: network archi- tecture search (NAS) [75], hyperparameter optimization (HPO) [83] and training objective selection (TOS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Table 2: Search algorithms and their applicable search targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Algorithm Target NAS HPO TOS References Grid/Random Search ✓ ✓ [84, 85] Evolutionary Algorithm ✓ ✓ ✓ [86, 87, 88, 89, 90] Reinforcement Learning ✓ ✓ [91, 81, 82, 92] Bayesian Optimization ✓ ✓ [93, 94, 95, 96] Gradient-based Method ✓ ✓ [97, 98, 99, 100, 101] Grid search is a brute-force algorithm that searches on a grid of configurations and evaluates all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' It is a good choice when the search space is small due to ease of imple- mentation and parallelization [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' However, most NAS and HPO problems in deep learning have extremely large search spaces and grid search can not scale well for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' More- over, grid search is used more popular in HPO and TOS than in NAS whose search space is difficult for grid layout except enumerating all possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Random search generates a number of candidate configura- tions using some stochastic sampling mechanisms, such as Monte Carlo or MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' It is very straightforward to im- plement and parallelize (mainly for independent sampling mechanisms such as Monte Carlo sampling or importance sampling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Random search is very flexible and can be used for NAS, HPO, and TOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Although random search is usually faster than grid search [84], it is still difficult to handle high- dimensional search space as the number of potential configu- rations grows exponentially with the number of hyperparam- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Evolutionary algorithm is an extension of random search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' It utilizes evolution mechanisms to improve model configura- tions iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' It encodes the architecture of network net- works as a population and performs the evolution steps on them to improve the model iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Specifically, the mod- els are first encoded according to their underlying computa- tion graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Then, a set of pre-defined evolutionary operators are applied to the encoded models, including architectural modifications such as inserting or deleting several operations and adding skipping connections, as well as hyperparameter- related operations such as learning rate adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' At each iteration, the best-performing models are selected via tourna- ments and combined via mutation and crossover operations to form the next generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Evolutionary algorithms inher- ently support weight inheriting among generations, which helps accelerate the convergence in training and increase the searching efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Reinforcement learning models the architecture search prob- lem as a Markov decision process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In each step, an RNN con- troller chooses an action to sample a new architecture and the corresponding deep neural network model is trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Then the performance of the model evaluated on the validation set is used as a reward which is forwarded to compute the policy gradient and update the RNN controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' This loop is iterated until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The reinforcement learning framework is very universal for most optimization problems and it could be used for NAS, HPO, and TOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Bayesian optimization explores the search space more effi- ciently by leveraging surrogate models to approximate the objective function that couldn’t be expressed explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Specif- ically, for a black-box objective function for HPO, Bayesian optimization initializes a prior distribution using a surrogate function such as Gaussian process or tree-structured Parzen estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Then it samples new data points from the prior dis- tribution (with importance) and calculates their values using the underlying objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Given these new samples and prior, the posterior function can be calculated and is used as an updated surrogate function to replace the original prior function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' This process is repeated until the optimal solution is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Traditionally, Bayesian optimization is used for continuous search tasks such as HPO, but recent works have extended it to NAS tasks as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Gradient-based methods is very efficient when the gradient of the objective function exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' However, for NAS, the search space is discrete and couldn’t define a gradient directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' One solution is to “soften” the architecture and define an over- parameterized “super-architecture” which covers all possi- ble candidates and is differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' A typical gradient-based NAS method is DARTS [97] which constructs an over-parameterized network where all types of candidate operations are present on the computation graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The resulting value is the weighted sum of the results of all the operations, where the weights are the softmax values of a parameterized probability vec- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Both model parameters and architectural parameters are trained via a bi-level optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In the inference process, the architecture and hyperparameters with the high- est probability are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' DARTS is substantially faster than random research and reinforcement learning in NAS and HPO tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Accelerating Evaluation The computational cost of automated model search comes from two parts: search algorithm and model evaluation, and the latter is usually the bottleneck of the computation because it is very time-consuming to train a deep neural network until convergence under a given configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Several methods are introduced in previous research to address this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Firstly, the training process of neural networks can be early-stopped before convergence to reduce the computational time for eval- uation [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Secondly, the model can select fewer samples to accelerate the training process [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Thirdly, warm-start model training can be used to leverage the information from existing selected models [102] or inherit the information from an over- parameterized “parent” model [103, 97, 98] to accelerate the search loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Automated One-click Deployment Model deployment is the task of transferring the developed model from offline research to online trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' It is not only simply transferring code and data, but also synchronizing data and factor dependency, adapting trading server and system, de- bugging model inference, testing computing latency, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In the following part, we focus on one important problem in model de- ployment: how to accelerate deep learning inference for high- frequency trading and algorithmic trading scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' We pro- pose an automated one-click deployment solution utilizing tech- niques such as model compilation [104] and model compres- sion [105, 106] to realize inference acceleration [104, 105, 106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The former makes the inference faster without changing the model itself, and the latter seeks smaller and lighter alternative models to save inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Acceleration by Model Compilation At the development stage, deep learning models’ function- ality is the top priority for the underlying framework which im- plements the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Hence, at this stage, the framework strictly maps all the operations to the computation graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' How- ever, such direct mapping introduces large room for optimiza- tion at the deployment stage where the computations are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Therefore, the model’s computation can be greatly simplified and adapted to hardware features without hurting its original semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Such optimization is one of the major topics in deep learning compilers [107, 108, 109], which can be categorized as front-end optimizations and back-end optimizations, which work on high-level and low-level intermediate representations (IRs) for deep learning models respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Following the sum- mary in [104], we will briefly introduce relevant optimization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Front-end optimization, as illustrated in Figure 18a, focuses on simplifying the structure of the computation graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' For example, algebraic simplification techniques such as constant folding [110] and strength reduction [111] convert expensive operations into cheaper ones via transformation or merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Common subexpression extraction (CSE) [112] techniques iden- tify repeated nodes in the computation graph and merge them into one node to avoid duplicate computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Back-end optimization, as illustrated in Figure 18b, is per- formed with an emphasis on the features of hardware archi- tectures, such as locality and memory latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' For example, pre-fetching techniques [113] load data from main memory to GPU before they are needed, and speed up fetch operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Loop-based optimizations [114] reorder, fuse and unroll oper- ations inside loops to enhance locality among neighboring in- structions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Memory latency hiding [115] techniques aim to in- crease instruction throughput so as to mitigate the problem of high latency in accessing memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Parallelization techniques such as loop splitting [116], automatic vectorization [117] and loop skewing [118, 119], can also be applied to maximize the parallelism provided by modern processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Acceleration by Model Compression Model compression aims to reduce model size for inference acceleration while minimizing drops in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In this way, the compressed model can be regarded as an approxima- tion of the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Generally speaking, model compres- sion can be performed at both micro- and macro-level, where the former focuses on the precision of individual model param- eters and the latter focuses on simplifying the overall model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' At the micro-level, pruning and quantization techniques can be applied to reduce both the number of parameters and the bit size of the individual parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Model pruning [120], as shown in Figure 19a, removes unimportant connections and neurons in neural networks that have little influence on the ac- tivation of the neural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The identification of candidate parame- ters is usually based on their weights, where those with smaller weights are considered for pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Model quantization [121], as shown in Figure 19b, converts the parameters from floating- point numbers to low-bit representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Specifically, a code- book is constructed to store the approximated values of the orig- inal parameters according to the distribution of all parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The parameters are then quantized according to the codebook and thus the bit size for the parameters is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Due to the inevitable performance drop induced by precision re- duction, the compressed model is usually re-trained to approach its original performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' At the macro-level, the model can be significantly com- pressed into a smaller model with simpler architectures via knowl- edge distillation [122] and low-rank factorization [123, 124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Knowledge distillation (Figure 19c) compresses a model by transferring useful knowledge from the original large model (called the teacher model) to a small and simple model (called the student model) with minimal knowledge loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Low-rank factorization techniques (Figure 19d) assume the sparsity of model parameters and then split the parameter matrix of the original neural networks into products of low-rank matrices [123], and thus reduce the model complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Explainable AI for Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0 XAI [126, 127], as an attractive research direction for decades, is critical to the trustworthiness and robustness of AI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In the case of quant, improvement in the explainability of AI can make the decision process more transparent and easy to ana- lyze, providing useful insights to researchers and investors and discovering potential risk exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In this section, we will discuss how to leverage XAI in Quant 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1 introduces common XAI techniques and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='2 connects these techniques to real quant scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Overview of Explainable AI XAI is an emerging interdisciplinary research area cover- ing machine/deep/reinforcement learning, statistics, game the- ory and visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Here we focus on two types of XAI: model-intrinsic explanation [128] and model-agnostic explana- tion [129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='�������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='����������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='��������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='���������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='��������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='���������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='��������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='��������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='����������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='����������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='���������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='��������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='��������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='��������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='����������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='���������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='��������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='��������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='�������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='����������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='���������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='����������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='���������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='����������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='�������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='���������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='���������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='����������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='��������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='��������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='�������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='���������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='��������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='����������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='����������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='����������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='��������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='���������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='�������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='�������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='��������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='�������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='�������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='��������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='��������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='���������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='(a) Front-end optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Hardware Intrinsic Mapping ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Memory Allocation & Fetching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Memory Latency Hiding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Loop Oriented Optimization Techniques ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Parallelization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Loop fusion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Slide windows ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Tiling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Reordering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Unrolling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Halide ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Polyhedral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='for i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='n Stmt1(i) for i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='n Stmt2(i) for i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='n Stmt1(i) Stmt2(i) for i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='4 Stmt1(i) Stmt1(1) Stmt1(2) Stmt1(3) Stmt1(4) for i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='n for j=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='m Stmt(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='j) for i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='m for j=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='n Stmt(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='j) Cache Fit in operator Hardware intrinsic gemm8x8(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' z) for i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='8 for j=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='8 for k=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='8 z(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='j)=x(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='i)*y(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='j) DRAM GPU Shared Memory GPU Memory ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Transfer Fetch Store Use Warp Data Data Data ld ex ld ex ld … ld ex ld ex ld … 0 1 0 ld ex ld ex 0 1 1 for i=1,n for j=1,n Stmt(i,j) for i=1,n for j=1,n/4 for k=1,4 Stmt(i,j*4+k) for i=1,n for j=1,n/4 vec (Stmt(i,j)) for i=min,max for j=1,n/4 vec (Stmt(i,j)) split vectorize parallelize Autotuner schedules Nested Polyhedral Parallelization / Vectorization (b) Back-end optimization Figure 18: Deep learning compiler optimization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Figure from [104] (a) Model pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Figure from [120] (b) Model quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Figure from [121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Data Teacher Model Student Model Knowledge Knowledge Distill Transfer Knowledge Transfer (c) Knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Figure from [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' 𝑐 channels W′ P 𝑑′ channels 𝑑 channels W (a) (b) (d) Low-rank approximation of CNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Figure from [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Figure 19: Model compression techniques 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Model-intrinsic Explanation in XAI Risk control and management is the top priority of financial industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' When AI models are deployed in real-world applica- tions, their decision process is usually required to be transparent by regulatory authorities for the safety of transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' More- over, model-intrinsic explainability is the requirement of many large financial institutions such as banks and insurance compa- nies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' A machine learning model is intrinsically explainable if its internal structure or mechanisms can be easily explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Some machine learning algorithms such as linear models and decision trees are inherently explainable, as many other algorithms such as deep neural networks and kernel learning methods (SVM, Gaussian process, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=') are black boxes with poor explainabili- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Figure 20 illustrates many popular machine learning meth- ods arranging along their general performance and explainabil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' We can see The increase in model-intrinsic explainability usually leads to a decrease in the model’s prediction perfor- mance, and therefore the selection of machine learning algo- rithms is essentially a trade-off between explainability and per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' We briefly introduce a few typical machine learning methods in terms of explainability and predictive performances and discuss their applicable scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Linear Models, such as linear regression, logistic regression, linear discriminant analysis, linear SVM and addition model, is a family of methods where features or transformation of a group of features are in an additive form and thus the perfor- mance of final prediction can be easily deposed to the effects from individual features or feature groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Therefore, linear models are intrinsically understandable and explainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' For example, linear regression explicitly encodes the importance of each feature in their corresponding regression coefficients (assuming every feature is normalized to eliminate the effect from scales and units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Although linear models are easy to explain, they are suffering the poor prediction performance since they couldn’t encode complicated nonlinear relation- ships between prediction outputs and features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Rule-based Learning is another type of easy-to-explain meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Different from a linear model which fits a linear deci- sion boundary, a rule-based learning method fits a stepwise decision boundary characterized by decision rules combin- ing a number of logical expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Examples of rule-based learning include decision tree [133] and symbolic regression, as well as ensemble models such as random forest [134] and boosting [135, 136, 137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Rule-based learning models are intrinsically explainable decision rules that are close to the logical thinking process of human beings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' However, to bet- ter fit the training data and improve prediction performance, the decision rule is usually complicated, and it reduces the explainability and increases the risk of overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Ensemble Learning combines multiple machine learning al- gorithms to achieve better decision performance than single models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Typical examples of ensemble learning include ran- dom forest and boosted trees that combine multiple tree mod- els and make predictions based on the aggregation of individ- ual decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Although there is controversy, in this article, we classify mixtures of experts (MoE) [138] as an ensemble method as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' MoE combines multiple expert networks in parallel in a layer and decides which expert (or experts) par- 17 before pruning after pruning （ pruning synapses pruning neuronsweights cluster index fine-tuned (32 bitfloat) (2 bit uint) centroids centroids 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='09 0 2 3: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='12 cluster 1 0 3 2: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='92 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='03 0 3 1 0 1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='87 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='49 1 2 2 0: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='00 Ir 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='97 gradient 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='12 group by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='02 reduce 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='02 个 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='03Gaussian Process ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Decision Tree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Residual Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Transformer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='HMM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Boosting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Random Forest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Mixture-of-Expert ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='LSTM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Kernel k-NN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Generalized Linear Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Kernel SVM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Deep Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Symbolic Regression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Bayesian Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='CRF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Vanilla RNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Model Explainability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Prediction Performance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Linear SVM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Kernel Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Sequence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Generalized Additive Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Linear Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='High ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='High ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Low ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content='Figure 20: Comparison of popular machine learning algorithms according to prediction performance and model explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Part of this figure is cited from [130, 131, 132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' ticipates in the decision of a specific data point via a gating mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Compared with other machine learning meth- ods, ensemble learning provides high-level explainability by demonstrating the relative importance of single models or ex- perts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Kernel Learning, also known as kernel method or kernel ma- chine, is a family of nonparametric learning methods which make predictions by computing the similarity between sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The similarity is characterized by a kernel function, which is a special inner product defined in a high-dimensional Hilbert space where original data samples are mapped [139].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' For example, kernel SVM [140] transforms original posi- tive/negative samples into another space where they can be easily separated using a linear decision boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' In prin- ciple, kernel functions can be of arbitrary forms that satisfy the Mercer’s condition [141], and they determine the non- linear relationships between inputs and outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Moreover, the idea of kernel functions is extended to the self-attention mechanism [142] used in neural networks such as Trans- former [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Traditionally, we think the kernel trick im- proves the performance of models but weakens their explain- ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' From another point of view, however, the definition of kernel itself encodes the prior insight of users and could help understand the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Sequence Learning refers to a family of machine learning methods that work with sequential data such as time series or sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' They are widely used in speech recognition, language understanding, DNA sequence analysis, and stock price prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Sequence learning methods characterize the underlying structure hidden in sequential data and dis- cover implicit patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' For example, hidden Markov model (HMM) [144] assumes that the underlying structure is a ho- mogeneous Markov chain determined by a transition matrix (or transition kernel for continuous state space) and assumes the observed sequence is randomly generated from this chain through emission probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The transition probabilities and emission probabilities are estimated during model train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Although HMM is generally a black-box model, its tran- sition probability matrix provides some insight into the auto- regressive structure in prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Conditional random field (CRF) [145] extends the first-order Markov assumption of HMM and characterizes longer-range time dependency us- ing graphical model for probability modeling, and this extra flexibility usually brings better prediction performance for CRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Recurrent neural networks (RNN) such as LSTM [146] and GRU [147] exhibit better performance in sequence pre- diction, but it is harder to explain their internal mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Deep Learning usually has superior prediction performance [148] but its shortage in explainability is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' Some special operators in deep neural networks such as convolution and attention provide partial and local explanations about their mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' For example, the self-attention layer in a Trans- former [143] characterizes the relative importance of each position in a sequence with respect to other positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' The model-intrinsic explainability for machine learning is al- ways a contradiction with its prediction capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' However, before the appearance of a brand-new machine learning model satisfying both high prediction accuracy and high explainabil- ity, we could rebuild and improve current machine learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfogjY/content/2301.04020v1.pdf'} +page_content=' We could either start from an explainable model such as a linear or rule-based model and improve its prediction per- formance by incorporating more local nonlinear structures with 18 root True node False a χ(Πr), then xr+1 has infinite order in Γ, and Πr+1 is the free product +of its subgroup Πr with its infinite cyclic subgroup ⟨xr+1⟩. This is proved by relating the +quantities χ(Πr) for 1 ≤ r ≤ m to the dimensions of certain varieties of representations. +For each r the representations of Πr in PSL2(C) are identified with the points of a complex +affine algebraic set, which for the purposes of this sketch will be denoted Rr. The inclusion +homomorphism Πr → PSL2(C), regarded as a representation, lies in a unique irreducible +component Vr of Rr, and has complex dimension 3χ(Πr) + 3. If for a given r < m the +group Πr+1 is not the free product of Πr with ⟨xr+1⟩ (or if xr+1 has finite order), it can +be shown that Vr+1 is isomorphic to a proper subvariety of Vr × PSL2(C); this implies that +dim Vr+1 < 3 + dim Vr, and hence that χ(Πr+1) ≤ χ(Πr). This completes the sketch of the +proof of Theorem B. (Parts of the argument sketched here are embodied in the statement +and proof of Lemma 3.1.) +The idea of using dimensions of representation varieties in this argument was suggested +by Ian Agol’s observation that one can use dimensions of representation varieties to give +alternative proofs of [11, Theorem VI.4.1], and its generalizations [4, Appendix, Theorem A] +and [3, Theorem 7.1], using dimensions of representation varieties. +Theorem A is proved by combining Theorem B with the so-called log(2k − 1) Theorem. +The latter result, which gives information about the lengths of loops α1, · · · , αk based at a +point p of an orientable hyperbolic 3-manifold M under the assumption that the elements +[α1], . . . , [αk] freely generate a free subgroup of π1(M, p), was proved under an additional +hypothesis in [3]. The general version of the log(2k − 1) Theorem is proved by combining + +EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS +3 +the results of [3] with the tameness theorem [1], [6] and the density theorem [17], [14] for +Kleinian groups. The general version of the log(2k − 1) Theorem is presented as Theorem +4.1 of [2], which we will quote in the proof of Theorem A. +The proofs of Theorems B and A will be given in Section 3, after needed background +about representations, representaion varieties, character varieties, and deformation spaces +of Kleinian groups, which occupies Section 2. +We will use the following standard conventions. A 3-manifold M is said to be irreducible if +M is connected and every (tame) 2-sphere in M bounds a 3-ball. To write Y ≤ X, where +X is a group, means that Y is a subgroup of X. If A is a subset of a group, ⟨A⟩ denotes the +subgroup generated by A. +I am grateful to Steve Boyer, Ken Bromberg and Dick Canary for their assistance. Boyer, +with tremendous patience, helped me navigate the material that is summarized in Subsection +2.9. Bromberg and Canary helped me locate the result from [8] which is used in the proof of +Lemma 2.4. Canary pointed out Theorem 8.44 of [13], which is used in the proof of Lemma +2.12. +2. +Preliminaries +Definition 2.1. A compact core of a 3-manifold M is a compact three-dimensional subman- +ifold N of M such that the inclusion N → M is a homotopy equivalence. +Although the following result is well known, we include a proof for clarity. +Proposition 2.2. Every orientable hyperbolic 3-manifold with finitely generated fundamental +group has a compact core. +Proof. If the given manifold M is isometric to H3, any ball in M is a compact core. If +M is not isometric to H3, we observe that since M admits H3 as a non-trivial covering +space, M is irreducible and has infinite fundamental group. We then apply [20, Proposition +3.8], a consequence of the main result of [18], which asserts that if M is an irreducible, +orientable 3-manifold with finitely generated fundamental group, then there is a compact, +irreducible submanifold N of M such that the inclusion homomorphism π1(N) → π1(M) +is an isomorphism. (The hypothesis of finite generation was unfortunately left out of the +statement of [20, Proposition 3.8], but the proof given there establishes the statement given +here.) In particular π1(N) is infinite; this, together with the irreducibility of N, implies that +N is aspherical (see [20, Proposition 3.6]). As the hyperbolic manifold M is also aspherical, +it now follows that the inclusion N → M is a homotopy equivalence. +□ +2.3. Let G be a finitely generated subgroup of the fundamental group of an orientable +hyperbolic 3-manifold M. The covering space � +M of M corresponding to the subgroup G has +a compact core N by Proposition 2.2; in particular N is aspherical and π1(N) ∼= G. Since +N is compact, it follows that G is homologically finite. + +EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS +4 +This establishes the fact, which was mentioned in the introduction and is necessary back- +ground for Theorems A and B, that a finitely generated subgroup G of the fundamental +group of an orientable hyperbolic 3-manifold is homologically finite, so that χ(G) is defined. +Lemma 2.4. If M is an orientable hyperbolic 3-manifold with finitely generated Kleinian +group, there exist an orientable hyperbolic 3-manifold M0 with no rank-2 cusps such that +χ(M0) = χ(M), and a surjective homomorphism from π1(M) to π1(M0). (Here χ(M) and +χ(M0) are defined in view of 2.3.) +Proof. It follows from Proposition 2.2 that dim H2(M; Q) < ∞. The number of ends of the +manifold M is at most 1 + dim H2(M; Q), and is in particular finite. We shall prove the +result by induction on the number of ends of M. +If the number of ends of M is 0, then M is compact and therefore has no cusps. +The +assertion is trivial in this case, since we can take M0 = M and choose the identity map as +required homomorphism. Now suppose that the number of ends of M is m > 0, and that +the assertion is true for manifolds with m − 1 ends. If M has no rank-2 cusps, the assertion +is again trivial. Now suppose that M has at least one rank-2 cusp, and fix a submanifold H +of M which is a standard neighborhood of a rank-2 cusp. +The 2-manifold ∂H is a torus, and inherits a Euclidean metric from the hyperbolic metric on +M. For each homotopically non-trivial simple closed curve s in ∂H, we denote by Z(s) the +orientable 3-manifold, defined up to homeomorphism and dependent only on the isotopy class +of s, which is obtained from the disjoint union M − H +⊨ +(D2×S1) by gluing ∂(M − H) = ∂H +to ∂(D2 × S1) = (∂D2) × S1 by a homeomorphism which maps s to (∂D2) × {pt}. (For any +s, the manifold Z(s) is said to be obtained from M − H by Dehn filling.) +For any closed geodesic s in ∂H, since M − H is homotopy equivalent to M, we have +χ(Z(s) = χ(M − H) − χ(∂H) + χ(D2 × S1) = χ(M) − 0 + 0 = χ(M). Furthermore, the +inclusion homomorphism π1(M − H) → π1(Z(s)) is surjective, and hence there is a surjective +homomorphism from π1(M) to π1(Z(s)). +We now apply the case k = 1 of [8, Theorem 1.4], which asserts that if the closed geodesic in +∂H representing the isotopy class of s has length strictly greater than 6, then Z(s) admits +a hyperbolic metric. Since there are only finitely many isotopy classes of geodesics in ∂H +whose lengths are bounded above by a given constant, we may fix a simple closed curve s1 +in ∂H such that Z(s1) is homeomorphic to a hyperbolic 3-manifold. +The manifold Z(s1) has m − 1 ends. Hence, by the induction hypothesis, there exist an +orientable hyperbolic 3-manifold M0 with no rank-2 cusps such that χ(M0) = χ(Z(s1)), and +a surjective homomorphism from π1(Z(s1)) to π1(M0). Since χ(Z(s1)) = χ(M), we have +χ(M0) = χ(M). Furthermore, since there is a surjective homomorphism from π1(M) to +π1(Z(s1)), we obtain by composition a surjective homomorphism from π1(M) to π1(M0). +This completes the induction. +□ +Lemma 2.5. If Γ is a finitely generated Kleinian group, there exist a finitely generated +Kleinian group Γ0 with no rank-2 cusp subgroups, with χ(Γ0) = χ(Γ), and a surjective +homomorphism η : Γ → Γ0. (Here χ(Γ) and χ(Γ0) are defined by 2.11.) + +EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS +5 +Proof. We apply Lemma 2.4 to the orientable hyperbolic 3-manifold M .= H3/Γ. The latter +lemma gives an orientable hyperbolic 3-manifold M0 with no rank-2 cusps such that χ(M0) = +χ(M), and a surjective homomorphism from π1(M) to π1(M0). We may write M0 = H3/Γ0 +where Γ is a Kleinian group without rank-2 cusp subgroups. +We have Γ ∼= π1(M) and +Γ0 ∼= π1(M0); and since the hyperbolic 3-manifolds M and M0 are aspherical, we have +χ(Γ0) = χ(M0) = χ(M) = χ(Γ). The assertion follows. +□ +The next two subsections involve the notion of index of freedom and minimal index of +freedom, which were defined in the introduction to this paper. +2.6. Note that if a finitely generated group G is non-trivial and torsion-free, then any gen- +erating set for G contains an element of infinite order, and hence miof(G) ≥ 1. +Lemma 2.7. Let G1 and G2 be groups, and suppose that there is a surjective homomorphism +η : G1 → G2. Then miof(G2) ≤ miof(G1). Furthermore, for any generating set ∆ for G1, +we have iof(η(∆)) ≤ iof(∆). +Proof. We prove the second assertion first. Write ∆ = {x1, . . . , xm}. Set k = iof(η(∆)). +Then after reindexing the xi we may assume that η(x1), . . . , η(xk) are independent. +It +follows that x1, . . . , xk are independent, and hence that iof(∆) ≥ k, as required. To prove +the first assertion, note that if ∆ is an arbitrary generating set for G1, the first assertion +gives iof(∆) ≥ iof(η(∆) ≥ miof(G2); hence miof(G1) ≥ miof(G2). +□ +The following lemma is a very special case of a result that is implicit in [16]. The latter +result has been refined in [21] and elsewhere, and was rediscovered in [19]. As an explicit +statement of the lemma is not easy to find, we have provided a simple, self-contained proof. +Lemma 2.8. Let Γ be a finitely generated subgroup of GL2(C) containing no non-trivial +scalar matrix. Then there is an element L of infinite order in SL2(C) such that the subgroup +⟨L, Γ⟩ of GL2(C)) is a free product of Γ with the infinite cyclic group ⟨L⟩, and contains no +non-trivial scalar matrix. +Proof. Since Γ contains no non-trivial scalar matrix, each non-trivial element of Γ has one or +two eigenvectors in C2. Since Γ is countable and C2 is not, there is a non-zero vector in C2 +which is not an eigenvector of any element of Γ. Hence after modifying Γ by a conjugation, +we may assume that Γ contains no upper triangular matrix except the identity. +Let F denote the countable subfield of C generated by the entries of the elements of Γ, and +fix an element t of C which is transcendental over F. We shall show that the conclusion of +the lemma holds if we set L = +�1 +t +0 +1 +� +. This is equivalent to the assertion that if F(X) +denotes the rational function field in one indeterminate over X, and if we set Λ = +�1 +X +0 +1 +� +, +then the subgroup of GL2(F[X]) ≤ GL2(F(X)) generated by Γ and Λ is a free product of Γ +with the infinite cyclic group ⟨Λ⟩, and contains no non-trivial scalar matrix. + +EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS +6 +We consider the abstract free product Γ ⋆ ⟨Λ⟩, and the homomorphism h : Γ ⋆ ⟨Λ⟩ → +GL2(F[X]) which is restricts to the inclusion on each factor; we must show that h does not +map any non-trivial element of Γ ⋆ ⟨Λ⟩ to a scalar matrix. Any non-trivial element of Γ ⋆ ⟨Λ⟩ +is conjugate either to an element of a factor or to an element of the form +(2.8.1) +w = γ1Λm1 · γkΛmk, +where k ≥ 1, the elements γ1, . . . , γk of Γ are non-trivial, and m1, . . . , mk are non-zero +integers. By hypothesis Γ contains no non-trivial scalar matrix, and it is clear from the +definition of Λ that ⟨Λ⟩ also contains no non-trivial scalar matrix. It therefore suffices to +prove that an element w of the form (2.8.1), where k ≥ 1 and the γi and mi satisfy the +conditions stated above, cannot be mapped by h onto a scalar matrix. By induction on +k ≥ 1 we shall establish the following stronger assertion: +2.8.2. If k is a strictly positive integer, and w ∈ Γ ⋆ ⟨Λ⟩ is given by 2.8.1, where k ≥ 1, the +elements γ1, . . . , γk of Γ are non-trivial, and m1, . . . , mk are non-zero integers, then h(w) has +the form +�A +B +C +D +� +∈ GL2(F[X]), where A and C are polynomials of degree at most k − 1, +while B is a polynomial of degree at most k and D is a polynomial of degree exactly k. (In +particular we have A ̸= D, so that h(w) cannot be a scalar matrix.) +In the statement of 2.8.2, the polynomial 0 is understood to have degree −∞. +In the case k = 1 of 2.8.2, we have w = γΛm, where m is a non-zero integer and γ = +�a +b +c +d +� +is a non-trivial element of Γ. This gives +h(w) = +�a +b +c +d +� �1 +mX +0 +1 +� += +�a +amX + b +c +cmX + d +� +. +Thus the entries in the first column of h(w) are elements of F, while the upper right-hand +entry is a polynomial of degree at most 1. Furthermore, since Γ contains no upper triangular +matrix except the identity, we have c ̸= 0, and hence the lower right-hand entry of h(w) has +degree exactly 1. This establishes the base case for the inductive proof of 2.8.2. +For the induction step, suppose that w = γ1Λm1 · · · γkΛmk satisfies the hypothesis of 2.8.2 for +a given k > 1, set w∗ = γ1Λm1 · · ·γk−1Λmk−1, and assume that h(w∗) = +�A∗ +B∗ +C∗ +D∗ +� +, where +A∗ and C∗ have degree at most k − 2, while B∗ has degree at most k − 1, and D∗ has degree +exactly k − 1. As the base case has been established, we may write h(γkΛmk) = +�A† +B† +C† +D† +� +, +where A† and C† have degree at most 0, while B† has degree at most 1, and D† has degree +exactly 1. Then +h(w) = h(w∗)h(γkΛmk) = +�A∗A† + B∗C† +A∗B† + B∗D† +C∗A† + D∗C† +C∗B† + D∗D† +� +. +It now follows that the entries in the first column of h(w) have degree at most k−1, while its +upper right-hand entry has degree at most k. In the lower right-hand entry C∗B† + D∗D†, + +EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS +7 +the first term has degree at most k − 1 and the second term has degree exactly k; hence this +lower right-hand entry has degree exactly k, and the induction is complete. +□ +2.9. The rest of this section will involve the use of representation varieties. We will take the +point of view used in [5]. +By a representation ρ of a group Γ in PSL2(C), we mean simply a homomorphism from Γ to +PSL2(C). A representation of Γ is said to be reducible if there is a 1-dimensional subspace +of C2 which is invariant under ρ(Γ). Otherwise it is said to be irreducible. +In [5, Section 3], it is shown how to identify PSL2(C) with an algebraic set in some complex +affine space; and it is pointed out that, if Γ is a group with a given finite generating set S, +and if one identifies an arbitrary representation ρ of Γ in PSL2(C) with the point (ρ(x))x∈S ∈ +PSL2(C)S, then the set of all PSL2(C)-representations of Γ is identified with a complex affine +algebraic subset R(Γ) of PSL2(C)S. +As in [5], we will also consider the PSL2(C)-character variety X(Γ) for an arbitrary finitely +generated group Γ. +This is the analogue of the SL2(C)-character variety of Γ that was +considered in [7], and we will review its definition and some of its useful properties here. +(We will need to use X(Γ) in this paper because Theorem 8.44 of [13], which is quoted in +the proof of Lemma 2.12, is stated in terms of X(Γ).) +There is a natural action (g, ρ) �→ g · ρ of PSL2(C) on R(Γ), where the representation g · ρ is +defined by setting g · ρ(x) = gρ(x)g−1 for each x ∈ Γ. This action will be referred to as the +action by conjugation, and its orbits will be called conjugacy classes of representations. In [5, +Section 3], X(Γ) is defined as the quotient, in the category of complex affine algebraic sets, of +R(Γ) by this action. There is a surjective morphism of affine algebraic sets t : R(Γ) → X(Γ) +which is constant on each conjugacy class of representations. For each ρ ∈ R(Γ), we call t(ρ) +the character of ρ. +(As in [5], the use of bars in R(Γ), X(Γ) and t, and the use of ρ as the default notation +for an element of R(Γ), are meant to emphasize that we are dealing with representations in +PSL2(C) rather than SL2(C). However, we have avoided using χρ to mean t(ρ), as is done in +[5], since the symbol χ figures prominently in the present paper with a different meaning.) +In the discussion at the beginning of Section 3 of [5], the authors quote [15] and [12] for the +precise definition and the basic properties of the quotient object X(Γ). The theory presented +in [15] applies to an arbitrary action (in the category of algebraic sets) of PSL2(C) on an +arbitrary algebraic set, whereas in [12] the emphasis is on the specific case of the action on +R(Γ) by conjugation, where Γ is a finitely generated group. (In both of these sources, the +role of PSL2(C) is played by a more general algebraic group, but we will implicitly specialize +to the case of PSL2(C) in the following discussion.) +According to the definition given on page 53 of [12], a point ρ of R(Γ) is stable if and only +if its conjugacy class is Zariski-closed and its stabilizer (isotopy subgroup) under the action +of PSL2(C) by conjugation is finite. +We will need the following properties of X(Γ) and t: + +EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS +8 +2.9.1. A point of ρ ∈ R(Γ) is stable if and only if ρ is an irreducible representation. +2.9.2. For each irreducible representation ρ ∈ R(Γ), the set t +−1(t(ρ)) is precisely the conju- +gacy class of the representation ρ. +2.9.3. Every Zariski-closed subset of R(Γ) which is invariant under the action of PSL2(C) +by conjugation is mapped by t onto a Zariski-closed subset of X(Γ). +2.9.4. The set of all characters of irreducible representations of Γ in PSL2(C) is a Zariski- +open subset of X(Γ). +Assertion 2.9.1 follows from [12, Theorem 1.1]. On p. 753 of [5], it is pointed out that +Assertion 2.9.2 follows from [15, Corollary 3.5.2] upon combining it with Assertion 2.9.1. In +the proof of [5, Lemma 4.1], it is pointed out that Assertion 2.9.3 follows from [15, Theorem +3.3.5(iv)]. +To prove 2.9.4, we first note that in view of 2.9.1 it is enough to prove that t maps the set S +of stable points of R(Γ) onto a Zariski-open subset of X(Γ). It is pointed out on page 54 of +[12], where it is deduced from Proposition 3.8 of [15], that S is Zariski-open in R(Γ). Hence +S′ = R(Γ)−S is Zariski-closed. Clearly S and S′ are invariant under the action of PSL2(C). +It follows from Assertion 2.9.2 that t(S) ∩ t(S′) = ∅; since t is surjective, it now follows that +t(S) = R(Γ) − t(S′). Since S′ is Zariski-closed in R(Γ), Assertion 2.9.3 implies that t(S′) is +Zariski-closed in X(Γ); hence t(S) is Zariski-closed in X(Γ), as required. +Lemma 2.10. Let Γ be a finitely generated group, and let ρ0 ∈ R(Γ) be an irreducible +representation of Γ in PSL2(C). Suppose that t(ρ0) lies in a unique component X0 of X(Γ), +and set d = dim X0. +Then ρ0 lies in a unique irreducible component R0 of R(Γ), and +dim R0 = d + 3. +Proof (cf. proof of Lemma 4.1 in [5]). Since X0 is an irreducible component of X(Γ) = +t(R(Γ)), there is an irreducible component R0 of R(Γ) such that t(R0) is a Zariski-dense sub- +set of X0. If we define a map h : R0 × PSL2(C) → R(Γ) by h(ρ, g) to be the representation +x → gρ(x)g−1, then since R0 × PSL2(C) is irreducible, h(R0 × PSL2(C)) must be contained +in a single component of R(Γ); but we have h(R0 ×PSL2(C)) ⊃ h(R0 ×{1} = R0, and hence +h(R0 × PSL2(C)) = R0. This shows that R0 is invariant under the action of PSL2(C) on +R(Γ) by conjugation. Since the irreducible component R0 of R(Γ) is also Zariski-closed in +R(Γ), It follows from 2.9.3 that t(R0) is a Zariski-closed subset of X(Γ). Hence t(R0) = X0. +Let X1 denote the union of all irreducible components of X(Γ) that are distinct from X0, +and set U1 = X(Γ) − X1. Then U1 is Zariski-open in X(Γ), and is contained in X0. On the +other hand, if U2 ⊂ X(Γ) denotes the set of all characters of irreducible representations of Γ +in PSL2(C), then U2 is Zariski-open in X(Γ) by 2.9.4. Hence U .= U1 ∩ U2 is Zariski-open in +X(Γ), and is contained in X0. +Set x0 = t(ρ0). Since by hypothesis ρ0 is an irreducible representation and X0 is the only +irreducible component of X(Γ) containing x0, we have x0 ∈ U. + +EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS +9 +Consider an arbitrary point x ∈ U. Since x ∈ U2, it follows from 2.9.2 that t +−1(x) is a single +conjugacy class of irreducible representations. Since x ∈ U1 ⊂ X0 = t(R0), and since R0 is +invariant under conjugation, it follows that t +−1(x) ⊂ R0. Hence we have t +−1(U) ⊂ R0. But +t +−1(U) is Zariski-open in R(Γ) since U is Zariski-open in X(Γ), and since t(ρ0) = x0 ∈ U, we +have ρ0 ∈ t +−1(U). Thus t +−1(U) is a Zariski neighborhood of ρ0 in R(Γ), and is contained in +R0; this shows that R0 is the only irreducible component of R(Γ) containing ρ0. It remains +to show that dim R0 = d + 3. +We have observed that for each x ∈ U, the fiber t +−1(x) is a single conjugacy class of irreducible +representations. If we fix a representation ρ ∈ t +−1(x), the algebraic set t +−1(x) is isomorphic +to the coset space PSL2(C)/C, where C denotes the stabilizer of ρ(Γ) under the action of +PSL2(C) by conjugation. Since ρ is a stable point by 2.9.1, it follows from the definition of +stability (see 2.9) that C is a finite group, and hence dim t +−1(x) = dim PSL2(C) = 3; that +is, all fibers of the map t|t +−1(U) : t +−1(U) → U are three-dimensional. Hence dim t +−1(U) = +3 + dim U. +But U is in particular a Zariski-open subset of X0, and is non-empty since +it contains x0; hence dim U = dim X0 = d. +and so dim t +−1(U) = d + 3. +Finally, since +t +−1(U) is in particular a Zariski-open subset of the irreducible variety R0, we have dim R0 = +dim t +−1(U) = d + 3. +□ +Definition and Remark 2.11. By a Kleinian group we mean a discrete subgroup of +PSL2(C). +If Γ is a finitely generated torsion-free Kleinian group, then Γ is isomorphic to the funda- +mental group of the orientable hyperbolic 3-manifold H3/Γ, and is therefore homologically +finite by 2.3. In particular, χ(Γ) is defined for every finitely generated torsion-free Kleinian +group Γ. +Lemma 2.12. Let Γ be a finitely generated, torsion-free group, and let ρ0 be a faithful +representation of Γ in PSL2(C) such that ρ0(Γ) is a Kleinian group (so that χ(Γ) is defined +by 2.11). Assume that Γ has no rank-2 cusp subgroups. Then the point ρ0 of R(Γ) lies in a +unique irreducible component R0 of R(Γ), and R0 has dimension 3χ(Γ) + 3. +Remark 2.13. It is likely that by using a little more geometric invariant theory, one can +prove that under the hypothesis of Lemma 2.12, the point ρ0 of R(Γ) is smooth. As we do +not need this stronger result, we have not attempted to include a proof of it. +Proof of Lemma 2.12. First consider the case in which the Kleinian group ρ0(Γ) ∼= Γ is +elementary. An elementary, torsion-free Kleinian group without rank-2 cusp subgroups is +either trivial or infinite cyclic. If Γ is trivial then χ(Γ) = −1 and R(Γ) is a single point; if +Γ is infinite cyclic, then χ(Γ) = 0 and the variety R(Γ) is isomorphic to PSL2(C), and is +therefore an irreducible complex affine variety of dimension 3. Thus the conclusions hold in +the elementary case. +Now suppose that Γ is non-elementary. According to [13, Theorem 8.44], t(ρ0) is a smooth +point of X(Γ), and hence lies in a unique irreducible component X0 of X(Γ); and furthermore, +we have dim X0 = 3χ(∂Mc)/2 = 3χ(Mc) = 3χ(Γ), where Mc denotes a compact core of M + +EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS +10 +(see Definition 2.1 and Proposition 2.2). (The quantity denoted τ in the statement of [13, +Theorem 8.44] is the number of conjugacy classes of rank-2 cusp subgroups of Γ, which +according to the hypothesis of the present lemma is equal to 0.) Furthermore, since ρ0(Γ) is +non-elementary, the representation ρ0 is irreducible. It now follows from Lemma 2.10 that +ρ0 lies in a unique component R0 of R(Γ), and that dim R0 = dim X0 + 3 = 3χ(Γ) + 3. +□ +3. Proofs of the main results +This section will include the proofs of Theorems A and B, which were stated in the Intro- +duction. +Lemma 3.1. Let Π be a finitely generated, torsion-free subgroup of a group Π′. Let T be +an element of Π′, and suppose that Π′ is generated by Π ∪ {T} (so that Π′ is in particular +finitely generated). Let ρ′ +0 be a representation of Π′ in PSL2(C) such that ρ0 +.= ρ′ +0|Π is a +faithful representation of Π. Suppose that ρ0 : Π → PSL2(C) admits a lift to PSL2(C). Let +V and V ′ be irreducible components of R(Π) and R(Π′) containing ρ0 and ρ′ +0 respectively, +and assume that V is the only irreducible component of R(Π) containing ρ0. Then either +(i) Π′ is the free product of the subgroups Π and ⟨T⟩, or +(ii) dim V ′ < (dim V ) + 3. +Proof. Let ⟨T0⟩ be an infinite cyclic group, and let Π′′ denote the abstract free product +Π ⋆ ⟨T0⟩. Since Π′ is generated by Π ∪ {T}, there is a unique surjective homomorphism +from h : Π′′ → Π′ which restricts to the inclusion homomorphism on Π and maps T0 to T. +Let W denote the subset of R(Π′′) consisting of all representations of Π′′ whose restrictions +to the kernel of h are trivial. Then W is a Zariski-closed subset of R(Π′′), and there is an +isomorphism of algebraic sets α : R(Π′) → W defined by α(ρ) = ρ ◦ h. There is also an +isomorphism of algebraic sets β : R(Π′′) → R(Π) × PSL2(C) defined by β(ρ) = (ρ|Π, ρ(T0)). +Thus β◦α is an isomorphism of R(Π′) onto the Zariski-closed subset β(W) of R(Π)×PSL2(C). +If we set E0 = ρ′ +0(T), we have β ◦ α(ρ′ +0) = (ρ′ +0|Π, ρ′ +0(T)) = (ρ0, E0). +Since V is the unique irreducible component of R(Π) containing ρ0, the unique irreducible +component of R(Π)×PSL2(C) containing (ρ0, E0) is V ×PSL2(C). Since V ′ is an irreducible +component of R(Π′) containing ρ′ +0, the set β ◦ α(V ′) is an irreducible component of β(W) ⊂ +R(Π) × PSL2(C) containing β ◦ α(ρ′ +0) = (ρ0, E0). It follows that β ◦ α(V ′) ⊂ V × PSL2(C). +Consider first the case in which β ◦ α(V ′) is a proper subset of V × PSL2(C). In this case, +since V × PSL2(C) is irreducible, we have dim V ′ = dim β ◦ α(V ′) < dim(V × PSL2(C)) = +(dim V ) + 3, so that Alternative (ii) of the conclusion of the lemma holds. +There remains the case in which β ◦ α(V ′) = V × PSL2(C). In particular we then have +{ρ0}×PSL2(C) ⊂ β◦α(V ′) ⊂ β(W). Thus if for every E ∈ PSL2(C) we set σE = β−1(ρ0, E), +we have σE ∈ W for every E ∈ PSL2(C). In view of the definition of W, it follows that: +3.1.1. ker σE ⊃ ker h for every E ∈ PSL2(C). +On the other hand, the definition of β implies: + +EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS +11 +3.1.2. For every E ∈ PSL2(C), the representation σE is the unique representation of Π′′ +which restricts to ρ0 on Π and maps T0 to E. +Now by hypothesis the representation ρ0 : Π → PSL2(C) admits a lift ρ0 : Π → SL2(C). +Since ρ0 is faithful, ρ0 is also faithful, and ρ0(Π) contains no non-trivial scalar matrix. +According to Lemma 2.8, there is an element L of infinite order in SL2(C) such that the +subgroup of SL2(C) generated by ρ0(Π) and L is a free product of ρ0(Π) with the infinite +cyclic group ⟨L⟩, and contains no non-trivial scalar matrix. If we denote by E1 the image +of L under the quotient map from SL2(C) to PSL2(C), it now follows that the subgroup of +PSL2(C) generated by ρ0(Π) and E1 is a free product of ρ0(Π) with the infinite cyclic group +⟨E1⟩. Applying 3.1.2 with E = E1, we deduce that σE1 is injective. But 3.1.1, applied with +E = E1, gives that ker h ⊂ ker σE1. Hence h is injective in this case, so that Alternative (ii) +of the conclusion of the lemma holds. +□ +Proposition 3.2. Let Γ be a finitely generated Kleinian group (so that χ(Γ) is defined by +2.11). Then χ(Γ) < miof(Γ). +Proof. We first consider the case in which Γ has no rank-2 cusp subgroups. +Set k = miof(π1(N)). By the definition of miof(Γ), there is a finite generating set ∆ for +Γ such that k = iof(∆). In particular, ∆ contains k independent elements. Hence we may +write ∆ = {x1, . . . , xm}, where m ≥ k and x1, . . . , xk are independent. +For r = k, . . . , m, let Πr denote the subgroup ⟨x1, . . . , xr⟩ of Γ. We claim that χ(Πr) < k +for r = k, . . . , m; the case r = m of this claim is the conclusion of the proposition, since +Πm = Γ. Note that Πk is free of rank k, and hence χ(Πk) = k − 1; thus the claim is true for +r = k. It therefore suffices to prove that χ(Πr+1) ≤ χ(Πr) whenever k ≤ r < m. +Since the Kleinian group Γ has no rank-2 cusp subgroup, its finitely generated subgroups Πr +are also Kleinian groups without rank-2 cusp subgroups. It therefore follows from Lemma +2.12 that, for any integer s with k ≤ s ≤ m, if we regard the inclusion homomorphism from +Πs ≤ Γ to PSL2(C) as a faithful representation of Πs in PSL2(C), then ρs lies in a unique +irreducible component Vs of R(Πs), and that +(3.2.1) +dim Vs = 3χ(Πs) + 3. +Suppose that r is an integer with k ≤ r < m. Since ρr is a discrete faithful representation, +it follows from [7, Proposition 3.1.1] that the representation ρr admits a lift to SL2(C). The +existence of such a lift, together with the fact that Vr is the only irreducible component of +R(Πr) containing ρr, allows us to apply Lemma 3.1, taking Π = Πr, Π′ = Πr+1, T = xr+1, +and letting ρr and ρr+1 play the respective roles of ρ0 and ρ′ +0. Lemma 3.1 gives that either +(i) Πr+1 is the free product of the subgroups Πr and ⟨xr+1⟩, or +(ii) dim Vr+1 < (dim Vr) + 3. +Thus for each r with k ≤ r < m, either (i) or (ii) holds. + +EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS +12 +If (i) holds for a given r, then since Πk ≤ Πr is generated by the independent elements +x1, . . . , xk, it follows that x1, . . . , xk+1 are independent. By the definition of index of freedom, +this means that iof(∆) ≥ k + 1, a contradiction to our choice of ∆. Hence for each r with +k ≤ r < m, Condition (ii) must hold. Combining (ii) with the cases s = r and s = r + 1 of +(3.2.1), we obtain 3χ(Πr+1) + 3 < 3χ(Πr) + 6, i.e. χ(Πr+1) < χ(Πr) + 1. Since χ(Πr+1) and +χ(Πr) are integers, this gives χ(Πr+1) ≤ χ(Πr), as required. This completes the proof of the +proposition in the case in which Γ has no rank-2 cusp subgroups. +We now turn to the general case. Let Γ be an arbitrary finitely generated Kleinian group. +According to Lemma 2.5, there exist a finitely generated Kleinian group Γ0 with no rank-2 +cusp subgroups, with χ(Γ0) = χ(Γ), and a surjective homomorphism η : Γ → Γ0. Applying +Lemma 2.7, with G1 = Γ and G1 = Γ0, and defining η as above, we deduce that miof(Γ0) ≤ +miof(Γ). +Since Γ0 has no rank-2 cusp subgroups, we may apply the special case of the +proposition that has already been proved to deduce that χ(Γ0) < miof(Γ0). Thus we have +χ(Γ) = χ(Γ0) < miof(Γ0) ≤ miof(Γ), and the proof is complete. +□ +We now prove Theorem B, which was stated in the introduction to this paper. +Proof of Theorem B. Since the finitely generated group G is a subgroup of the fundamental +group of an orientable hyperbolic 3-manifold, it is isomorphic to a Kleinian group. Thus the +theorem follows immediately from Proposition 3.2. +□ +We are now in a position to prove Theorem A, which was stated in the introduction. +Proof of Theorem A. By Theorem B we have χ(G) ≤ miof(G) − 1. It therefore suffices to +show that miof(G) ≤ k − 1. +Assume that miof(G) ≥ k. Set ∆ = {[α1], . . . , [αm]}. Since ∆ is a generating set for G, +we have iof(∆) ≥ miof(G) ≥ k. By definition this means that ∆ contains k independent +elements. Thus we have m ≥ k, and after possibly re-indexing the αi we may assume that +[α1], . . . , [αk] are independent. +Let us write M = H3/Γ, where Γ is a torsion-free Kleinian group, let q : H3 → M denote +the quotient map, and let us fix a point z ∈ q−1(p). Then z defines an isomorphism j : +π1(M, p) → Γ. +Set ξi = j([αi]) for i = 1, . . . , k. +Since [α1], . . . , [αk] are independent, +ξ1, . . . , ξk freely generate a free subgroup of Γ, which may be regarded as a Kleinian group. +If we set di = dist(z, ξi · z) for i = 1, . . . , k, Theorem 4.1 of [2] now asserts that +k +� +i=1 +1 +1 + edi ≤ 1 +2, +so that in particular di ≥ log(2k − 1) for some i ∈ {1, . . . , k}. But for each i ∈ {1, . . . , k} we +have di ≤ length αi, which with the hypothesis of the theorem gives di < log(2k − 1). This +contradiction completes the proof. +□ + +EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS +13 +References +[1] Ian Agol. Tameness of hyperbolic 3-manifolds. arXiv:math.GT/0405568. +[2] Ian Agol, Marc Culler, and Peter B. Shalen. Dehn surgery, homology and hyperbolic volume. Algebr. +Geom. Topol., 6:2297–2312, 2006. +[3] James W. Anderson, Richard D. Canary, Marc Culler, and Peter B. Shalen. Free Kleinian groups and +volumes of hyperbolic 3-manifolds. J. Differential Geom., 43(4):738–782, 1996. +[4] Gilbert Baumslag and Peter B. Shalen. Groups whose three-generator subgroups are free. Bull. Austral. +Math. Soc., 40(2):163–174, 1989. +[5] S. Boyer and X. Zhang. On Culler-Shalen seminorms and Dehn filling. Ann. of Math. (2), 148(3):737– +801, 1998. +[6] Danny Calegari and David Gabai. Shrinkwrapping and the taming of hyperbolic 3-manifolds. J. Amer. +Math. Soc., 19(2):385–446 (electronic), 2006. +[7] Marc Culler and Peter B. Shalen. Varieties of group representations and splittings of 3-manifolds. Ann. +of Math. (2), 117(1):109–146, 1983. +[8] David Futer, Jessica S. Purcell, and Saul Schleimer. Effective drilling and filling of tame hyperbolic +3-manifolds. Comment. Math. Helv., 97(3):457–512, 2022. +[9] Rosemary K. Guzman and Peter B. Shalen. The ratio of homology rank to hyperbolic volume, I. Journal +of Topology and Analysis, To appear. arXiv:2207.00040. +[10] Rosemary K. Guzman and Peter B. Shalen. The ratio of homology rank to hyperbolic volume, II: The +role of the Four Color Theorem. Journal of Topology and Analysis, To appear. arXiv:2110.14847. +[11] William H. Jaco and Peter B. Shalen. Seifert fibered spaces in 3-manifolds. Mem. Amer. Math. Soc., +21(220):viii+192, 1979. +[12] Dennis Johnson and John J. Millson. Deformation spaces associated to compact hyperbolic manifolds. +In Discrete groups in geometry and analysis (New Haven, Conn., 1984), volume 67 of Progr. Math., +pages 48–106. Birkh¨auser Boston, Boston, MA, 1987. +[13] Michael Kapovich. Hyperbolic manifolds and discrete groups. Modern Birkh¨auser Classics. Birkh¨auser +Boston, Ltd., Boston, MA, 2009. Reprint of the 2001 edition. +[14] Hossein Namazi and Juan Souto. Non-realizability and ending laminations: proof of the density conjec- +ture. Acta Math., 209(2):323–395, 2012. +[15] P. E. Newstead. Introduction to moduli problems and orbit spaces, volume 51 of Tata Institute of Fun- +damental Research Lectures on Mathematics and Physics. Tata Institute of Fundamental Research, +Bombay; Narosa Publishing House, New Delhi, 1978. +[16] V. L. Nisnewitsch. ¨Uber Gruppen, die durch Matrizen ¨uber einem kommutativen Feld isomorph darstell- +bar sind. Rec. Math. [Mat. Sbornik] N.S., 8 (50):395–403, 1940. +[17] Ken’ichi Ohshika. Realising end invariants by limits of minimally parabolic, geometrically finite groups. +Geom. Topol., 15(2):827–890, 2011. +[18] G. P. Scott. Compact submanifolds of 3-manifolds. J. London Math. Soc. (2), 7:246–250, 1973. +[19] Peter B. Shalen. Linear representations of certain amalgamated products. J. Pure Appl. Algebra, +15(2):187–197, 1979. +[20] Peter B. Shalen. Small optimal Margulis numbers force upper volume bounds. Trans. Amer. Math. Soc., +365(2):973–999, 2013. +[21] B. A. F. Wehrfritz. Generalized free products of linear groups. Proc. London Math. Soc. (3), 27:402–424, +1973. +Department of Mathematics, Statistics, and Computer Science (M/C 249), University of +Illinois at Chicago, 851 S. Morgan St., Chicago, IL 60607-7045 +Email address: shalen@math.uic.edu + diff --git a/FtE4T4oBgHgl3EQffw3Q/content/tmp_files/load_file.txt b/FtE4T4oBgHgl3EQffw3Q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4d58e10fe0a80c3b2c662c17e62c65cbce732634 --- /dev/null +++ b/FtE4T4oBgHgl3EQffw3Q/content/tmp_files/load_file.txt @@ -0,0 +1,748 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf,len=747 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='05111v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='GT] 12 Jan 2023 EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS IN HYPERBOLIC 3-MANIFOLDS PETER B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' SHALEN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let p be a point of an orientable hyperbolic 3-manifold M, and let m ≥ 1 and k ≥ 2 be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Suppose that α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , αm are loops based at p having length less than log(2k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We show that if G denotes the subgroup of π1(M, p) generated by [α1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , [αm], then χ(G) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='= −χ(G) ≤ k − 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' here χ(G) denotes the Euler characteristic of the group G, which is always defined in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' This result is deduced from a result about an arbitrary finitely generated subgroup G of the fundamental group of an orientable hyperbolic 3-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' If ∆ is a finite generating set for G, we define the index of freedom iof(∆) to be the largest integer k such that ∆ contains k elements that freely generate a rank-k free subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We define the minimum index of freedom miof(G) to be min∆ iof(∆), where ∆ ranges over all finite generating sets for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The result is that χ(G) < miof(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Introduction Definitions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We shall say that a space X is homologically finite if the abelian group �∞ i=1 Hi(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Z) is finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' A homologically finite space has a well-defined Euler characteristic, which we will denote by χ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We will write χ(X) = −χ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' A group Π is said to be homologically finite if K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='= K(Π, 1) is homologically finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' In this case we write χ(Π) = χ(K) and χ(Π) = χ(K) = −χ(Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' It is a well-known fact, which will be explained in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='3 below, that if G is a finitely generated subgroup of the fundamental group of a hyperbolic 3-manifold, then G is homologically finite, so that χ(G) is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' One of the main theorems of this paper is: Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let M be an orientable hyperbolic 3-manifold, let p be a point of M, and let m ≥ 1 and k ≥ 2 be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Suppose that α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , αm are loops based at p having length less than log(2k − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let G denote the subgroup of π1(M, p) generated by [α1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , [αm] (so that χ(G) is defined by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Then χ(G) ≤ k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' This theorem will be used in a forthcoming paper with Rosemary Guzman to obtain a result that gives a new bound on the ratio of the dimension of the mod 2 homology of a closed, orientable 3-manifold to the volume;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' this result is strictly stronger than the one established in [9], and is essentially stronger than the one established in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' 1 EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS 2 The proof of Theorem A depends on our other main result, Theorem B below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Theorem B gives a purely group-theoretical property of Kleinian groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' the statement of Theorem B involves a few other concepts which we now define.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Definitions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let G be a finitely generated group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We will say that elements x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , xk of G are independent if x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , xk freely generate a free subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' If ∆ is a finite generating set for G, I will define the index of freedom of ∆, denoted iof(∆), to be the largest integer k such that ∆ contains k independent elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We define the minimal index of freedom of G, denoted miof(G), by miof(G) = min∆ iof(∆), where ∆ ranges over all finite generating sets for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let M be an orientable hyperbolic 3-manifold, and let G be a finitely generated subgroup of π1(M) (so that χ(G) is defined by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Then χ(G) < miof(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Under the hypothesis of Theorem B, the group G is isomorphic to a Kleinian group, and the proof of the theorem immediately reduces to a statement about a Kleinian group Γ, which is Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Using a Dehn surgery trick, one can reduce the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2 to the case in which Γ has no rank-2 cusp subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The essential step in the proof of this case of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2 is to show that if x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , xm are generators for the group Γ, if we set Πr = ⟨x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , xr⟩ ≤ Γ for r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , m, and if for a given r ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , m − 1} we have χ(Πr+1) > χ(Πr), then xr+1 has infinite order in Γ, and Πr+1 is the free product of its subgroup Πr with its infinite cyclic subgroup ⟨xr+1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' This is proved by relating the quantities χ(Πr) for 1 ≤ r ≤ m to the dimensions of certain varieties of representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' For each r the representations of Πr in PSL2(C) are identified with the points of a complex affine algebraic set, which for the purposes of this sketch will be denoted Rr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The inclusion homomorphism Πr → PSL2(C), regarded as a representation, lies in a unique irreducible component Vr of Rr, and has complex dimension 3χ(Πr) + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' If for a given r < m the group Πr+1 is not the free product of Πr with ⟨xr+1⟩ (or if xr+1 has finite order), it can be shown that Vr+1 is isomorphic to a proper subvariety of Vr × PSL2(C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' this implies that dim Vr+1 < 3 + dim Vr, and hence that χ(Πr+1) ≤ χ(Πr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' This completes the sketch of the proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' (Parts of the argument sketched here are embodied in the statement and proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=') The idea of using dimensions of representation varieties in this argument was suggested by Ian Agol’s observation that one can use dimensions of representation varieties to give alternative proofs of [11, Theorem VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1], and its generalizations [4, Appendix, Theorem A] and [3, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1], using dimensions of representation varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Theorem A is proved by combining Theorem B with the so-called log(2k − 1) Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The latter result, which gives information about the lengths of loops α1, · · · , αk based at a point p of an orientable hyperbolic 3-manifold M under the assumption that the elements [α1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , [αk] freely generate a free subgroup of π1(M, p), was proved under an additional hypothesis in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The general version of the log(2k − 1) Theorem is proved by combining EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS 3 the results of [3] with the tameness theorem [1], [6] and the density theorem [17], [14] for Kleinian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The general version of the log(2k − 1) Theorem is presented as Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1 of [2], which we will quote in the proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The proofs of Theorems B and A will be given in Section 3, after needed background about representations, representaion varieties, character varieties, and deformation spaces of Kleinian groups, which occupies Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We will use the following standard conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' A 3-manifold M is said to be irreducible if M is connected and every (tame) 2-sphere in M bounds a 3-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' To write Y ≤ X, where X is a group, means that Y is a subgroup of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' If A is a subset of a group, ⟨A⟩ denotes the subgroup generated by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' I am grateful to Steve Boyer, Ken Bromberg and Dick Canary for their assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Boyer, with tremendous patience, helped me navigate the material that is summarized in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Bromberg and Canary helped me locate the result from [8] which is used in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Canary pointed out Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='44 of [13], which is used in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Preliminaries Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' A compact core of a 3-manifold M is a compact three-dimensional subman- ifold N of M such that the inclusion N → M is a homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Although the following result is well known, we include a proof for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Every orientable hyperbolic 3-manifold with finitely generated fundamental group has a compact core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' If the given manifold M is isometric to H3, any ball in M is a compact core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' If M is not isometric to H3, we observe that since M admits H3 as a non-trivial covering space, M is irreducible and has infinite fundamental group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We then apply [20, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='8], a consequence of the main result of [18], which asserts that if M is an irreducible, orientable 3-manifold with finitely generated fundamental group, then there is a compact, irreducible submanifold N of M such that the inclusion homomorphism π1(N) → π1(M) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' (The hypothesis of finite generation was unfortunately left out of the statement of [20, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='8], but the proof given there establishes the statement given here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=') In particular π1(N) is infinite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' this, together with the irreducibility of N, implies that N is aspherical (see [20, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' As the hyperbolic manifold M is also aspherical, it now follows that the inclusion N → M is a homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let G be a finitely generated subgroup of the fundamental group of an orientable hyperbolic 3-manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The covering space � M of M corresponding to the subgroup G has a compact core N by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' in particular N is aspherical and π1(N) ∼= G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since N is compact, it follows that G is homologically finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS 4 This establishes the fact, which was mentioned in the introduction and is necessary back- ground for Theorems A and B, that a finitely generated subgroup G of the fundamental group of an orientable hyperbolic 3-manifold is homologically finite, so that χ(G) is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' If M is an orientable hyperbolic 3-manifold with finitely generated Kleinian group, there exist an orientable hyperbolic 3-manifold M0 with no rank-2 cusps such that χ(M0) = χ(M), and a surjective homomorphism from π1(M) to π1(M0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' (Here χ(M) and χ(M0) are defined in view of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' It follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2 that dim H2(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Q) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The number of ends of the manifold M is at most 1 + dim H2(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Q), and is in particular finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We shall prove the result by induction on the number of ends of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' If the number of ends of M is 0, then M is compact and therefore has no cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The assertion is trivial in this case, since we can take M0 = M and choose the identity map as required homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Now suppose that the number of ends of M is m > 0, and that the assertion is true for manifolds with m − 1 ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' If M has no rank-2 cusps, the assertion is again trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Now suppose that M has at least one rank-2 cusp, and fix a submanifold H of M which is a standard neighborhood of a rank-2 cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The 2-manifold ∂H is a torus, and inherits a Euclidean metric from the hyperbolic metric on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' For each homotopically non-trivial simple closed curve s in ∂H, we denote by Z(s) the orientable 3-manifold, defined up to homeomorphism and dependent only on the isotopy class of s, which is obtained from the disjoint union M − H ⊨ (D2×S1) by gluing ∂(M − H) = ∂H to ∂(D2 × S1) = (∂D2) × S1 by a homeomorphism which maps s to (∂D2) × {pt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' (For any s, the manifold Z(s) is said to be obtained from M − H by Dehn filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=') For any closed geodesic s in ∂H, since M − H is homotopy equivalent to M, we have χ(Z(s) = χ(M − H) − χ(∂H) + χ(D2 × S1) = χ(M) − 0 + 0 = χ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Furthermore, the inclusion homomorphism π1(M − H) → π1(Z(s)) is surjective, and hence there is a surjective homomorphism from π1(M) to π1(Z(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We now apply the case k = 1 of [8, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='4], which asserts that if the closed geodesic in ∂H representing the isotopy class of s has length strictly greater than 6, then Z(s) admits a hyperbolic metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since there are only finitely many isotopy classes of geodesics in ∂H whose lengths are bounded above by a given constant, we may fix a simple closed curve s1 in ∂H such that Z(s1) is homeomorphic to a hyperbolic 3-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The manifold Z(s1) has m − 1 ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Hence, by the induction hypothesis, there exist an orientable hyperbolic 3-manifold M0 with no rank-2 cusps such that χ(M0) = χ(Z(s1)), and a surjective homomorphism from π1(Z(s1)) to π1(M0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since χ(Z(s1)) = χ(M), we have χ(M0) = χ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Furthermore, since there is a surjective homomorphism from π1(M) to π1(Z(s1)), we obtain by composition a surjective homomorphism from π1(M) to π1(M0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' This completes the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' If Γ is a finitely generated Kleinian group, there exist a finitely generated Kleinian group Γ0 with no rank-2 cusp subgroups, with χ(Γ0) = χ(Γ), and a surjective homomorphism η : Γ → Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' (Here χ(Γ) and χ(Γ0) are defined by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=') EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='4 to the orientable hyperbolic 3-manifold M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='= H3/Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The latter lemma gives an orientable hyperbolic 3-manifold M0 with no rank-2 cusps such that χ(M0) = χ(M), and a surjective homomorphism from π1(M) to π1(M0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We may write M0 = H3/Γ0 where Γ is a Kleinian group without rank-2 cusp subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We have Γ ∼= π1(M) and Γ0 ∼= π1(M0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' and since the hyperbolic 3-manifolds M and M0 are aspherical, we have χ(Γ0) = χ(M0) = χ(M) = χ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' □ The next two subsections involve the notion of index of freedom and minimal index of freedom, which were defined in the introduction to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Note that if a finitely generated group G is non-trivial and torsion-free, then any gen- erating set for G contains an element of infinite order, and hence miof(G) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let G1 and G2 be groups, and suppose that there is a surjective homomorphism η : G1 → G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Then miof(G2) ≤ miof(G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Furthermore, for any generating set ∆ for G1, we have iof(η(∆)) ≤ iof(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We prove the second assertion first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Write ∆ = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , xm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Set k = iof(η(∆)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Then after reindexing the xi we may assume that η(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , η(xk) are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' It follows that x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , xk are independent, and hence that iof(∆) ≥ k, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' To prove the first assertion, note that if ∆ is an arbitrary generating set for G1, the first assertion gives iof(∆) ≥ iof(η(∆) ≥ miof(G2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' hence miof(G1) ≥ miof(G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' □ The following lemma is a very special case of a result that is implicit in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The latter result has been refined in [21] and elsewhere, and was rediscovered in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' As an explicit statement of the lemma is not easy to find, we have provided a simple, self-contained proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let Γ be a finitely generated subgroup of GL2(C) containing no non-trivial scalar matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Then there is an element L of infinite order in SL2(C) such that the subgroup ⟨L, Γ⟩ of GL2(C)) is a free product of Γ with the infinite cyclic group ⟨L⟩, and contains no non-trivial scalar matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since Γ contains no non-trivial scalar matrix, each non-trivial element of Γ has one or two eigenvectors in C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since Γ is countable and C2 is not, there is a non-zero vector in C2 which is not an eigenvector of any element of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Hence after modifying Γ by a conjugation, we may assume that Γ contains no upper triangular matrix except the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let F denote the countable subfield of C generated by the entries of the elements of Γ, and fix an element t of C which is transcendental over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We shall show that the conclusion of the lemma holds if we set L = �1 t 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' This is equivalent to the assertion that if F(X) denotes the rational function field in one indeterminate over X, and if we set Λ = �1 X 0 1 � , then the subgroup of GL2(F[X]) ≤ GL2(F(X)) generated by Γ and Λ is a free product of Γ with the infinite cyclic group ⟨Λ⟩, and contains no non-trivial scalar matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS 6 We consider the abstract free product Γ ⋆ ⟨Λ⟩, and the homomorphism h : Γ ⋆ ⟨Λ⟩ → GL2(F[X]) which is restricts to the inclusion on each factor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' we must show that h does not map any non-trivial element of Γ ⋆ ⟨Λ⟩ to a scalar matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Any non-trivial element of Γ ⋆ ⟨Λ⟩ is conjugate either to an element of a factor or to an element of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1) w = γ1Λm1 · γkΛmk, where k ≥ 1, the elements γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , γk of Γ are non-trivial, and m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , mk are non-zero integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' By hypothesis Γ contains no non-trivial scalar matrix, and it is clear from the definition of Λ that ⟨Λ⟩ also contains no non-trivial scalar matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' It therefore suffices to prove that an element w of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1), where k ≥ 1 and the γi and mi satisfy the conditions stated above, cannot be mapped by h onto a scalar matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' By induction on k ≥ 1 we shall establish the following stronger assertion: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' If k is a strictly positive integer, and w ∈ Γ ⋆ ⟨Λ⟩ is given by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1, where k ≥ 1, the elements γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , γk of Γ are non-trivial, and m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , mk are non-zero integers, then h(w) has the form �A B C D � ∈ GL2(F[X]), where A and C are polynomials of degree at most k − 1, while B is a polynomial of degree at most k and D is a polynomial of degree exactly k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' (In particular we have A ̸= D, so that h(w) cannot be a scalar matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=') In the statement of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2, the polynomial 0 is understood to have degree −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' In the case k = 1 of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2, we have w = γΛm, where m is a non-zero integer and γ = �a b c d � is a non-trivial element of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' This gives h(w) = �a b c d � �1 mX 0 1 � = �a amX + b c cmX + d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Thus the entries in the first column of h(w) are elements of F, while the upper right-hand entry is a polynomial of degree at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Furthermore, since Γ contains no upper triangular matrix except the identity, we have c ̸= 0, and hence the lower right-hand entry of h(w) has degree exactly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' This establishes the base case for the inductive proof of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' For the induction step, suppose that w = γ1Λm1 · · · γkΛmk satisfies the hypothesis of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2 for a given k > 1, set w∗ = γ1Λm1 · · ·γk−1Λmk−1, and assume that h(w∗) = �A∗ B∗ C∗ D∗ � , where A∗ and C∗ have degree at most k − 2, while B∗ has degree at most k − 1, and D∗ has degree exactly k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' As the base case has been established, we may write h(γkΛmk) = �A† B† C† D† � , where A† and C† have degree at most 0, while B† has degree at most 1, and D† has degree exactly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Then h(w) = h(w∗)h(γkΛmk) = �A∗A† + B∗C† A∗B† + B∗D† C∗A† + D∗C† C∗B† + D∗D† � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' It now follows that the entries in the first column of h(w) have degree at most k−1, while its upper right-hand entry has degree at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' In the lower right-hand entry C∗B† + D∗D†, EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS 7 the first term has degree at most k − 1 and the second term has degree exactly k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' hence this lower right-hand entry has degree exactly k, and the induction is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The rest of this section will involve the use of representation varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We will take the point of view used in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' By a representation ρ of a group Γ in PSL2(C), we mean simply a homomorphism from Γ to PSL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' A representation of Γ is said to be reducible if there is a 1-dimensional subspace of C2 which is invariant under ρ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Otherwise it is said to be irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' In [5, Section 3], it is shown how to identify PSL2(C) with an algebraic set in some complex affine space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' and it is pointed out that, if Γ is a group with a given finite generating set S, and if one identifies an arbitrary representation ρ of Γ in PSL2(C) with the point (ρ(x))x∈S ∈ PSL2(C)S, then the set of all PSL2(C)-representations of Γ is identified with a complex affine algebraic subset R(Γ) of PSL2(C)S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' As in [5], we will also consider the PSL2(C)-character variety X(Γ) for an arbitrary finitely generated group Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' This is the analogue of the SL2(C)-character variety of Γ that was considered in [7], and we will review its definition and some of its useful properties here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' (We will need to use X(Γ) in this paper because Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='44 of [13], which is quoted in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='12, is stated in terms of X(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=') There is a natural action (g, ρ) �→ g · ρ of PSL2(C) on R(Γ), where the representation g · ρ is defined by setting g · ρ(x) = gρ(x)g−1 for each x ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' This action will be referred to as the action by conjugation, and its orbits will be called conjugacy classes of representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' In [5, Section 3], X(Γ) is defined as the quotient, in the category of complex affine algebraic sets, of R(Γ) by this action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' There is a surjective morphism of affine algebraic sets t : R(Γ) → X(Γ) which is constant on each conjugacy class of representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' For each ρ ∈ R(Γ), we call t(ρ) the character of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' (As in [5], the use of bars in R(Γ), X(Γ) and t, and the use of ρ as the default notation for an element of R(Γ), are meant to emphasize that we are dealing with representations in PSL2(C) rather than SL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' However, we have avoided using χρ to mean t(ρ), as is done in [5], since the symbol χ figures prominently in the present paper with a different meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=') In the discussion at the beginning of Section 3 of [5], the authors quote [15] and [12] for the precise definition and the basic properties of the quotient object X(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The theory presented in [15] applies to an arbitrary action (in the category of algebraic sets) of PSL2(C) on an arbitrary algebraic set, whereas in [12] the emphasis is on the specific case of the action on R(Γ) by conjugation, where Γ is a finitely generated group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' (In both of these sources, the role of PSL2(C) is played by a more general algebraic group, but we will implicitly specialize to the case of PSL2(C) in the following discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=') According to the definition given on page 53 of [12], a point ρ of R(Γ) is stable if and only if its conjugacy class is Zariski-closed and its stabilizer (isotopy subgroup) under the action of PSL2(C) by conjugation is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We will need the following properties of X(Γ) and t: EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' A point of ρ ∈ R(Γ) is stable if and only if ρ is an irreducible representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' For each irreducible representation ρ ∈ R(Γ), the set t −1(t(ρ)) is precisely the conju- gacy class of the representation ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Every Zariski-closed subset of R(Γ) which is invariant under the action of PSL2(C) by conjugation is mapped by t onto a Zariski-closed subset of X(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The set of all characters of irreducible representations of Γ in PSL2(C) is a Zariski- open subset of X(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Assertion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1 follows from [12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' On p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' 753 of [5], it is pointed out that Assertion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2 follows from [15, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2] upon combining it with Assertion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' In the proof of [5, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1], it is pointed out that Assertion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='3 follows from [15, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='5(iv)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' To prove 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='4, we first note that in view of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1 it is enough to prove that t maps the set S of stable points of R(Γ) onto a Zariski-open subset of X(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' It is pointed out on page 54 of [12], where it is deduced from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='8 of [15], that S is Zariski-open in R(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Hence S′ = R(Γ)−S is Zariski-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Clearly S and S′ are invariant under the action of PSL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' It follows from Assertion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2 that t(S) ∩ t(S′) = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' since t is surjective, it now follows that t(S) = R(Γ) − t(S′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since S′ is Zariski-closed in R(Γ), Assertion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='3 implies that t(S′) is Zariski-closed in X(Γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' hence t(S) is Zariski-closed in X(Γ), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let Γ be a finitely generated group, and let ρ0 ∈ R(Γ) be an irreducible representation of Γ in PSL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Suppose that t(ρ0) lies in a unique component X0 of X(Γ), and set d = dim X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Then ρ0 lies in a unique irreducible component R0 of R(Γ), and dim R0 = d + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Proof (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1 in [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since X0 is an irreducible component of X(Γ) = t(R(Γ)), there is an irreducible component R0 of R(Γ) such that t(R0) is a Zariski-dense sub- set of X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' If we define a map h : R0 × PSL2(C) → R(Γ) by h(ρ, g) to be the representation x → gρ(x)g−1, then since R0 × PSL2(C) is irreducible, h(R0 × PSL2(C)) must be contained in a single component of R(Γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' but we have h(R0 ×PSL2(C)) ⊃ h(R0 ×{1} = R0, and hence h(R0 × PSL2(C)) = R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' This shows that R0 is invariant under the action of PSL2(C) on R(Γ) by conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since the irreducible component R0 of R(Γ) is also Zariski-closed in R(Γ), It follows from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='3 that t(R0) is a Zariski-closed subset of X(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Hence t(R0) = X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let X1 denote the union of all irreducible components of X(Γ) that are distinct from X0, and set U1 = X(Γ) − X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Then U1 is Zariski-open in X(Γ), and is contained in X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' On the other hand, if U2 ⊂ X(Γ) denotes the set of all characters of irreducible representations of Γ in PSL2(C), then U2 is Zariski-open in X(Γ) by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Hence U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='= U1 ∩ U2 is Zariski-open in X(Γ), and is contained in X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Set x0 = t(ρ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since by hypothesis ρ0 is an irreducible representation and X0 is the only irreducible component of X(Γ) containing x0, we have x0 ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS 9 Consider an arbitrary point x ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since x ∈ U2, it follows from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2 that t −1(x) is a single conjugacy class of irreducible representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since x ∈ U1 ⊂ X0 = t(R0), and since R0 is invariant under conjugation, it follows that t −1(x) ⊂ R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Hence we have t −1(U) ⊂ R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' But t −1(U) is Zariski-open in R(Γ) since U is Zariski-open in X(Γ), and since t(ρ0) = x0 ∈ U, we have ρ0 ∈ t −1(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Thus t −1(U) is a Zariski neighborhood of ρ0 in R(Γ), and is contained in R0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' this shows that R0 is the only irreducible component of R(Γ) containing ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' It remains to show that dim R0 = d + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We have observed that for each x ∈ U, the fiber t −1(x) is a single conjugacy class of irreducible representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' If we fix a representation ρ ∈ t −1(x), the algebraic set t −1(x) is isomorphic to the coset space PSL2(C)/C, where C denotes the stabilizer of ρ(Γ) under the action of PSL2(C) by conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since ρ is a stable point by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1, it follows from the definition of stability (see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='9) that C is a finite group, and hence dim t −1(x) = dim PSL2(C) = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' that is, all fibers of the map t|t −1(U) : t −1(U) → U are three-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Hence dim t −1(U) = 3 + dim U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' But U is in particular a Zariski-open subset of X0, and is non-empty since it contains x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' hence dim U = dim X0 = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' and so dim t −1(U) = d + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Finally, since t −1(U) is in particular a Zariski-open subset of the irreducible variety R0, we have dim R0 = dim t −1(U) = d + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' □ Definition and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' By a Kleinian group we mean a discrete subgroup of PSL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' If Γ is a finitely generated torsion-free Kleinian group, then Γ is isomorphic to the funda- mental group of the orientable hyperbolic 3-manifold H3/Γ, and is therefore homologically finite by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' In particular, χ(Γ) is defined for every finitely generated torsion-free Kleinian group Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let Γ be a finitely generated, torsion-free group, and let ρ0 be a faithful representation of Γ in PSL2(C) such that ρ0(Γ) is a Kleinian group (so that χ(Γ) is defined by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Assume that Γ has no rank-2 cusp subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Then the point ρ0 of R(Γ) lies in a unique irreducible component R0 of R(Γ), and R0 has dimension 3χ(Γ) + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' It is likely that by using a little more geometric invariant theory, one can prove that under the hypothesis of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='12, the point ρ0 of R(Γ) is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' As we do not need this stronger result, we have not attempted to include a proof of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' First consider the case in which the Kleinian group ρ0(Γ) ∼= Γ is elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' An elementary, torsion-free Kleinian group without rank-2 cusp subgroups is either trivial or infinite cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' If Γ is trivial then χ(Γ) = −1 and R(Γ) is a single point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' if Γ is infinite cyclic, then χ(Γ) = 0 and the variety R(Γ) is isomorphic to PSL2(C), and is therefore an irreducible complex affine variety of dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Thus the conclusions hold in the elementary case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Now suppose that Γ is non-elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' According to [13, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='44], t(ρ0) is a smooth point of X(Γ), and hence lies in a unique irreducible component X0 of X(Γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' and furthermore, we have dim X0 = 3χ(∂Mc)/2 = 3χ(Mc) = 3χ(Γ), where Mc denotes a compact core of M EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS 10 (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' (The quantity denoted τ in the statement of [13, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='44] is the number of conjugacy classes of rank-2 cusp subgroups of Γ, which according to the hypothesis of the present lemma is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=') Furthermore, since ρ0(Γ) is non-elementary, the representation ρ0 is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' It now follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='10 that ρ0 lies in a unique component R0 of R(Γ), and that dim R0 = dim X0 + 3 = 3χ(Γ) + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Proofs of the main results This section will include the proofs of Theorems A and B, which were stated in the Intro- duction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let Π be a finitely generated, torsion-free subgroup of a group Π′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let T be an element of Π′, and suppose that Π′ is generated by Π ∪ {T} (so that Π′ is in particular finitely generated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let ρ′ 0 be a representation of Π′ in PSL2(C) such that ρ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='= ρ′ 0|Π is a faithful representation of Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Suppose that ρ0 : Π → PSL2(C) admits a lift to PSL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let V and V ′ be irreducible components of R(Π) and R(Π′) containing ρ0 and ρ′ 0 respectively, and assume that V is the only irreducible component of R(Π) containing ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Then either (i) Π′ is the free product of the subgroups Π and ⟨T⟩, or (ii) dim V ′ < (dim V ) + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let ⟨T0⟩ be an infinite cyclic group, and let Π′′ denote the abstract free product Π ⋆ ⟨T0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since Π′ is generated by Π ∪ {T}, there is a unique surjective homomorphism from h : Π′′ → Π′ which restricts to the inclusion homomorphism on Π and maps T0 to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let W denote the subset of R(Π′′) consisting of all representations of Π′′ whose restrictions to the kernel of h are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Then W is a Zariski-closed subset of R(Π′′), and there is an isomorphism of algebraic sets α : R(Π′) → W defined by α(ρ) = ρ ◦ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' There is also an isomorphism of algebraic sets β : R(Π′′) → R(Π) × PSL2(C) defined by β(ρ) = (ρ|Π, ρ(T0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Thus β◦α is an isomorphism of R(Π′) onto the Zariski-closed subset β(W) of R(Π)×PSL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' If we set E0 = ρ′ 0(T), we have β ◦ α(ρ′ 0) = (ρ′ 0|Π, ρ′ 0(T)) = (ρ0, E0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since V is the unique irreducible component of R(Π) containing ρ0, the unique irreducible component of R(Π)×PSL2(C) containing (ρ0, E0) is V ×PSL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since V ′ is an irreducible component of R(Π′) containing ρ′ 0, the set β ◦ α(V ′) is an irreducible component of β(W) ⊂ R(Π) × PSL2(C) containing β ◦ α(ρ′ 0) = (ρ0, E0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' It follows that β ◦ α(V ′) ⊂ V × PSL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Consider first the case in which β ◦ α(V ′) is a proper subset of V × PSL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' In this case, since V × PSL2(C) is irreducible, we have dim V ′ = dim β ◦ α(V ′) < dim(V × PSL2(C)) = (dim V ) + 3, so that Alternative (ii) of the conclusion of the lemma holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' There remains the case in which β ◦ α(V ′) = V × PSL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' In particular we then have {ρ0}×PSL2(C) ⊂ β◦α(V ′) ⊂ β(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Thus if for every E ∈ PSL2(C) we set σE = β−1(ρ0, E), we have σE ∈ W for every E ∈ PSL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' In view of the definition of W, it follows that: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' ker σE ⊃ ker h for every E ∈ PSL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' On the other hand, the definition of β implies: EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' For every E ∈ PSL2(C), the representation σE is the unique representation of Π′′ which restricts to ρ0 on Π and maps T0 to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Now by hypothesis the representation ρ0 : Π → PSL2(C) admits a lift ρ0 : Π → SL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since ρ0 is faithful, ρ0 is also faithful, and ρ0(Π) contains no non-trivial scalar matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' According to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='8, there is an element L of infinite order in SL2(C) such that the subgroup of SL2(C) generated by ρ0(Π) and L is a free product of ρ0(Π) with the infinite cyclic group ⟨L⟩, and contains no non-trivial scalar matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' If we denote by E1 the image of L under the quotient map from SL2(C) to PSL2(C), it now follows that the subgroup of PSL2(C) generated by ρ0(Π) and E1 is a free product of ρ0(Π) with the infinite cyclic group ⟨E1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Applying 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2 with E = E1, we deduce that σE1 is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' But 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1, applied with E = E1, gives that ker h ⊂ ker σE1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Hence h is injective in this case, so that Alternative (ii) of the conclusion of the lemma holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let Γ be a finitely generated Kleinian group (so that χ(Γ) is defined by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Then χ(Γ) < miof(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We first consider the case in which Γ has no rank-2 cusp subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Set k = miof(π1(N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' By the definition of miof(Γ), there is a finite generating set ∆ for Γ such that k = iof(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' In particular, ∆ contains k independent elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Hence we may write ∆ = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , xm}, where m ≥ k and x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , xk are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' For r = k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , m, let Πr denote the subgroup ⟨x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , xr⟩ of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We claim that χ(Πr) < k for r = k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' the case r = m of this claim is the conclusion of the proposition, since Πm = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Note that Πk is free of rank k, and hence χ(Πk) = k − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' thus the claim is true for r = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' It therefore suffices to prove that χ(Πr+1) ≤ χ(Πr) whenever k ≤ r < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since the Kleinian group Γ has no rank-2 cusp subgroup, its finitely generated subgroups Πr are also Kleinian groups without rank-2 cusp subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' It therefore follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='12 that, for any integer s with k ≤ s ≤ m, if we regard the inclusion homomorphism from Πs ≤ Γ to PSL2(C) as a faithful representation of Πs in PSL2(C), then ρs lies in a unique irreducible component Vs of R(Πs), and that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1) dim Vs = 3χ(Πs) + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Suppose that r is an integer with k ≤ r < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since ρr is a discrete faithful representation, it follows from [7, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1] that the representation ρr admits a lift to SL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The existence of such a lift, together with the fact that Vr is the only irreducible component of R(Πr) containing ρr, allows us to apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1, taking Π = Πr, Π′ = Πr+1, T = xr+1, and letting ρr and ρr+1 play the respective roles of ρ0 and ρ′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1 gives that either (i) Πr+1 is the free product of the subgroups Πr and ⟨xr+1⟩, or (ii) dim Vr+1 < (dim Vr) + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Thus for each r with k ≤ r < m, either (i) or (ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS 12 If (i) holds for a given r, then since Πk ≤ Πr is generated by the independent elements x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , xk, it follows that x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , xk+1 are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' By the definition of index of freedom, this means that iof(∆) ≥ k + 1, a contradiction to our choice of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Hence for each r with k ≤ r < m, Condition (ii) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Combining (ii) with the cases s = r and s = r + 1 of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1), we obtain 3χ(Πr+1) + 3 < 3χ(Πr) + 6, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' χ(Πr+1) < χ(Πr) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since χ(Πr+1) and χ(Πr) are integers, this gives χ(Πr+1) ≤ χ(Πr), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' This completes the proof of the proposition in the case in which Γ has no rank-2 cusp subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' We now turn to the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let Γ be an arbitrary finitely generated Kleinian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' According to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='5, there exist a finitely generated Kleinian group Γ0 with no rank-2 cusp subgroups, with χ(Γ0) = χ(Γ), and a surjective homomorphism η : Γ → Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='7, with G1 = Γ and G1 = Γ0, and defining η as above, we deduce that miof(Γ0) ≤ miof(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since Γ0 has no rank-2 cusp subgroups, we may apply the special case of the proposition that has already been proved to deduce that χ(Γ0) < miof(Γ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Thus we have χ(Γ) = χ(Γ0) < miof(Γ0) ≤ miof(Γ), and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' □ We now prove Theorem B, which was stated in the introduction to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since the finitely generated group G is a subgroup of the fundamental group of an orientable hyperbolic 3-manifold, it is isomorphic to a Kleinian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Thus the theorem follows immediately from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' □ We are now in a position to prove Theorem A, which was stated in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' By Theorem B we have χ(G) ≤ miof(G) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' It therefore suffices to show that miof(G) ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Assume that miof(G) ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Set ∆ = {[α1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , [αm]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since ∆ is a generating set for G, we have iof(∆) ≥ miof(G) ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' By definition this means that ∆ contains k independent elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Thus we have m ≥ k, and after possibly re-indexing the αi we may assume that [α1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , [αk] are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Let us write M = H3/Γ, where Γ is a torsion-free Kleinian group, let q : H3 → M denote the quotient map, and let us fix a point z ∈ q−1(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Then z defines an isomorphism j : π1(M, p) → Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Set ξi = j([αi]) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Since [α1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , [αk] are independent, ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , ξk freely generate a free subgroup of Γ, which may be regarded as a Kleinian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' If we set di = dist(z, ξi · z) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , k, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='1 of [2] now asserts that k � i=1 1 1 + edi ≤ 1 2, so that in particular di ≥ log(2k − 1) for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' But for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' , k} we have di ≤ length αi, which with the hypothesis of the theorem gives di < log(2k − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' This contradiction completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' □ EULER CHARACTERISTICS, FREE SUBGROUPS, AND LENGTHS OF LOOPS 13 References [1] Ian Agol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Tameness of hyperbolic 3-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' arXiv:math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='GT/0405568.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [2] Ian Agol, Marc Culler, and Peter B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Shalen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Dehn surgery, homology and hyperbolic volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Algebr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=', 6:2297–2312, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [3] James W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Anderson, Richard D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Canary, Marc Culler, and Peter B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Shalen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Free Kleinian groups and volumes of hyperbolic 3-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Differential Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=', 43(4):738–782, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [4] Gilbert Baumslag and Peter B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Shalen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Groups whose three-generator subgroups are free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Austral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=', 40(2):163–174, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Boyer and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' On Culler-Shalen seminorms and Dehn filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' (2), 148(3):737– 801, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [6] Danny Calegari and David Gabai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Shrinkwrapping and the taming of hyperbolic 3-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=', 19(2):385–446 (electronic), 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [7] Marc Culler and Peter B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Shalen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Varieties of group representations and splittings of 3-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' (2), 117(1):109–146, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [8] David Futer, Jessica S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Purcell, and Saul Schleimer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Effective drilling and filling of tame hyperbolic 3-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Helv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=', 97(3):457–512, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [9] Rosemary K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Guzman and Peter B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Shalen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The ratio of homology rank to hyperbolic volume, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Journal of Topology and Analysis, To appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='00040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [10] Rosemary K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Guzman and Peter B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Shalen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' The ratio of homology rank to hyperbolic volume, II: The role of the Four Color Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Journal of Topology and Analysis, To appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='14847.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [11] William H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Jaco and Peter B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Shalen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Seifert fibered spaces in 3-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=', 21(220):viii+192, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [12] Dennis Johnson and John J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Millson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Deformation spaces associated to compact hyperbolic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' In Discrete groups in geometry and analysis (New Haven, Conn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=', 1984), volume 67 of Progr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=', pages 48–106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Birkh¨auser Boston, Boston, MA, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [13] Michael Kapovich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Hyperbolic manifolds and discrete groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Modern Birkh¨auser Classics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Birkh¨auser Boston, Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=', Boston, MA, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Reprint of the 2001 edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [14] Hossein Namazi and Juan Souto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Non-realizability and ending laminations: proof of the density conjec- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=', 209(2):323–395, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [15] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Newstead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Introduction to moduli problems and orbit spaces, volume 51 of Tata Institute of Fun- damental Research Lectures on Mathematics and Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Tata Institute of Fundamental Research, Bombay;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Narosa Publishing House, New Delhi, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [16] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Nisnewitsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' ¨Uber Gruppen, die durch Matrizen ¨uber einem kommutativen Feld isomorph darstell- bar sind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Sbornik] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=', 8 (50):395–403, 1940.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [17] Ken’ichi Ohshika.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Realising end invariants by limits of minimally parabolic, geometrically finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=', 15(2):827–890, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [18] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Scott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Compact submanifolds of 3-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' (2), 7:246–250, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [19] Peter B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Shalen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Linear representations of certain amalgamated products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Algebra, 15(2):187–197, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [20] Peter B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Shalen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Small optimal Margulis numbers force upper volume bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=', 365(2):973–999, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' [21] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Wehrfritz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Generalized free products of linear groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' (3), 27:402–424, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=' Morgan St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content=', Chicago, IL 60607-7045 Email address: shalen@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='uic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQffw3Q/content/2301.05111v1.pdf'} diff --git a/G9E5T4oBgHgl3EQfWA_j/vector_store/index.pkl b/G9E5T4oBgHgl3EQfWA_j/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..12edc5f1c8a4859b240afe640363470e0fa057fb --- /dev/null +++ b/G9E5T4oBgHgl3EQfWA_j/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:87393d7923ded8fae93eb98775fa47d3207ec1ed87ff382e6c3194162afc40a7 +size 175184 diff --git a/GNAzT4oBgHgl3EQfHPvT/vector_store/index.faiss b/GNAzT4oBgHgl3EQfHPvT/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..b1ccd4e186842d9ae94bbdda819f1d24b8b6a22e --- /dev/null +++ b/GNAzT4oBgHgl3EQfHPvT/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5d725545aaea65056c4b44aac47a75665ce24984c51a91697d7c8fbc41ea53e8 +size 1507373 diff --git a/JdE1T4oBgHgl3EQfsAXy/content/2301.03362v1.pdf b/JdE1T4oBgHgl3EQfsAXy/content/2301.03362v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..01923c7abfc056b4c53ee5c4f4e6ef40504434bb --- /dev/null +++ b/JdE1T4oBgHgl3EQfsAXy/content/2301.03362v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5658249e91ac38dd0933d141adfa1bdf7d0306d5e944c47281547df340c7579a +size 13995207 diff --git a/JdE1T4oBgHgl3EQfsAXy/vector_store/index.faiss b/JdE1T4oBgHgl3EQfsAXy/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..85a1293d3e9d4e579c41b51074723f2f7a375c25 --- /dev/null +++ b/JdE1T4oBgHgl3EQfsAXy/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b8d315f8e06b88051f37d9ac3d848414bc7bcf93408d2df7604ab5e729edcf6f +size 14352429 diff --git a/JdFRT4oBgHgl3EQfCjdP/content/tmp_files/2301.13469v1.pdf.txt b/JdFRT4oBgHgl3EQfCjdP/content/tmp_files/2301.13469v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7ad286cd7de8083f3ed9df795e1402aa2e04d7cf --- /dev/null +++ b/JdFRT4oBgHgl3EQfCjdP/content/tmp_files/2301.13469v1.pdf.txt @@ -0,0 +1,281 @@ +arXiv:2301.13469v1 [astro-ph.GA] 31 Jan 2023 +Proceedings of the 7th Chile-Cologne-Bonn-Symposium: Physics and Chemistry of Star Formation +V. Ossenkopf-Okada, R. Schaaf, I. Breloy (eds.) +STAR FORMATION IN THE EXTREME GALACTIC CENTER ENVIRONMENT +Mark R. Morris1 +Abstract. Copious star formation occurs in the dense Central Molecular Zone (CMZ) of our Galaxy, +but at a much smaller rate than occurs in a comparable mass of molecular gas in the Galactic disk. +The combination of large turbulent velocity dispersions, a relatively strong magnetic field, and +a strong tidal field all contribute to inhibiting star formation (SF) in different ways in different +CMZ locations. Nonetheless, there are spectacular displays of recent and ongoing SF in the CMZ, +including massive young stellar clusters, sites of abundant SF in progress, and numerous spots of +protostellar or YSO activity. The presence of giant molecular clouds in the CMZ that are almost +entirely devoid of SF indicates that SF requires a trigger that is not present everywhere. +The +dominant provocation of SF is likely to be cloud compression, either by large-scale shocks or by +orbital motion of clouds into a region of enhanced tidal compression and/or enhanced external +pressure. Recent hypotheses for where and how SF takes place in the CMZ are constrained by the +recent orbital determinations of the massive Arches and Quintuplet clusters. Star formation in the +central parsec is subject to a very different set of physical conditions, and is less well understood, +but is important for the co-evolution of the central black hole and the nuclear star cluster. +1 +Introduction +The discussion of SF in the Galactic center divides naturally into that taking place in the gravitational +domain of the Galactic black hole (GBH, i.e., roughly within a parsec) and that taking place in the rest of +the CMZ. The very strong tidal forces exerted by the GBH in the central parsec raise the threshold for SF +considerably, and it remains debatable whether SF has occurred very recently there, but the presence of the +young nuclear cluster within half a parsec of the GBH shows that a dramatic SF event did occur several +million years ago, unavoidably accompanied by a major episode of accretion onto the GBH. Here, the two +regimes – CMZ and central parsec – are discussed separately. For complementary recent overviews of star +formation in the CMZ, the reader is referred to Krumholz (2021) and Henshaw et al. (2022). +2 +The Large-Scale Arena: the Central Molecular Zone +The CMZ contains on the order of 5 × 107 M⊙ of molecular gas lying within ∼ ±200 pc of the Galactic +center (Dahmen et al. 1998; Ferrière et al. 2007; Longmore et al. 2013a), the majority of which is organized +into a coherent structure, or structures, surrounding the nucleus. That structure might be be a twisted, +elongated closed ring (Molinari et al. 2011), several gas streams following open orbits (Kruijssen et al. 2015; +Henshaw et al. 2016b), or relatively tightly wound spiral "arms" of gas (Sofue 1995) that more or less coincide +with the most prominent orbiting streams. In any case, molecular "clouds" in the CMZ tend to be denser +portions of tidally stretched molecular gas streams or of a ring (Molinari et al. 2011; Butterfield et al. 2018; +Kruijssen et al. 2019). Simulations show that closed rings of dense gas form naturally in the presence of a bar +potential (Kim et al. 2011; Krumholz & Kruijssen 2015; Sormani et al. 2018; Salas et al. 2020; Sormani et al. +2020; Tress et al. 2020), generally occupying the region of the outermost X2 orbits (c.f., Morris & Serabyn +1996). +Because most of the dense CMZ mass is in the ∼ 100 pc ring (or the component streams of the ring +distribution), one should expect Galactic center star formation to predominantly take place there. However, +although the star formation rate is substantial (∼ 0.1 M⊙ yr−1; c.f., Barnes et al. 2017; Henshaw et al. 2022, +1 Department of Physics & Astronomy, University of California, Los Angeles, CA, USA 90095-1547 +e-mail: morris@astro.ucla.edu +©Universität zu Köln 2022 + +2 +Proceedings of the 7th Chile-Cologne-Bonn-Symposium: Physics and Chemistry of Star Formation +and references therein), the rate of star formation per unit mass of gas is much smaller in the CMZ than in the +Galactic disk (Morris 1989; Longmore et al. 2013a; Longmore 2014). The dominant reason for the inhibition +of star formation in the CMZ is likely to be the large energy density of turbulence, or microturbulence, +which is implied by the large linewidths observed with even the highest possible spatial resolution (typically +∼ 10 km s−1 or greater). This is attributable to several causes: supernovae and other feedback from star +formation, supersonic turbulence induced by MHD instabilities in the differentially rotating medium, and +by instabilities accompanying bar-driven angular momentum transport (Krumholz & Kruijssen 2015). The +large velocity dispersions associated with turbulence effectively raise the Jeans mass to values akin to the +masses of clusters rather than individual stars, which could be relevant to the formation of the massive +Arches and Quintuplet clusters in the CMZ. Another contributing reason for the relatively low rate of SF is +the relatively strong magnetic field in clouds of the CMZ (Morris 1993; Chuss et al. 2003), which provides a +pressure that anisotropically counteracts gravitational collapse. The magnetic field in CMZ clouds is likely +to be closely linked to, and in energy equipartition with, the turbulence. Galactic tidal shear has been +considered as another factor inhibiting star formation, but its effect is likely to be negligible except in the +central parsec (Kruijssen et al. 2014). +The supersonic turbulence within CMZ clouds must inevitably produce an unresolved network of shocks. +Indeed, observations of molecular shock tracers indicate that shocks are present throughout the CMZ. +Those tracers include SiO (Riquelme et al. 2010; Tsuboi et al. 2011, and Riquelme, this conference), HNCO +(Henshaw et al. 2016b), and hot NH3 (Mills & Morris 2013). Density enhancements and cooling in post- +shock regions might lead to conditions favoring star formation, but given the strong magnetic fields in CMZ +clouds, the shocks are likely to be of type C, so these effects are probably somewhat muted. In any case, +there is currently no evidence for a correlation between the locations of shocks and sites of star formation. +Indeed, the internal shocks in CMZ clouds undoubtedly play a strong role in maintaining a relatively high +gas temperature in the molecular gas (Immer et al. 2016), which contributes to the impediments to star +formation. +An important environmental factor affecting star formation in the CMZ is that the pressure there, +including turbulent gas pressure and magnetic pressure, P/k ∼ 108 K cm−3, is several orders of magnitude +higher than in the Galactic disk (Spergel & Blitz 1992; Rathborne et al. 2014). According to the analysis of +Rathborne et al., this translates into much higher surface densities of CMZ clouds compared to Galactic disk +clouds, and therefore much higher volume densities to be in hydrostatic equilibrium with the hydrostatic +pressure from self-gravity. Indeed, Rathborne et al. (2014) find that the physical conditions and structures +of CMZ clouds are close to those representing hydrostatic equilibrium, which accounts for their relatively +high densities throughout the clouds, typically ≳ 104 cm−3. +3 +Where and Why Does Star Formation Take Place in the CMZ? +There are a few places in the CMZ where star formation has apparently been provoked by some local +compressive event. An example is G-0.02-0.07, a linear string of 4 HII regions lying adjacent to a dense ridge +in the 50 km s−1 molecular cloud (Serabyn et al. 1992; Yusef-Zadeh et al. 2010; Mills et al. 2011). +However, there are two hypotheses for how most of the star formation in the CMZ is initiated in +a more global manner. +The first hypothesis, which has been referred to as the "conveyor belt" model +(Longmore et al. 2013b, 2014; Kruijssen et al. 2015; Henshaw et al. 2016a; Kruijssen et al. 2019), notes that +the trajectories of the streams and their component clouds constituting the molecular ring are elongated, +so that the clouds pass through a pericenter position where they orbit closest to the GBH. At that point, +the clouds are maximally compressed by the Galactic tidal force imposed by the GBH plus the nuclear +stellar cluster, and at least portions of those clouds are pushed over the threshold of gravitational stability. +The proponents of this hypothesis point out that clouds distributed along the gas streams following passage +through the pericenter location display a sequence of evolutionary stages for star formation, from clouds that +are almost entirely devoid of star formation (e.g., G0.253+0.016, the "Brick"; Kauffmann et al. (2013)), to +the massively star-forming cloud, Sgr B2 (Ginsburg et al. 2018). Kruijssen et al. (2015) note that the orbital +time between the locations of those clouds corresponds to the free-fall time for star formation, once the col- +lapse has been initiated. Another phenomenon that adds to cloud compression at the pericenter position is +the fact that there is probably a radial gradient in the pressure of the interstellar medium, which is greatest +near the GBH. +The second hypothesis for large-scale triggering of star formation is that cloud compression occurs where +gas flowing inward along the bar from outside the CMZ encounters and shocks the gas in the CMZ, triggering + +Galactic Center Star Formation +3 +star formation at the apocenters of the CMZ cloud orbits (Sormani et al. 2020; Tress et al. 2020). In this +model, which is based on detailed hydrodynamical simulations, the sequence of star formation begins at the +apocenters of the cloud orbits, and the newborn stars are presumably arrayed as "beads on a string" at +downstream positions along the X2 orbit ring. +Assessment of these contrasting hypotheses would be facilitated by having line-of-sight distance mea- +surements or well-determined orbits for recently-formed stars in the CMZ. The massive, young Arches and +Quintuplet clusters can be helpful in this regard. Sormani et al. (2020) suggest that these clusters origi- +nated near the collision sites where the bar-driven inflow accretes onto the CMZ. This view is supported by +the recent work of Hosek et al. (2022), who determined the absolute proper motions of these clusters, and +combined them with radial velocities and 2D locations to strongly constrain their Galactic orbits. Although +the line-of-sight distances and the precise cluster ages remain undetermined, the constraining information +at hand allowed Hosek et al. (2022) to show that the probability maps for the cluster birth locations favor +the hypothesis that they originated at the apocenters of the gas ring. +Numerous studies have revealed the presence of many individual massive stars, HII regions, and embedded +protostars spread throughout the CMZ (e.g., Yusef-Zadeh et al. 2009; Mauerhan et al. 2010b,a; An et al. +2011; Dong et al. 2012; Hankins et al. 2019; Lu et al. 2019; Clark et al. 2021). Clark et al. (2021) enumerate +≥ 320 spectroscopically classified stars that would be expected to undergo core collapse within the next ∼ 20 +Myr, and state that this number is a substantial underestimate. Work on the mid-infrared SOFIA survey by +Hankins et al. (2020) is continuing and will add considerably to the known population of embedded stars and +protostars. The overall collection of such objects will eventually be invaluable for determining the temporal +and 3-dimensional spatial distribution of star formation in the CND. +Some of the massive stars distributed within the CMZ could be escapees from the massive Arches and +Quintuplet clusters, distributed along the tidal tails of these clusters (Mauerhan et al. 2010a; Habibi et al. +2014). In addition to tidal evaporation, supernovae in binaries and 3-body gravitational interactions cause the +occasional expulsion of massive stars from such massive, dense clusters (Kim et al. 1999, 2000; Portegies Zwart et al. +2002). It is therefore possible that a sizeable fraction of the population of isolated, massive field stars could +have originally formed in massive clusters. However, in order to assess this possibility, considerable work +remains to determine the dynamics of the isolated stars to see whether their orbits trace back to the clusters. +In any case, the fact that massive clusters contain, or formerly contained, such a significant fraction of all +massive stars in the CMZ provides an extremely salient clue to a key mode of star formation there. +4 +The Central Parsec +The Young Nuclear Cluster (YNC), lying within a radius of 0.5 pc of the GBH, and having an age of 3 - 8 +×106 years (Lu et al. 2013; Lu 2018), illustrates that star formation is possible within the immediate vicinity +of the GBH, despite the extreme tidal force exerted by the GBH, once considered an insurmountable barrier +to star formation (Sanders 1992). The possibility that the YNC could have migrated in from elsewhere as +a result of dynamical friction has been considered (Gerhard 2001), but this would only be possible for a +cluster having a dense core that is much more massive than the YNC, even if the cluster is centered on +an intermediate-mass black hole (Kim & Morris 2003; Kim et al. 2004). However, Fujii et al. (2008) found +that the inspiral timescale for clusters onto the GBH is reduced by core collapse of massive clusters, so it +might have been possible for the YNC to migrate into the center from a distance of a few parsecs within +the lifetime of the YNC. In any case, the prevailing opinion now seems to be that the YNC formed in situ, +but that raises many questions about how local gravitational collapse to form stars could have overcome the +extreme tidal shear and probable strong magnetic fields, amplified by the shear, in that region. Furthermore, +formation of a few × 104 M⊙ cluster that close to the GBH would almost certainly have been accompanied +by strong accretion activity that would have added considerably to the heating and turbulence. +A fraction (≥ 20%) of the stars in the YNC form a disk around the GBH, albeit with a relatively large +distribution of eccentricities (Levin & Beloborodov 2003; Yelda et al. 2014, and references therein). This +suggests that at least some of the star formation leading to the YNC occurred in an accretion disk (although +an inspiralling cluster would also leave stars distributed in a disk, once it dynamically relaxed at the center). +A numerical model produced by Nayakshin et al. (2007) showed that a stellar disk could be formed starting +from a gravitationally unstable gaseous disk surrounding the GBH. It would be interesting and timely to +repeat such a calculation with additional physics (magnetic fields, black hole feedback), and higher spatial +resolution to capture the disk instabilities. The physical conditions in the accretion disk maelstrom that +would have produced the YNC are far from ordinary, so it is no surprise that the mass function of the YNC + +4 +Proceedings of the 7th Chile-Cologne-Bonn-Symposium: Physics and Chemistry of Star Formation +is somewhat top-heavy, with a slope of 1.7 ± 0.2 (Bartko et al. 2010; Lu et al. 2013). (Note that a top-heavy +mass function would also be expected for the surviving members of a cluster that has migrated to the center +as a result of tidal friction (Fujii et al. 2008).) +An alternative model for in situ production of the YNC that has been very popular invokes an in- +falling molecular cloud or tidal disruption of a cloud approaching the GBH on a highly eccentric orbit +(e.g. Bonnell & Rice 2008; Generozov et al. 2022, and many others cited by Dinh et al., 2021). However, +Dinh et al. (2021) argue that the phase-space volume accessible to such low-angular-momentum clouds is +very small, and there is no obvious way of producing them, so they should be quite rare. Put another way, +it is very difficult to scatter clouds onto approximately radial orbits because of their typically large size, +because they participate in Galactic rotation, and because of the likely absence of sufficiently massive per- +turbers. Invoking cloud collisions to produce infalling clouds (e.g., Hobbs & Nayakshin 2009) faces similar +constraints (Dinh et al. 2021). Another alternative model for in situ formation of the YNC is one in which +the circumnuclear gas disk, fed quasi-continuously from the outside by gas migrating inward from the CMZ, +underwent viscous evolution prior to the formation of the YNC, and extended much further in toward the +center than it does now, perhaps all the way to the GBH. At that point, some trigger, perhaps enhanced +pressure from accretion activity, started a runaway process of star formation within the central half parsec +of the disk. In such a circumstance the combined energy produced by the rapid formation of a massive +cluster and the near-Eddington accretion onto the GBH could have expelled much of the central portions +of the CND, creating a central cavity such as that which is present today (Morris et al. 1999). Noting that +accretion onto the CND is likely continuing (e.g., Tress et al. 2020), and that the lifetimes of the massive +stars in the YNC are only a few ×107 years, one can speculate that the formation of a YNC follows a limit +cycle, repetitively recreating such a cluster on a time scale limited by the CND growth rate and by the +viscous evolution timescale of the CND. +Evidence for more recent star formation at various places in the central parsec has been presented (e.g., +Eckart et al. 2013; Yusef-Zadeh et al. 2013, 2015a,b, 2017), but has been questioned (Morris 20211), largely +because there are alternative explanations for the phenomena presented as evidence. Furthermore, obser- +vations of the infrared counterparts to the putative protostars and YSOs, or the supporting spectroscopic +evidence, have not yet been acquired. The issue of whether star formation in the central parsec is a continuing +and current process will probably soon be resolved by ongoing observations with JWST. +References +An, D., Ramírez, S. V., Sellgren, K., et al. 2011, ApJ, 736, 133 +Barnes, A. T., Longmore, S. N., Battersby, C., et al. 2017, MNRAS, 469, 2263 +Bartko, H., Martins, F., Trippe, S., et al. 2010, ApJ, 708, 834 +Bonnell, I. A. & Rice, W. K. M. 2008, Science, 321, 1060 +Butterfield, N., Lang, C. C., Morris, M., Mills, E. A. C., & Ott, J. 2018, ApJ, 852, 11 +Chuss, D. T., Davidson, J. A., Dotson, J. L., et al. 2003, ApJ, 599, 1116 +Clark, J. S., Patrick, L. R., Najarro, F., Evans, C. J., & Lohr, M. 2021, A&A, 649, A43 +Dahmen, G., Huttemeister, S., Wilson, T. L., & Mauersberger, R. 1998, A&A, 331, 959 +Dinh, C. K., Salas, J. M., Morris, M. R., & Naoz, S. 2021, ApJ, 920, 79 +Dong, H., Wang, Q. D., & Morris, M. R. 2012, MNRAS, 425, 884 +Eckart, A., Mužić, K., Yazici, S., et al. 2013, A&A, 551, A18 +Ferrière, K., Gillard, W., & Jean, P. 2007, A&A, 467, 611 +Fujii, M., Iwasawa, M., Funato, Y., & Makino, J. 2008, ApJ, 686, 1082 +Generozov, A., Nayakshin, S., & Madigan, A. M. 2022, MNRAS, 512, 4100 +Gerhard, O. 2001, ApJL, 546, L39 +Ginsburg, A., Bally, J., Barnes, A., et al. 2018, ApJ, 853, 171 +Habibi, M., Stolte, A., & Harfst, S. 2014, A&A, 566, A6 +Hankins, M. J., Lau, R. M., Mills, E. A. C., Morris, M. R., & Herter, T. L. 2019, ApJ, 877, 22 +Hankins, M. J., Lau, R. M., Radomski, J. T., et al. 2020, ApJ, 894, 55 +Henshaw, J. D., Barnes, A. T., Battersby, C., et al. 2022, arXiv e-prints, arXiv:2203.11223 +Henshaw, J. D., Longmore, S. N., & Kruijssen, J. M. D. 2016a, MNRAS, 463, L122 +Henshaw, J. D., Longmore, S. N., Kruijssen, J. M. D., et al. 2016b, MNRAS, 457, 2675 +1https://www.youtube.com/watch?v=8ezDLs_MJxw + +Galactic Center Star Formation +5 +Hobbs, A. & Nayakshin, S. 2009, MNRAS, 394, 191 +Hosek, M. W., Do, T., Lu, J. R., et al. 2022, ApJ, 939, 68 +Immer, K., Kauffmann, J., Pillai, T., Ginsburg, A., & Menten, K. M. 2016, A&A, 595, A94 +Kauffmann, J., Pillai, T., & Zhang, Q. 2013, ApJL, 765, L35 +Kim, S. S., Figer, D. F., Lee, H. M., & Morris, M. 2000, ApJ, 545, 301 +Kim, S. S., Figer, D. F., & Morris, M. 2004, ApJL, 607, L123 +Kim, S. S. & Morris, M. 2003, ApJ, 597, 312 +Kim, S. S., Morris, M., & Lee, H. M. 1999, ApJ, 525, 228 +Kim, S. S., Saitoh, T. R., Jeon, M., et al. 2011, ApJL, 735, L11 +Kruijssen, J. M. D., Dale, J. E., & Longmore, S. N. 2015, MNRAS, 447, 1059 +Kruijssen, J. M. D., Dale, J. E., Longmore, S. N., et al. 2019, MNRAS, 484, 5734 +Kruijssen, J. M. D., Longmore, S. N., Elmegreen, B. G., et al. 2014, MNRAS, 440, 3370 +Krumholz, M. R. 2021, in Astronomical Society of the Pacific Conference Series, Vol. 528, New Horizons in +Galactic Center Astronomy and Beyond, ed. M. Tsuboi & T. Oka, 19 +Krumholz, M. R. & Kruijssen, J. M. D. 2015, MNRAS, 453, 739 +Levin, Y. & Beloborodov, A. M. 2003, ApJL, 590, L33 +Longmore, S. N. 2014, in IAU Symposium, Vol. 303, IAU Symposium, ed. L. O. Sjouwerman, C. C. Lang, +& J. Ott, 132–138 +Longmore, S. N., Bally, J., Testi, L., et al. 2013a, MNRAS, 429, 987 +Longmore, S. N., Kruijssen, J. M. D., Bally, J., et al. 2013b, MNRAS, 433, L15 +Longmore, S. N., Kruijssen, J. M. D., Bastian, N., et al. 2014, in Protostars and Planets VI, ed. H. Beuther, +R. S. Klessen, C. P. Dullemond, & T. Henning, 291–314 +Lu, J. R. 2018, Astrophysics and Space Science Library, Vol. 424, Massive Young Clusters Near the Galactic +Center, ed. S. Stahler, 69 +Lu, J. R., Do, T., Ghez, A. M., et al. 2013, ApJ, 764, 155 +Lu, X., Mills, E. A. C., Ginsburg, A., et al. 2019, ApJS, 244, 35 +Mauerhan, J. C., Cotera, A., Dong, H., et al. 2010a, ApJ, 725, 188 +Mauerhan, J. C., Muno, M. P., Morris, M. R., Stolovy, S. R., & Cotera, A. 2010b, ApJ, 710, 706 +Mills, E., Morris, M. R., Lang, C. C., et al. 2011, ApJ, 735, 84 +Mills, E. A. C. & Morris, M. R. 2013, ApJ, 772, 105 +Molinari, S., Bally, J., Noriega-Crespo, A., et al. 2011, ApJL, 735, L33+ +Morris, M. 1989, in IAU Symposium, Vol. 136, The Center of the Galaxy, ed. M. Morris, 171 +Morris, M. 1993, ApJ, 408, 496 +Morris, M., Ghez, A. M., & Becklin, E. E. 1999, Advances in Space Research, 23, 959 +Morris, M. & Serabyn, E. 1996, ARAA, 34, 645 +Nayakshin, S., Cuadra, J., & Springel, V. 2007, MNRAS, 379, 21 +Portegies Zwart, S. F., Makino, J., McMillan, S. L. W., & Hut, P. 2002, ApJ, 565, 265 +Rathborne, J. M., Longmore, S. N., Jackson, J. M., et al. 2014, ApJL, 795, L25 +Riquelme, D., Bronfman, L., Mauersberger, R., May, J., & Wilson, T. L. 2010, A&A, 523, A45+ +Salas, J. M., Naoz, S., & Morris, M. R. 2020, arXiv e-prints, arXiv:2010.04170 +Sanders, R. H. 1992, Nature, 359, 131 +Serabyn, E., Lacy, J. H., & Achtermann, J. M. 1992, ApJ, 395, 166 +Sofue, Y. 1995, PASJ, 47, 527 +Sormani, M. C., Sobacchi, E., Fragkoudi, F., et al. 2018, MNRAS, 481, 2 +Sormani, M. C., Tress, R. G., Glover, S. C. O., et al. 2020, MNRAS, 497, 5024 +Spergel, D. N. & Blitz, L. 1992, Nature, 357, 665 +Tress, R. G., Sormani, M. C., Glover, S. C. O., et al. 2020, MNRAS, 499, 4455 +Tsuboi, M., Tadaki, K.-I., Miyazaki, A., & Handa, T. 2011, PASJ, 63, 763 +Yelda, S., Ghez, A. M., Lu, J. R., et al. 2014, ApJ, 783, 131 +Yusef-Zadeh, F., Cotton, B., Wardle, M., et al. 2017, MNRAS, 467, 922 +Yusef-Zadeh, F., Hewitt, J. W., Arendt, R. G., et al. 2009, ApJ, 702, 178 +Yusef-Zadeh, F., Lacy, J. H., Wardle, M., et al. 2010, ApJ, 725, 1429 +Yusef-Zadeh, F., Roberts, D. A., Wardle, M., et al. 2015a, ApJL, 801, L26 +Yusef-Zadeh, F., Royster, M., Wardle, M., et al. 2013, ApJL, 767, L32 +Yusef-Zadeh, F., Wardle, M., Sewilo, M., et al. 2015b, ApJ, 808, 97 + diff --git a/JdFRT4oBgHgl3EQfCjdP/content/tmp_files/load_file.txt b/JdFRT4oBgHgl3EQfCjdP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ab7f5a017ae0e3ab29a6ca3b2c7f59fa842b8007 --- /dev/null +++ b/JdFRT4oBgHgl3EQfCjdP/content/tmp_files/load_file.txt @@ -0,0 +1,625 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf,len=624 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='13469v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='GA] 31 Jan 2023 Proceedings of the 7th Chile-Cologne-Bonn-Symposium: Physics and Chemistry of Star Formation V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Ossenkopf-Okada, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Schaaf, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Breloy (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=') STAR FORMATION IN THE EXTREME GALACTIC CENTER ENVIRONMENT Mark R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Morris1 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Copious star formation occurs in the dense Central Molecular Zone (CMZ) of our Galaxy, but at a much smaller rate than occurs in a comparable mass of molecular gas in the Galactic disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' The combination of large turbulent velocity dispersions, a relatively strong magnetic field, and a strong tidal field all contribute to inhibiting star formation (SF) in different ways in different CMZ locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Nonetheless, there are spectacular displays of recent and ongoing SF in the CMZ, including massive young stellar clusters, sites of abundant SF in progress, and numerous spots of protostellar or YSO activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' The presence of giant molecular clouds in the CMZ that are almost entirely devoid of SF indicates that SF requires a trigger that is not present everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' The dominant provocation of SF is likely to be cloud compression, either by large-scale shocks or by orbital motion of clouds into a region of enhanced tidal compression and/or enhanced external pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Recent hypotheses for where and how SF takes place in the CMZ are constrained by the recent orbital determinations of the massive Arches and Quintuplet clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Star formation in the central parsec is subject to a very different set of physical conditions, and is less well understood, but is important for the co-evolution of the central black hole and the nuclear star cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 1 Introduction The discussion of SF in the Galactic center divides naturally into that taking place in the gravitational domain of the Galactic black hole (GBH, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', roughly within a parsec) and that taking place in the rest of the CMZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' The very strong tidal forces exerted by the GBH in the central parsec raise the threshold for SF considerably, and it remains debatable whether SF has occurred very recently there, but the presence of the young nuclear cluster within half a parsec of the GBH shows that a dramatic SF event did occur several million years ago, unavoidably accompanied by a major episode of accretion onto the GBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Here, the two regimes – CMZ and central parsec – are discussed separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' For complementary recent overviews of star formation in the CMZ, the reader is referred to Krumholz (2021) and Henshaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2 The Large-Scale Arena: the Central Molecular Zone The CMZ contains on the order of 5 × 107 M⊙ of molecular gas lying within ∼ ±200 pc of the Galactic center (Dahmen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Ferrière et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Longmore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2013a), the majority of which is organized into a coherent structure, or structures, surrounding the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' That structure might be be a twisted, elongated closed ring (Molinari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2011), several gas streams following open orbits (Kruijssen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Henshaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2016b), or relatively tightly wound spiral "arms" of gas (Sofue 1995) that more or less coincide with the most prominent orbiting streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' In any case, molecular "clouds" in the CMZ tend to be denser portions of tidally stretched molecular gas streams or of a ring (Molinari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Butterfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Kruijssen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Simulations show that closed rings of dense gas form naturally in the presence of a bar potential (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Krumholz & Kruijssen 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Sormani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Salas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Sormani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Tress et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2020), generally occupying the region of the outermost X2 orbits (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Morris & Serabyn 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Because most of the dense CMZ mass is in the ∼ 100 pc ring (or the component streams of the ring distribution), one should expect Galactic center star formation to predominantly take place there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' However, although the star formation rate is substantial (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='1 M⊙ yr−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Henshaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2022, 1 Department of Physics & Astronomy, University of California, Los Angeles, CA, USA 90095-1547 e-mail: morris@astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='edu ©Universität zu Köln 2022 2 Proceedings of the 7th Chile-Cologne-Bonn-Symposium: Physics and Chemistry of Star Formation and references therein), the rate of star formation per unit mass of gas is much smaller in the CMZ than in the Galactic disk (Morris 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Longmore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2013a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Longmore 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' The dominant reason for the inhibition of star formation in the CMZ is likely to be the large energy density of turbulence, or microturbulence, which is implied by the large linewidths observed with even the highest possible spatial resolution (typically ∼ 10 km s−1 or greater).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' This is attributable to several causes: supernovae and other feedback from star formation, supersonic turbulence induced by MHD instabilities in the differentially rotating medium, and by instabilities accompanying bar-driven angular momentum transport (Krumholz & Kruijssen 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' The large velocity dispersions associated with turbulence effectively raise the Jeans mass to values akin to the masses of clusters rather than individual stars, which could be relevant to the formation of the massive Arches and Quintuplet clusters in the CMZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Another contributing reason for the relatively low rate of SF is the relatively strong magnetic field in clouds of the CMZ (Morris 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Chuss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2003), which provides a pressure that anisotropically counteracts gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' The magnetic field in CMZ clouds is likely to be closely linked to, and in energy equipartition with, the turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Galactic tidal shear has been considered as another factor inhibiting star formation, but its effect is likely to be negligible except in the central parsec (Kruijssen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' The supersonic turbulence within CMZ clouds must inevitably produce an unresolved network of shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Indeed, observations of molecular shock tracers indicate that shocks are present throughout the CMZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Those tracers include SiO (Riquelme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Tsuboi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2011, and Riquelme, this conference), HNCO (Henshaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2016b), and hot NH3 (Mills & Morris 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Density enhancements and cooling in post- shock regions might lead to conditions favoring star formation, but given the strong magnetic fields in CMZ clouds, the shocks are likely to be of type C, so these effects are probably somewhat muted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' In any case, there is currently no evidence for a correlation between the locations of shocks and sites of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Indeed, the internal shocks in CMZ clouds undoubtedly play a strong role in maintaining a relatively high gas temperature in the molecular gas (Immer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2016), which contributes to the impediments to star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' An important environmental factor affecting star formation in the CMZ is that the pressure there, including turbulent gas pressure and magnetic pressure, P/k ∼ 108 K cm−3, is several orders of magnitude higher than in the Galactic disk (Spergel & Blitz 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Rathborne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' According to the analysis of Rathborne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', this translates into much higher surface densities of CMZ clouds compared to Galactic disk clouds, and therefore much higher volume densities to be in hydrostatic equilibrium with the hydrostatic pressure from self-gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Indeed, Rathborne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' (2014) find that the physical conditions and structures of CMZ clouds are close to those representing hydrostatic equilibrium, which accounts for their relatively high densities throughout the clouds, typically ≳ 104 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 3 Where and Why Does Star Formation Take Place in the CMZ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' There are a few places in the CMZ where star formation has apparently been provoked by some local compressive event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' An example is G-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='02-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='07, a linear string of 4 HII regions lying adjacent to a dense ridge in the 50 km s−1 molecular cloud (Serabyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Yusef-Zadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Mills et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' However, there are two hypotheses for how most of the star formation in the CMZ is initiated in a more global manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' The first hypothesis, which has been referred to as the "conveyor belt" model (Longmore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2013b, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Kruijssen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Henshaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Kruijssen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2019), notes that the trajectories of the streams and their component clouds constituting the molecular ring are elongated, so that the clouds pass through a pericenter position where they orbit closest to the GBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' At that point, the clouds are maximally compressed by the Galactic tidal force imposed by the GBH plus the nuclear stellar cluster, and at least portions of those clouds are pushed over the threshold of gravitational stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' The proponents of this hypothesis point out that clouds distributed along the gas streams following passage through the pericenter location display a sequence of evolutionary stages for star formation, from clouds that are almost entirely devoid of star formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='253+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='016, the "Brick";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' (2013)), to the massively star-forming cloud, Sgr B2 (Ginsburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Kruijssen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' (2015) note that the orbital time between the locations of those clouds corresponds to the free-fall time for star formation, once the col- lapse has been initiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Another phenomenon that adds to cloud compression at the pericenter position is the fact that there is probably a radial gradient in the pressure of the interstellar medium, which is greatest near the GBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' The second hypothesis for large-scale triggering of star formation is that cloud compression occurs where gas flowing inward along the bar from outside the CMZ encounters and shocks the gas in the CMZ, triggering Galactic Center Star Formation 3 star formation at the apocenters of the CMZ cloud orbits (Sormani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Tress et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' In this model, which is based on detailed hydrodynamical simulations, the sequence of star formation begins at the apocenters of the cloud orbits, and the newborn stars are presumably arrayed as "beads on a string" at downstream positions along the X2 orbit ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Assessment of these contrasting hypotheses would be facilitated by having line-of-sight distance mea- surements or well-determined orbits for recently-formed stars in the CMZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' The massive, young Arches and Quintuplet clusters can be helpful in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Sormani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' (2020) suggest that these clusters origi- nated near the collision sites where the bar-driven inflow accretes onto the CMZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' This view is supported by the recent work of Hosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' (2022), who determined the absolute proper motions of these clusters, and combined them with radial velocities and 2D locations to strongly constrain their Galactic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Although the line-of-sight distances and the precise cluster ages remain undetermined, the constraining information at hand allowed Hosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' (2022) to show that the probability maps for the cluster birth locations favor the hypothesis that they originated at the apocenters of the gas ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Numerous studies have revealed the presence of many individual massive stars, HII regions, and embedded protostars spread throughout the CMZ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Yusef-Zadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Mauerhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2010b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Hankins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' (2021) enumerate ≥ 320 spectroscopically classified stars that would be expected to undergo core collapse within the next ∼ 20 Myr, and state that this number is a substantial underestimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Work on the mid-infrared SOFIA survey by Hankins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' (2020) is continuing and will add considerably to the known population of embedded stars and protostars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' The overall collection of such objects will eventually be invaluable for determining the temporal and 3-dimensional spatial distribution of star formation in the CND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Some of the massive stars distributed within the CMZ could be escapees from the massive Arches and Quintuplet clusters, distributed along the tidal tails of these clusters (Mauerhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2010a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Habibi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' In addition to tidal evaporation, supernovae in binaries and 3-body gravitational interactions cause the occasional expulsion of massive stars from such massive, dense clusters (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 1999, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Portegies Zwart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' It is therefore possible that a sizeable fraction of the population of isolated, massive field stars could have originally formed in massive clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' However, in order to assess this possibility, considerable work remains to determine the dynamics of the isolated stars to see whether their orbits trace back to the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' In any case, the fact that massive clusters contain, or formerly contained, such a significant fraction of all massive stars in the CMZ provides an extremely salient clue to a key mode of star formation there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 4 The Central Parsec The Young Nuclear Cluster (YNC), lying within a radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='5 pc of the GBH, and having an age of 3 - 8 ×106 years (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Lu 2018), illustrates that star formation is possible within the immediate vicinity of the GBH, despite the extreme tidal force exerted by the GBH, once considered an insurmountable barrier to star formation (Sanders 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' The possibility that the YNC could have migrated in from elsewhere as a result of dynamical friction has been considered (Gerhard 2001), but this would only be possible for a cluster having a dense core that is much more massive than the YNC, even if the cluster is centered on an intermediate-mass black hole (Kim & Morris 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' However, Fujii et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' (2008) found that the inspiral timescale for clusters onto the GBH is reduced by core collapse of massive clusters, so it might have been possible for the YNC to migrate into the center from a distance of a few parsecs within the lifetime of the YNC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' In any case, the prevailing opinion now seems to be that the YNC formed in situ, but that raises many questions about how local gravitational collapse to form stars could have overcome the extreme tidal shear and probable strong magnetic fields, amplified by the shear, in that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Furthermore, formation of a few × 104 M⊙ cluster that close to the GBH would almost certainly have been accompanied by strong accretion activity that would have added considerably to the heating and turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' A fraction (≥ 20%) of the stars in the YNC form a disk around the GBH, albeit with a relatively large distribution of eccentricities (Levin & Beloborodov 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Yelda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2014, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' This suggests that at least some of the star formation leading to the YNC occurred in an accretion disk (although an inspiralling cluster would also leave stars distributed in a disk, once it dynamically relaxed at the center).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' A numerical model produced by Nayakshin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' (2007) showed that a stellar disk could be formed starting from a gravitationally unstable gaseous disk surrounding the GBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' It would be interesting and timely to repeat such a calculation with additional physics (magnetic fields, black hole feedback), and higher spatial resolution to capture the disk instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' The physical conditions in the accretion disk maelstrom that would have produced the YNC are far from ordinary, so it is no surprise that the mass function of the YNC 4 Proceedings of the 7th Chile-Cologne-Bonn-Symposium: Physics and Chemistry of Star Formation is somewhat top-heavy, with a slope of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='2 (Bartko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' (Note that a top-heavy mass function would also be expected for the surviving members of a cluster that has migrated to the center as a result of tidal friction (Fujii et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=') An alternative model for in situ production of the YNC that has been very popular invokes an in- falling molecular cloud or tidal disruption of a cloud approaching the GBH on a highly eccentric orbit (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Bonnell & Rice 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Generozov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2022, and many others cited by Dinh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' However, Dinh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' (2021) argue that the phase-space volume accessible to such low-angular-momentum clouds is very small, and there is no obvious way of producing them, so they should be quite rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Put another way, it is very difficult to scatter clouds onto approximately radial orbits because of their typically large size, because they participate in Galactic rotation, and because of the likely absence of sufficiently massive per- turbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Invoking cloud collisions to produce infalling clouds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Hobbs & Nayakshin 2009) faces similar constraints (Dinh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Another alternative model for in situ formation of the YNC is one in which the circumnuclear gas disk, fed quasi-continuously from the outside by gas migrating inward from the CMZ, underwent viscous evolution prior to the formation of the YNC, and extended much further in toward the center than it does now, perhaps all the way to the GBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' At that point, some trigger, perhaps enhanced pressure from accretion activity, started a runaway process of star formation within the central half parsec of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' In such a circumstance the combined energy produced by the rapid formation of a massive cluster and the near-Eddington accretion onto the GBH could have expelled much of the central portions of the CND, creating a central cavity such as that which is present today (Morris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Noting that accretion onto the CND is likely continuing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Tress et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2020), and that the lifetimes of the massive stars in the YNC are only a few ×107 years, one can speculate that the formation of a YNC follows a limit cycle, repetitively recreating such a cluster on a time scale limited by the CND growth rate and by the viscous evolution timescale of the CND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Evidence for more recent star formation at various places in the central parsec has been presented (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Eckart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Yusef-Zadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2013, 2015a,b, 2017), but has been questioned (Morris 20211), largely because there are alternative explanations for the phenomena presented as evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Furthermore, obser- vations of the infrared counterparts to the putative protostars and YSOs, or the supporting spectroscopic evidence, have not yet been acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' The issue of whether star formation in the central parsec is a continuing and current process will probably soon be resolved by ongoing observations with JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' References An, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Ramírez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Sellgren, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2011, ApJ, 736, 133 Barnes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Longmore, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Battersby, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2017, MNRAS, 469, 2263 Bartko, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Martins, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Trippe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2010, ApJ, 708, 834 Bonnell, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' & Rice, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2008, Science, 321, 1060 Butterfield, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Lang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Mills, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Ott, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2018, ApJ, 852, 11 Chuss, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Davidson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Dotson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2003, ApJ, 599, 1116 Clark, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Patrick, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Najarro, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Evans, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Lohr, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2021, A&A, 649, A43 Dahmen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Huttemeister, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Wilson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Mauersberger, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 1998, A&A, 331, 959 Dinh, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Salas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Naoz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2021, ApJ, 920, 79 Dong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Wang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2012, MNRAS, 425, 884 Eckart, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Mužić, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Yazici, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2013, A&A, 551, A18 Ferrière, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Gillard, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Jean, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2007, A&A, 467, 611 Fujii, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Iwasawa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Funato, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Makino, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2008, ApJ, 686, 1082 Generozov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Nayakshin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Madigan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2022, MNRAS, 512, 4100 Gerhard, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2001, ApJL, 546, L39 Ginsburg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Bally, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Barnes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2018, ApJ, 853, 171 Habibi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Stolte, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Harfst, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2014, A&A, 566, A6 Hankins, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Lau, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Mills, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Herter, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2019, ApJ, 877, 22 Hankins, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Lau, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Radomski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2020, ApJ, 894, 55 Henshaw, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Barnes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Battersby, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='11223 Henshaw, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Longmore, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Kruijssen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2016a, MNRAS, 463, L122 Henshaw, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Longmore, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Kruijssen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2016b, MNRAS, 457, 2675 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='v=8ezDLs_MJxw Galactic Center Star Formation 5 Hobbs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' & Nayakshin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2009, MNRAS, 394, 191 Hosek, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Do, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2022, ApJ, 939, 68 Immer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Kauffmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Pillai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Ginsburg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Menten, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2016, A&A, 595, A94 Kauffmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Pillai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Zhang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2013, ApJL, 765, L35 Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Figer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2000, ApJ, 545, 301 Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Figer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2004, ApJL, 607, L123 Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' & Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2003, ApJ, 597, 312 Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 1999, ApJ, 525, 228 Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Saitoh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Jeon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2011, ApJL, 735, L11 Kruijssen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Dale, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Longmore, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2015, MNRAS, 447, 1059 Kruijssen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Dale, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Longmore, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2019, MNRAS, 484, 5734 Kruijssen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Longmore, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Elmegreen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2014, MNRAS, 440, 3370 Krumholz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2021, in Astronomical Society of the Pacific Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 528, New Horizons in Galactic Center Astronomy and Beyond, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Tsuboi & T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Oka, 19 Krumholz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' & Kruijssen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2015, MNRAS, 453, 739 Levin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' & Beloborodov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2003, ApJL, 590, L33 Longmore, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2014, in IAU Symposium, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 303, IAU Symposium, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Sjouwerman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Lang, & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Ott, 132–138 Longmore, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Bally, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Testi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2013a, MNRAS, 429, 987 Longmore, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Kruijssen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Bally, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2013b, MNRAS, 433, L15 Longmore, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Kruijssen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Bastian, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2014, in Protostars and Planets VI, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Beuther, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Klessen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Dullemond, & T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Henning, 291–314 Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2018, Astrophysics and Space Science Library, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 424, Massive Young Clusters Near the Galactic Center, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Stahler, 69 Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Do, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Ghez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2013, ApJ, 764, 155 Lu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Mills, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Ginsburg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2019, ApJS, 244, 35 Mauerhan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Cotera, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Dong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2010a, ApJ, 725, 188 Mauerhan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Muno, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Stolovy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Cotera, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2010b, ApJ, 710, 706 Mills, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Lang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2011, ApJ, 735, 84 Mills, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' & Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2013, ApJ, 772, 105 Molinari, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Bally, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Noriega-Crespo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2011, ApJL, 735, L33+ Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 1989, in IAU Symposium, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 136, The Center of the Galaxy, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' Morris, 171 Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 1993, ApJ, 408, 496 Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Ghez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Becklin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 1999, Advances in Space Research, 23, 959 Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' & Serabyn, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 1996, ARAA, 34, 645 Nayakshin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Cuadra, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Springel, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2007, MNRAS, 379, 21 Portegies Zwart, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Makino, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', McMillan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Hut, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2002, ApJ, 565, 265 Rathborne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Longmore, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Jackson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2014, ApJL, 795, L25 Riquelme, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Bronfman, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Mauersberger, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', May, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Wilson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2010, A&A, 523, A45+ Salas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Naoz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2020, arXiv e-prints, arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='04170 Sanders, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 1992, Nature, 359, 131 Serabyn, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Lacy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Achtermann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 1992, ApJ, 395, 166 Sofue, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 1995, PASJ, 47, 527 Sormani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Sobacchi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Fragkoudi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2018, MNRAS, 481, 2 Sormani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Tress, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Glover, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2020, MNRAS, 497, 5024 Spergel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' & Blitz, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 1992, Nature, 357, 665 Tress, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Sormani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Glover, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2020, MNRAS, 499, 4455 Tsuboi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Tadaki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content='-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Miyazaki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', & Handa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2011, PASJ, 63, 763 Yelda, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Ghez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2014, ApJ, 783, 131 Yusef-Zadeh, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Cotton, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Wardle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2017, MNRAS, 467, 922 Yusef-Zadeh, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Hewitt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Arendt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2009, ApJ, 702, 178 Yusef-Zadeh, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Lacy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Wardle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2010, ApJ, 725, 1429 Yusef-Zadeh, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Roberts, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Wardle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2015a, ApJL, 801, L26 Yusef-Zadeh, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Royster, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Wardle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2013, ApJL, 767, L32 Yusef-Zadeh, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Wardle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', Sewilo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} +page_content=' 2015b, ApJ, 808, 97' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFRT4oBgHgl3EQfCjdP/content/2301.13469v1.pdf'} diff --git a/JtE5T4oBgHgl3EQfXg-b/content/2301.05567v1.pdf b/JtE5T4oBgHgl3EQfXg-b/content/2301.05567v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..afd4f6e8fc528439e4c482baf92850d007bae60d --- /dev/null +++ b/JtE5T4oBgHgl3EQfXg-b/content/2301.05567v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2a731a57f5cf78d718d1800d911ce7d2cd962384c27cf9d45dfcd4f650a8f5c7 +size 654179 diff --git a/JtE5T4oBgHgl3EQfXg-b/vector_store/index.faiss b/JtE5T4oBgHgl3EQfXg-b/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..1bdfcccb7f57568df656944eecb764c359ef720d --- /dev/null +++ b/JtE5T4oBgHgl3EQfXg-b/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3a7c9651a747187651b92611d9fcae1b82f09c60570b30853c9fa81f80314162 +size 2818093 diff --git a/JtE5T4oBgHgl3EQfXg-b/vector_store/index.pkl b/JtE5T4oBgHgl3EQfXg-b/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..395556775465a086b65ade4f86200b0fda589a90 --- /dev/null +++ b/JtE5T4oBgHgl3EQfXg-b/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:14af16b4b5ed78cc982031b5294cb2d92162258a3a7ef31db3913577b898375e +size 98713 diff --git a/K9A0T4oBgHgl3EQfC_82/vector_store/index.faiss b/K9A0T4oBgHgl3EQfC_82/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..961162f2a61b23625a35556d8d4928cc9c01621c --- /dev/null +++ b/K9A0T4oBgHgl3EQfC_82/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2dc7402d07d73b5b5c5aa9c7c7e9180d07a63afe8b3462d938f3e034df0c595a +size 2162733 diff --git a/K9A0T4oBgHgl3EQfC_82/vector_store/index.pkl b/K9A0T4oBgHgl3EQfC_82/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..c36ac6e34efbf85ada3957e26cfd83f44dfa4f00 --- /dev/null +++ b/K9A0T4oBgHgl3EQfC_82/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8335427ec86f6c595638a15b50bfa5f467ed7e7615af741c4cefe58d2498a822 +size 79382 diff --git a/LNFRT4oBgHgl3EQf1jiU/vector_store/index.faiss b/LNFRT4oBgHgl3EQf1jiU/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..3bbc8fa36db9a7a8e91c91e96c0f320bc3b728b0 --- /dev/null +++ b/LNFRT4oBgHgl3EQf1jiU/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8aa18ef9a01c8eea2da7fcd838e14f0084418ae7c97dd6b30eb0966e1796af1e +size 4325421 diff --git a/MNE3T4oBgHgl3EQfwQsY/content/2301.04700v1.pdf b/MNE3T4oBgHgl3EQfwQsY/content/2301.04700v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ee1e474b9adbe4c90e29fbefb5e291ccd2458554 --- /dev/null +++ b/MNE3T4oBgHgl3EQfwQsY/content/2301.04700v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:347a5cff2bd34c75419701b2e5c02a8a4e844747f905081f584737b06157b10b +size 468047 diff --git a/MNE3T4oBgHgl3EQfwQsY/vector_store/index.pkl b/MNE3T4oBgHgl3EQfwQsY/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..baf2c710f574823f5c5b56ef06dc694ac4dbd4d3 --- /dev/null +++ b/MNE3T4oBgHgl3EQfwQsY/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:72d727032239c8705c78af2a9676db44b0c2fe62386b48cb8e8507ace464b42a +size 173920 diff --git a/MNFLT4oBgHgl3EQfMy8Y/content/2301.12017v1.pdf b/MNFLT4oBgHgl3EQfMy8Y/content/2301.12017v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..065f82ce4d324ccf167640dd25d958ea9513939b --- /dev/null +++ b/MNFLT4oBgHgl3EQfMy8Y/content/2301.12017v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7f3dc060933ca6441eb51b512dfd6570b008647f14a2a620004072f09c9f020b +size 1492625 diff --git a/MNFLT4oBgHgl3EQfMy8Y/vector_store/index.faiss b/MNFLT4oBgHgl3EQfMy8Y/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..3ecc16b9df08e436a6fdcc843980e41f8d3bae45 --- /dev/null +++ b/MNFLT4oBgHgl3EQfMy8Y/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5bf15149ddea03a44a2506e45ace7818ecaf3297c9f5353dfa4f00fe40aac409 +size 6946861 diff --git a/MtFIT4oBgHgl3EQfcSt1/content/tmp_files/2301.11265v1.pdf.txt b/MtFIT4oBgHgl3EQfcSt1/content/tmp_files/2301.11265v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f45f6c81ce9ecb8b81f7ef1e5cec9cd8fec4da59 --- /dev/null +++ b/MtFIT4oBgHgl3EQfcSt1/content/tmp_files/2301.11265v1.pdf.txt @@ -0,0 +1,875 @@ +Quantum and quantum-inspired optimization +for solving the minimum bin packing problem +A.A Bozhedarov,1 A.S. Boev,1 S.R. Usmanov,1 G.V. Salahov,1 E.O. Kiktenko,1 and A.K. Fedorov1 +1Russian Quantum Center, Skolkovo, Moscow 143025, Russia +Quantum computing devices are believed to be powerful in solving hard computational tasks, +in particular, combinatorial optimization problems. In the present work, we consider a particular +type of the minimum bin packing problem, which can be used for solving the problem of filling +spent nuclear fuel in deep-repository canisters that is relevant for atomic energy industry. We first +redefine the aforementioned problem it in terms of quadratic unconstrained binary optimization. +Such a representation is natively compatible with existing quantum annealing devices as well as +quantum-inspired algorithms. We then present the results of the numerical comparison of quantum +and quantum-inspired methods. Results of our study indicate on the possibility to solve industry- +relevant problems of atomic energy industry using quantum and quantum-inspired optimization. +I. +INTRODUCTION +Optimization is a primary tool with numerous ap- +plications across various industries [1]. +Specific atten- +tion is traditionally paid to combinatorial optimization +problems, which are especially difficult in the view of +the so-called curse of dimensionality — a dramatic in- +crease of the complexity with increasing problem size. +One of the notable classes of combinatorial optimiza- +tion problems is quadratic unconstrained binary opti- +mization (QUBO) [2, 3], which appears in various ap- +plications. +Quantum computing devices, both univer- +sal and specialized, are considered to be useful in solv- +ing such computational problems [3–7]. An idea behind, +generally speaking, is to encode a cost function in a +quantum Hamiltonian [2], so that its low-energy state +corresponds to the minimum of the cost function. Sev- +eral architectures of quantum computing devices, which +are of interests for solving optimization problems, have +been developed [3]. Specifically, quantum annealing de- +vices based on superconducting qubits, which are able +to solve problems of a non-trivial size [8], have been +used to tackle various industry-relevant tasks, includ- +ing quantum chemistry calculations [9, 10], (lattice) pro- +tein folding [11, 12], genome assembly [13, 14], solving +polynomial [15] and linear systems of equations [15], fi- +nancial optimization [16–24], traffic optimization [25–27], +scheduling [28–33], railway conflict management [32, 33], +and many others (for a review, see Ref. [3]). An alter- +native approach is to use programmable quantum sim- +ulators based on atomic arrays [34], where the most re- +cent advances include a demonstration of a superlinear +quantum speedup in finding exact solutions for the hard- +est maximum independent set graphs [35]. One may also +note that gate-based running variational optimization al- +gorithms, mainly quantum approximate optimization al- +gorithm [7], also offer interesting possibilities for com- +binatorial optimization [36, 37]. Although such devices +in principle are able to demonstrate quantum computa- +tional advantage in near future, still various limitations +make it challenging to use them for solving problems of +industry relevant sizes. +The problem of a clear comparison between quantum +and classical algorithms, which can be used to highlight +the quantum origin of the speed up, is also nontrivial [38]. +As a result of such a comparison, a new class of algo- +rithms and techniques, know as quantum-inspired, has +been developed [39, 40]. +As soon as these algorithms +are compatible with currently existing (classical) hard- +ware, analyzing their limiting capabilities and advantages +over classical approaches are required towards their use +in practice. Recently, solving the wavelength assignment +problem in telecommunication using quantum-inspired +algorithm SimCIM [39] has been demonstrated [41]. For +a wide range of benchmark of quantum-inspired heuristic +solvers for quadratic unconstrained binary optimization, +namely D-Wave Hybrid Solver Service, Toshiba Simu- +lated Bifurcation Machine, Fujitsu Digital Annealer, and +simulated annealing, see Ref. [42]. +A specific class of a combinatorial optimization prob- +lem that appear across many industry application is the +minimum bin packing problem, where items of different +sizes must be allocated into a finite number of bins (con- +tainers), each of a fixed given capacity, in a way that +minimizes the number of bins [1]; this problem is known +to be NP-hard. A particular application of this problem +is optimization of spent nuclear fuel (SNF) filling in can- +isters for the deep repository. According to existing stan- +dards, the deposing should be realized by using special +(deep-repository) canisters, so that the maximum heat +output per canister does not exceed the limiting value. +The tasks of the optimization of the SNF using canis- +ter filling (CF) is then clearly linked to the aforemen- +tioned minimum bin packing problem [43]. The use of +combinatorial methods to optimize the filling of SNF in +metal canisters for the final deep repository, according to +the maximal allowed thermal power per canister and the +limit in the number of spent-fuel assemblies per canister, +has been demonstrated [43]. In this context, quantum +and quantum-inspired tools are now considered as a way +to solve this problem for larger sizes; in particular, op- +timization of fuel arrangements in nuclear power plants +using quantum tools has been considered [44]. +In this work, we present a method for solving the +arXiv:2301.11265v1 [quant-ph] 26 Jan 2023 + +2 +SNF management problem using quantum and quantum- +inspired annealing. We first formulate the problem in the +QUBO form, which allows solving this problem using var- +ious annealing tools, including quantum annealing. We +then benchmark its solution using quantum annealing +device from D-Wave1, quantum-inspired algorithm Sim- +CIM [39], and quantum-inspired Simulated Bifurcation +Machine (SBM)2 [45]. +Our results indicate the possi- +bility to solve such an industry-relevant problem using +quantum and quantum-inspired annealing. +Our work is organized as follows. In Sec. II, we for- +mulate the CF optimization problem in the QUBO form, +which makes it suitable for solving this using quantum +and quantum-inspired annealing. Sec. III, describes the +numerical analysis setup; there we also benchmark a so- +lution of the CF problem using available the quantum an- +nealer and quantum-inspired annealing algorithms. We +summarize our results and conclude in Sec. IV. +II. +CANISTER FILLING OPTIMIZATION +PROBLEM +The SNF is a subject of the deposition for further +safe keeping. Existing industrial standards require that +the deposing should be realized using special (deep- +repository) canisters, so that the total heat output per +canister does not exceed the limiting value Pmax. At the +same time, there is a minimum number of spent fuel el- +ements that can be stored in one canister Nmin. This is +a subject of the SNF management problem, which is im- +portant for the optimal use of existing canisters without +violating standards. The SNF management problem can +be formulated as a combinatorial optimization problem +as follows. +Let n be the total number of spent fuel elements, m +is the total number of available canisters, and pi is the +heat output of the i-th fuel element. Let us introduce +additional variables for indication of fuel element location +and canister usage as following: +xij = +� +1, +if i-th fuel element is in j-th canister, +0, +otherwise; +(1) +yj = +� +1, +if j-th canister is being used, +0, +otherwise. +(2) +Then, one may formulate optimization problem in the +1 The results of the present paper are based on the data that have +been collected during the availability of the device. +2 The results of the present paper are based on the data that have +been collected during the availability of the algorithm. +following way: +M = +m +� +j=1 +yj → min, +(3) +such that +n +� +i=1 +pixij ≤ Pmax +∀j ∈ {1, . . . , m}, +(4) +m +� +j=1 +xij = 1 +∀i ∈ {1, . . . , n}, +(5) +n +� +i=1 +xij ≥ Nminyj +∀j ∈ {1, . . . , m}, +(6) +xij ≤ yj +∀i ∈ {1, . . . , n}, ∀j ∈ {1, . . . , m}, +(7) +where condition (4) restricts the maximum heat output +per one canister, condition (5) implies that every fuel ele- +ment placed only in one canister, condition (6) stands for +minimal filling of every used canister, and condition (7) +binds the variables xij and yj so that the placement of the +fuel elements matches the vector of the used canisters. +The main step in solving an optimization problem us- +ing quantum and quantum-inspired annealing is to map +the problem of interest to the energy Hamiltonian, so the +quantum device could find the ground state that corre- +sponds to the optimum value of the objective function. +The natural way of mathematical description of a quan- +tum annealer is the Ising spin Hamiltonian that can be +transformed into QUBO problem in a straightforward +way. We are to formulate mapping of the CF problem +into QUBO form. In general, a QUBO problem may be +formulated using matrix notation as following: +zT Qz → min, +(8) +where z is the vector of binary decision variables and Q +is a square symmetric matrix of constants. +It is necessary to include optimization constraints by +adding penalty terms to the objective function. Let us +represent optimization constraints (4)–(7) in the QUBO +form. Constraint (4) can be represented as +H1 = +m +� +j=1 +� n +� +i=1 +pixij + +s−1 +� +l=0 +2lalj − Pmax +�2 +, +(9) +where s = ⌈log2 Pmax⌉ and alj denotes auxiliary bi- +nary variables, which are required to represent (4) in the +form of equality: �s +l=0 2lalj is certain non-negative in- +teger that corresponds to a difference between Pmax and +�n +i=1 pixij. Constraints (5) and (6) take the following +form: +H2 = +n +� +i=1 +� +� +m +� +j=1 +xij − 1 +� +� +2 +, +(10) + +3 +and +H3 = +m +� +j=1 +� n +� +i=1 +xij − +k−1 +� +l=0 +2lblj − Nminyj +�2 +, +(11) +correspondingly. +Here k = ⌈log2 Nmin⌉ and blj are +another auxiliary binary variables used to represent +non-negative difference �k +l=0 2lblj between �n +i=1 xij and +Nminyj. The final constraint (7) takes the form +H4 = +n +� +i=1 +m +� +j=1 +(xij − xijyj) +(12) +The problem Hamiltonian then consist of two main +components: +H = A +m +� +j=1 +yj + B +4 +� +r=1 +Hr +(13) +where A is a positive constant and B stands for a positive +penalty value. Parameters A and B should be set man- +ually, using the following criteria. Penalty value should +be high enough to keep final solution from violating con- +straints. At the same time, too large penalty value may +overwhelm the objective function so it becomes hard to +distinguish solutions of different quality. Therefore, the +solution of the optimization problem requires finding op- +timal values of the variables xij, yj, alj and blj. +More +details about the total number of binary variables may +be found in Subsec. III A. +One may transform QUBO problem into Ising Hamil- +tonian using following approach: +zi = σZ +i + 1 +2 +∈ {0, 1}, +(14) +where σZ +i = ±1 and vector z contains all optimized vari- +ables xij, yj, alj, and blj. +III. +BENCHMARKING PROCEDURE +In order to evaluate the feasibility of the proposed +scheme of solving the SNF problem in the QUBO form, +we conduct a comparison of existing quantum annealing +device and quantum-inspired annealing simulator as in- +struments to solving CF combinatorial problem. +A. +Generating of synthetic dataset +We first prepare a synthetic dataset of 80 problem in- +stances with number of fuel elements ranging from 3 to +10 and the maximum number of canisters equal to 3. For +each problem size, 10 different cell configurations with +various heat output are prepared (plus single trivial case +with 2 elements). The optimal allocation for all instances +is known in advance and requires at most 2 canisters. +The general idea of dataset is to create problem with +minimal possible QUBO sizes for guaranteed best so- +lution achievement by annealers. +Using notation from +Eqs.(9)–(12), the total number of logical transformation +variables may be represented as follows: +D = m (1 + n + s + k) , +(15) +where m is the available number of canisters, n is the +number of fuel elements, s = ⌈log2 Pmax⌉ and k = +⌈log2 Nmin⌉ is the number of auxiliary variables used in +equalities (9) and (11), correspondingly. As a result, the +smallest-size problem with m = 2, n = 2, s = 2, k = 0 +requires 10 logical variables, we mark it as a trivial case +(see Fig. 1). +We use only fixed configurations, where the optimal +number of canisters is 2 and the feasible solution without +constraint violation can have 3 canisters, in other words, +we restrict problem samples to cases, where m = 3 and +M = 2. The maximum capacities of canisters are equal +with limit 2s=4 −1, where k is also fixed and equals zero, +since we do not use minimum elements constraint when +Nmin = 1. This compression allows us to compute model +problems on present quantum hardware and evaluate the +dependence of problem size and performance. Tasks with +the same elements quantity are also have identical QUBO +size for avoiding additional deviation in data; see Table I. +We use public access to D-Wave 5000 Advantage system +with Pegasus topology processor [46] to run our quantum +annealing algorithm. +Elements +quantity QUBO size +Physical qubits +Number +of qubits +Heuristic, +mean +Heuristic, +std +2 +10 +20 +18.4 +2.3 +3 +21 +80 +64.0 +3.5 +4 +24 +90 +85.0 +1.6 +5 +27 +102 +103.8 +8.6 +6 +33 +157 +153.6 +5.5 +7 +36 +170 +185.6 +7.6 +8 +39 +185 +226.6 +15.6 +9 +42 +246 +246.0 +10.6 +10 +45 +262 +280.0 +5.3 +TABLE I. Synthetic dataset scheme. Heuristic embeddings +on physical qubits of D-Wave device were found via D-Wave +Leap SDK for 5 different random samples. +B. +Benchmarking +Each problem instance has been further transformed +into the QUBO matrix and run through both quantum +and quantum-inspired instruments, specifically, the D- +Wave quantum annealer and two quantum-inspired al- +gorithms (SimCIM [39] and SBM [45]). SimCIM algo- +rithm [39] is based on the method of efficient simulation + +4 +2.46e-01 (D-Wave) +6.44e+02 +8.81e+02 +2.12e+03 +5.67e+00 +5.95e+00 +6.64e+00 +1.06e+01 +4.08e+01 +2.56e+02 +3.21e+02 +4.81e+02 +8.26e+02 +2.30e-01 (SBM) +4.21e+00 +2.48e+01 +8.76e+01 +3.13e+02 +4.35e+03 +1.29e+04 +9.86e+03 +3.06e+04 +2 +3 +4 +5 +6 +7 +8 +9 +10 +2 +5 +1 +2 +5 +10 +2 +5 +100 +2 +5 +1000 +2 +5 +10k +2 +5 +100k +D-Wave +SimCIM +SBM +Number of elements +Time-to-Solution +FIG. 1. Comparison of the performance of quantum and quantum-inspired methods for bin packing problem based on synthetic +data: we compare TTS (mean and standard deviation) for quantum device D-Wave and two quantum-inspired optimization +algorithms, SimCIM and SBM. +of Coherent Ising Machine [47] using classical computer. +As it has been shown, SimCIM outperforms Coherent +Ising Machine in terms of samples quality and speed of +computation, and that is why it has been chosen as a +benchmarking tool for comparative analysis. +Simulated Bifurcation algorithm (SBM) [45] is a +heuristic algorithm for combinatorial optimization. Its +workflow is inspired by quantum bifurcation machine [48] +that is based on nonlinear oscillators and implements +quantum adiabatic algorithm for solving optimization +problems. +We run D-Wave experiments in a pure quantum mode +using the Advantage chip featuring 5000 qubits with 15- +way connectivity. +In order to embed QUBO problem +into physical qubit layout we utilized clique embedding +supported by D-Wave Leap SDK. SimCIM was run on +Xeon E3-1230v5 4x3,4GHz, 16 GB DDR4, GeForce GTX +1080. Comparison results are shown in Fig. 1. +C. +Analysis +As a figure of merit in our benchmarking procedure, +we use time-to-solution (TTS). The TTS means a time +that is needed to a heuristic algorithm to find the solution +(ground state energy) with 99% success probability. It is +given by +TTS = taR99, +(16) +where ta is the annealing time (default value of ta for +D-Wave is 20µs), and R99 stands for the number of rep- +etition that is needed for the desired success probabil- +ity [31, 49]. It can be calculated as follows: +R99 = log(1 − 0.99) +log(1 − θ) , +(17) +where θ is the estimated success probability of each run. +All tasks were grouped by fuel elements quantity for +demonstrating results (see Fig. 1). We note that the D- +Wave annealer shows good result in problem solving in +the small-size cases (2 possible canisters and 2 elements), +while optimal solution was not found for 6 and more ele- +ments problem. The standard deviation of TTS is signif- +icantly increase with elements quantity for all methods. +This is caused by an exponential growth of the space of +possible solutions, leading to a decrease in the probability +of finding the optimal solution. As a result, the annealing +process often terminates at a suboptimal point instead +of the ground state. This is especially true for complex +problems that require a large number of variables to be +taken into consideration. +While the main obstacle of quantum-inspired optimiza- +tion methods is complexity and the size of the space of +possible solutions, for quantum annealing a very impor- +tant parameter is the gap between the ground state and +the first excited state of the Hamiltonian. The smaller +the gap, the slower the adiabatic evolution of the quan- +tum system should proceed in order to stay in the ground + +5 +Annealing time, µs Success probability TTS, µs +20 +0.277 +284 +40 +0.303 +511 +80 +0.343 +878 +TABLE II. Dependence of success probability and TTS for +trivial case (2 fuel elements) for D-Wave device. +state. However, a long evolution time increases the influ- +ence of quantum decoherence and can lead to incorrect +solutions. +IV. +CONCLUSION +Quantum computing is a promising technique for solv- +ing combinatorial optimization problems. In our work, +we have demonstrated the potential of quantum and +quantum-inspired tools to solve computational problems +of the minimum bin packing problem, which is formu- +lated as the problem of atomic energy industry, As a tar- +get problem, we have chosen the optimization of spent +nuclear fuel filling in canisters for the deep repository. +The CF problem has been formulated in a QUBO ma- +trix form (see Eqs. (9)–(12)) and was solved using exist- +ing quantum annealer and quantum-inspired annealing +algorithms. We note that the current development level +of quantum computing devices does not allow to solve +large-scale practical problems, however, it is possible to +scale a size of the problem for the next generation of +quantum computers. Moreover, such research helps to +identify practical-oriented tasks that may be solved more +efficiently by quantum computing. +ACKNOWLEDGEMENTS +We acknowledge use of the D-Wave quantum annealer +and Toshiba Simulated Bifurcation Machine for this +work; the views expressed are those of the authors and +do not reflect the official policy or position of D-Wave +and Toshiba. The results of the present paper are based +on the data that have been collected during the availabil- +ity of the D-Wave quantum annealer and the Simulated +Bifurcation algorithm. This work was supported by Rus- +sian Science Foundation (19-71-10092). +APPENDIX +In our benchmarking procedure, we use various accessi- +ble parameters (particularly, annealing time, embedding +type and chain strength) of the quantum hardware on +the final solution quality. For the simplest task, we have +observed that increasing annealing time gives a better +success probability (see Table II), but in the same time +TTS is getting worse, so annealing time was set to 20 +µs (the. default value). The number of annealing runs +is set to 104 (maximum possible value). Comparing dif- +ferent embeddings (see Table I), we analyze deviation +of physical qubits number on five different random sam- +ples and decided that for better experiment performance +using stable stable clique embedding is more preferable. +According to Ref. [31] we choose custom optimized chain +strength instead of other variants such as maximum ab- +solute value in QUBO. Thereby we have used mostly the +standard configuration of the D-Wave processor during +our experiments, so we do not have any specific require- +ments on the weights/couplers in the model. We use only +pure quantum regime to obtain solution for each task. +[1] V. T. Paschos, ed., Paradigms of combinatorial optimiza- +tion, 2nd ed., ISTE. (John Wiley & Sons, Inc., London : +Hoboken, 2014) p. 815. +[2] A. Lucas, Frontiers in Physics 2, 5 (2014). +[3] A. K. Fedorov, N. Gisin, S. M. Beloussov, +and A. I. +Lvovsky, “Quantum computing at the quantum advan- +tage threshold: a down-to-business review,” (2022). +[4] E. Farhi, J. Goldstone, S. Gutmann, +and M. Sipser, +“Quantum computation by adiabatic evolution,” (2000). +[5] A. Das and B. K. Chakrabarti, Rev. Mod. Phys. 80, 1061 +(2008). +[6] T. Albash and D. A. Lidar, Rev. Mod. Phys. 90, 015002 +(2018). +[7] E. Farhi, J. Goldstone, +and S. Gutmann, “A quan- +tum approximate optimization algorithm,” +(2014), +arXiv:1411.4028 [quant-ph]. +[8] A. +D. +King, +J. +Raymond, +T. +Lanting, +S. +V. +Isakov, M. Mohseni, G. Poulin-Lamarre, S. Ejtemaee, +W. Bernoudy, I. Ozfidan, A. Y. Smirnov, M. Reis, +F. Altomare, M. Babcock, C. Baron, A. J. Berkley, +K. Boothby, P. I. Bunyk, H. Christiani, C. Enderud, +B. Evert, R. Harris, E. Hoskinson, S. Huang, K. Jooya, +A. Khodabandelou, N. Ladizinsky, R. Li, P. A. Lott, +A. J. R. MacDonald, D. Marsden, G. Marsden, T. Med- +ina, R. Molavi, R. Neufeld, M. Norouzpour, T. Oh, +I. Pavlov, I. Perminov, T. Prescott, C. Rich, Y. Sato, +B. Sheldan, G. Sterling, L. J. Swenson, N. Tsai, M. H. +Volkmann, J. D. Whittaker, W. Wilkinson, J. Yao, +H. Neven, J. P. Hilton, E. Ladizinsky, M. W. John- +son, and M. H. Amin, Nature Communications 12, 1113 + +6 +(2021). +[9] M. Streif, F. Neukart, and M. Leib, in Quantum Tech- +nology and Optimization Problems, edited by S. Feld and +C. Linnhoff-Popien (Springer International Publishing, +Cham, 2019) pp. 111–122. +[10] D. A. Chermoshentsev, A. O. Malyshev, E. S. Tiunov, +D. Mendoza, A. Aspuru-Guzik, A. K. Fedorov, +and +A. I. Lvovsky, “Polynomial unconstrained binary op- +timisation inspired by optical simulation,” +(2021), +arXiv:2106.13167 [quant-ph]. +[11] A. Perdomo-Ortiz, N. Dickson, M. Drew-Brook, G. Rose, +and A. Aspuru-Guzik, Scientific Reports 2, 571 (2012). +[12] T. Babej, C. Ing, +and M. Fingerhuth, “Coarse-grained +lattice protein folding on a quantum annealer,” (2018), +arXiv:1811.00713 [quant-ph]. +[13] A. S. Boev, A. S. Rakitko, S. R. Usmanov, A. N. +Kobzeva, I. V. Popov, V. V. Ilinsky, E. O. Kiktenko, +and A. K. Fedorov, Scientific Reports 11, 13183 (2021). +[14] A. Sarkar, Z. Al-Ars, and K. Bertels, PLOS ONE 16, 1 +(2021). +[15] C. C. Chang, A. Gambhir, T. S. Humble, and S. Sota, +Scientific Reports 9, 10258 (2019). +[16] R. Or´us, S. Mugel, and E. Lizaso, Reviews in Physics 4, +100028 (2019). +[17] S. Mugel, C. Kuchkovsky, E. Sanchez, S. Fernandez- +Lorenzo, J. Luis-Hita, E. Lizaso, +and R. Orus, “Dy- +namic portfolio optimization with real datasets using +quantum processors and quantum-inspired tensor net- +works,” (2020), arXiv:2007.00017 [quant-ph]. +[18] E. Grant, T. S. Humble, and B. Stump, Phys. Rev. Ap- +plied 15, 014012 (2021). +[19] D. Herman, C. Googin, X. Liu, A. Galda, I. Safro, Y. Sun, +M. Pistoia, and Y. Alexeev, “A survey of quantum com- +puting for finance,” (2022). +[20] R. Or´us, S. Mugel, +and E. Lizaso, Phys. Rev. A 99, +060301 (2019). +[21] G. Rosenberg, P. Haghnegahdar, P. Goddard, P. Carr, +K. Wu, and M. L. de Prado, IEEE Journal of Selected +Topics in Signal Processing 10, 1053 (2016). +[22] G. Rosenberg (2016). +[23] M. R. Andrew Milne and P. Goddard (2017). +[24] M. Vesely, “Application of quantum computers in foreign +exchange reserves management,” (2022). +[25] F. Neukart, G. Compostella, C. Seidel, D. von Dollen, +S. Yarkoni, and B. Parney, Frontiers in ICT 4, 29 (2017). +[26] D. Inoue, A. Okada, T. Matsumori, K. Aihara, +and +H. Yoshida, Scientific Reports 11, 3303 (2021). +[27] H. Hussain, M. B. Javaid, F. S. Khan, A. Dalal, +and +A. Khalique, Quantum Information Processing 19, 312 +(2020). +[28] D. Venturelli, D. J. J. Marchand, and G. Rojo, “Quan- +tum annealing implementation of job-shop scheduling,” +(2016), arXiv:1506.08479 [quant-ph]. +[29] K. Ikeda, Y. Nakamura, +and T. S. Humble, Scientific +Reports 9, 12837 (2019). +[30] A. Sadhu, +S. Zaman, +K. Das, +A. Banerjee, +and +F. Khan, “Quantum annealing for solving a nurse- +physician scheduling problem in covid-19 clinics,” +(2020). +[31] T. Stollenwerk, +V. Michaud, +E. Lobe, +M. Picard, +A. Basermann, and T. Botter, “Image acquisition plan- +ning for earth observation satellites with a quantum an- +nealer,” (2020), arXiv:2006.09724 [quant-ph]. +[32] K. Domino, M. Koniorczyk, K. Krawiec, K. Ja�lowiecki, +and B. Gardas, “Quantum computing approach to rail- +way dispatching and conflict management optimization +on single-track railway lines,” (2021), arXiv:2010.08227 +[cs.ET]. +[33] K. Domino, M. Koniorczyk, K. Krawiec, K. Ja�lowiecki, +S. Deffner, +and B. Gardas, “Quantum annealing in +the nisq era: +railway conflict management,” +(2021), +arXiv:2112.03674 [quant-ph]. +[34] L. +Henriet, +L. +Beguin, +A. +Signoles, +T. +Lahaye, +A. Browaeys, G.-O. Reymond, +and C. Jurczak, Quan- +tum 4, 327 (2020). +[35] S. +Ebadi, +A. +Keesling, +M. +Cain, +T. +T. +Wang, +H. +Levine, +D. +Bluvstein, +G. +Semeghini, +A. +Om- +ran, J.-G. Liu, R. Samajdar, X.-Z. Luo, B. Nash, +X. Gao, B. Barak, E. Farhi, S. Sachdev, N. Gemelke, +L. Zhou, S. Choi, H. Pichler, S.-T. Wang, M. Greiner, +V. Vuleti´c, and M. D. Lukin, Science 376, 1209 (2022), +https://www.science.org/doi/pdf/10.1126/science.abo6587. +[36] M. P. Harrigan, K. J. Sung, M. Neeley, K. J. Satzinger, +F. Arute, K. Arya, J. Atalaya, J. C. Bardin, R. Barends, +S. Boixo, M. Broughton, B. B. Buckley, D. A. Buell, +B. Burkett, N. Bushnell, Y. Chen, Z. Chen, B. Chiaro, +R. Collins, W. Courtney, S. Demura, A. Dunsworth, +D. Eppens, A. Fowler, B. Foxen, C. Gidney, M. Giustina, +R. Graff, S. Habegger, A. Ho, S. Hong, T. Huang, +L. B. Ioffe, S. V. Isakov, E. Jeffrey, Z. Jiang, C. Jones, +D. Kafri, K. Kechedzhi, J. Kelly, S. Kim, P. V. Klimov, +A. N. Korotkov, F. Kostritsa, D. Landhuis, P. Laptev, +M. Lindmark, M. Leib, O. Martin, J. M. Martinis, +J. R. McClean, +M. McEwen, +A. Megrant, +X. Mi, +M. Mohseni, W. Mruczkiewicz, J. Mutus, O. Naa- +man, C. Neill, F. Neukart, M. Y. Niu, T. E. O’Brien, +B. O’Gorman, E. Ostby, A. Petukhov, H. Putterman, +C. Quintana, P. Roushan, N. C. Rubin, D. Sank, A. Sko- +lik, V. Smelyanskiy, D. Strain, M. Streif, M. Szalay, +A. Vainsencher, T. White, Z. J. Yao, P. Yeh, A. Zal- +cman, L. Zhou, H. Neven, D. Bacon, E. Lucero, E. Farhi, +and R. Babbush, Nature Physics 17, 332 (2021). +[37] K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, +S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, +J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.- +C. Kwek, +and A. Aspuru-Guzik, Rev. Mod. Phys. 94, +015004 (2022). +[38] S. Boixo, +T. F. Rønnow, +S. V. Isakov, +Z. Wang, +D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, +Nature Physics 10, 218 (2014). +[39] E. S. Tiunov, A. E. Ulanov, +and A. I. Lvovsky, Opt. +Express 27, 10288 (2019). +[40] N. Killoran, T. R. Bromley, J. M. Arrazola, M. Schuld, +N. Quesada, and S. Lloyd, Phys. Rev. Research 1, 033063 +(2019). +[41] A. S. Boev, S. R. Usmanov, A. M. Semenov, M. M. +Ushakova, G. V. Salahov, A. S. Mastiukova, E. O. Kik- +tenko, and A. K. Fedorov, “Quantum-inspired optimiza- +tion for routing and wavelength assignment,” (2022). +[42] H. Oshiyama and M. Ohzeki, Scientific Reports 12, 2146 +(2022). +[43] G. +ˇZerovnik, +L. +Snoj, +and +M. +Ravnik, +Nu- +clear +Science +and +Engineering +163, +183 +(2009), +https://doi.org/10.13182/NSE163-183. +[44] Whyte, Andy and Parks, Geoff, EPJ Web Conf. 247, +06028 (2021). + +7 +[45] H. +Goto, +K. +Tatsumura, +and +A. +R. +Dixon, +Science +Advances +5, +eaav2372 +(2019), +https://www.science.org/doi/pdf/10.1126/sciadv.aav2372. +[46] K. Boothby, P. Bunyk, J. Raymond, and A. Roy, “Next- +generation topology of d-wave quantum processors,” +(2020). +[47] Y. Yamamoto, K. Aihara, T. Leleu, K.-i. Kawarabayashi, +S. Kako, M. Fejer, K. Inoue, and H. Takesue, npj Quan- +tum Information 3, 49 (2017). +[48] H. Goto, Scientific Reports 6, 21686 (2016). +[49] M. Aramon, G. Rosenberg, E. Valiante, T. Miyazawa, +H. Tamura, and H. G. Katzgraber, Frontiers in Physics +7 (2019), 10.3389/fphy.2019.00048. + diff --git a/MtFIT4oBgHgl3EQfcSt1/content/tmp_files/load_file.txt b/MtFIT4oBgHgl3EQfcSt1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..169f9b3001c7d954fd2a62c284f58af62363246d --- /dev/null +++ b/MtFIT4oBgHgl3EQfcSt1/content/tmp_files/load_file.txt @@ -0,0 +1,783 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf,len=782 +page_content='Quantum and quantum-inspired optimization for solving the minimum bin packing problem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='A Bozhedarov,1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Boev,1 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Usmanov,1 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Salahov,1 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Kiktenko,1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Fedorov1 1Russian Quantum Center, Skolkovo, Moscow 143025, Russia Quantum computing devices are believed to be powerful in solving hard computational tasks, in particular, combinatorial optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' In the present work, we consider a particular type of the minimum bin packing problem, which can be used for solving the problem of filling spent nuclear fuel in deep-repository canisters that is relevant for atomic energy industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' We first redefine the aforementioned problem it in terms of quadratic unconstrained binary optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Such a representation is natively compatible with existing quantum annealing devices as well as quantum-inspired algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' We then present the results of the numerical comparison of quantum and quantum-inspired methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Results of our study indicate on the possibility to solve industry- relevant problems of atomic energy industry using quantum and quantum-inspired optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' INTRODUCTION Optimization is a primary tool with numerous ap- plications across various industries [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Specific atten- tion is traditionally paid to combinatorial optimization problems, which are especially difficult in the view of the so-called curse of dimensionality — a dramatic in- crease of the complexity with increasing problem size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' One of the notable classes of combinatorial optimiza- tion problems is quadratic unconstrained binary opti- mization (QUBO) [2, 3], which appears in various ap- plications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Quantum computing devices, both univer- sal and specialized, are considered to be useful in solv- ing such computational problems [3–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' An idea behind, generally speaking, is to encode a cost function in a quantum Hamiltonian [2], so that its low-energy state corresponds to the minimum of the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Sev- eral architectures of quantum computing devices, which are of interests for solving optimization problems, have been developed [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' quantum annealing de- vices based on superconducting qubits,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' which are able to solve problems of a non-trivial size [8],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' have been used to tackle various industry-relevant tasks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' includ- ing quantum chemistry calculations [9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' 10],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' (lattice) pro- tein folding [11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' 12],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' genome assembly [13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' 14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' solving polynomial [15] and linear systems of equations [15],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' fi- nancial optimization [16–24],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' traffic optimization [25–27],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' scheduling [28–33],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' railway conflict management [32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' 33],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' and many others (for a review,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' An alter- native approach is to use programmable quantum sim- ulators based on atomic arrays [34], where the most re- cent advances include a demonstration of a superlinear quantum speedup in finding exact solutions for the hard- est maximum independent set graphs [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' One may also note that gate-based running variational optimization al- gorithms, mainly quantum approximate optimization al- gorithm [7], also offer interesting possibilities for com- binatorial optimization [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Although such devices in principle are able to demonstrate quantum computa- tional advantage in near future, still various limitations make it challenging to use them for solving problems of industry relevant sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' The problem of a clear comparison between quantum and classical algorithms, which can be used to highlight the quantum origin of the speed up, is also nontrivial [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' As a result of such a comparison, a new class of algo- rithms and techniques, know as quantum-inspired, has been developed [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' As soon as these algorithms are compatible with currently existing (classical) hard- ware, analyzing their limiting capabilities and advantages over classical approaches are required towards their use in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Recently, solving the wavelength assignment problem in telecommunication using quantum-inspired algorithm SimCIM [39] has been demonstrated [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' For a wide range of benchmark of quantum-inspired heuristic solvers for quadratic unconstrained binary optimization, namely D-Wave Hybrid Solver Service, Toshiba Simu- lated Bifurcation Machine, Fujitsu Digital Annealer, and simulated annealing, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' A specific class of a combinatorial optimization prob- lem that appear across many industry application is the minimum bin packing problem, where items of different sizes must be allocated into a finite number of bins (con- tainers), each of a fixed given capacity, in a way that minimizes the number of bins [1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' this problem is known to be NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' A particular application of this problem is optimization of spent nuclear fuel (SNF) filling in can- isters for the deep repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' According to existing stan- dards, the deposing should be realized by using special (deep-repository) canisters, so that the maximum heat output per canister does not exceed the limiting value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' The tasks of the optimization of the SNF using canis- ter filling (CF) is then clearly linked to the aforemen- tioned minimum bin packing problem [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' The use of combinatorial methods to optimize the filling of SNF in metal canisters for the final deep repository, according to the maximal allowed thermal power per canister and the limit in the number of spent-fuel assemblies per canister, has been demonstrated [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' In this context, quantum and quantum-inspired tools are now considered as a way to solve this problem for larger sizes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' in particular, op- timization of fuel arrangements in nuclear power plants using quantum tools has been considered [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' In this work, we present a method for solving the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='11265v1 [quant-ph] 26 Jan 2023 2 SNF management problem using quantum and quantum- inspired annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' We first formulate the problem in the QUBO form, which allows solving this problem using var- ious annealing tools, including quantum annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' We then benchmark its solution using quantum annealing device from D-Wave1, quantum-inspired algorithm Sim- CIM [39], and quantum-inspired Simulated Bifurcation Machine (SBM)2 [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Our results indicate the possi- bility to solve such an industry-relevant problem using quantum and quantum-inspired annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Our work is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' II, we for- mulate the CF optimization problem in the QUBO form, which makes it suitable for solving this using quantum and quantum-inspired annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' III, describes the numerical analysis setup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' there we also benchmark a so- lution of the CF problem using available the quantum an- nealer and quantum-inspired annealing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' We summarize our results and conclude in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' CANISTER FILLING OPTIMIZATION PROBLEM The SNF is a subject of the deposition for further safe keeping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Existing industrial standards require that the deposing should be realized using special (deep- repository) canisters, so that the total heat output per canister does not exceed the limiting value Pmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' At the same time, there is a minimum number of spent fuel el- ements that can be stored in one canister Nmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' This is a subject of the SNF management problem, which is im- portant for the optimal use of existing canisters without violating standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' The SNF management problem can be formulated as a combinatorial optimization problem as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Let n be the total number of spent fuel elements, m is the total number of available canisters, and pi is the heat output of the i-th fuel element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Let us introduce additional variables for indication of fuel element location and canister usage as following: xij = � 1, if i-th fuel element is in j-th canister, 0, otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' (1) yj = � 1, if j-th canister is being used, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' (2) Then, one may formulate optimization problem in the 1 The results of the present paper are based on the data that have been collected during the availability of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' 2 The results of the present paper are based on the data that have been collected during the availability of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' following way: M = m � j=1 yj → min, (3) such that n � i=1 pixij ≤ Pmax ∀j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' , m}, (4) m � j=1 xij = 1 ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' , n}, (5) n � i=1 xij ≥ Nminyj ∀j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' , m}, (6) xij ≤ yj ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' , n}, ∀j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' , m}, (7) where condition (4) restricts the maximum heat output per one canister, condition (5) implies that every fuel ele- ment placed only in one canister, condition (6) stands for minimal filling of every used canister, and condition (7) binds the variables xij and yj so that the placement of the fuel elements matches the vector of the used canisters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' The main step in solving an optimization problem us- ing quantum and quantum-inspired annealing is to map the problem of interest to the energy Hamiltonian, so the quantum device could find the ground state that corre- sponds to the optimum value of the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' The natural way of mathematical description of a quan- tum annealer is the Ising spin Hamiltonian that can be transformed into QUBO problem in a straightforward way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' We are to formulate mapping of the CF problem into QUBO form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' In general, a QUBO problem may be formulated using matrix notation as following: zT Qz → min, (8) where z is the vector of binary decision variables and Q is a square symmetric matrix of constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' It is necessary to include optimization constraints by adding penalty terms to the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Let us represent optimization constraints (4)–(7) in the QUBO form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Constraint (4) can be represented as H1 = m � j=1 � n � i=1 pixij + s−1 � l=0 2lalj − Pmax �2 , (9) where s = ⌈log2 Pmax⌉ and alj denotes auxiliary bi- nary variables, which are required to represent (4) in the form of equality: �s l=0 2lalj is certain non-negative in- teger that corresponds to a difference between Pmax and �n i=1 pixij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Constraints (5) and (6) take the following form: H2 = n � i=1 � � m � j=1 xij − 1 � � 2 , (10) 3 and H3 = m � j=1 � n � i=1 xij − k−1 � l=0 2lblj − Nminyj �2 , (11) correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Here k = ⌈log2 Nmin⌉ and blj are another auxiliary binary variables used to represent non-negative difference �k l=0 2lblj between �n i=1 xij and Nminyj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' The final constraint (7) takes the form H4 = n � i=1 m � j=1 (xij − xijyj) (12) The problem Hamiltonian then consist of two main components: H = A m � j=1 yj + B 4 � r=1 Hr (13) where A is a positive constant and B stands for a positive penalty value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Parameters A and B should be set man- ually, using the following criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Penalty value should be high enough to keep final solution from violating con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' At the same time, too large penalty value may overwhelm the objective function so it becomes hard to distinguish solutions of different quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Therefore, the solution of the optimization problem requires finding op- timal values of the variables xij, yj, alj and blj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' More details about the total number of binary variables may be found in Subsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' One may transform QUBO problem into Ising Hamil- tonian using following approach: zi = σZ i + 1 2 ∈ {0, 1}, (14) where σZ i = ±1 and vector z contains all optimized vari- ables xij, yj, alj, and blj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' BENCHMARKING PROCEDURE In order to evaluate the feasibility of the proposed scheme of solving the SNF problem in the QUBO form, we conduct a comparison of existing quantum annealing device and quantum-inspired annealing simulator as in- struments to solving CF combinatorial problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Generating of synthetic dataset We first prepare a synthetic dataset of 80 problem in- stances with number of fuel elements ranging from 3 to 10 and the maximum number of canisters equal to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' For each problem size, 10 different cell configurations with various heat output are prepared (plus single trivial case with 2 elements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' The optimal allocation for all instances is known in advance and requires at most 2 canisters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' The general idea of dataset is to create problem with minimal possible QUBO sizes for guaranteed best so- lution achievement by annealers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Using notation from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' (9)–(12), the total number of logical transformation variables may be represented as follows: D = m (1 + n + s + k) , (15) where m is the available number of canisters, n is the number of fuel elements, s = ⌈log2 Pmax⌉ and k = ⌈log2 Nmin⌉ is the number of auxiliary variables used in equalities (9) and (11), correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' As a result, the smallest-size problem with m = 2, n = 2, s = 2, k = 0 requires 10 logical variables, we mark it as a trivial case (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' We use only fixed configurations, where the optimal number of canisters is 2 and the feasible solution without constraint violation can have 3 canisters, in other words, we restrict problem samples to cases, where m = 3 and M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' The maximum capacities of canisters are equal with limit 2s=4 −1, where k is also fixed and equals zero, since we do not use minimum elements constraint when Nmin = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' This compression allows us to compute model problems on present quantum hardware and evaluate the dependence of problem size and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Tasks with the same elements quantity are also have identical QUBO size for avoiding additional deviation in data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' see Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' We use public access to D-Wave 5000 Advantage system with Pegasus topology processor [46] to run our quantum annealing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Elements quantity QUBO size Physical qubits Number of qubits Heuristic, mean Heuristic, std 2 10 20 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='3 3 21 80 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='5 4 24 90 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='6 5 27 102 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='6 6 33 157 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='5 7 36 170 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='6 8 39 185 226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='6 9 42 246 246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='6 10 45 262 280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='3 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Synthetic dataset scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Heuristic embeddings on physical qubits of D-Wave device were found via D-Wave Leap SDK for 5 different random samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Benchmarking Each problem instance has been further transformed into the QUBO matrix and run through both quantum and quantum-inspired instruments, specifically, the D- Wave quantum annealer and two quantum-inspired al- gorithms (SimCIM [39] and SBM [45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' SimCIM algo- rithm [39] is based on the method of efficient simulation 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='46e-01 (D-Wave) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='44e+02 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='81e+02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='12e+03 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='67e+00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='95e+00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='64e+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='06e+01 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='08e+01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='56e+02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='21e+02 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='81e+02 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='26e+02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='30e-01 (SBM) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='21e+00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='48e+01 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='76e+01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='13e+02 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='35e+03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='29e+04 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='86e+03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='06e+04 2 3 4 5 6 7 8 9 10 2 5 1 2 5 10 2 5 100 2 5 1000 2 5 10k 2 5 100k D-Wave SimCIM SBM Number of elements Time-to-Solution FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Comparison of the performance of quantum and quantum-inspired methods for bin packing problem based on synthetic data: we compare TTS (mean and standard deviation) for quantum device D-Wave and two quantum-inspired optimization algorithms, SimCIM and SBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' of Coherent Ising Machine [47] using classical computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' As it has been shown, SimCIM outperforms Coherent Ising Machine in terms of samples quality and speed of computation, and that is why it has been chosen as a benchmarking tool for comparative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Simulated Bifurcation algorithm (SBM) [45] is a heuristic algorithm for combinatorial optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Its workflow is inspired by quantum bifurcation machine [48] that is based on nonlinear oscillators and implements quantum adiabatic algorithm for solving optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' We run D-Wave experiments in a pure quantum mode using the Advantage chip featuring 5000 qubits with 15- way connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' In order to embed QUBO problem into physical qubit layout we utilized clique embedding supported by D-Wave Leap SDK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' SimCIM was run on Xeon E3-1230v5 4x3,4GHz, 16 GB DDR4, GeForce GTX 1080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Comparison results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Analysis As a figure of merit in our benchmarking procedure, we use time-to-solution (TTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' The TTS means a time that is needed to a heuristic algorithm to find the solution (ground state energy) with 99% success probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' It is given by TTS = taR99, (16) where ta is the annealing time (default value of ta for D-Wave is 20µs), and R99 stands for the number of rep- etition that is needed for the desired success probabil- ity [31, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' It can be calculated as follows: R99 = log(1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='99) log(1 − θ) , (17) where θ is the estimated success probability of each run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' All tasks were grouped by fuel elements quantity for demonstrating results (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' We note that the D- Wave annealer shows good result in problem solving in the small-size cases (2 possible canisters and 2 elements), while optimal solution was not found for 6 and more ele- ments problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' The standard deviation of TTS is signif- icantly increase with elements quantity for all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' This is caused by an exponential growth of the space of possible solutions, leading to a decrease in the probability of finding the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' As a result, the annealing process often terminates at a suboptimal point instead of the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' This is especially true for complex problems that require a large number of variables to be taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' While the main obstacle of quantum-inspired optimiza- tion methods is complexity and the size of the space of possible solutions, for quantum annealing a very impor- tant parameter is the gap between the ground state and the first excited state of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' The smaller the gap, the slower the adiabatic evolution of the quan- tum system should proceed in order to stay in the ground 5 Annealing time, µs Success probability TTS, µs 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='277 284 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='303 511 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='343 878 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Dependence of success probability and TTS for trivial case (2 fuel elements) for D-Wave device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' However, a long evolution time increases the influ- ence of quantum decoherence and can lead to incorrect solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' CONCLUSION Quantum computing is a promising technique for solv- ing combinatorial optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' In our work, we have demonstrated the potential of quantum and quantum-inspired tools to solve computational problems of the minimum bin packing problem, which is formu- lated as the problem of atomic energy industry, As a tar- get problem, we have chosen the optimization of spent nuclear fuel filling in canisters for the deep repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' The CF problem has been formulated in a QUBO ma- trix form (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' (9)–(12)) and was solved using exist- ing quantum annealer and quantum-inspired annealing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' We note that the current development level of quantum computing devices does not allow to solve large-scale practical problems, however, it is possible to scale a size of the problem for the next generation of quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Moreover, such research helps to identify practical-oriented tasks that may be solved more efficiently by quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We acknowledge use of the D-Wave quantum annealer and Toshiba Simulated Bifurcation Machine for this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' the views expressed are those of the authors and do not reflect the official policy or position of D-Wave and Toshiba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' The results of the present paper are based on the data that have been collected during the availabil- ity of the D-Wave quantum annealer and the Simulated Bifurcation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' This work was supported by Rus- sian Science Foundation (19-71-10092).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' APPENDIX In our benchmarking procedure, we use various accessi- ble parameters (particularly, annealing time, embedding type and chain strength) of the quantum hardware on the final solution quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' For the simplest task, we have observed that increasing annealing time gives a better success probability (see Table II), but in the same time TTS is getting worse, so annealing time was set to 20 µs (the.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' default value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' The number of annealing runs is set to 104 (maximum possible value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Comparing dif- ferent embeddings (see Table I), we analyze deviation of physical qubits number on five different random sam- ples and decided that for better experiment performance using stable stable clique embedding is more preferable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' According to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [31] we choose custom optimized chain strength instead of other variants such as maximum ab- solute value in QUBO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Thereby we have used mostly the standard configuration of the D-Wave processor during our experiments, so we do not have any specific require- ments on the weights/couplers in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' We use only pure quantum regime to obtain solution for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Paschos, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=', Paradigms of combinatorial optimiza- tion, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=', ISTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' (John Wiley & Sons, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=', London : Hoboken, 2014) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' 815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Lucas, Frontiers in Physics 2, 5 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Fedorov, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Gisin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Beloussov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Lvovsky, “Quantum computing at the quantum advan- tage threshold: a down-to-business review,” (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Farhi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Goldstone, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Gutmann, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Sipser, “Quantum computation by adiabatic evolution,” (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Das and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Chakrabarti, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' 80, 1061 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [6] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Albash and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Lidar, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' 90, 015002 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [7] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Farhi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Goldstone, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Gutmann, “A quan- tum approximate optimization algorithm,” (2014), arXiv:1411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='4028 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' King, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Raymond, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Lanting, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Isakov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Mohseni, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Poulin-Lamarre, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Ejtemaee, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Bernoudy, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Ozfidan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Smirnov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Reis, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Altomare, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Babcock, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Baron, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Berkley, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Boothby, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Bunyk, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Christiani, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Enderud, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Evert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Harris, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Hoskinson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Huang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Jooya, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Khodabandelou, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Ladizinsky, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Li, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Lott, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' MacDonald, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Marsden, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Marsden, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Med- ina, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Molavi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Neufeld, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Norouzpour, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Oh, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Pavlov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Perminov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Prescott, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Rich, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Sato, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Sheldan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Sterling, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Swenson, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Tsai, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Volkmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Whittaker, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Wilkinson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Yao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Neven, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Hilton, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Ladizinsky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' John- son, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Amin, Nature Communications 12, 1113 6 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Streif, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Neukart, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Leib, in Quantum Tech- nology and Optimization Problems, edited by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Feld and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Linnhoff-Popien (Springer International Publishing, Cham, 2019) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' 111–122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Chermoshentsev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Malyshev, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Tiunov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Mendoza, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Aspuru-Guzik, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Fedorov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Lvovsky, “Polynomial unconstrained binary op- timisation inspired by optical simulation,” (2021), arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='13167 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Perdomo-Ortiz, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Dickson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Drew-Brook, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Rose, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Aspuru-Guzik, Scientific Reports 2, 571 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [12] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Babej, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Ing, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Fingerhuth, “Coarse-grained lattice protein folding on a quantum annealer,” (2018), arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='00713 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Boev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Rakitko, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Usmanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Kobzeva, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Popov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Ilinsky, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Kiktenko, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Fedorov, Scientific Reports 11, 13183 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Sarkar, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Al-Ars, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Bertels, PLOS ONE 16, 1 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [15] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Chang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Gambhir, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Humble, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Sota, Scientific Reports 9, 10258 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [16] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Or´us, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Mugel, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Lizaso, Reviews in Physics 4, 100028 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [17] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Mugel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Kuchkovsky, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Sanchez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Fernandez- Lorenzo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Luis-Hita, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Lizaso, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Orus, “Dy- namic portfolio optimization with real datasets using quantum processors and quantum-inspired tensor net- works,” (2020), arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='00017 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [18] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Grant, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Humble, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Stump, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Ap- plied 15, 014012 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [19] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Herman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Googin, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Liu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Galda, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Safro, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Sun, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Pistoia, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Alexeev, “A survey of quantum com- puting for finance,” (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Or´us, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Mugel, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Lizaso, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' A 99, 060301 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [21] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Rosenberg, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Haghnegahdar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Goddard, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Carr, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Wu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' de Prado, IEEE Journal of Selected Topics in Signal Processing 10, 1053 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [22] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Rosenberg (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [23] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Andrew Milne and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Goddard (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Vesely, “Application of quantum computers in foreign exchange reserves management,” (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [25] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Neukart, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Compostella, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Seidel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' von Dollen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Yarkoni, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Parney, Frontiers in ICT 4, 29 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [26] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Inoue, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Okada, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Matsumori, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Aihara, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Yoshida, Scientific Reports 11, 3303 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [27] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Hussain, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Javaid, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Khan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Dalal, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Khalique, Quantum Information Processing 19, 312 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [28] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Venturelli, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Marchand, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Rojo, “Quan- tum annealing implementation of job-shop scheduling,” (2016), arXiv:1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='08479 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [29] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Ikeda, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Nakamura, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Humble, Scientific Reports 9, 12837 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Sadhu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Zaman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Das, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Banerjee, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Khan, “Quantum annealing for solving a nurse- physician scheduling problem in covid-19 clinics,” (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [31] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Stollenwerk, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Michaud, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Lobe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Picard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Basermann, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Botter, “Image acquisition plan- ning for earth observation satellites with a quantum an- nealer,” (2020), arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='09724 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [32] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Domino, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Koniorczyk, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Krawiec, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Ja�lowiecki, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Gardas, “Quantum computing approach to rail- way dispatching and conflict management optimization on single-track railway lines,” (2021), arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='08227 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='ET].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [33] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Domino, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Koniorczyk, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Krawiec, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Ja�lowiecki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Deffner, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Gardas, “Quantum annealing in the nisq era: railway conflict management,” (2021), arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='03674 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [34] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Henriet, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Beguin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Signoles, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Lahaye, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Browaeys, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='-O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Reymond, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Jurczak, Quan- tum 4, 327 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [35] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Ebadi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Keesling, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Cain, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Levine, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Bluvstein, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Semeghini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Om- ran, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Liu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Samajdar, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Luo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Nash, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Gao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Barak, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Farhi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Sachdev, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Gemelke, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Zhou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Choi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Pichler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Greiner, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Vuleti´c, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Lukin, Science 376, 1209 (2022), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='org/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='abo6587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Harrigan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Sung, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Neeley, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Satzinger, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Arute, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Arya, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Atalaya, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Bardin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Barends, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Boixo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Broughton, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Buckley, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Buell, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Burkett, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Bushnell, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Chen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Chiaro, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Collins, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Courtney, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Demura, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Dunsworth, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Eppens, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Fowler, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Foxen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Gidney, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Giustina, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Graff, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Habegger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Ho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Hong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Huang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Ioffe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Isakov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Jeffrey, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Jiang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Jones, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Kafri, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Kechedzhi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Kelly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Kim, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Klimov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Korotkov, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Kostritsa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Landhuis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Laptev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Lindmark, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Leib, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Martin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Martinis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' McClean, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' McEwen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Megrant, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Mi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Mohseni, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Mruczkiewicz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Mutus, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Naa- man, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Neill, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Neukart, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Niu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' O’Brien, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' O’Gorman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Ostby, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Petukhov, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Putterman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Quintana, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Roushan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Rubin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Sank, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Sko- lik, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Smelyanskiy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Strain, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Streif, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Szalay, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Vainsencher, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' White, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Yao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Yeh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Zal- cman, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Zhou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Neven, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Bacon, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Lucero, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Farhi, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Babbush, Nature Physics 17, 332 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [37] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Bharti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Cervera-Lierta, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Kyaw, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Haug, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Alperin-Lea, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Anand, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Degroote, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Heimonen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Kottmann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Menke, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Mok, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Sim, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='- C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Kwek, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Aspuru-Guzik, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' 94, 015004 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [38] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Boixo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Rønnow, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Isakov, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Wecker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Lidar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Martinis, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Troyer, Nature Physics 10, 218 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [39] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Tiunov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Ulanov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Lvovsky, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Express 27, 10288 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [40] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Killoran, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Bromley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Arrazola, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Schuld, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Quesada, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Lloyd, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Research 1, 033063 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [41] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Boev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Usmanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Semenov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Ushakova, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Salahov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Mastiukova, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Kik- tenko, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Fedorov, “Quantum-inspired optimiza- tion for routing and wavelength assignment,” (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [42] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Oshiyama and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Ohzeki, Scientific Reports 12, 2146 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [43] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' ˇZerovnik, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Snoj, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Ravnik, Nu- clear Science and Engineering 163, 183 (2009), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='13182/NSE163-183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [44] Whyte, Andy and Parks, Geoff, EPJ Web Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' 247, 06028 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' 7 [45] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Goto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Tatsumura, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Dixon, Science Advances 5, eaav2372 (2019), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='org/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='1126/sciadv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='aav2372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [46] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Boothby, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Bunyk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Raymond, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Roy, “Next- generation topology of d-wave quantum processors,” (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [47] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Yamamoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Aihara, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Leleu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Kawarabayashi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Kako, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Fejer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Inoue, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Takesue, npj Quan- tum Information 3, 49 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [48] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Goto, Scientific Reports 6, 21686 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' [49] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Aramon, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Rosenberg, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Valiante, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Miyazawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Tamura, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content=' Katzgraber, Frontiers in Physics 7 (2019), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='3389/fphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} +page_content='00048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFIT4oBgHgl3EQfcSt1/content/2301.11265v1.pdf'} diff --git a/N9FKT4oBgHgl3EQffC6n/content/tmp_files/2301.11828v1.pdf.txt b/N9FKT4oBgHgl3EQffC6n/content/tmp_files/2301.11828v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fbf5d649d58b6f7aa28f6da3368693b23f0aaf4d --- /dev/null +++ b/N9FKT4oBgHgl3EQffC6n/content/tmp_files/2301.11828v1.pdf.txt @@ -0,0 +1,2552 @@ +A fast computational framework for the linear bond-based peridynamic +model +Chenguang Liua +Hao Tiana +Wai Sun Dona +Hong Wangb +a School of Mathematical Science, Ocean University of China, Qingdao, Shandong 266100, China +bDepartment of Mathematics, University of South Carolina, Columbia, South Carolina 29208, USA +Abstract +Peridynamic (PD) theory is significant and promising in engineering and materials science; how- +ever, it imposes challenges owing to the enormous computational cost caused by its nonlocality. +Our main contribution, which overcomes the restrictions of the existing fast method, is a general +computational framework for the linear bond-based peridynamic models based on the meshfree +method, called the matrix-structure-based fast method (MSBFM), which is suitable for the gen- +eral case, including 2D/3D problems, and static/dynamic issues, as well as problems with general +boundary conditions, in particular, problems with crack propagation. Consequently, we provide a +general calculation flow chart. The proposed computational framework is practical and easily em- +bedded into the existing computational algorithm. With this framework, the computational cost +is reduced from O(N2) to O(N log N), and the storage request is reduced from O(N2) to O(N), +where N is the degree of freedom. Finally, the vast reduction of the computational and memory +requirement is verified by numerical examples. +Keywords: +bond-based peridynamics, matrix-structure-based fast method, computational +framework, crack propagation +1. Introduction +Classical continuum mechanics are expressed as a partial differential equation, which is chal- +lenging to describe models with discontinuities. As a result, peridynamics(PD), as proposed by +Silling[1], is an integral-type nonlocal model and can provide a general theory for solving problems +in the form of discontinuities. Over the past few decades, the effectiveness of PD has attracted +extensive research conducted on modeling methods, numeral techniques, and applications. In this +paper, we focus on the bond-based peridyanmics [2], which is an early version of peridynamics and +can be applied to isotropic materials, with Poisson’s ratio of 1/4 for plane strain and 1/3 for plane +stress. Later the ordinary state-based and non-ordinary state-based PD were proposed to elimi- +nate the constraints of fixed Poisson’s ratios on materials[3]. The PD models have been frequently +used in many practical problems. A range of cutting-edge applications can be found in composite +material deformation[4, 5, 6], corrosion[7, 8, 9, 10, 11], damage prediction[12, 13, 14, 15, 16] and +crack simulation for a variety of materials[17, 18, 19, 20, 21]. +There have been much research directed at developing numerical methods that can solve the +PD models, including meshfree methods, finite difference, finite element methods, and collocation +methods[22, 23, 24, 25, 26, 27, 28].Among other research in the literature, explored in[22, 29, 30], the +asymptotically compatible schemes retained a limit behavior that makes the limit of the zero-horizon +Preprint submitted to Computer Methods in Applied and Mechanical Engineering +January 30, 2023 +arXiv:2301.11828v1 [math.NA] 27 Jan 2023 + +of the nonlocal operator become the local differential operator, providing a consistency between local +and non-local models. However, these methods are very restrictive due to the vast computational +cost caused by nonlocality of PD. The increasing computation cost limits the application of PD +theory, especially for multidimensional cases. Lots of efforts have been made to overcome this issue. +A class of coupled method was introduced to accelerate the PD simulation, which utilizes PD on the +area around the cracks and classical mechanics on the rest area [31, 32, 33, 34, 35, 36]. A fast method +based on the convolution structure of PD models[38, 39] is proposed to accelerate the simulation. +A super-fast peridynamic model[40] based on decreasing the number of inner loop operations is +also introduced to overcome this difficulty. The work above also gives us some inspiration for this +article. +In the literature, a class of fast methods utilizing the structure of stiff matrix are booming, +which can reduce the computational cost from O(N2) to O(N log N) without loss of accuracy. In +2010, a fast method[41] based on the Toeplitz structure of stiff matrices was proposed, hence solving +the 1D static linear bond-based peridynamics. Subsequently, a fast collocation method based on +TBT matrix structure was given for 2D nonlocal diffusion models in reference[42], which can be +thought of a approximation model of scalar-valued. A fast collocation method for the 2D static +linear bond-based peridynamics with volume boundary conditions was investigated in [43] , where +we use an equivalent but more effective way to evaluate. In 2017, a fast method was also presented +to solve nonlocal diffusion models with variable coefficients[44], and a discontinuous method was +discussed to solve linear bond-based PD models with discontinuous solves[45]. +In 2020, a fast +algorithm with preconditioned processing was proposed[46, 47], accelerating the convergence of the +iterative method. Although the research above has significantly contribute to applying fast matrix +structure based methods in the simulation of PD, there are still several pending issues. The first +one is that all this research is proposed for 1D or 2D static models. The second one is that there +need to be methods taking into account volume constrained boundary conditions. The third one +is that all these research is developed for models with no cracks. A natural question comes into +the work here. To fill this gap, we give a fast matrix-based method(MSBFM), which is the main +contribution of this article. +In this paper, we offer an insightful look at a very general setting. We propose a fast matrix- +based method(MSBFM) for solving 2D/3D linear bond-based peridynamic models. By establishing +the relationship between the stiffness matrix and the Toeplitz-Block-Toeplitz(TBT) matrix, we +focos on the matrix decomposition, stiff matrix, one for TBT matrix and another for sparse matrix. +This reduces the structural limitations of the matrices and hence a more efficient method applied +to general boundary and cracks, and dramatically reduces the amount of computation and storage +from O(N2) to O(N log N) and from O(N2) to O(N), respectively. Meanwhile, the results are for +models in 2D as well as in 3D. Numerical experiments verify the accuracy of the method. +The following articles are organized as follows. In section 2, we reviewed the linear model of +the peridynamic and meshfree methods. In section 3, we analyze the matrix structure of the two- +dimensional problem and give an accelerated process based on the matrix structure. In section +4, the matrix structure of the 3D model is discussed, and the MSBFM method is introduced. In +section 5, the accuracy and acceleration effect of the MSBFM method are shown by numerical +examples. +2 + +2. Fundamentals of linear bond-based peridynamics and its discretization +The bond-based peridynamics, is a reformulation for classical continuum solid mechanics by the +integral form instead of partial differential equations. Typically, with sufficiently small displace- +ment, bond-based peridynamics can be approximated as linear bond-based peridynamics. In this +section, we mainly review the linearized version of the bond-based peridynamic mode, addressed +in this study, and the discretization to solve the model. +2.1. Linear bond-based peridynamics +The equation of motion for the linear bond-based peridynamics with prescribed volume bound- +ary condition can be defined as follows: +� +� +� +ρ¨u(x, t) = +� +Bδ(x)∩Ω +µC(x +′ − x)(u +� +x′, t +� +− u(x, t))dVx′ + b(x, t) +x ∈ Ωs, +u(x, t) = h(x, t) +x ∈ Ωc. +(1) +where ρ is the mass density, Ω = Ωs ∪ Ωc is the spatial domain, Bδ(x) is the horizon which is +usually taken as a disk or ball of radius δ. u(x, t) is displacement vector field, and b(x, t) is a body +force density field. h(x, t) is the prescribed displacement data imposed on the volume constrained +boundary Ωc. +C(x′ − x) is the micromodulus tensor which can be written as[24]: +C(x +′ − x) = α(x +′ − x) ⊗ (x +′ − x) +|x +′ − x|3 +, +(2) +where α is a scalar parameter introduced to keep the energy of peridynamic model and classic +elastic model equal, which is determined by δ and the elastic modulus E: +α = +� +9E +πδ3h +2D plane stress or plane strain, +12E +πδ4 +3D. +(3) +with h being the plate thickness in the 2D model. +µ is a history-dependent scalar-valued function which can be written as: +µ(s, t) = +� +1 +if the bond is not broken s δ2 +(19) +It’s easy to see that cj,j +′ +i,i′ = 0 if |j +′ − j| > M or |i +′ − i| > M, M = δ/h. Each matrix block satisfies +Bi,i′ = 0 if |i +′ − i| > M. Furthermore, if |i +′ − i| ≤ M and |j +′ − j| > M, the entries in the matrix +block Bi,i′ also satisfies cj,j +′ +i,i′ = 0. Therefore A is a block-banded-block matrix, as shown in Fig. 3. +Figure 3: The block-banded-block structure of matrix A for 2D PD +3.2. Analysis of Ap,q for any xp ∈ Ωin +In most cases, Ωin occupies most part of the Ω. Different from the material points in Ωe and +Ωbr, The influence area of these material points in Ωin is a complete disk , in which none of the +bonds inside Ωin are broken. +If xp ∈ Ωin, we have Bδ(xp) ∩ Ω = Bδ(xp), µ = 1, then fi,j can be expressed as follows: +fi,j = +� +xq∈Bδ(xp) +α +(xi′ − xi)2 +((xq − xp)2 + (yq − yp)2) +3 +2 +� +ui′,j′ − ui,j +� +λi′,j′h2, +∀xp ∈ Ωin +(20) +8 + +With the relations xi = xl + (i − 1/2)h and yj = yb + (j − 1/2)h, matrix entries Ap,q can be +reorganized as follows when xp ∈ Ωin and q ̸= p: +Ap,q = cj,j +′ +i,i′ = α +(xi′ − xi)2 +((xi′ − xi)2 + (yi′ − yi)2) +3 +2 +λi′,j′h2 += α +(i′ − i)2 +((i′ − i)2 + (j′ − j)2) +3 +2 +λi′,j′h. +(21) +where +λi′,j′ = +� +� +� +� +� +1 +when ∥xq − xp∥ ≤ δ − h/2 +M − +� +(i′ − i)2 + (j′ − j)2/2 + 1/2 +when δ − h/2<∥xq − xp∥ ≤ δ +0 +otherwise +(22) +When q = p, we can also get: +Ap,p = cj,j +i,i = − +� +xq∈Bδ(xp) +cj,j +′ +i,i′ = − +� +xq∈Bδ(xp) +α +(i′ − i)2 +((i′ − i)2 + (j′ − j)2) +3 +2 +λi′,j′h +(23) +Thus the entries Ap,q does not depend on the position of xp or xq, but on the distance xp − xq. +Moreover with a uniform mesh on a rectangular plate, Ap,q is directly related to the difference i +′ −i +and j +′ − j, according to Eq.(23). Since −M ≤ i +′ − i, j +′ − j ≤ M, for every p− th row if xp ∈ Ωin, +there are (2M +1)2 non-zero entries and they are the same . Therefore, we only need to store these +matrix entries in a 2M + 1-by-2M + 1 matrix instead of a N-by-(2M + 1)2 matrix, which greatly +reduce memory size. and footprint. The 2M + 1-by-2M + 1 matrix is termed as the kernel matrix, +which can be defined as: +Km,n = Ki′−i+M+1,j′−j+M+1 = cj,j +′ +i,i′ , +0 ≤ m, n ≤ 2M + 1, +∀xp ∈ Ωin +(24) +In the actual calculation, we can obtain Km,n by computing the interaction between xp and (2M + +1)2 material points around. Here xp is an arbitrary material point in Ωin, as shown in Fig. 4. +Specifically, KM+1,M+1 = cj,j +i,i = Ap,p. +Assuming that all material points are in Ωin, which means all material points satisfy the Eqs.(21) +and (23), we can get a matrix ˆA defined by: +ˆA = +� +�� +ˆB1,1 +· · · +ˆB1,Ny +... +... +... +ˆBNy,1 +· · · +ˆBNy,Ny +� +�� , ˆBi,i′ = +� +��� +ˆc1,1 +i,i′ +· · · +ˆc1,Nx +i,i′ +... +... +... +ˆcNx,1 +i,i′ +· · · +ˆcNx,Nx +i,i′ +� +��� , +(25) +This is a block-banded-block matrix generated by K, which means: +ˆAp,q = ˆcj,j +′ +i,i′ = Ki′−i+M+1,j′−j+M+1 = Km,n, +∀xp ∈ Ω +(26) +The value of ˆAp,q depends only on i +′ − i and j +′ − j. Hence, for every matrix block ˆBi,i′, entries +ˆcj,j +′ +i,i′ on each diagonal are equal. For the matrix ˆA, blocks ˆBi,i′ on each diagonal are equal. A +matrix satisfies the above properties is called a Toeplitz-Block-Toeplitz(TBT) matrix. +9 + +(a) +(b) +Figure 4: Description of the keneral matrix: (a) Entries Km,n in the kernel matrix (b) Contact +between material points and matrix entries +To construct the fast method , we decompose the matrix A = ˆA + (A − ˆA). Thus we have: +f = f +′ + (A − ˆA)u, +(27) +where f +′ = ˆAu. Based on the TBT structure of ˆA, The matrix-vector multiplication ˆAu can be +accelerated by a fast matrix-vector multiplication(FMVM)[43] in Algorithm 3: +Algorithm 3: FMVM +1 ˆG = FFT2(G), ˆU = FFT2(U) +2 ˆH = ˆG ◦ ˆU +3 H = FFT2−1( ˆH) +Here we use FFT2 and FFT2−1 to denote two-dimension FFT and iFFT operations. +ˆH +represents the Hadamard product of ˆG and ˆU. H is a 2Nx-by-2Ny matrix and we can obtain f by +fjNx+i = Hi,j, if i ≤ Nx and j ≤ Ny. G is the first column of a extended matrix embedded by K. +Here we write it as a 2Nx-by-2Ny matrix , namely: +G := +� +� +� +� +� +� +� +� +� +� +� +� +� +� +KM+1,M+1 +··· +KM+1,2M+1 +0 +··· +0 +KM+1,1 +··· +KM+1,M +... +... +... +... ... ... +... +... +... +K2M+1,M+1 ··· K2M+1,2M+1 +0 ... +0 +K2M+1,1 ··· K2M+1,M +0 +··· +0 +0 +··· +0 +0 +··· +0 +... +... +... +... ... ... +... +... +... +0 +··· +0 +0 +··· +0 +0 +··· +0 +K1,M+1 +··· +K1,2M+1 +0 ... +0 +K1,1 +··· K2M+1,M +... +... +... +... ... ... +... +... +... +KM,M+1 +··· +KM,2M+1 +0 +··· +0 +KM,1 +··· +KM,M +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(28) +10 + +KSW+I'I +KW+I'W+I +KSW+I'SW+I +... +... +... +... +.. +KS'T +SW+J +KI'T +KI'W+I +I'SW+IKSWI+I'SW+I ++I'V+Iwhich means each matrix entries Gi,j can be expressed as follows: +Gi,j = +� +0 +i ∈ [M + 2, 2Nx − M] or j ∈ [M + 2, 2Ny − M] +Km,n +otherwise +(29) +where +m = +� +i + M +i ∈ [1, M + 1] +i − 2Nx + M +i ∈ [2Nx − M + 1, 2Nx] +n = +� +j + M +j ∈ [1, M + 1] +j − 2Ny + M +j ∈ [2Ny − M + 1, 2Ny] +(30) +U is a extended matrix embedded by displacement vector u, which can be expressed as: +U := +� +� +� +� +� +� +� +� +� +� +� +� +u1,1 +· · · +uNx,1 +0 +· · · +0 +u1,2 +· · · +uNx,2 +0 +· · · +0 +... +... +... +... +... +... +u1,Ny−1 +· · · +uNx,Ny−1 +0 +· · · +0 +0 +· · · +0 +0 +· · · +0 +... +... +... +... +... +... +0 +· · · +0 +0 +· · · +0 +� +� +� +� +� +� +� +� +� +� +� +� +(31) +which means each entries Ui,j can be defined by: +Ui,j = +� +ui,j +if i ≤ Nx and j ≤ Ny +0 +otherwise +(32) +By Algorithm 3, The calculation of form f +′ = ˆAu can be decreased from O(N2) to O(N log N). +The calculation of (A − ˆA)u needs to be considered additionally, which will be discussed below. +Algorithm 3 is implemented with the Matlab code. The codes of the two-dimensional Fourier +transform and the two-dimensional inverse transformation are called as ˆG = fft2(G) and H = +ifft2( ˆH). +3.3. Analysis of Ap,q for any xp ∈ Ωc +In many applications it is usually desired/needed to apply local boundary conditions, the prop- +erties of material points xp ∈ Ωc are not considered in the matrix, so that the actual matrix is not +a square matrix. However, the FMVM algorithm requires that the matrix be a square matrix, so +we need to consider it in the form f = Au. +According to Eq.(1) and Eq.(6), up can be expressed as: +up = h(xp), +xp ∈ Ωc +(33) +Equation (33) means up is given by the prescribed displacement data rather than the form f = Au. +Thus Ap,q for xp ∈ Ωc is meaningless and can be arbitrarily chosen. Here we let Ap,q = ˆAp,q, which +means Ap,q − ˆAp,q = 0, so that these entries do not have to repeat operations in the form (A− ˆA)u. +. When the computation domain Ω = Ωc ∪ Ωin, the stiff matrix A = ˆA. +11 + +Considering that the displacement up is obtained through h(xp) for xp ∈ Ωc , we need to +replace the displacement with h(xp) after obtaining the displacement through f in each time step +of Algorithm 1 or Algorithm 2. +This will bring additional calculation, but it will not exceed +O(N log N) in general. +3.4. Analysis of Ap,q for any xp ∈ Ωe +For the material points xp ∈ Ωe, the influence area is not a complete disk. Hence, according +to (18), if xp ∈ Ωe, most of the matrix entries in p−th row are equal for the matrix entries +corresponding to the material points on Ωin except the matrix entries on the main diagonal. +When xp ∈ Ωe, we have Bδ(xp) ∩ Ω ̸= Bδ(xp) and µ = 1. Thus fp can be written as: +fi,j = +� +xq∈Bδ(xp)∩Ω +α +(xi′ − xi)2 +((xi′ − xi)2 + (yi′ − yi)2) +3 +2 +λi′,j′h2 � +ui′,j′ − ui,j +� +λi′,j′h2, +∀xp ∈ Ωe +(34) +Then entries Ap,q can be represented as: +Ap,q = +� +� +� +� +� +� +� +� +� +α +(yj′ − yj)(xi′ − xi) +((xi′ − xi)2 + (yi′ − yi)2) +3 +2 +λi′,j′h2 +q ̸= p +− +� +xq∈Bδ(xp)∩Ω +α +(yj′ − yj)(xi′ − xi) +((xi′ − xi)2 + (yi′ − yi)2) +3 +2 +λi′,j′h2 +q = p +(35) +Notice that each entry in p−th is equal to that in q−th row expect one on the diagonal for xp ∈ Ωe +and xq ∈ Ωin, which means: +� +� +� +Ap,q = ˆAp,q, +q ̸= p +Ap,p = − +� +xq∈Bδ(xp)∩Ω +Ap,q = − +� +xq∈Bδ(xp)∩Ω +ˆAp,q ̸= − +� +xq∈Bδ(xp) +ˆAp,q = ˆAp,p +(36) +Hence, Ap,p can be expressed as follows: +Ap,p = ˆAp,p + +� +xq∈Bδ(xp\(Bδ(xp)∩Ω) +Ap,q +(37) +Then the matrix A can be decomposed into following form by introducing a diagonal matrix D: +A = ˆA + De +(38) +Here matrix entries in De can be written as: +De +p,p = +� +� +� +� +xq∈Bδ(xp)\(Bδ(xp)∩Ω) +Ap,q +if xp ∈ Ωe +0 +otherwise +(39) +Then f can be decomposed as follows: +fp = +� +f +′ +p + De +p,pup +if xp ∈ Ωe +f +′ +p +otherwise +(40) +12 + +The total number of material points on Ωe do not exceed the total number of material points N. +Thus the calculation and storage memory brought by (40) is O(N). +Since the displacement of the material point on Ωc is not affected by the form Au = f, the +problem of incomplete horizon on Ωc do not need to be considered, which means that Ωe ∩ Ωc = ∅ +in most cases. +Remark 2. A special case is that a material point is affected by two constraints, which means: (a) +The displacement constraint conditions is applied in the x-direction, so it is regarded as a material +point on Ωc, and the displacement ux is replaced when calculating fx = Axxux + Axyuy; (b) It is +affected by the incomplete horizon in the y-direction, so it is considered as a matter point on Ωe +and the corresponding Du is subtracted when calculating Ayxuy, Ayyuy. +Remark 3. Surface correction algorithms are often used on material points xp ∈ Ωe to calculate +accurately in engineering problems[50]. In this algorithm, a coefficient vp,q is introduced to increase +the micromodule of each bond in Bδ(xp) ∩ Ω, which means Ap,q = vp,q ˆAp,q for xp ∈ Ωe and p ̸= q. +This way, the impact that Bδ(xp) ∩ Ω is not a complete disk is eliminated. However, this coefficient +breaks the above matrix structure, so the form Ap,quq for xp ∈ Ωe needs to be recalculated. We will +mention this part in numerical examples. +3.5. Analysis of Ap,q for xp ∈ Ωbr +For the sub-domain Ωbr, the interaction between the material points on the broken bond is +considered as 0. Hence, the structure of the matrix mentioned above is destroyed, which requires +special treatment. +Since all sub-domains with cracks are called Ωbr for the domain Ω, Ωbr does not exist alone. The +sub-domain always intersects one of sub-domains Ωin, Ωe, and Ωc, which means Ωbr∩(Ωs∪Ωc) ̸= ∅. +In this case, we first consider the effect of broken bonds on the matrix. For xp ∈ Ωbr, fp can be +expressed as follows: +fi,j = +� +xq∈Bδ(xp)∩Ω +αµ +(xi′ − xi)2 +((xi′ − xi)2 + (yi′ − yi)2) +3 +2 +� +ui′,j′ − ui,j +� +λi′,j′h2 +(41) +Here the history-dependent scalar valued function µ is defined in (4). +As observed from Fig. 5, when the bond between two material points xp and xq is broken, the +history-dependent scalar valued function µ = 0. Thus Ap,q = 0 ̸= ˆAp,q for xp ∈ Ωbr. Note that Ap,p +is the sum of Ap,q, thus matrix entries ˆAp,p are also affected by the broken bonds. For the purpose +of explaining Ap,p, we introduce a set vp to store xq on the broken bond, which can be expressed +as vp = {xq : the bonds between xp and xq are broken}. The entry Ap,p can be given by: +Ap,p = − +� +xq∈Bδ(xp)∩Ω +Ap,q + +� +xq∈vp +Ap,q +(42) +Then the matrix A can be decomposed into a form: +A = ˆA + Df, +(43) +13 + +p +q +p +q +Figure 5: Schematic diagram of cracks when there is a fracture on bond between xp and xq: the +left part is peridynamic discretization, and the right part is geometry details +where +Df +p,q = +� +� +� +� +� +� +� +− ˆAp,q +if xp ∈ Ωbr, xq ∈ vp +� +xs∈vp +ˆAp,s +if xp ∈ Ωbr, q = p +0, +otherwise +(44) +Thus f can be expressed as: +fp = +� +� +� +f +′ +p + +� +xq∈vp +ˆAp,q(up − uq), +if xp ∈ Ωbr, +f +′ +p +otherwise +(45) +The calculation of form Dfu is related to the number of broken bonds. In fact, the cracks are +lower dimensional manifolds compared to the domain’s dimension, which means that the number of +material points on Ωbr will not exceed N +d−1 +d . Thus the calculation is generally obtained by N +d−1 +d N, +which is O(N +2d−1 +d ). +After calculating the matrix Df, we will consider the influence of Ωe, Ωin, which means that +stiff matrix A is decomposed into the form A = ˆA + Df + De. If Ωc ∩ Ωbr ̸= ∅, we also replace +the corresponding displacement in the iteration. +4. A fast matrix-based method for a 3D linear bond-based peridynamic model +To develop the MSBFM on the 3D model, a linear bond-based peridynamics in three spaces +dimensions on a block Ω = Ωc ∩Ωs = [xl, xr]×[yb, yt]×[zc, zd] is introduced in this section. Here Ωc +represents the area affected by volume constrained boundary conditions. Identical to the form (11), +xp, up and fp can be denoted as xp = [xp, yp, zp]T , up = [ux +p, uy +p, uz +p]T , fp = [fx +p , fy +p , fz +p ]T , 1 ≤ p ≤ N. +Then f obtained by Eq.(9) can be expressed as: +� +� +fx +fy +fz +� +� = +� +� +Axx +Axy +Axz +Axy +Ayy +Ayz +Axz +Ayz +Azz +� +� +� +� +ux +uy +uz +� +� +(46) +14 + +物Here fz = [fz +1 , . . . , fz +p ]T , uz = [uz +1, . . . , uz +p]T , and Axz, Ayz, Azz are defined as: +Axx +p,q = +� +� +� +� +� +� +� +� +� +αµ +(xq − xp)2 +((xq − xp)2 + (yq − yp)2 + (zq − zp)2) +3 +2 +λqVq +q ̸= p, ∀xp ∈ Ωs +− +� +xq∈Bδ(xp)∩Ω +αµ +(xq − xp)2 +((xq − xp)2 + (yq − yp)2 + (zq − zp)2) +3 +2 +λqVq +q = p, ∀xp ∈ Ωs +Axy +p,q = +� +� +� +� +� +� +� +� +� +αµ +(xq − xp)(yq − yp) +((xq − xp)2 + (yq − yp)2 + (zq − zp)2) +3 +2 +λqVq +q ̸= p, ∀xp ∈ Ωs +− +� +xq∈Bδ(xp)∩Ω +αµ +(xq − xp)(yq − yp) +((xq − xp)2 + (yq − yp)2 + (zq − zp)2) +3 +2 +λqVq +q = p, ∀xp ∈ Ωs +Axz +p,q = +� +� +� +� +� +� +� +� +� +αµ +(xq − xp)(zq − zp) +((xq − xp)2 + (yq − yp)2 + (zq − zp)2) +3 +2 +λqVq +q ̸= p, ∀xp ∈ Ωs +− +� +xq∈Bδ(xp)∩Ω +αµ +(xq − xp)(zq − zp) +((xq − xp)2 + (yq − yp)2 + (zq − zp)2) +3 +2 +λqVq +q = p, ∀xp ∈ Ωs +Ayy +p,q = +� +� +� +� +� +� +� +� +� +αµ +(yq − yp)2 +((xq − xp)2 + (yq − yp)2 + (zq − zp)2) +3 +2 +λqVq +q ̸= p, ∀xp ∈ Ωs +− +� +xq∈Bδ(xp)∩Ω +αµ +(yq − yp)2 +((xq − xp)2 + (yq − yp)2 + (zq − zp)2) +3 +2 +λqVq +q = p, ∀xp ∈ Ωs +Ayz +p,q = +� +� +� +� +� +� +� +� +� +αµ +(yq − yp)(zq − zp) +((xq − xp)2 + (yq − yp)2 + (zq − zp)2) +3 +2 +λqVq +q ̸= p, ∀xp ∈ Ωs +− +� +xq∈Bδ(xp)∩Ω +αµ +(yq − yp)(zq − zp) +((xq − xp)2 + (yq − yp)2 + (zq − zp)2) +3 +2 +λqVq +q = p, ∀xp ∈ Ωs +Azz +p,q = +� +� +� +� +� +� +� +� +� +αµ +(zq − zp)2 +((xq − xp)2 + (yq − yp)2 + (zq − zp)2) +3 +2 +λqVq +q ̸= p, ∀xp ∈ Ωs +− +� +xq∈Bδ(xp)∩Ω +αµ +(zq − zp)2 +((xq − xp)2 + (yq − yp)2 + (zq − zp)2) +3 +2 +λqVq +q = p, ∀xp ∈ Ωs +(47) +According to the symmetry of the kernel function, we can get Ayx = Axy, Azx = Axz and +Azy = Ayz. Here we only consider fx = Axxux, and record it as f = Au. +This model can also be discretized by uniform mesh, as shown in 2D model. Here the material +points can be expressed as xp = [xi, yj, zk]T , where xi = xl + (i − 1/2)hx, yj = yb + (j − 1/2)hy, +zk = zc + (k − 1/2)hz. hx, hy, hz are positive constants representing the grid spacing, and we let +hx = hy = hz = h. z is the index of the layer, and Nz denotes the numbers of intervals in the z +directions. Node number p can be obtained by p = (k − 1)NxNy + (j − 1)Nx + i. Thus f and u can +be rewritten as: +u = [u1,1,1, . . . , uNx,1,1, . . . , u1,Ny,1, . . . , uNx,Ny,1, . . . , u1,1,Nz, . . . , uNx,Ny,Nz]T +f = [f1,1,1, . . . , fNx,1,1, . . . , f1,Ny,1, . . . , fNx,Ny,1, . . . , f1,1,Nz, . . . , fNx,Ny,Nz]T +(48) +15 + +where the matrix entry Ap,q can be written as: +Ap,q = +� +� +� +� +� +� +� +� +� +αµ +(xi′ − xi)2 +((xi′ − xi)2 + (yj′ − yj)2 + (zk′ − zk)2) +3 +2 +λi′,j′,k′Vi′,j′,k′ +q ̸= p, ∀xp ∈ Ωs +− +� +xq∈Bδ(xp)∩Ω +αµ +(xi′ − xi)2 +((xi′ − xi)2 + (yj′ − yj)2 + (zk′ − zk)2) +3 +2 +λi′,j′,k′Vi′,j′,k′ +q = p, ∀xp ∈ Ωs +(49) +Here Vi′,j′,k′ = h3. +Following the spatial partition in the 2D model, we divide the region Ωs into Ωin, Ωe and Ωbr. +They represent the internal area, the area with the incomplete disk, and the area with broken +bonds, respectively. +Then A becomes a stiff matrix with a block-banded-block-banded-block structure, which means +A is a matrix composed of N2 +z matrix blocks ¯Bi,j. Each matrix block ¯Bi,j represents the action of +the j−th layer on the i−th layer , and the structure of ¯Bi,j is a block-banded-block matrix, which +is as same as the form (17), see Fig. 6. +Figure 6: Illustration of the matrix structure in 3D model:The left side is the matrix A composed +of matrix block ¯Bi,j, and the right side is the structure of matrix block ¯Bi,j +Similar to Eq.(21) and Eq.(26), Ap,q can be proved to satisfy the following properties without +considering broken bonds: +Ap,q = α +(xi′ − xi)2 +((xi′ − xi)2 + (yj′ − yj)2 + (zk′ − zk)2) +3 +2 +λi′,j′h3 += α +(i′ − i)2 +((i′ − i)2 + (j′ − j)2 + (k′ − k)2) +3 +2 +λi′,j′h2 +(50) +16 + +where xq = [xi′, yj′, zk′]T , q ̸= p. When q = p, we can also get: +Ap,p = − +� +xq∈Bδ(xp)∩Ω +α +(xi′ − xi)2 +((xi′ − xi)2 + (yj′ − yj)2 + (zk′ − zk)2) +3 +2 +λi′,j′h3 += − +� +xq∈Bδ(xp)∩Ω +α +(i′ − i)2 +((i′ − i)2 + (j′ − j)2 + (k′ − k)2) +3 +2 +λi′,j′h2 +(51) +Identical to the 2D model, we can prove that matrix A is only related to i +′ − i, j +′ − j, k +′ − k and +introduce a 2M + 1-by-2M + 1-by-2M + 1 tensor K to store entries Ap,q, namely: +Ap,q = Km,n,l, +0 ≤ m, n, l ≤ 2M + 1, +∀xp = (xi, yj, zk) ∈ Ωin +(52) +Here m = i +′ − i + M + 1, n = j +′ − j + M + 1, l = k +′ − k + M + 1. Then a Toeplitz-Block-Toeplitz- +Block-Toeplitz(TBTBT) matrix ˆA can also be defined as: +ˆAp,q = Km,n,l, +∀xp ∈ Ω +(53) +The form f +′ = ˆAu can be solved by a fast tensor-tensor multiplication(FTTM), as shown in +Algorithm 4: +Algorithm 4: FMVM +1 ˆG = FFT3(G), ˆU = FFT3(U) +2 ˆH = ˆG ◦ ˆU +3 H = FFT3−1( ˆH) +Here we use FFT3 and FFT3−1 to denote three-dimension FFT and iFFT operations. H +is a 2Nx-by-2Ny-by-2Nz tensor and we can obtain f by fkNxNy+jNx+i = Hi,j,k, if 1 ≤ i ≤ Nx, +1 ≤ j ≤ Ny, 1 ≤ k ≤ Nz. For the first column G of the extended matrix embedded by the tensor +K, we define it as a 2Nx-by-2Ny-by-2Nz tensor, namely: +Gi,j,k = +� +0 +i ∈ [M + 2, 2Nx − M], j ∈ [M + 2, 2Ny − M], k ∈ [M + 2, 2Nz − M] +Km,n,l +otherwise +(54) +where +m = +� +i + M +i ∈ [1, M + 1] +i − 2Nx + M +i ∈ [2Nx − M + 1, 2Nx] +n = +� +j + M +j ∈ [1, M + 1] +j − 2Ny + M +j ∈ [2Ny − M + 1, 2Ny] +l = +� +k + M +k ∈ [1, M + 1] +k − 2Nz + M +k ∈ [2Nz − M + 1, 2Nz] +(55) +The expansion vector U is also a 2Nx-by-2Ny-by-2Nz tensor, which can be expressed as: +Ui,j,k = +� +u(i+1)/2,j,k, +if i ≤ Nx, j ≤ Ny, k ≤ Nz +0, +otherwise +(56) +17 + +The codes of the three-dimensional Fourier transform and the inverse Fourier transform in Matlab +are called as ˆG = fftn(G) and G = ifftn( ˆG). +For the material points in Ωc, Ωe, and Ωbr, they are treated in the same way as the matrix +mentioned in the form (17), which means the matrix A of the 3D model is decomposed into +A = ˆA + De + Df and the displacement up is replaced with h(xp) in each time iteration if xp ∈ Ωc. +The above analysis shows that MSBFM is an algorithm based on the structure of matrix A. +Most entries in matrix A satisfy the TBT or TBTBT structure; thus, the form f = Au can be +accelerated by FFT. For problems in 2D and 3D, broken bonds, incomplete disks, and volume +constrained boundary conditions break this structure and need special steps to deal with them. +However, the calculation is O(N log N) because most material points are in Ωin. Here a flowchart +is introduced to illustrate these steps as shown in Fig. 7. +Start +Initialize The Model +Input the horizon size δ, grid size N , +find the horizon Bδ(x),Obtain the scalar +s0, Identify boundary +data h(x) and +the initialization of the matrix + assemble matrix K,and Obtain matrix +D, G by K +Apply the initial condition. +Set the Initial displacement +u(x,0),maximum Nt, and time step tn. +Time iteration +Compute f in the step tn. + Obtain and + Compute , by algorithm 3 + and get , +I=xx,xy,xz,yx,yy,yz,zx,zy,zz + If + If +NO +Obtabin disp u(x,t) by +algorithm 1 or algorithm 2 + Is +If tn arrives Nt? +End +Yes +Yes +NO +NO +Yes +yes +nt=nt+1 + , J=x or y +Figure 7: The flowchart of MSBFM +5. Numerical results +In this section, the MSBFM is verified by comparing the meshfree method on four examples +built on the peridynamic model, including 2D/3D models with various boundary conditions and +18 + +Table 1: Performance of meshfree method and MSBFM in 2D model +Mesh +400 × 200 +800 × 400 +1600 × 800 +3200 × 1600 +6400 × 3200 +Time steps +3000 +4000 +5000 +6000 +7000 +Meshfree +Matrix assembly +1m1s +16m40s +4h38m +3d3h +- +Time stepping +4m35s +4h28m +4d18h +- +- +MSBFM +Matrix assembly +24s +3m9s +40m5s +14h40m +6d6h +Time stepping +2m30s +11m42s +52m29s +3h40m +1d4h +cracks. We implement these methods in Matlab and run all experiments on a workstation with +Intel Xeon Gold 6240(2.6GHz/18C) logical processors and 2048G installed memory. +5.1. Peridynamic in 2D body with external loading +As a first illustrative example, let us consider a 2D peridynamic model on a plate with external +loading in this section. +As shown in Fig. (8), the plate has the width W = 0.5 m, the length L = 1.0 m, and thickness +h = 0.0025 m. The material properties are E = 2 × 105Mpa( elastic modulus), ν = 1/3(Poisson’s +(a) +(b) +Figure 8: Geometry of a plate with exteral loads and its discretization (a)plate with external +loading; (b)a simple example of uniform grid structure. +ratio) and ρ = 7850kg/ m3(density ). The external loading bp = p0W/h is applied to the boundary +layer Dc, where the uniaxial tension loading p0 is chosen as 200Mpa, and the width of Dc is h. D +We selected the horizon size δ as 0.03 m and discretize the model by considering Nx = 400 and +Ny = 200. We can consider implementing MSBFM and the meshfree method in this model since it +can be written as a matrix-vector multiplication due to the Eq.(13) and the ADR method is used +for temporal integration(See Section 2.1). +Fig. 9(a) shows the displacement variations obtained by two algorithms when the total time +step equals 3000. It is noticed from Fig. 9(a) that the displacement variations by our algorithm +has a good match with the results obtained by meshfree method when 400 × 200 mesh elements +were employed. +To perform the simulations by using various discretization sizes, we gradually increase the +number of grids from 400×200 to 1600×800. Table 1 compares the computational time required to +perform the simulations using MSBFM and meshfree method . We automatically stop a numerical +run if it takes more than 10 days of CPU time. Even in various mesh elements, the results obtained +19 + +(a) +(b) +Figure 9: Displacement variation with respect to step number in x direction and y direction for +xp = (0.255 m, 0.125 m): (a) The displacement variations obtained by MSBFM and meshfree +method under the mesh 400×200, where the blue solid line represents the data obtained by FMEM, +and the blue dotted line represents the data obtained by the meshfree method; (b)comparison of +displacements in the x-direction under grids is 400 × 200(green), 800 × 400(blue), 1600 × 800(red) +by MSBFM. +by the two algorithms are the same, so we can only consider one of them when analyzing the +properties of material. +In fact, as the mesh elements increases, the number of time steps required by the ADR method +to achieve stability also increases. As shown in the Fig. 9(b), when the mesh number increases +from 400×200 to 1600×800, the number of time steps to achieve stability also increases from 1000 +to 3000. Therefore, we select various time steps according to various mesh elements. +For the non fracture problems, the calculation is mainly divided into two parts: Phase I: +Matrix assembly. For the meshfree method, we need to traverse the horizon of all material points +and initialize an N-by-N matrix A in this phase. In the MSBFM method, matrices K and De are +used instead of matrices A, thus reducing the computational time. Phase II: Time stepping. The +form Au and ADR methods are calculated in this phase for meshfree method, while the MSBFM +method uses FMVM mentioned in (3) and form Deu to replace the calculation of Au. +For the part I, the advantages of MSBFM are mainly reflected in two aspects: one is the +traversal of horizons of the material points. +In this example, we use the method of traversing +all material points and comparing distances to find points within the horizon, so the calculation +amount is usually O(N). For the MSBFM method, we only need to traverse the information of +the material point affected by the incomplete horizon and a complete horizon material point, so +we can reduce the calculation to O(N +3 +2 ). The other is the assembly of stiffness matrix. For the +meshfree method, the assembly of stiffness matrix A need to be considered before the calculation +of time integral equation, and the computational complexity of this part is usually O(N2). The +MSBFM method replaces the stiffness matrix A with matrices K and De, and the calculation +can be obtained by (2M + 1)2 = O(N) for matrix K, where the calculation of matrix De can be +obtained by N(4δ/h) = O(N +3 +2 ), neither of which will exceed O(N2). Table 1 illustrates the time +20 + +0 +0001 +5000 +3000 +2.0- +0 +2.0 +J +2.1 +.S +X10-40 +0001 +S000 +3000 +4000 +0 +2.0 +1 +1'2 +S +2.S +3 +X10-4Table 2: Performance of meshfree method and MSBFM with surface correction algorithm in 2D +model +Mesh +400 × 200 +800 × 400 +1600 × 800 +3200 × 1600 +6400 × 3200 +Time steps +3000 +4000 +5000 +6000 +7000 +Meshfree +Matrix assembly +1m1s +16m40s +4h38m +3d3h +- +Time stepping +4m35s +4h28m +4d18h +- +- +MSBFM +Matrix assembly +24s +3m9s +40m5s +14h40m +6d6h +Time stepping +2m40s +22m50s +6h27m +12h48m +- +cost for the matrix assembly in meshfree method and MSBFM. +The computational time of the phase II depends on the complexity of the form f = Au, which +is O(N2) and O(N log N) for meshfree methods and MSBFM. When the number of grids increases +from Nx, Ny to βNx, βNy, the mesh free time will increase by β4 times since N2 = (β2NxNy)2 = +β4N. However, the MSBFM time only increases by β4 times because β2NxNy log(β2NxNy) = β2 = +β2N log N + 2 log β. +However, the calculation is not strict O(N log N) for the part II in MSBFM algorithm if we use +the surface correction algorithm. In the surface correction algorithm, we need to recalculate fp for +xp ∈ Ωe, which is: +fp = +� +xq∈Bδ(xp)∩Ω +vp,qAp,q(uq − up) +(57) +This means that we need to recalculate the form Au for the point on Ωe by the form (57). In the +meshfree method, this part of the calculation is not omitted, but in the MSBFM method, the form +ˆAu and Deu is used instead of the original form of Au, thus it will bring an additional calculation, +which is O(N2), for the MSBFM method. In order to store the material points that need to be +affected by surface correction, the storage amount will increase to O(N2), which means that the +time of matrix assembly will also increase. +In most cases, the material points on Ωe only account for a part of the total material points. +Thus the simulation speed of our algorithm is significantly faster, see Table 2. +5.2. Peridynamic in 2D body with a pre-existing crack +A 2D model with pre-existing crack is considered in Fig. 10. The length L of this plate is +0.05 m, the width W is 0.05 m, and the crack is created as the length 2c = 0.01 m. Destiny ρ, +horizon size δ, elastic modulus E and Poisson ratio ν are chosen to be consistent with Section 5.1. +Since the PD equation of motion do not contain any spatial derivatives, the constraints often +do not affect the solutions of the integro-differential equations. However, the constraint conditions +can still be imposed by introducing a virtual boundary layer, and the displacement on this virtual +boundary layer will not be affected by the material points on the actual material area. In this +example, virtual boundary layer D1 +c and D2 +c with depth δ is introduced at the upper and lower ends +of the actual material area D, and the velocity constraints is applied on D1 +c and D2 +c, which can be +expressed as follows: +˙uy(xp, t) = 20.0 m/s +xp ∈ D1 +c +˙uy(xp, t) = −20.0 m/s +xp ∈ D2 +c +(58) +21 + +(a) +(b) +Figure 10: Geometry of a 2D model with pre-existing crack under velocity constraints and its +discretization:(a)plate with pre-existing crack; (b) a simple example of uniform grid structure. +Although the displacement of the material point on the virtual material layer is independent of +the actual problem, we still consider its displacement to ensure that the stiffness matrix A calculated +in the MSBFM algorithm is a square matrix. For the material point xp on D1 +c and D2 +c, displacement +up is calculated from two aspects: (a) In the x direction, xp is treated as the material point on Ωc, +which means that we need to replace the displacement ux +p after computing fx = Axxux + Axyux. +(b) In the y direction, we treat xp as the material point on Ωe, which means that we need to +subtract the corresponding De mentioned in (39) when calculating fxx = Ayxuy + Ayyuy. +We discretize the model with a grid size of 600×600, and the Velocity Verlet algorithm is chosen +for time discretization because this is a time-dependent problem. We collect data from from t = 0 +to t = 1350 with a time-step size of ∆t = 1.3367×10−8s. Fig. 11 shows the crack simulation under +two algorithms. +Table 3: Performance of meshfree method and MSBFM with in 2D model +Mesh +600 × 600 1200 × 1200 1800 × 1800 2400 × 2400 3000 × 3000 +Time +1250 +1250 +1250 +1250 +1250 +Meshfree +Matrix assembly +21m36s +5h48m +1d1h +3d11h +- +Time stepping +13m55s +5h50m +1d12h +4d14h +- +Crack factor +8m23s +3h30m +21h4m +2d18h +- +MSBFM +Matrix assembly +2m11s +28m54s +2h28m +8h11m +18h46m +Time stepping +3m22s +16m20s +1h5ms +4h50m +18h36m +Crack factor +8m23s +3h30m +21h4m +2d18h +4d24m +The computational time by using various discretization sizes is shown in Table 3, and we only +22 + +(a) +(b) +Figure 11: Crack simulation results at 1.6708 × 10−3s with two methods: (a) MSBFM (b)meshfree +method +calculate the results within 10 days. +Here the simulation process is divided into three phases. +Phase I: matrix assembly; Phase II:Time stepping; Phase III: Crack factor. The phase I and the +phase II are the same as the non-fracture problem mentioned in Section 5.1, and Phase III mainly +includes the calculation of history-dependent scalar-valued µ and matrix Df. At each phase, there +are differences in the time of non fracture problems and fracture problems. +In Phase I, the advantages of constructing the MSBFM stiffness matrix were retained. Although +an additional matrix Df is introduced, the elements of matrix Df can be obtained from the entries +of matrix K, so no additional assembly is required. But we still need to consider the information of +material points near the fracture to calculate s when traversing the horizon. Since the crack shape +is impossible to estimate, we often need to consider all the material point information in this part, +which leads to MSBFM can not save the calculation amount in this part. In actual calculation, we +can only consider a preset region and a region with incomplete horizon if all cracks do not exceed +this preset region, thus reducing the traversal time. +In Phase II, the matrix Df mentioned in (44) is computed in each time step, which causes +that the computational time of MSBFM in phase I is not strict O(N log N). Df will also affect the +time of matrix assembly, but compared with the meshfree method, MSBFM still has computational +advantages in the fracture problem because the computational complexity of Df does not exceed +O(N2) according to the above analysis. +In the non fracture problem, we do not need to consider Part III. But in fracture problem, the +calculation time for solving µ, which is the main part of part III takes up a large part, which is mainly +caused by the elongation s. s is obtained by a nonlinear form s = (|x′+u′−x−u|−|x +′−x|)/(|x +′−x|), +so it cannot be solved by FFT, which causes that the calculation of this part is usually O(N2). In +the meshfree and MSBFM methods, the calculation amount for solving s is the same. +5.3. Peridynamic in 3D body under displacement constraints +We perform PD simulations using a 3D model with incomplete horizons and displacement +constraints in this section. +23 + +001 +S00 +300 +400 +eoo +0 +100 +S.0 +S00 +0'4 +300 +a.0 +400 +8.0 +000 +eoo001 +S00 +300 +400 +eoo +0 +100 +S.0 +S00 +0'4 +300 +a.0 +400 +8.0 +000 +eooAs shown in Fig. 12, a block with length L = 1.0 m, width W = 0.3 m, and thickness H = 0.3 m +is introduced . Horizon size δ is chosen as 0.00315 m. External loading bp = p0W/h2 is applied to +the area Ds, and the value of p0 is the same as that in Section 5.1. +The displacement constraints is imposed on the virtual boundary layer Dc, which means: +ux(xp, t) = uy(xp, t) = uz(xp, t) = 0, +xp ∈ Ac +(59) +Elastic modulus E and Poisson’s ratio ν are taken as 2 × 105Mpa and 1/4, respectively. +The +material points affected by displacement constraints are treated as material points on Ωc. +(a) +(b) +Figure 12: Geometry of a block under displacement constraints and its discretization: (a)block +with external loading and displacement constraints; (b)a simple example of uniform grid structure. +The grid of size 100×30×30 is used to discretize the model and perform time stepping through +the ADR method. The displacement variations of the two algorithms are provided in in Fig. 13, +which verify the accuracy of MSBFM in 3D problems. +(a) +(b) +Figure 13: Displacement variation with respect to step number in x direction (blue), y direc- +tion(red), z direction(green): (a)MSBFM (b)meshfree method +Table 4 compares the computational time required to perform the simulations using MSBFM +and the meshfree PD method, and Only results not exceeding 10 days are considered. Similar to the +two-dimensional problem, the computational time of the three-dimensional non fracture problem +can also be divided into two phases: phase I: Compute f, including the calculation of fx, fy, fz +and the form Du; B: Matrix assembly, including searching the horizon of the material point and +initialization of matrices Axx, Axy, Axz, Ayx, Ayy, Ayz, Azx, Azy, Azz and corresponding matrices +De. +24 + +0 +S00 +400 +eoo +008 +0001 +-S +0 +4 +e +8 +JO +X10-40 +S00 +400 +eoo +008 +0001 +-S +0 +4 +e +8 +JO +X10-4Table 4: Performance of meshfree method and MSBFM in 3D model +Mesh +100 × 30 × 30 +200 × 60 × 60 +300 × 90 × 90 +Time +1000 +2000 +3000 +Meshfree +Matrix assembly +1m40s +1h47m +4d17h +Time stepping +35m5s +1d13h +- +MSBFM +Matrix assembly +37s +20m12s +3h20m +Time stepping +3m2s +49m15s +9h10m +For the part I, the calculation amount of De can be obtained from N2(4δ/h) = O(N +7 +3 ). In the +part of traversing horizon of material points, the ratio of the computational time is the same for the +3D model and the 2D model. In fact, the the computational time of part I depends on the number +of material points rather than the dimension of the point, and in three-dimensional problems, the +number of material points tends to be more than in two-dimensional problems, thus MSBFM will +be more advantageous in part I in three-dimensional problems. +We observed a high rate between MSBFM and meshfree method in part II, and this is the +result of the differences in dimensions between the 2D model and the 3D model. When the Nx, +Ny, Nz increases to βNx, βNy, βNz, N increases to β3N. Similar to the analysis in section 5.1, +the computational time of our algorithm will increase by β3 times, while the meshfree method will +increase by β6 times. This is why the time ratio of part II is larger than that of 2D model. +One problem that needs to be noted is that the proportion of material points on Ωe in the +total material points in the 3D model will also increase. Therefore, when the surface correction +algorithm is adopted, the computational advantage of algorithms may not be obvious. +5.4. Kalthoff-Winkler experiment +To simulate the calculation rate of MSBFM algorithm on 3D fracture model, a KW example is +introduced in this subsections, as shown in Fig. (14). +The problem description is as follows: A block of length L = 0.2 m, width W = 0.1 m, and +thickness h = 0.009 m with two thin notches is subjected to the incomplete horizons and impactor. +The notch in this model has width h0 = 0.0015 m, length a0 = 0.05 m, and distance between +notches d are 0.05 m and the impactor’s diameter D and height H are all 0.05 m. Horizon size δ, +density ρ, elastic modulus E and Poisson’s rate ν considered are chosen as in Section 5.3. Initial +velocity v0 = 32 m/s is imposed to ensure crack propagation. Critical stretch s0 is chosen for +judging fracture, which is taken as 0.01. We conduct this simulations by using Velocity Verlet, and +each time step is 1.3367 × 10−8. +The crack propagations by our algorithm and meshfree method are provided in Fig. 15. we +can obverse that the cracking angle is 135◦, which is is consistent with the results obtained in the +standard experiment. The crack propagations in our algorithm at different time steps shown in +Fig. 16 and the computational time is shown in Table 5. For more complex meshes, both FMBM +and meshfree methods have exceeded the time limit, which is mainly caused by point arrangement. +If other point arrangement methods are used, the calculation time will be greatly reduced. +As expected, the KW experiment retains the computational advantage of MSBFM in the process +of computing f. However, the number of broken bonds in 3D model will also increase, which mainly +affects two aspects: first, the construction of matrix Df, whose calculation is usually O(N2), which +25 + +L +W +h +L +D +H +a +d +0 +h0 +(a) +(b) +Figure 14: Geometric of Kalthoff-Winkler test and its discretization(a)block used in Kalthoff- +Winkler test; (b)a simple example of uniform grid structure. +(a) +(b) +Figure 15: crack simulation results at 1.1754 × 10−8s with two methods: (a)MSBFM (b)meshfree +method +(a) +(b) +(c) +Figure 16: crack simulation results by MSBFM (a)at 350 time steps(b)at 650 time steps(c)at 1350 +time steps +will affect the process of computing f and matrix assembly. The other is the calculation of µ, +because of the increase of broken bonds, the calculation of fracture simulation will also increase, +which occupies the main part in each time step. +26 + +SO +40 +eo +08 +001 +0 +S.0 +0'4 +001 +a.0 +8.0 +S00SO +40 +eo +08 +001 +0 +S.0 +0'4 +001 +a.0 +8.0 +S00SO +40 +eo +08 +001 +0 +S.0 +0'4 +001 +a.0 +8.0 +S00SO +40 +eo +08 +001 +0 +S.0 +0'4 +001 +a.0 +8.0 +S00SO +40 +eo +08 +001 +0 +S.0 +0'4 +001 +a.0 +8.0 +S00Table 5: Performance of meshfree method and MSBFM in KW experiment +Mesh +201 × 101 × 9 +402 × 202 × 18 +Time +1350 +1350 +Meshfree +Matrix assembly +9m25s +11h10m +Time stepping +1h16m +1d4h +Crack factor +40m30s +17h5m +MSBFM +Matrix assembly +6m29s +7h39m +Time stepping +13m27s +2h5m +Crack factor +40m25s +17h5m +In fact, since the number of broken bonds cannot exceed the total number of bonds, the com- +putational advantages of MSBFM can be maintained in most fracture models, especially in 3D +models. The fewer broken bonds, the more obvious the computational advantage of the MSBFM +algorithm. +Comparisons between the computational efficiency of the new MSBFM method for PD models +with that of the original meshfree discretization of PD formulations by four examples showed the +computational and storage advantages of our algorithm, especially in 3D problems. One can now +easily simulate fracture problems by selecting some material points instead of all material points +by using MSBFM, which reduces memory allocation and maintains high accuracy compared with +the mesh free method +6. Conclusions +In this paper, we introduce a matrix-structure-based fast method(MSBFM). In this framework, +the stiff matrix is decomposed into a summation of several matrices according to the model’s other +boundary conditions and fracture conditions. Following these decompositions, FFT and its inverse +operation are used to calculate the PD integral with the cost of O(N log N) instead of O(N2) +required by the usual meshless or FEM discretization methods of the PD model. Because of the +Fourier transform, storing all the information about the material points and their horizon is no +longer necessary, thus reducing the storage cost from O(N2) to O(N) of the meshfree or FEM +discrete method. Therefore, the time for initializing the matrix is also reduced. For the time- +dependent problems and quasi-static problems, the time marching schemes are used to simulate. +The method mentioned in this paper applies to most nonlocal models as long as their dis- +crete forms can be written in a matrix. This paper focuses on the bond-based PD model, and +the following numerical test are performed: two-dimensional non-fracture problems with loading +two-dimensional fracture problems with displacement constraints, three-dimensional non-fracture +problems with two kinds of boundary conditions. The results are in good agreement with the the- +ory. The comparison with the computational speed of the meshfree method shows that MSBFM +can reduce the computational time of tens of days in this method to several hours. This means +that for complex fracture problems, the selection of PD nodes and the computational cost is no +longer the main obstacles for complex fracture problems. +The algorithm still depends on the matrix structure to some extent, which means it must be +a quasi-Toeplitz structure. In other words, most entries satisfy the Toeplitz structure. Efforts are +27 + +underway to extend the application of the SFPD algorithm to more PD problems with a complex +matrix structure, including state-based problems, coupling problems, and nonlinear problems. +Acknowledgements +The work was carried out at Marine Big Data Center of Institute for Advanced Ocean Study +of Ocean University of China. +References +References +[1] S.Silling, Reformulation of elasticity theory for discontinuous and long-range forces, J. Mech. +Phys. Solids 48 (2000) 175–209, https://doi.org/10.1016/S0022-5096(99)00029-0 +[2] J.Xu, A.Askari, O.Weckner, S.Silling, Peridynamic analysis of impact damage in composite +laminates, J. Aerosp. Eng. 21 (2008) 187–194, https://doi.org/10.1061/ASCE0893-1321 +[3] S.Silling, M.Epton, O Weckner, J.Xu, E. Askari, Peridynamic States and Constitutive Modeling, +J. Elasticity (2007) 88:151–184, DOI 10.1007/s10659-007-9125-1 +[4] G.Ongaro, R.Bertani, U.Galvanetto, A.Pontefisso, M.Zaccariotto, A multiscale peridynamic +framework for modelling mechanical properties of polymer-based nanocomposite, Eng. Fract. +Mech 274 (2022) 108751, https://doi.org/10.1016/j.engfracmech.2022.108751 +[5] Y. L. Hu, N. V. De Carvalho, E. Madenci, Peridynamic modeling of delamination growth in com- +posite laminates, Compos. Strut. 132 (2015) 610–620, http://dx.doi.org/10.1016/j.compstruct +.2015.05.079 +[6] V. A. Buryachenko, Generalized effective fields method in peridynamic micromechanics of ran- +dom structure composites, Int. J. Solids. Struct. 202 (2020) 765–786, https://doi.org/10.1016/j.ij +solstr.2020.06.022 +[7] S. Jafarzadeh, Z. Chen and F. Bobaru, Computational modeling of pitting corrosion, Corros. +Rev. 37(5)(2019) 419–439, https://doi.org/10.1515/corrrev-2019-0049 +[8] Z. Chen, S. Jafarzadeh, J. Zhao, F. Bobaru, A coupled mechano-chemical peridynamic model +for pit-to-crack transition in stress-corrosion cracking, J. Mech. Phys. Solids. 146 (2021) 104203, +https://doi.org/10.1016/j.jmps.2020.104203 +[9] J. +Zhao, +S. +Jafarzadeh, +M. +Rahmani, +Z. +Chen, +Y. +Kim, +F. +Bobarua, +A +peridy- +namic model for galvanic corrosion and fracture, Electrochim. Acta. 391 (2021) 138968, +https://doi.org/10.1016/j.electacta.2021.138968 +[10] S.Jafarzadeh, +Z. +Chen, +S. +Li, +F. +Bobaru, +A +peridynamic +mechano-chemical +dam- +age +model +for +stress-assisted +corrosion, +Electrochim. +Acta. +323 +(2019) +134795, +https://doi.org/10.1016/j.electacta.2019.134795 +[11] S. Jafarzadeh, Z. Chen, J. Zhao, F. Bobaru, Pitting, lacy covers, and pit merger in stainless +steel: 3D peridynamic models, Corros. Sci. 150 (2019) 17–31. +28 + +[12] C. T. Nguyen, S. Oterkus, E. Oterkus, An energy-based peridynamic model for fatigue crack- +ing, Eng. Fract Mech. 241 (2021) 107373, https://doi.org/10.1016/j.engfracmech.2020.107373 +[13] X. Li, X. Gu, X. Xia, E. Madenci, X. Chen, Q. Zhang, Effect of water-cement ratio and size on +tensile damage in hardened cement paste: Insight from peridynamic simulations, Constr. Build. +Mater. 356 (2022) 129256, https://doi.org/10.1016/j.conbuildmat.2022.129256 +[14] D. +Jin, +W. +Liu, +A +peridynamic +modeling +approach +of +solid +state +impact +bond- +ing and simulation of interface morphologies, +Appl. Math. Model. 92 (2021) 466–485, +https://doi.org/10.1016/j.apm.2020.11.014 +[15] M. Nowak, K. Mulewska, A. Azarov, L. Kurpaska , A. Ustrzyck, A peridynamic elasto- +plastic damage model for ion-irradiated materials, Int. J. Mech. Sci. 237 (2023) 107806, +https://doi.org/10.1016/j.ijmecsci.2022.107806 +[16] P. Wu, Z. Chen, Peridynamic Electromechanical Modeling of Damaging and Cracking in +Conductive Composites: A Stochastically Homogenized Approach, Compos. Struct. 305 (2023) +116528, https://doi.org/10.1016/j.compstruct.2022.116528 +[17] W. Gerstle, N. Sau, S. Silling, Peridynamic modeling of concrete structures, Nucl. Eng. Des. +237 (2007) 1250–1258, http://dx.doi.org/ 10.1016/j.nucengdes.2006.10.002. +[18] Y. Xia, X. Meng, G. Shen, G. Zheng, P. Hu, Isogeometric analysis of cracks with peridynam- +ics, Comput. Methods Appl. Mech. Engrg. 377 (2021) 113700, http://dx.doi.org/10.1016/j.cma. +2021.113700. +[19] S. Silling, E. Askari, A meshfree method based on the peridynamic model of solid mechanics, +Comput. Struct. 83 (2005) 1526–1535, http://dx.doi.org/10.1016/j.compstruc.2004.11.026. +[20] S. Li, Y . Jin, X. Huang, L. Zhai, An extended bond-based peridynamic approach +for analysis on fracture in brittle materials, +Math. Probl. Eng. 2020 (2020) 9568015, +http://dx.doi.org/10.1155/2020/9568015. +[21] P. Diehl, S. Prudhomme, M. L´evesque, A review of benchmark experiments for the +validation of peridynamics models, +J. Peridynamics Nonlocal Model. 1 (2019) 14–35, +http://dx.doi.org/10.1007/s42102-018-0004-x. +[22] X. Tian, Q. Du, Analysis and comparison of different approximations to nonlocal dif- +fusion and linear peridynamic equations, +SIAM J. Numer. Anal.51 (2013) 3458–3482, +http://www.siam.org/journals/sinum/51-6 +[23] T. Bode1, C. Weiβenfels, P. Wriggers, Peridynamic Galerkin method: an attractive alternative +to finite elements, Comput. Mech. (2022) 70 723-743, +[24] X. Chen, M. Gunzburger, Continuous and discontinuous finite element methods for a peri- +dynamics model of mechanics, Comput.Methods Appl. Mech. Engrg. 200 (2011) 1237–1250, +https://doi.org/10.1016/j.cma.2010.10.014 +[25] S. Silling and E. Askari, A meshfree method based on the peridynamic model of solid mechan- +ics, Comput. Struct, 83 (2005) 1526-1535, https://doi.org/10.1007/s00466-022-02202-w +29 + +[26] J. Lu, Y. Nie, A reduced-order fast reproducing kernel collocation method for nonlocal mod- +els with inhomogeneous volume constraints, Comput. Math. with Appl. 121 (2022) 52–61, +https://doi.org/10.1016/j.camwa.2022.06.024 +[27] J. Lu, M. Yang, Y. Nie, Convergence analysis of Jacobi spectral collocation methods for weakly +singular nonlocal diffusion equations with volume constraints, Appl. Math. Comput. 431 (2022) +127345, https://doi.org/10.1016/j.amc.2022.127345 +[28] S. Zhang, Y. Nie, Localized Chebyshev and MLS collocation methods for solving 2D steady +state nonlocal diffusion and peridynamic equations, Math. Comput. Simulat. 206 (2023) 264–285, +https://doi.org/10.1016/j.matcom.2022.11.018 +[29] X.Tian, Q. Du, Asymptotically compatible schemes and applications to robust discretization +of nonlocal models, SIAM.J. Numer. Anal. 52 (2014) 1641–1665, https://doi:10.1137/130942644. +[30] Q. Du, J. Yang, Asymptotically compatible fourier spectral approximations of nonlocal allen- +cahn equation, SIAM. J. Numer. Anal. 54(3):1899–1919, https://doi.org/10.1137/15M1039857 +[31] M. Zaccariotto, T. Mudric, D. Tomasi, A. Shojaei, U. Galvanetto, Coupling of FEM +meshes with peridynamic grids, Comput. Methods. Appl. Mech. Engrg. 330 (2018) 471–497, +https://doi.org/10.1016/j.cma.2017.11.011 +[32] T. Ni, +M. Zaccariotto, +Q.Z. Zhu, +U. Galvanetto, +Static solution of crack propaga- +tion problems in peridynamics, Comput. Methods Appl. Mech. Engrg. 346 (2019) 126–151, +http://dx.doi.org/10.1016/j.cma.2018.11.028. +[33] J. Zhang, F. Han, Z. Yang, J. Cui, Coupling of an atomistic model and bond-based peridynamic +model using an extended Arlequin framework, Comput. Methods Appl. Mech. Engrg. 403 (2023) +115663, https://doi.org/10.1016/j.cma.2022.115663 +[34] F. Han, G. Lubineau, Y. Azdoud, Adaptive coupling between damage mechanics and peridy- +namics: A route for objective simulation of material degradation up to complete failure, J. Mech. +Phys. Solids. 94 (2016) 453–472, https://doi.org/10.1016/j.jmps.2016.05.017 +[35] H. Zhang, H.Li, H. Ye, Y. Zheng, A coupling peridynamic approach for the consolida- +tion and dynamic analysis of saturated porous media, Comput. Mech. 64(2019) 1097-1113, +https://doi.org/10.1007/s00466-019-01695-2 +[36] M. Birner, P. Diehl, R. Lipton, M. Alexander Schweitzer, A fracture multiscale model for +peridynamic enrichment within the partition of unity method, Adv. Eng. Softw. 176 (2023) +103360 ,https://doi.org/10.1016/j.advengsoft.2022.103360 +[37] S. Jafarzadeh, A. Larios, F. Bobaru, Efficient solutions for nonlocal diffusion problems +via boundary-adapted spectral methods, J.Peridynamics Nonlocal Model. 2 (2020) 85–110, +https://doi.org/10.1007/s42102-019-00026-6 +[38] S. Jafarzadeha, F. Mousavia, A. Lariosb, F. Bobarua, A general and fast convolution-based +method for peridynamics: Applications to elasticity and brittle fracture, Comput. Methods Appl. +Mech. Engrg. 392 (2022) 114666, https://doi.org/10.1016/j.cma.2022.114666 +30 + +[39] S. Jafarzadeh, L. Wang, A. Lariosb, F. Bobarua, A fast convolution-based method for peridy- +namic transient diffusion in arbitrary domains, Comput. Methods Appl. Mech. Engrg. 375 (2021) +113633, https://doi.org/10.1016/j.cma.2020.113633 +[40] D.A. Abdoh, B.B. Yin, V. K. R. Kodur, K.M. Liew, Computationally efficient and effective +peridynamic model for cracks and fractures in homogeneous and heterogeneous materials, Com- +put. Methods Appl. Mech. Engrg. 399 (2022) 115318, https://doi.org/10.1016/j.cma.2022.115318 +[41] H. +Wang +, +H. +Tian, +A +fast +Galerkin +method +with +efficient +matrix +assembly +and +storage +for +a +peridynamic +model, +J. +Comput. +Phys. +231 +(2012) +7730–7738, +https://doi.org/10.1016/j.jcp.2012.06.009 +[42] H. Wang, H. Tian, A fast and faithful collocation method with efficient matrix assembly for +a two-dimensional nonlocal diffusion model, Comput. Methods Appl. Mech. Engrg. 273 (2014) +19–36, https://doi.org/10.1016/j.cma.2014.01.026 +[43] X. +Zhang, +H. +Wang, +A +fast +collocation +method +for +a +static +bond-based +lin- +ear +peridynamic +model, +Comput. +Methods +Appl. +Mech. +Engrg. +311 +(2016) +280–303, +https://doi.org/10.1016/j.cma.2016.08.020 +[44] C. Wang, H. Wang, A fast collocation method for a variable-coefficient nonlocal diffusion +model, J.Comput. Phys 330 (2017) 114–126, https://doi.org/10.1016/j.jcp.2016.11.003 +[45] H. Liu, A. Cheng, H. Wang, A Fast Discontinuous Galerkin Method for a Bond-Based Linear +Peridynamic Model Discretized on a Locally Refined Composite Mesh, J. Sci. Comput (2018) +76:913–942, https://doi.org/10.1007/s10915-018-0645-6 +[46] X. Zhang, X. Li, A. Cheng, H. Wang, A preconditioned fast collocation method for a linear +bond-based peridynamic model, Adv. Differ. Equ-Ny (2020) 244, https://doi.org/10.1186/s13662 +-020-02700-2 +[47] X. +Zhang, +A. +Cheng, +H. +Wang, +Preconditioned +Fast +Collocation +Method +for +a +Linear +Nonlocal +Diffusion +Model +in +Convex +Domains, +IEEE. +Access. +Vol.2, +2020, +https://doi.org/10.1109/ACCESS.2020.3027247 +[48] B.Kilic, +E.Madenci, +An +adaptive +dynamic +relaxation +method +for +quasi-static +simu- +lations using the peridynamic theory, +Theor. Appl. Fract. Mech. 53 (2010) 194–204, +https://doi.org/10.1016/j.tafmec.2010.08.001 +[49] E. Madenci, E. Oterku, Peridynamic theory and its applications. New York: Springer; 2014. +https://doi.org/10.1007/978-1-4614-8465-3. +[50] Q.V.Le, F.Bobaru, Surface corrections for peridynamic models in elasticity, Comput. Mech. +(2018) 61:499–518, https://doi.org/10.1007/s00466-017-1469-1 +31 + diff --git a/N9FKT4oBgHgl3EQffC6n/content/tmp_files/load_file.txt b/N9FKT4oBgHgl3EQffC6n/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1a89248da4d7e71482e8e621b5c34c483a5c909e --- /dev/null +++ b/N9FKT4oBgHgl3EQffC6n/content/tmp_files/load_file.txt @@ -0,0 +1,1488 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf,len=1487 +page_content='A fast computational framework for the linear bond-based peridynamic model Chenguang Liua Hao Tiana Wai Sun Dona Hong Wangb a School of Mathematical Science, Ocean University of China, Qingdao, Shandong 266100, China bDepartment of Mathematics, University of South Carolina, Columbia, South Carolina 29208, USA Abstract Peridynamic (PD) theory is significant and promising in engineering and materials science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' how- ever, it imposes challenges owing to the enormous computational cost caused by its nonlocality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' Our main contribution, which overcomes the restrictions of the existing fast method, is a general computational framework for the linear bond-based peridynamic models based on the meshfree method, called the matrix-structure-based fast method (MSBFM), which is suitable for the gen- eral case, including 2D/3D problems, and static/dynamic issues, as well as problems with general boundary conditions, in particular, problems with crack propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' Consequently, we provide a general calculation flow chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' The proposed computational framework is practical and easily em- bedded into the existing computational algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' With this framework, the computational cost is reduced from O(N2) to O(N log N), and the storage request is reduced from O(N2) to O(N), where N is the degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' Finally, the vast reduction of the computational and memory requirement is verified by numerical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' Keywords: bond-based peridynamics, matrix-structure-based fast method, computational framework, crack propagation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' Introduction Classical continuum mechanics are expressed as a partial differential equation, which is chal- lenging to describe models with discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' As a result, peridynamics(PD), as proposed by Silling[1], is an integral-type nonlocal model and can provide a general theory for solving problems in the form of discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' Over the past few decades, the effectiveness of PD has attracted extensive research conducted on modeling methods, numeral techniques, and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' In this paper, we focus on the bond-based peridyanmics [2], which is an early version of peridynamics and can be applied to isotropic materials, with Poisson’s ratio of 1/4 for plane strain and 1/3 for plane stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' Later the ordinary state-based and non-ordinary state-based PD were proposed to elimi- nate the constraints of fixed Poisson’s ratios on materials[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' The PD models have been frequently used in many practical problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' A range of cutting-edge applications can be found in composite material deformation[4, 5, 6], corrosion[7, 8, 9, 10, 11], damage prediction[12, 13, 14, 15, 16] and crack simulation for a variety of materials[17, 18, 19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' There have been much research directed at developing numerical methods that can solve the PD models, including meshfree methods, finite difference, finite element methods, and collocation methods[22, 23, 24, 25, 26, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content='Among other research in the literature, explored in[22, 29, 30], the asymptotically compatible schemes retained a limit behavior that makes the limit of the zero-horizon Preprint submitted to Computer Methods in Applied and Mechanical Engineering January 30, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content='11828v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content='NA] 27 Jan 2023 of the nonlocal operator become the local differential operator, providing a consistency between local and non-local models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' However, these methods are very restrictive due to the vast computational cost caused by nonlocality of PD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' The increasing computation cost limits the application of PD theory, especially for multidimensional cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' Lots of efforts have been made to overcome this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' A class of coupled method was introduced to accelerate the PD simulation, which utilizes PD on the area around the cracks and classical mechanics on the rest area [31, 32, 33, 34, 35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' A fast method based on the convolution structure of PD models[38, 39] is proposed to accelerate the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' A super-fast peridynamic model[40] based on decreasing the number of inner loop operations is also introduced to overcome this difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' The work above also gives us some inspiration for this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' In the literature, a class of fast methods utilizing the structure of stiff matrix are booming, which can reduce the computational cost from O(N2) to O(N log N) without loss of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' In 2010, a fast method[41] based on the Toeplitz structure of stiff matrices was proposed, hence solving the 1D static linear bond-based peridynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' Subsequently, a fast collocation method based on TBT matrix structure was given for 2D nonlocal diffusion models in reference[42], which can be thought of a approximation model of scalar-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' A fast collocation method for the 2D static linear bond-based peridynamics with volume boundary conditions was investigated in [43] , where we use an equivalent but more effective way to evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' In 2017, a fast method was also presented to solve nonlocal diffusion models with variable coefficients[44], and a discontinuous method was discussed to solve linear bond-based PD models with discontinuous solves[45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' In 2020, a fast algorithm with preconditioned processing was proposed[46, 47], accelerating the convergence of the iterative method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' Although the research above has significantly contribute to applying fast matrix structure based methods in the simulation of PD, there are still several pending issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' The first one is that all this research is proposed for 1D or 2D static models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' The second one is that there need to be methods taking into account volume constrained boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' The third one is that all these research is developed for models with no cracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' A natural question comes into the work here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' To fill this gap, we give a fast matrix-based method(MSBFM), which is the main contribution of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' In this paper, we offer an insightful look at a very general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' We propose a fast matrix- based method(MSBFM) for solving 2D/3D linear bond-based peridynamic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' By establishing the relationship between the stiffness matrix and the Toeplitz-Block-Toeplitz(TBT) matrix, we focos on the matrix decomposition, stiff matrix, one for TBT matrix and another for sparse matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' This reduces the structural limitations of the matrices and hence a more efficient method applied to general boundary and cracks, and dramatically reduces the amount of computation and storage from O(N2) to O(N log N) and from O(N2) to O(N), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' Meanwhile, the results are for models in 2D as well as in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' Numerical experiments verify the accuracy of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' The following articles are organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' In section 2, we reviewed the linear model of the peridynamic and meshfree methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' In section 3, we analyze the matrix structure of the two- dimensional problem and give an accelerated process based on the matrix structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' In section 4, the matrix structure of the 3D model is discussed, and the MSBFM method is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' In section 5, the accuracy and acceleration effect of the MSBFM method are shown by numerical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' Fundamentals of linear bond-based peridynamics and its discretization The bond-based peridynamics, is a reformulation for classical continuum solid mechanics by the integral form instead of partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' Typically, with sufficiently small displace- ment, bond-based peridynamics can be approximated as linear bond-based peridynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' In this section, we mainly review the linearized version of the bond-based peridynamic mode, addressed in this study, and the discretization to solve the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' Linear bond-based peridynamics The equation of motion for the linear bond-based peridynamics with prescribed volume bound- ary condition can be defined as follows: � � � ρ¨u(x, t) = � Bδ(x)∩Ω µC(x ′ − x)(u � x′, t � − u(x, t))dVx′ + b(x, t) x ∈ Ωs, u(x, t) = h(x, t) x ∈ Ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' (1) where ρ is the mass density, Ω = Ωs ∪ Ωc is the spatial domain, Bδ(x) is the horizon which is usually taken as a disk or ball of radius δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' u(x, t) is displacement vector field, and b(x, t) is a body force density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' h(x, t) is the prescribed displacement data imposed on the volume constrained boundary Ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' C(x′ − x) is the micromodulus tensor which can be written as[24]: C(x ′ − x) = α(x ′ − x) ⊗ (x ′ − x) |x ′ − x|3 , (2) where α is a scalar parameter introduced to keep the energy of peridynamic model and classic elastic model equal, which is determined by δ and the elastic modulus E: α = � 9E πδ3h 2D plane stress or plane strain, 12E πδ4 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' (3) with h being the plate thickness in the 2D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FKT4oBgHgl3EQffC6n/content/2301.11828v1.pdf'} +page_content=' µ is a history-dependent scalar-valued function which can be written as: µ(s, t) = � 1 if the bond is not broken s (i, 1) > (i, 0) < (i + 1, 2) for every i ∈ Zn. +It is easy to see that +such poset satisfies Aut(P) ≃ Zn. Suppose then that n ≥ 8. We take two copies of Zn: +A = Zn = {0, 1, . . . , n − 1} and A′ = {0′, 1′, . . . , (n − 1)′}. Let S = {0, 1, 2, 4} ⊆ Zn. For +i ∈ A and j′ ∈ A′ we set i < j′ if j − i ∈ S. Any two elements in the same copy of Zn +are not comparable (see Figure 1). We will prove that the automorphism group of this +poset P is Zn. It is clear that G = Zn acts regularly on each copy of Zn by multiplication +(addition), and this gives a faithful action G → Aut(P) on P. So G can be seen as a +subgroup of Aut(P). Since each automorphism of P maps 0 ∈ A to another minimal +element of P, then the order of the Aut(P)-orbit of 0 ∈ P is n. If we prove that the +Aut(P)-stabilizer of 0 ∈ P is trivial, then |Aut(P)| = n, so Aut(P) is isomorphic to G. +Let h ∈ Aut(P) be such that h(0) = 0. +We define the double neighborhood B(i) of i ∈ A as the set of those j ∈ A such that +#(P>i ∩ P>j) ≥ 2, that is, there are at least two points in A′ greater than both, i and j. +The reduced double neighborhood of i ∈ A is ˆB(i) = B(i)∖{i}. Since h is an automorphism, +B(h(i)) = h(B(i)) and ˆB(h(i)) = h( ˆB(i)). Given k ≥ 1, we say that two points i, j ∈ A are + +SMALLEST POSETS WITH GIVEN CYCLIC AUTOMORPHISM GROUP +3 +Figure 1. The Hasse diagram of P for n = 8. +k-adjacent if #(B(i) ∩ B(j)) = k, and they are reduced k-adjacent if #( ˆB(i) ∩ ˆB(j)) = k. +Clearly, h preserves k-adjacency and reduced k-adjacency. Suppose first that n ≥ 9. Then +for each i ∈ A, B(i) = {i − 2, i − 1, i, i + 1, i + 2}. It is easy to see that i, j are 4-adjacent +if and only if i − j = ±1. Thus, h induces an automorphism of the cyclic graph on A with +edges given by 4-adjacency. Since h(0) = 0, h is either the identity 1Zn or −1Zn. The +second case cannot occur as {0, 2, 3, 4} has an upper bound while {0, −2, −3, −4} does +not. Thus every point of A is fixed by h. If j′ ∈ A′, then j′ is the unique upper bound of +{j, j − 1, j − 2, j − 4}. Thus h(j′) = j′. This proves that h = 1P . +Finally, suppose n = 8. Given i ∈ A, we have now ˆB(i) = {i−2, i−1, i+1, i+2, i+4} and +i, j ∈ A are reduced 4-adjacent if and only if i−j = ±3. Thus, h induces an automorphism +in the cyclic graph on A with edges given by reduced 4-adjacency. Then h = 1Zn or −1Zn. +The second case cannot occur for the same reason as before. Since each point in A′ is +determined by the set of smaller points, h = 1P . +□ +Example 3. There exists a poset P with 20 points and automorphism group isomorphic +to Z12. +Take two copies A = {0, 1, 2, 3, 4, 5}, A′ = {0′, 1′, 2′, 3′, 4′, 5′} of Z6 and two copies +B = {0′′, 1′′, 2′′, 3′′}, B′ = {0′′′, 1′′′, 2′′′, 3′′′} of Z4. The underlying set of P is the union of +these four sets. Let S = {0, 1, 3} ⊆ Z6, T = {0, 1} ⊆ Z4. Define the following order in P: +i < j′ if j − i ∈ S, i′′ < j′′′ if j − i ∈ T, i′′′ < j′ if j − i is even, i′′ < j if j − i is even, +i′′ < j′ for every i, j (see Figure 2). +0 +1 +2 +3 +4 +5 +0' +1' +2' +3' +4' +5' +0'' +1'' +2'' +3'' +0''' +1''' +2''' +3''' +Figure 2. A poset P of 20 points and Aut(P) ≃ Z12. +It is clear that G = Z12 acts in each copy of Z6 and of Z4 by multiplication (addition). +This induces a faithful action of G on P. If h ∈ Aut(P), h(0′′) must be a minimal point i′′ +and h(0′) must be a maximal point j′. However i, j cannot have different parity. Indeed, + +0 +2 +3' +4 +5' +2 +YQ +2 +5' +6 +4 +5 +2 +3 +4 +64 +J.A. BARMAK AND A.N. BARRETO +among the points 0, 2, 4, 0′′′, 1′′′ which cover 0′′, there are just two 0, 0′′′ smaller that 0′. +However, if i ∈ Z4 and j ∈ Z6 have different parity, among the points covering i′′ (k ∈ A +with k ≡ i(2) and i′′′, (i + 1)′′′) there are three smaller that j′: both j − 1, j − 3, and +one of i′′′, (i + 1)′′′. Thus i ≡ j(2), which implies that the Aut(P)-orbit of the set {0′, 0′′} +has at most 12 elements. If we prove that the Aut(P)-stabilizer of {0′, 0′′} is trivial, then +|Aut(P)| ≤ 12 = |G|, so Aut(P) is isomorphic to G. Let h be an automorphism of P +which fixes 0′ and 0′′. +Note that 2′′ is the unique minimal point different from 0′′ which is covered by three +points that cover 0′′. Thus h(2′′) = 2′′. Now, the points of B′ are the unique points of P +which cover exactly one of 0′′, 2′′. Thus B′ is invariant. This implies that h restricts to an +automorphism of the subposet R with underlying set B ∪ B′ and of the subposet Q with +set A ∪ A′. Since R is a cycle, there are only two automorphisms of R fixing 0′′. One is +the identity and the other maps 0′′′ to 1′′′. However, 0′′′ < 0′ while 1′′′ ≮ 0′. Thus 0′′′ is +fixed by h and then h is the identity of R. +Suppose that i′ ∈ A′ is a fixed point. Among the points i, i − 1, i − 3 in A covered by +i′, only i − 1 and i − 3 share a lower bound. Thus h(i) = i. Similarly, among the points +(i − 4)′, (i − 2)′, (i − 1)′ of A′ not covering i, only (i − 4)′ and (i − 2)′ share a lower bound +in B′. Thus (i−1)′ is fixed. In conclusion, we showed that i′ fixed implies that both i and +(i − 1)′ are fixed. Since 0′ is fixed, this implies that every point of A and of A′ is fixed. +Thus h = 1P . +We say that a prime power pr (r ≥ 1) exactly divides an integer n, and write pr ∥ n, if +pr|n and pr+1 ∤ n. +Theorem 4. Let n = pr1 +1 pr2 +2 . . . prk +k where the pi are different primes and ri ≥ 1 for every i. +Then there exists a poset with automorphism group isomorphic to Zn and +k� +i=1 +b(pri +i )pri +i − 1 +points if 3 ∥ n and 4 ∥ n, and with +k� +i=1 +b(pri +i )pri +i points otherwise. +Proof. By Proposition 2, for each 1 ≤ i ≤ k there exists a poset Pi with b(pri +i )pri +i points and +Aut(Pi) ≃ Zpri +i . The non-Hausdorff join or ordinal sum P = P1⊕P2⊕. . .⊕Pk is constructed +by taking a copy of each poset and keeping the given ordering in each copy, while setting +x < y for each x ∈ Pi and y ∈ Pj if i < j. Since each automorphism of P preserves +heights (the maximum length of a chain with a given maximum element), it restricts to +automorphisms of each Pi. Thus Aut(P) = Aut(P1) ⊕ Aut(P2) ⊕ . . . ⊕ Aut(Pk) = Zn. +If pri +i += 3 and prj +j += 4, instead of Pi and Pj we take the poset in Example 3 of 20 = +b(3)3 + b(4)4 − 1 points and automorphism group Z12. +□ +3. Lemmas +Let X be a finite set, n ≥ 1 and x0, x1, . . . , xn−1 pairwise different elements of X. The +cycle α = (x0, x1, . . . , xn−1) is the permutation which maps xi to xi+1 (indices considered +modulo n) and fixes every other point of X. The number n is the order or length of the +cycle, which we denote by |α|. A cycle of order n is also called an n-cycle. A cycle α +is non-trivial if |α| ≥ 2. The representation (x0, x1, . . . , xn−1) of a non-trivial n-cycle is +unique up to cyclic permutation of the n-tuple x0, x1, . . . , xn−1. The underlying set of +a non-trivial cycle (x0, x1, . . . , xn−1) is {x0, x1, . . . , xn−1}. Many times we will identify +a non-trivial cycle with its underlying set. +Two non-trivial cycles are disjoint if their + +SMALLEST POSETS WITH GIVEN CYCLIC AUTOMORPHISM GROUP +5 +underlying sets are. Any permutation g of X can be written as a composition α1α2 . . . αk +of disjoint non-trivial cycles. This representation is unique up to reordering of the cycles. +If a cycle α appears in the factorization of g, we say that α is contained in g and write +α ∈ g. The orbits of g, or of the action of the cyclic group ⟨g⟩ on X, are the underlying +sets of the cycles in g and the singletons consisting of fixed points. Disjoint non-trivial +cycles commute. Thus, if g is a composition α1α2 . . . αk of disjoint non-trivial cycles and +m ∈ Z, then gm = αm +1 αm +2 . . . αm +k . If α is a cycle of length n and m ∈ Z, the permutation +αm is a composition of (n, m) =gcd{n, m} cycles of length +n +(n,m). In particular, αm is a +cycle with the same underlying set as α if n and m are coprime. Moreover, the order of +g is the least common multiple of the lengths of its cycles and if a cycle of g has order n, +and m ∈ Z, then gm fixes every point of the cycle if n|m, and fixes no point of the cycle +otherwise. +If g is an automorphism of a poset P, then each orbit of g is discrete, as a < b would +imply that a < gk(a) for some k ∈ Z and then {gnk(a)}n≥0 would be an infinite chain. +If A and B are two different orbits of g we cannot have an element a ∈ A smaller than +another b ∈ B and at the same time an element b′ ∈ B smaller than another a′ ∈ A, as +this would imply that a < b = gk(b′) < gk(a′) for some k ∈ Z, contradicting the fact that +A is discrete, or the antisymmetry of the order. +Remark 5. Let P be a poset and let g be an automorphism of P. Let Q be the subposet +of points which are not fixed by g. Let A0, A1, . . . , Ak be the orbits of the automorphism +induced by g on Q. If h is an automorphism of Q such that h(Ai) = Ai for every i, then +it extends to an automorphism of P which fixes every element not in Q. +Indeed, if x ∈ P ∖ Q, y ∈ Ai and x < y, then h(y) ∈ Ai, so there exists r ≥ 0 such that +gr(y) = h(y). Then x = gr(x) < gr(y) = h(y). Similarly, if x > y, then x > h(y). +Lemma 6. Let n ≥ 1 and let pr ̸= 2 be a prime power which exactly divides n. Let P be a +poset with Aut(P) cyclic of order n, and let g be a generator of Aut(P). Then g contains +at least two cycles of length divisible by pr. +Proof. Since g has order n, it contains at least one cycle α of length divisible by pr. Assume +there is no other cycle of length divisible by pr. The automorphism g +n +p fixes then every +point not in α. Let x be an element of α and let τ be the transposition of the underlying +set of α which permutes x and g +n +p (x) ̸= x. By Remark 5, τ extends to an automorphism +h of P which is a transposition. But any power of g either fixes each point in α or fixes no +point of α. Since the order of α is at least pr > 2, h /∈ ⟨g⟩ = Aut(P), a contradiction. +□ +If a group G acts on a poset P, an automorphism of P is said to be induced by the +action if it is in the image of the homomorphism G → Aut(P). +Lemma 7. Let p = 3, 5 or 7. Let P be a poset on which Zp acts with exactly two orbits, +both of order p. Then there exists an automorphism of P not induced by the action for +which each orbit of the action is invariant. +Proof. Let g = αβ ∈ Aut(P) be the automorphism induced by a generator of Zp, where +α = (0, 1, . . . , p − 1) and β = (0′, 1′, . . . , (p − 1)′). If no element of α is comparable with +an element of β, then the transposition (0, 1) is an automorphism which is different to gk +for any k ∈ Z, that is, not induced by the action. +Without loss of generality we can assume then that 0 and 0′ are comparable, and +moreover, that 0 < 0′. Then no element in β can be smaller than another in α. Since + +6 +J.A. BARMAK AND A.N. BARRETO +g is an automorphism, i < i′ for every 0 ≤ i ≤ p − 1. If no other pair of elements are +comparable, then (0, 1)(0′, 1′) is an automorphism not induced by the action (it has order +2, for example). If i < j′ for every 0 ≤ i, j ≤ p−1, then (0, 1) satisfies the desired property. +This completes the proof of the case p = 3 by the following argument. The case we did +not analyze is when P has exactly 6 edges. In that case, let P c be the complement of +P, defined as the poset P c with the same underlying set and setting i < j′ if and only if +i ≮ j′ in P, while i, j are not comparable and i′, j′ are not comparable for every i ̸= j. +Since P and P c are non-discrete, they have the same automorphisms. As P c has only 3 +edges, there is an automorphism of P c not induced by the action, so this is the required +automorphism of P. +For p = 5 we need to consider the case that P has 10 edges. +By the complement +argument, this will complete the p = 5 case. So, suppose 0 < k′ for some 1 ≤ k ≤ 4 (and +then i < (i + k)′ for every i, where i + k is considered modulo 5). Note that gk is induced +by another generator of Zp and it maps i′ to (i + k)′. Thus, for each 0 ≤ i ≤ 4, i < i′ and +i < gk(i′). Therefore we can assume that k = 1. We have then the “symmetry about the +axis 03′”, which maps i to −i and j′ to (1 − j)′ (see Figure 3). This is an automorphism +of P which is different to any power of g (it has order 2). +0' +1' +2' +3' +4' +1 +2 +3 +4 +Figure 3. The underlying undirected graph of a poset with 10 points and +edges i′ > i < (i + 1)′, and the axis 03′. +For p = 7, if P has 14 edges, then by the argument above we can assume i′ > i < (i+1)′ +for every 0 ≤ i ≤ 6 and there is then a symmetry about 04′. By the complement argument +it only remains to analyze the case that P has exactly 21 edges. Here i < i′, (i+k)′, (i+l)′ +for certain 1 ≤ k ̸= l ≤ 6 and again we can assume k = 1 by replacing g by gk. Finally, +by replacing g by g−1, it suffices to consider the cases l = 2, 3 and 4 (Figure 4). +For l = 2 we have the involution that maps i to −i and j′ to (2 − j)′. For l = 3 we +have the following automorphism of order 3: (142)(356)(0′3′1′)(2′4′5′) (see Figure 5). For +l = 4, there is again the symmetry about 04′. +□ +Lemma 8. Let P be a poset on which Z4 acts with exactly two orbits of order 4 or exactly +three orbits: two of order 4 and one of order 2. Then there exists an automorphism of P +not induced by the action for which each orbit of the action is invariant. +Proof. Let g be an automorphism induced by a generator of the action and suppose first +that g = (0, 1, 2, 3)(0′, 1′, 2′, 3′). If P is discrete, (0, 1) satisfies the required conditions. + +0 +0' +4 +2' +2 +3 +3'SMALLEST POSETS WITH GIVEN CYCLIC AUTOMORPHISM GROUP +7 +Figure 4. Posets with two Z7-regular orbits and S = {0, 1, l} for l = 2, 3, 4. +Figure 5. The underlying graph of the poset P of 14 points and edges +i < i′, (i + 1)′, (i + 3)′. An automorphism of order 3 is given by a rotation +of angle 2π +3 . +If P has exactly 4 edges, then as in the proof of Lemma 7 we can assume i < i′ for +every 0 ≤ i ≤ 3, and (0, 1)(0′, 1′) works. By the complement argument we can assume P +has exactly 8 edges and that it is determined by the relations i′ > i < (i + k)′ for some +1 ≤ k ≤ 3. The case k = 3 reduces to the case k = 1 by replacing g by g3. If k = 1, +the symmetry (1, 3)(0′, 1′)(2′, 3′) about 02 satisfies the required conditions. If k = 2, then +(0, 2) works. +Suppose then that g = αβγ with α = (0, 1, 2, 3), β = (0′, 1′, 2′, 3′), γ = (0′′, 1′′). Let Q be +the subposet of points in α and β. Since g2 = (0, 2)(1, 3)(0′, 2′)(1′, 3′), every automorphism +of the poset Q which has {0, 2}, {1, 3}, {0′, 2′}, {1′, 3′} as invariant sets, extends to P by +Remark 5. If Q is discrete or if Q has 16 edges, then (0, 2) is an automorphism of Q which +extends to P and this extension is not induced by the action. If Q has exactly 4 edges, +we may assume i < i′ for every i and then (0, 2)(0′, 2′) extends to an automorphism of P +different to any power of g. If Q has exactly 12 edges, the complement argument can be +used. Suppose then Q has exactly 8 edges. By relabelling we can assume the relations +are (a) i < j′ for i ≡ j(2) or (b) i′ > i < (i + 1)′ for every i. In case (a), (0, 2) is again + +4 +5' +2' +0 +5 +2 +3' +1 +4' +33 +5' +6 +4 +6 +3 +2 +3 +4 +6 +0 +2 +5 +1 +3 +4 +5 +6 +0 +1 +3 +5' +4' +6' +. +2 +3 +4 +5 +68 +J.A. BARMAK AND A.N. BARRETO +an automorphism which has every nontrivial orbit of g2 as an invariant set. In the rest of +the proof we assume we are in case (b). +If the points of γ are not comparable with any point of Q, then the symmetry about 02 +which maps i to −i and j′ to (1 − j)′, is an automorphism of Q which extends to P, and +this extension satisfies the required conditions. +By considering the opposite order, we can assume a point of γ is comparable with a +point of α. Moreover, by relabelling if needed we can assume 0′′ is comparable with 0. +Suppose first that 0′′ < 0. Since g is an automorphism, then 0′′ < 2 and 1′′ < 1, 3. If +0′′ ≮ 1, then 0′′ ≮ 3 and 1′′ ≮ 0, 2. If 0′′ < 1, then 0′′ < 3 and 1′′ < 0, 2. In either case, the +symmetry of Q about 02 extends by the identity to an automorphim of P which is not +induced by the action, even though this automorphism of Q does not have the orbits of +g2 as invariant sets. Finally suppose 0′′ > 0. Then 0′′ > 2 and 1′′ > 1, 3. We can assume +no element in β is smaller than an element in γ, by the previous case and the duality +argument. Also, we cannot have an element of γ being smaller than another j′ of β, since +this would imply that i < j′ > i + 2, modulo 4, for certain 0 ≤ i ≤ 3, which is absurd. In +any case, if 0′′ ≯ 1 or if 0′′ > 1, we have that the symmetry of Q about 02 extends to an +automorphism of P. +□ +Lemma 9. Let p = 3, 5 or 7. Let P be a poset with cyclic automorphism group of order +n ≥ 1, and let g ∈ Aut(P) be a generator. Suppose g contains a p-cycle α and a pk-cycle +β ̸= α for some p ∤ k ≥ 1. Then it contains a third cycle whose length is divisible by p. +Proof. Suppose β = (0, 1, . . . , pk − 1). Let Q be the subposet of P whose points are those +of α and β. Assume that there is no other cycle in g whose length is divisible by p. In +particular p ∥ n. Since the order of any cycle of g different from α and β divides n +p, the +automorphism g +n +p fixes every point not in Q. Moreover g +n +p has k + 1 orbits of order p, +which are the underlying set of α and Ai = {0 ≤ j ≤ pk − 1| j ≡ i(k)} for 0 ≤ i ≤ k − 1. +In particular, by Remark 5 every automorphism of Q for which these sets are invariant +extends to an automorphism of P. +Let Q′ be the subposet of Q whose points are those of α and A0. Since gk induces an +automorphism of Q′ with two orbits of order p, by Lemma 7 there is an automorphism h +of Q′ not induced by a power of gk for which the underlying set of α and A0 are invariant. +We extend h to an automorphism h of Q as follows. Let j be a point of β, 0 ≤ j ≤ kp − 1. +Let 0 ≤ i ≤ k −1 be such that j ∈ Ai. Since p ∤ k, there exists a unique 0 ≤ t ≤ k −1 such +that k|j +tp, in other words j +tp, considered modulo kp, lies in A0. Then h(j +tp) ∈ A0. +Define h(j) = h(j + tp) − tp ∈ Ai. We claim that h is an automorphism of Q. It is clearly +bijective. Two different points of β cannot be comparable as they are in the same orbit. +Suppose j in β and a in α are comparable, say a < j. Let 0 ≤ t ≤ k − 1 be such that +k|j +tp. Then a = gtp(a) < gtp(j) = j +tp. Since h is a morphism, h(a) < h(j +tp). Thus +h(a) = h(a) = g−tp(h(a)) < g−tp(h(j + tp)) = h(j + tp) − tp = h(j). Since the underlying +set of α and each Ai are h-invariant, h extends to an automorphism of P, which must be +a power gr of g. Since gr leaves A0 invariant, in particular r = gr(0) ∈ A0, so k|r and h +is then induced by a power of gk, a contradiction. +□ +Lemma 10. Let P be a poset with cyclic automorphism group of order n ≥ 1, and let +g ∈ Aut(P) be a generator. Suppose that g contains two 4-cycles α, β. Then it contains a +third cycle of length divisible by 4 or two more cycles of even length. + +SMALLEST POSETS WITH GIVEN CYCLIC AUTOMORPHISM GROUP +9 +Proof. The proof is very similar to that of Lemma 9, so we omit details. If α and β are +the unique two cycles of even length in g, then by Lemma 8 there is an automorphism +h of the poset of points of these two cycles which is not induced by a power of g, and +moreover has the underlying sets of α and β as invariant sets. Since the non-trivial orbits +of g +n +4 ∈ Aut(P) are the underlying sets of α and β, h extends to an automorphism of P, +a contradiction. +Suppose then there exists a third cycle γ = (1, 2, . . . , 2k) in g with k odd, and that +there is no other cycle of even length. We define Q to be the subposet whose points are +those of α, β and γ. Then g +n +4 fixes every point not in Q. The other orbits of g +n +4 are the +underlying sets of α and β, and Ai = {i, k + i} for 0 ≤ i ≤ k − 1. Let Q′ be the subposet +whose points are those of α, β and A0. Then gk induces an automorphism of Q′ and by +Lemma 8 there is an automorphism h of Q′ which is not induced by a power of gk, and for +which the underlying sets of α, β and A0 are invariant. We extend it to an automorphism +h of Q by defining h(j) = h(j + 4t) − 4t, where t is such that k|j + 4t. Then h is bijective, +it is a morphism and leaves each Ai invariant. It extends to an automorphism of P, say +gr. Since gr leaves A0 invariant, then k|r, which implies that h is induced by a power of +gk, a contradiction. +□ +4. Weights and the lower bound +Let g be a permutation of order n of a finite set X. Let α be a cycle in g of length +l = pr1 +1 pr2 +2 . . . prk +k , where the pi are distinct prime integers, ri ≥ 1 for every i. For each +prime power pr we will define a weight wpr(α) ∈ R≥0 which depends on pr, l and n, in +such a way that � +pr wpr(α)pr = l, where the sum is taken over all prime powers dividing n. +In particular #X ≥ � +pr∥n +( � +α∈g +wpr(α))pr. For each l ≥ 2 we will assign the weight of every +prime power pr in a cycle α of length |α| = l according to a series of rules. In every case, +if the weight wpr(α) is not explicitly defined for some prime power, we assume it is 0. +Exception 6. Suppose l = 6. If 3 ∥ n then w3(α) = 2. If 3 ∦ n and 2 ∥ n, then w2(α) = 3. +If 3 ∦ n and 2 ∦ n, then w4(α) = 3 +2. +Exception 12. Suppose l = 12. If 3 ∥ n then w3(α) = 4. If 3 ∦ n, then w4(α) = 3. +Exception 10-14. Suppose l = 2p for p = 5 or 7. If 2 ∥ n, w2(α) = 1. Otherwise +w4(α) = 1 +2. In any case wp(α) = 2(p−1) +p +. +General case. Suppose l = pr1 +1 pr2 +2 . . . prk +k ̸= 6, 12, 10, 14, where the pi are different primes +and each ri ≥ 1. For each 1 ≤ i ≤ k, we define wpri +i (α) = +� +j̸=i +p +rj +j +k +, unless pri +i = 2 and 2 ∦ n. +In that case, w2(α) = 0, while w4(α) = +� +j̸=i +p +rj +j +2k +. In particular, if l = pr ≥ 3 is a prime +power, wpr(α) = 1. +Note that, as we required, the sum � +pr|n +wpr(α) over all the prime powers dividing n is +the length l of α. Note also that if l = pr1 +1 pr2 +2 . . . prk +k , then wpr(α) ̸= 0 only if pr = pri +i for +some 1 ≤ i ≤ k or pr = 4. + +10 +J.A. BARMAK AND A.N. BARRETO +Theorem 11. Let n ≥ 1. Let P be a poset with Aut(P) cyclic of order n generated by g. +Let pr be a prime power which exactly divides n. If pr ̸= 2, 4 then � +α∈g +wpr(α) ≥ b(pr). If +3 ∦ n and pr = 2 or pr = 4, � +α∈g +wpr(α) ≥ b(pr) as well. If 3 ∥ n and 2 ∥ n, � +α∈g +(2w2(α) + +3w3(α)) ≥ 2b(2) + 3b(3) = 11. Finally, if 3 ∥ n and 4 ∥ n, � +α∈g +(4w4(α) + 3w3(α)) ≥ +4b(4) + 3b(3) − 1 = 20. +Proof. If pr ̸= 2, 3, 4, 5, 7, by Lemma 6, there are at least two cycles of length divisible by +pr. By hypothesis their lengths are not multiples of pr+1. But if α is a cycle of g whose +length is a multiple of pr, then wpr(α) ≥ 1. Indeed, the weights in α are assigned according +to the General case. If the length of α is l = pr1 +1 pr2 +2 . . . prk +k , we can assume pr = pr1 +1 and +then wpr(α) = +k� +j=2 +p +rj +j +k +≥ 2k−1 +k +≥ 1. Thus, � +α∈g +wpr(α) ≥ 2 = b(pr). +Suppose now pr = 5. If α is a cycle of g of length l = 5, then w5(α) = 1. If l = 10, +then w5(α) = 8 +5 ≥ 3 +2 (Exception 10-14). If l = 5s with s = pr2 +2 pr3 +3 . . . prk +k ≥ 3 not divisible +by 5, then either k = 2, or k ≥ 3. In the first case w5(α) = s +2 ≥ 3 +2, and in the second case +w5(α) = +k� +j=2 +p +rj +j +k +≥ 2k−2.3 +k +≥ 2 ≥ 3 +2. +By Lemma 6, there are at least two cycles of length divisible by 5 (and not by 52). +Suppose first there exactly two such cycles, α and α′. None of them can be of length 5 +by Lemma 9. Thus w5(α) + w5(α′) ≥ 2.3 +2 = 3 = b(5). Finally, if there are at least three +cycles in g of length divisible by 5, then � +α∈g +w5(α) ≥ 3 = b(5). +The case pr = 7 is similar to the previous one, with the observation that for length +l = 14, w7(α) = 12 +7 ≥ 3 +2 (Exception 10-14). So, also in this case � +α∈g +w7(α) ≥ 3 = b(7). +Let pr = 3. If the length of a cycle α in g is l = 3, w3(α) = 1. If l = 6, w3(α) = 2 +(Exception 6). If l = 12, w3(α) = 4 (Exception 12). If l = 3s with s = pr2 +2 pr3 +3 . . . prk +k ≥ 5, +then either k = 2, or k ≥ 3. In the first case w3(α) = s +2 ≥ 5 +2, and in the second case +w3(α) = +k� +j=2 +p +rj +j +k +≥ 2k−2.3 +k +≥ 2. +By Lemma 6 there are at least two cycles in g of length divisible by 3 (and not by 32). +Suppose first there are exactly two such cycles α and α′. None of them can have length 3 +by Lemma 9. Then w3(α)+w3(α′) ≥ 2.2 = 4 ≥ 3 = b(3). Finally, if there are at least three +cycles in g of length divisible by 3, then � +α∈g +w3(α) ≥ 3 = b(3). Note that � +α∈g +w3(α) ≥ 4 +unless there are exactly three cycles of length 3 and no other cycle of length divisible by +3. +We analyze now the case that 3 ∦ n and pr = 2 or 4. In the first situation, there is at +least one cycle α of even length l (not divisible by 4). If l = 2, w2(α) = 1 (General case). +If l = 6, w2(α) = 3 (Exception 6). If l = 10 or l = 14, then w2(α) = 1 (Exception 10-14). +If l = 2s with s = pr2 +2 pr3 +3 . . . prk +k ̸= 1, 3, 5, 7 (odd), then w2(α) = +k� +j=2 +p +rj +j +k +≥ 3k−1 +k +≥ 3 +2. Thus +� +α∈g +w2(α) ≥ 1 = b(2). We consider the second situation, pr = 4. If α has length l = 4, then +w4(α) = 1. If l = 12, w4(α) = 3 (Exception 12). If l = 4s with s = pr2 +2 pr3 +3 . . . prk +k ≥ 5 (odd), + +SMALLEST POSETS WITH GIVEN CYCLIC AUTOMORPHISM GROUP +11 +then k = 2 or k ≥ 3. For k = 2 we have w4(α) = s +2 ≥ 5 +2. For k ≥ 3, w4(α) ≥ 3k−1 +k +≥ 3. By +Lemma 6, g contains at least two cycles of lengths divisible by 4 (and not by 8). Suppose +first there are exactly two such cycles, α and α′, of lengths l, l′. If l = l′ = 4, then by +Lemma 10, there exists a third and a fourth cycle β, β′ of lengths 2m and 2m′ for some +odd m, m′. The weights w4(β) that we obtain for each m are the halves of the weights that +we obtained for 2 in cycles of the same length when 2 ∥ n. Namely, if m = 1, w4(β) = 1 +2 +(General case); if m = 3, w4(β) = 3 +2 (Exception 6); if m = 5, 7, w4(β) = 1 +2 (Exception +10-14); if m = pr2 +2 pr3 +3 . . . prk +k ̸= 1, 3, 5, 7 then w4(β) ≥ 3k−1 +2k +≥ 3 +4 (General case). +The same happens with β′. Thus w4(α) + w4(α′) + w4(β) + w4(β′) ≥ 1 + 1 + 1 +2 + 1 +2 = +3 = b(4). If instead l = 4 and l′ = 12, then w4(α) + w4(α′) = 1 + 3 = 4 > 3. If l = 4 +and l′ = 4s for some odd s ≥ 5, then w4(α) + w4(α′) ≥ 1 + 5 +2 > 3. If both l and l′ are +greater than 4, then w4(α) + w4(α′) ≥ 5 +2 + 5 +2 > 3. Finally, if there are at least three cycles +of length divisible by 4, then � +α∈g +w4(α) ≥ 3. Thus, in any case � +α∈g +w4(α) ≥ 3 = b(4). +It only remains to analyze the case 3 ∥ n and 2 ∥ n and the case 3 ∥ n and 4 ∥ n. +If 3 ∥ n and 2 ∥ n, recall that we have already proved that � +α∈g +w3(α) ≥ 4 or there are +exactly three cycles of length 3 and no other cycle of length divisible by 3. In the first +case � +α∈g +(2w2(α) + 3w3(α)) ≥ � +α∈g +3w3(α) ≥ 12. In the second case, there exists a cycle β +in g of even length m ̸= 6, so w2(β) ≥ 1. Thus � +α∈g +(2w2(α) + 3w3(α)) ≥ 2.1 + 3.3 = 11. +The last case is 3 ∥ n and 4 ∥ n. Note that if there are no cycles of length 6 nor 12 in g, +then the computation � +α∈g +w4(α) ≥ 3 remains valid as Exceptions 6 and 12 do not occur. +Thus � +α∈g +(3w3(α)+4w4(α)) ≥ 3.3+4.3 = 21 > 20. If there are at least two 12-cycles, then +� +α∈g +(3w3(α) + 4w4(α)) ≥ 2.3.4 = 24 > 20. If there is no 12-cycle in g and � +α∈g +w4(α) < 3, +then we must be in the case that there is a 6-cycle. This already implies � +α∈g +w3(α) ≥ 4, +while the existence of two cycles of length divisible by 4 implies � +α∈g +w4(α) ≥ 2. Thus +� +α∈g +(3w3(α) + 4w4(α)) ≥ 3.4 + 4.2 = 20. +Thus we may assume g has a unique 12-cycle. By Lemma 6 there is another cycle of +length divisible by 4, so � +α∈g +w4(α) ≥ 1. On the other hand, � +α∈g +w3(α) ≥ 4 + 2 = 6, as the +weight of 3 in a 12-cycle is 4 and by Lemmas 6 and 9 there are either two more cycles +of lengths divisible by 3 or just one, but of length not 3. Thus � +α∈g +(3w3(α) + 4w4(α)) ≥ +3.6 + 4.1 = 22 > 20. +□ +Corollary 12. Let n = pr1 +1 pr2 +2 . . . prk +k , where the pi are different primes and ri ≥ 1 for +every i. Then the minimum number β(Zn) of points in a poset with cyclic automorphism +group of order n is +k� +i=1 +b(pri +i )pri +i − 1 if 3 ∥ n and 4 ∥ n, and +k� +i=1 +b(pri +i )pri +i otherwise. +Proof. If P is a poset with Aut(P) ≃ Zn generated by g, then the number of points in P is +at least � +α∈g +|α| = � +α∈g +� +pr|n +wpr(α)pr ≥ +k� +i=1 +( � +α∈g +wpri +i (α))pri +i . If both 3 and 4 exactly divide n, + +12 +J.A. BARMAK AND A.N. BARRETO +by Theorem 11 this is +� +pri +i ̸=3,4 +( � +α∈g +wpri +i (α))pri +i + � +α∈g +(3w3(α) + 4w4(α)) ≥ +� +pri +i ̸=3,4 +b(pri +i )pri +i + +3b(3) + 4b(4) − 1 = +k� +i=1 +b(pri +i )pri +i − 1. Otherwise, the bound is one more than this number. +The bound is attained by Theorem 4. +□ +References +[1] W.C. Arlinghaus. The structure of minimal graphs with given abelian automorphism group. Ph.D. +Thesis, Wayne State University, 1979. +[2] W.C. Arlinghaus. The classification of minimal graphs with given abelian automorphism group. Mem. +Amer. Math. Soc. 57(1985), no. 330, viii+86. +[3] L. Babai. On the minimum order of graphs with given group. Canad. Math. Bull. 17(1974), no. 4, +467-470. +[4] L. Babai. Finite digraphs with given regular automorphism groups. Period. Math. Hungar. 11(1980), +no. 4, 257-270. +[5] A.N. Barreto. Sobre los posets m´as chicos con grupo de automorfismos abeliano dado. Tesis de Licen- +ciatura, Universidad de Buenos Aires, 2021. +[6] G. Birkhoff. Sobre los grupos de automorfismos. Rev. Un. Mat. Argentina 11(1946), 155-157. +[7] R. Frucht. Herstellung von Graphen mit vorgegebener abstrakter Gruppe. Compositio Math. 6(1939), +239-250. +[8] R. Frucht. Graphs of degree 3 with given abstract group. Canad. J. Math. 1(1949) 305-378. +[9] R. Frucht. On the construction of partially ordered systems with a given group of automorphisms. +Amer. J. Math. 72(1950), 195-199. +[10] R. L. Meriwether. Smallest graphs with a given cyclic group. 1963, unpublished. (1967) +[11] G. Sabidussi. On the minimum order of graphs with given automorphism group. Monatsh. Math. +63(1959), 124-127. +Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento +de Matem´atica. Buenos Aires, Argentina. +CONICET-Universidad de Buenos Aires. Instituto de Investigaciones Matem´aticas Luis +A. Santal´o (IMAS). Buenos Aires, Argentina. +Email address: jbarmak@dm.uba.ar +Email address: abarreto@dm.uba.ar + diff --git a/QNFAT4oBgHgl3EQf0B4_/content/tmp_files/load_file.txt b/QNFAT4oBgHgl3EQf0B4_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e94ca50a6c4708b8e20ce4e61ef63925917eef5c --- /dev/null +++ b/QNFAT4oBgHgl3EQf0B4_/content/tmp_files/load_file.txt @@ -0,0 +1,658 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf,len=657 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='08701v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='CO] 20 Jan 2023 SMALLEST POSETS WITH GIVEN CYCLIC AUTOMORPHISM GROUP JONATHAN ARIEL BARMAK AND AGUST´IN NICOL´AS BARRETO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' For each n ≥ 1 we determine the minimum number of points in a poset with cyclic automorphism group of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Introduction In 1938 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Frucht [7] proved that any finite group can be realized as the automorphism group of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Moreover, the graph can be taken with 3d|G| vertices, where d is the cardinality of any generator set of G ([8, Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In 1959 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Sabidussi [11] showed that in fact O(|G|log(d)) vertices suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In 1974 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Babai proved that the number of generators is not relevant, and with exception of the cyclic groups Z3, Z4 and Z5, the graph can be taken with just 2|G| vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Sabbidussi claims in [11] that he was able to compute the smallest number of vertices α(G) in a graph with automorphism group G in the case that G is cyclic of prime power order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Also, he asserts that for n = pr1 1 pr2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' prk k , α(Zn) = k� i=1 α(Zpri i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Unfortunately both his computations for Zpr and the assertion are wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Meriwether rectifies these errors and correctly determines α(Zn) for any n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' However, he commits similar mistakes when trying to extend this computation to arbitrary finite abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In [1, 2] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Arlinghaus provides a complete calculation of α(G) for G finite abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The proof follows these steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' First compute α(G) for G cyclic of prime power order, then for arbitrary finite cyclic groups, then for abelian p-groups and finally, the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In parallel, the analogous problem was studied for partially ordered sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In 1946 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Birkhoff [6] proved that for any finite group G there is a poset of |G|(|G| + 1) points and automorphism group isomorphic to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then Frucht [9] improved this to (d+2)|G| points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In 1980 Babai [4] proved that 3|G| points are enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' However, the smallest number β(G) of points of a poset with an arbitrary finite abelian group G of automorphisms has not yet been determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In this paper we compute β(G) for every finite cyclic group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' This result was first announced in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In [5] we computed first β(G) for G cyclic of prime power order, then for arbitrary finite cyclic and for finite abelian p-groups with p ≥ 11, following the steps of the proof of the graph case exposed by Arlinghaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The calculation of β(Zn) in this paper is more direct than the original we gave in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The case of p-groups will not 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' 06A11, 20B25, 06A07, 05E18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Posets, Automorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Both authors were supported by CONICET and partially supported by grant UBACyT 20020190100099BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The first named author was also partially supported by grants CONICET PIP 11220170100357CO, ANPCyT PICT-2017-2806 and ANPCyT PICT-2019-02338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' 1 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' BARMAK AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' BARRETO be addressed in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Just as in graphs, the bound β(Zn) ≤ k� i=1 β(Zpri i ) holds for n = pr1 1 pr2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' prk k , but not the equality, in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' For instance β(Z12) = β(3) + β(4) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In Section 2 we construct explicit examples which provide an upper bound for β(Zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In Section 3 we prove some lemmas concerning the cyclic structure of a generator of Aut(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In the last section we introduce the notion of weight of a prime power in a cycle, which we use in the proof of the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Construction of the examples A poset is a set with a partial order ≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The elements of the underlying set of a poset are called points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' All posets are assumed to be finite, that is, their underlying set is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If P is a poset and x, y ∈ P, we write x < y if x ≤ y and x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' We say that y covers x if x < y and there is no x < z < y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The edges of P are the pairs (x, y) such that y covers x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The Hasse diagram of P is the digraph whose vertices are the points of P and the edges are the edges of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If the orientation of an arrow is not indicated in the graphical representation, we assume it points upwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' A morphism P → Q of posets is an order-preserving map, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' a function f between the underlying sets such that for every pair x, y ∈ P with x ≤ y we have f(x) ≤ f(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If P is a poset, since it is finite, an automorphism of P is just a permutation of the underlying set which is a morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' A subposet of a poset P is a subset of the underlying set with the inherited order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Given an automorphism g of a poset P, we say that a subset A of the underlying set of P is invariant or g-invariant if g(A) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In this case, g induces an automorphism on the subposet with underlying set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Define b(1) = 0, b(2) = 1, b(3) = b(4) = b(5) = b(7) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' For any other prime power pr, define b(pr) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let n = pr, where p ≥ 2 is a prime and r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then there exists a poset P with b(n)n points and automorphism group Aut(P) isomorphic to Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' For n = 1 we take the empty poset and for n = 2 we take the discrete poset on 2 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' By discrete we mean an antichain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' a poset of pairwise incomparable elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If n = 3, 4, 5, 7 we use the well-known general construction [9]: P = Zn × {0, 1, 2} with the order (i, 2) > (i, 1) > (i, 0) < (i + 1, 2) for every i ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' It is easy to see that such poset satisfies Aut(P) ≃ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Suppose then that n ≥ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' We take two copies of Zn: A = Zn = {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' , n − 1} and A′ = {0′, 1′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' , (n − 1)′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let S = {0, 1, 2, 4} ⊆ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' For i ∈ A and j′ ∈ A′ we set i < j′ if j − i ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Any two elements in the same copy of Zn are not comparable (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' We will prove that the automorphism group of this poset P is Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' It is clear that G = Zn acts regularly on each copy of Zn by multiplication (addition), and this gives a faithful action G → Aut(P) on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' So G can be seen as a subgroup of Aut(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since each automorphism of P maps 0 ∈ A to another minimal element of P, then the order of the Aut(P)-orbit of 0 ∈ P is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If we prove that the Aut(P)-stabilizer of 0 ∈ P is trivial, then |Aut(P)| = n, so Aut(P) is isomorphic to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let h ∈ Aut(P) be such that h(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' We define the double neighborhood B(i) of i ∈ A as the set of those j ∈ A such that #(P>i ∩ P>j) ≥ 2, that is, there are at least two points in A′ greater than both, i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The reduced double neighborhood of i ∈ A is ˆB(i) = B(i)∖{i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since h is an automorphism, B(h(i)) = h(B(i)) and ˆB(h(i)) = h( ˆB(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Given k ≥ 1, we say that two points i, j ∈ A are SMALLEST POSETS WITH GIVEN CYCLIC AUTOMORPHISM GROUP 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The Hasse diagram of P for n = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' k-adjacent if #(B(i) ∩ B(j)) = k, and they are reduced k-adjacent if #( ˆB(i) ∩ ˆB(j)) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Clearly, h preserves k-adjacency and reduced k-adjacency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Suppose first that n ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then for each i ∈ A, B(i) = {i − 2, i − 1, i, i + 1, i + 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' It is easy to see that i, j are 4-adjacent if and only if i − j = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus, h induces an automorphism of the cyclic graph on A with edges given by 4-adjacency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since h(0) = 0, h is either the identity 1Zn or −1Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The second case cannot occur as {0, 2, 3, 4} has an upper bound while {0, −2, −3, −4} does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus every point of A is fixed by h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If j′ ∈ A′, then j′ is the unique upper bound of {j, j − 1, j − 2, j − 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus h(j′) = j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' This proves that h = 1P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Finally, suppose n = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Given i ∈ A, we have now ˆB(i) = {i−2, i−1, i+1, i+2, i+4} and i, j ∈ A are reduced 4-adjacent if and only if i−j = ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus, h induces an automorphism in the cyclic graph on A with edges given by reduced 4-adjacency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then h = 1Zn or −1Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The second case cannot occur for the same reason as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since each point in A′ is determined by the set of smaller points, h = 1P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' There exists a poset P with 20 points and automorphism group isomorphic to Z12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Take two copies A = {0, 1, 2, 3, 4, 5}, A′ = {0′, 1′, 2′, 3′, 4′, 5′} of Z6 and two copies B = {0′′, 1′′, 2′′, 3′′}, B′ = {0′′′, 1′′′, 2′′′, 3′′′} of Z4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The underlying set of P is the union of these four sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let S = {0, 1, 3} ⊆ Z6, T = {0, 1} ⊆ Z4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Define the following order in P: i < j′ if j − i ∈ S, i′′ < j′′′ if j − i ∈ T, i′′′ < j′ if j − i is even, i′′ < j if j − i is even, i′′ < j′ for every i, j (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=" 0 1 2 3 4 5 0' 1' 2' 3' 4' 5' 0'' 1'' 2'' 3'' 0''' 1''' 2''' 3''' Figure 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' A poset P of 20 points and Aut(P) ≃ Z12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' It is clear that G = Z12 acts in each copy of Z6 and of Z4 by multiplication (addition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' This induces a faithful action of G on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If h ∈ Aut(P), h(0′′) must be a minimal point i′′ and h(0′) must be a maximal point j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' However i, j cannot have different parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=" Indeed, 0 2 3' 4 5' 2 YQ 2 5' 6 4 5 2 3 4 64 J." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' BARMAK AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' BARRETO among the points 0, 2, 4, 0′′′, 1′′′ which cover 0′′, there are just two 0, 0′′′ smaller that 0′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' However, if i ∈ Z4 and j ∈ Z6 have different parity, among the points covering i′′ (k ∈ A with k ≡ i(2) and i′′′, (i + 1)′′′) there are three smaller that j′: both j − 1, j − 3, and one of i′′′, (i + 1)′′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus i ≡ j(2), which implies that the Aut(P)-orbit of the set {0′, 0′′} has at most 12 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If we prove that the Aut(P)-stabilizer of {0′, 0′′} is trivial, then |Aut(P)| ≤ 12 = |G|, so Aut(P) is isomorphic to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let h be an automorphism of P which fixes 0′ and 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Note that 2′′ is the unique minimal point different from 0′′ which is covered by three points that cover 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus h(2′′) = 2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Now, the points of B′ are the unique points of P which cover exactly one of 0′′, 2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus B′ is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' This implies that h restricts to an automorphism of the subposet R with underlying set B ∪ B′ and of the subposet Q with set A ∪ A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since R is a cycle, there are only two automorphisms of R fixing 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' One is the identity and the other maps 0′′′ to 1′′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' However, 0′′′ < 0′ while 1′′′ ≮ 0′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus 0′′′ is fixed by h and then h is the identity of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Suppose that i′ ∈ A′ is a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Among the points i, i − 1, i − 3 in A covered by i′, only i − 1 and i − 3 share a lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus h(i) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Similarly, among the points (i − 4)′, (i − 2)′, (i − 1)′ of A′ not covering i, only (i − 4)′ and (i − 2)′ share a lower bound in B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus (i−1)′ is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In conclusion, we showed that i′ fixed implies that both i and (i − 1)′ are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since 0′ is fixed, this implies that every point of A and of A′ is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus h = 1P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' We say that a prime power pr (r ≥ 1) exactly divides an integer n, and write pr ∥ n, if pr|n and pr+1 ∤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let n = pr1 1 pr2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' prk k where the pi are different primes and ri ≥ 1 for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then there exists a poset with automorphism group isomorphic to Zn and k� i=1 b(pri i )pri i − 1 points if 3 ∥ n and 4 ∥ n, and with k� i=1 b(pri i )pri i points otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' By Proposition 2, for each 1 ≤ i ≤ k there exists a poset Pi with b(pri i )pri i points and Aut(Pi) ≃ Zpri i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The non-Hausdorff join or ordinal sum P = P1⊕P2⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='⊕Pk is constructed by taking a copy of each poset and keeping the given ordering in each copy, while setting x < y for each x ∈ Pi and y ∈ Pj if i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since each automorphism of P preserves heights (the maximum length of a chain with a given maximum element), it restricts to automorphisms of each Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus Aut(P) = Aut(P1) ⊕ Aut(P2) ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' ⊕ Aut(Pk) = Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If pri i = 3 and prj j = 4, instead of Pi and Pj we take the poset in Example 3 of 20 = b(3)3 + b(4)4 − 1 points and automorphism group Z12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Lemmas Let X be a finite set, n ≥ 1 and x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' , xn−1 pairwise different elements of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The cycle α = (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' , xn−1) is the permutation which maps xi to xi+1 (indices considered modulo n) and fixes every other point of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The number n is the order or length of the cycle, which we denote by |α|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' A cycle of order n is also called an n-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' A cycle α is non-trivial if |α| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The representation (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' , xn−1) of a non-trivial n-cycle is unique up to cyclic permutation of the n-tuple x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' , xn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The underlying set of a non-trivial cycle (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' , xn−1) is {x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' , xn−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Many times we will identify a non-trivial cycle with its underlying set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Two non-trivial cycles are disjoint if their SMALLEST POSETS WITH GIVEN CYCLIC AUTOMORPHISM GROUP 5 underlying sets are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Any permutation g of X can be written as a composition α1α2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' αk of disjoint non-trivial cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' This representation is unique up to reordering of the cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If a cycle α appears in the factorization of g, we say that α is contained in g and write α ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The orbits of g, or of the action of the cyclic group ⟨g⟩ on X, are the underlying sets of the cycles in g and the singletons consisting of fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Disjoint non-trivial cycles commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus, if g is a composition α1α2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' αk of disjoint non-trivial cycles and m ∈ Z, then gm = αm 1 αm 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' αm k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If α is a cycle of length n and m ∈ Z, the permutation αm is a composition of (n, m) =gcd{n, m} cycles of length n (n,m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In particular, αm is a cycle with the same underlying set as α if n and m are coprime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Moreover, the order of g is the least common multiple of the lengths of its cycles and if a cycle of g has order n, and m ∈ Z, then gm fixes every point of the cycle if n|m, and fixes no point of the cycle otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If g is an automorphism of a poset P, then each orbit of g is discrete, as a < b would imply that a < gk(a) for some k ∈ Z and then {gnk(a)}n≥0 would be an infinite chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If A and B are two different orbits of g we cannot have an element a ∈ A smaller than another b ∈ B and at the same time an element b′ ∈ B smaller than another a′ ∈ A, as this would imply that a < b = gk(b′) < gk(a′) for some k ∈ Z, contradicting the fact that A is discrete, or the antisymmetry of the order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let P be a poset and let g be an automorphism of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let Q be the subposet of points which are not fixed by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let A0, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' , Ak be the orbits of the automorphism induced by g on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If h is an automorphism of Q such that h(Ai) = Ai for every i, then it extends to an automorphism of P which fixes every element not in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Indeed, if x ∈ P ∖ Q, y ∈ Ai and x < y, then h(y) ∈ Ai, so there exists r ≥ 0 such that gr(y) = h(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then x = gr(x) < gr(y) = h(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Similarly, if x > y, then x > h(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let n ≥ 1 and let pr ̸= 2 be a prime power which exactly divides n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let P be a poset with Aut(P) cyclic of order n, and let g be a generator of Aut(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then g contains at least two cycles of length divisible by pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since g has order n, it contains at least one cycle α of length divisible by pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Assume there is no other cycle of length divisible by pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The automorphism g n p fixes then every point not in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let x be an element of α and let τ be the transposition of the underlying set of α which permutes x and g n p (x) ̸= x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' By Remark 5, τ extends to an automorphism h of P which is a transposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' But any power of g either fixes each point in α or fixes no point of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since the order of α is at least pr > 2, h /∈ ⟨g⟩ = Aut(P), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' □ If a group G acts on a poset P, an automorphism of P is said to be induced by the action if it is in the image of the homomorphism G → Aut(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let p = 3, 5 or 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let P be a poset on which Zp acts with exactly two orbits, both of order p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then there exists an automorphism of P not induced by the action for which each orbit of the action is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let g = αβ ∈ Aut(P) be the automorphism induced by a generator of Zp, where α = (0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' , p − 1) and β = (0′, 1′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' , (p − 1)′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If no element of α is comparable with an element of β, then the transposition (0, 1) is an automorphism which is different to gk for any k ∈ Z, that is, not induced by the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Without loss of generality we can assume then that 0 and 0′ are comparable, and moreover, that 0 < 0′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then no element in β can be smaller than another in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' BARMAK AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' BARRETO g is an automorphism, i < i′ for every 0 ≤ i ≤ p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If no other pair of elements are comparable, then (0, 1)(0′, 1′) is an automorphism not induced by the action (it has order 2, for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If i < j′ for every 0 ≤ i, j ≤ p−1, then (0, 1) satisfies the desired property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' This completes the proof of the case p = 3 by the following argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The case we did not analyze is when P has exactly 6 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In that case, let P c be the complement of P, defined as the poset P c with the same underlying set and setting i < j′ if and only if i ≮ j′ in P, while i, j are not comparable and i′, j′ are not comparable for every i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since P and P c are non-discrete, they have the same automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' As P c has only 3 edges, there is an automorphism of P c not induced by the action, so this is the required automorphism of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' For p = 5 we need to consider the case that P has 10 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' By the complement argument, this will complete the p = 5 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' So, suppose 0 < k′ for some 1 ≤ k ≤ 4 (and then i < (i + k)′ for every i, where i + k is considered modulo 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Note that gk is induced by another generator of Zp and it maps i′ to (i + k)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus, for each 0 ≤ i ≤ 4, i < i′ and i < gk(i′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Therefore we can assume that k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' We have then the “symmetry about the axis 03′”, which maps i to −i and j′ to (1 − j)′ (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' This is an automorphism of P which is different to any power of g (it has order 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=" 0' 1' 2' 3' 4' 1 2 3 4 Figure 3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The underlying undirected graph of a poset with 10 points and edges i′ > i < (i + 1)′, and the axis 03′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' For p = 7, if P has 14 edges, then by the argument above we can assume i′ > i < (i+1)′ for every 0 ≤ i ≤ 6 and there is then a symmetry about 04′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' By the complement argument it only remains to analyze the case that P has exactly 21 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Here i < i′, (i+k)′, (i+l)′ for certain 1 ≤ k ̸= l ≤ 6 and again we can assume k = 1 by replacing g by gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Finally, by replacing g by g−1, it suffices to consider the cases l = 2, 3 and 4 (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' For l = 2 we have the involution that maps i to −i and j′ to (2 − j)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' For l = 3 we have the following automorphism of order 3: (142)(356)(0′3′1′)(2′4′5′) (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' For l = 4, there is again the symmetry about 04′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' □ Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let P be a poset on which Z4 acts with exactly two orbits of order 4 or exactly three orbits: two of order 4 and one of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then there exists an automorphism of P not induced by the action for which each orbit of the action is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let g be an automorphism induced by a generator of the action and suppose first that g = (0, 1, 2, 3)(0′, 1′, 2′, 3′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If P is discrete, (0, 1) satisfies the required conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=" 0 0' 4 2' 2 3 3'SMALLEST POSETS WITH GIVEN CYCLIC AUTOMORPHISM GROUP 7 Figure 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Posets with two Z7-regular orbits and S = {0, 1, l} for l = 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The underlying graph of the poset P of 14 points and edges i < i′, (i + 1)′, (i + 3)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' An automorphism of order 3 is given by a rotation of angle 2π 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If P has exactly 4 edges, then as in the proof of Lemma 7 we can assume i < i′ for every 0 ≤ i ≤ 3, and (0, 1)(0′, 1′) works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' By the complement argument we can assume P has exactly 8 edges and that it is determined by the relations i′ > i < (i + k)′ for some 1 ≤ k ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The case k = 3 reduces to the case k = 1 by replacing g by g3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If k = 1, the symmetry (1, 3)(0′, 1′)(2′, 3′) about 02 satisfies the required conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If k = 2, then (0, 2) works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Suppose then that g = αβγ with α = (0, 1, 2, 3), β = (0′, 1′, 2′, 3′), γ = (0′′, 1′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let Q be the subposet of points in α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since g2 = (0, 2)(1, 3)(0′, 2′)(1′, 3′), every automorphism of the poset Q which has {0, 2}, {1, 3}, {0′, 2′}, {1′, 3′} as invariant sets, extends to P by Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If Q is discrete or if Q has 16 edges, then (0, 2) is an automorphism of Q which extends to P and this extension is not induced by the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If Q has exactly 4 edges, we may assume i < i′ for every i and then (0, 2)(0′, 2′) extends to an automorphism of P different to any power of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If Q has exactly 12 edges, the complement argument can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Suppose then Q has exactly 8 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' By relabelling we can assume the relations are (a) i < j′ for i ≡ j(2) or (b) i′ > i < (i + 1)′ for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=" In case (a), (0, 2) is again 4 5' 2' 0 5 2 3' 1 4' 33 5' 6 4 6 3 2 3 4 6 0 2 5 1 3 4 5 6 0 1 3 5' 4' 6' ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' 2 3 4 5 68 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' BARMAK AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' BARRETO an automorphism which has every nontrivial orbit of g2 as an invariant set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In the rest of the proof we assume we are in case (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If the points of γ are not comparable with any point of Q, then the symmetry about 02 which maps i to −i and j′ to (1 − j)′, is an automorphism of Q which extends to P, and this extension satisfies the required conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' By considering the opposite order, we can assume a point of γ is comparable with a point of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Moreover, by relabelling if needed we can assume 0′′ is comparable with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Suppose first that 0′′ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since g is an automorphism, then 0′′ < 2 and 1′′ < 1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If 0′′ ≮ 1, then 0′′ ≮ 3 and 1′′ ≮ 0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If 0′′ < 1, then 0′′ < 3 and 1′′ < 0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In either case, the symmetry of Q about 02 extends by the identity to an automorphim of P which is not induced by the action, even though this automorphism of Q does not have the orbits of g2 as invariant sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Finally suppose 0′′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then 0′′ > 2 and 1′′ > 1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' We can assume no element in β is smaller than an element in γ, by the previous case and the duality argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Also, we cannot have an element of γ being smaller than another j′ of β, since this would imply that i < j′ > i + 2, modulo 4, for certain 0 ≤ i ≤ 3, which is absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In any case, if 0′′ ≯ 1 or if 0′′ > 1, we have that the symmetry of Q about 02 extends to an automorphism of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' □ Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let p = 3, 5 or 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let P be a poset with cyclic automorphism group of order n ≥ 1, and let g ∈ Aut(P) be a generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Suppose g contains a p-cycle α and a pk-cycle β ̸= α for some p ∤ k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then it contains a third cycle whose length is divisible by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Suppose β = (0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' , pk − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let Q be the subposet of P whose points are those of α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Assume that there is no other cycle in g whose length is divisible by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In particular p ∥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since the order of any cycle of g different from α and β divides n p, the automorphism g n p fixes every point not in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Moreover g n p has k + 1 orbits of order p, which are the underlying set of α and Ai = {0 ≤ j ≤ pk − 1| j ≡ i(k)} for 0 ≤ i ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In particular, by Remark 5 every automorphism of Q for which these sets are invariant extends to an automorphism of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let Q′ be the subposet of Q whose points are those of α and A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since gk induces an automorphism of Q′ with two orbits of order p, by Lemma 7 there is an automorphism h of Q′ not induced by a power of gk for which the underlying set of α and A0 are invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' We extend h to an automorphism h of Q as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let j be a point of β, 0 ≤ j ≤ kp − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let 0 ≤ i ≤ k −1 be such that j ∈ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since p ∤ k, there exists a unique 0 ≤ t ≤ k −1 such that k|j +tp, in other words j +tp, considered modulo kp, lies in A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then h(j +tp) ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Define h(j) = h(j + tp) − tp ∈ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' We claim that h is an automorphism of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' It is clearly bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Two different points of β cannot be comparable as they are in the same orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Suppose j in β and a in α are comparable, say a < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let 0 ≤ t ≤ k − 1 be such that k|j +tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then a = gtp(a) < gtp(j) = j +tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since h is a morphism, h(a) < h(j +tp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus h(a) = h(a) = g−tp(h(a)) < g−tp(h(j + tp)) = h(j + tp) − tp = h(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since the underlying set of α and each Ai are h-invariant, h extends to an automorphism of P, which must be a power gr of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since gr leaves A0 invariant, in particular r = gr(0) ∈ A0, so k|r and h is then induced by a power of gk, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' □ Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let P be a poset with cyclic automorphism group of order n ≥ 1, and let g ∈ Aut(P) be a generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Suppose that g contains two 4-cycles α, β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then it contains a third cycle of length divisible by 4 or two more cycles of even length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' SMALLEST POSETS WITH GIVEN CYCLIC AUTOMORPHISM GROUP 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The proof is very similar to that of Lemma 9, so we omit details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If α and β are the unique two cycles of even length in g, then by Lemma 8 there is an automorphism h of the poset of points of these two cycles which is not induced by a power of g, and moreover has the underlying sets of α and β as invariant sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since the non-trivial orbits of g n 4 ∈ Aut(P) are the underlying sets of α and β, h extends to an automorphism of P, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Suppose then there exists a third cycle γ = (1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' , 2k) in g with k odd, and that there is no other cycle of even length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' We define Q to be the subposet whose points are those of α, β and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then g n 4 fixes every point not in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The other orbits of g n 4 are the underlying sets of α and β, and Ai = {i, k + i} for 0 ≤ i ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let Q′ be the subposet whose points are those of α, β and A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then gk induces an automorphism of Q′ and by Lemma 8 there is an automorphism h of Q′ which is not induced by a power of gk, and for which the underlying sets of α, β and A0 are invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' We extend it to an automorphism h of Q by defining h(j) = h(j + 4t) − 4t, where t is such that k|j + 4t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then h is bijective, it is a morphism and leaves each Ai invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' It extends to an automorphism of P, say gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Since gr leaves A0 invariant, then k|r, which implies that h is induced by a power of gk, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Weights and the lower bound Let g be a permutation of order n of a finite set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let α be a cycle in g of length l = pr1 1 pr2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' prk k , where the pi are distinct prime integers, ri ≥ 1 for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' For each prime power pr we will define a weight wpr(α) ∈ R≥0 which depends on pr, l and n, in such a way that � pr wpr(α)pr = l, where the sum is taken over all prime powers dividing n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In particular #X ≥ � pr∥n ( � α∈g wpr(α))pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' For each l ≥ 2 we will assign the weight of every prime power pr in a cycle α of length |α| = l according to a series of rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In every case, if the weight wpr(α) is not explicitly defined for some prime power, we assume it is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Exception 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Suppose l = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If 3 ∥ n then w3(α) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If 3 ∦ n and 2 ∥ n, then w2(α) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If 3 ∦ n and 2 ∦ n, then w4(α) = 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Exception 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Suppose l = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If 3 ∥ n then w3(α) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If 3 ∦ n, then w4(α) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Exception 10-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Suppose l = 2p for p = 5 or 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If 2 ∥ n, w2(α) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Otherwise w4(α) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In any case wp(α) = 2(p−1) p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' General case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Suppose l = pr1 1 pr2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' prk k ̸= 6, 12, 10, 14, where the pi are different primes and each ri ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' For each 1 ≤ i ≤ k, we define wpri i (α) = � j̸=i p rj j k , unless pri i = 2 and 2 ∦ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In that case, w2(α) = 0, while w4(α) = � j̸=i p rj j 2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In particular, if l = pr ≥ 3 is a prime power, wpr(α) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Note that, as we required, the sum � pr|n wpr(α) over all the prime powers dividing n is the length l of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Note also that if l = pr1 1 pr2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' prk k , then wpr(α) ̸= 0 only if pr = pri i for some 1 ≤ i ≤ k or pr = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' 10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' BARMAK AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' BARRETO Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let P be a poset with Aut(P) cyclic of order n generated by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let pr be a prime power which exactly divides n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If pr ̸= 2, 4 then � α∈g wpr(α) ≥ b(pr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If 3 ∦ n and pr = 2 or pr = 4, � α∈g wpr(α) ≥ b(pr) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If 3 ∥ n and 2 ∥ n, � α∈g (2w2(α) + 3w3(α)) ≥ 2b(2) + 3b(3) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Finally, if 3 ∥ n and 4 ∥ n, � α∈g (4w4(α) + 3w3(α)) ≥ 4b(4) + 3b(3) − 1 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If pr ̸= 2, 3, 4, 5, 7, by Lemma 6, there are at least two cycles of length divisible by pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' By hypothesis their lengths are not multiples of pr+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' But if α is a cycle of g whose length is a multiple of pr, then wpr(α) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Indeed, the weights in α are assigned according to the General case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If the length of α is l = pr1 1 pr2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' prk k , we can assume pr = pr1 1 and then wpr(α) = k� j=2 p rj j k ≥ 2k−1 k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus, � α∈g wpr(α) ≥ 2 = b(pr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Suppose now pr = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If α is a cycle of g of length l = 5, then w5(α) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If l = 10, then w5(α) = 8 5 ≥ 3 2 (Exception 10-14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If l = 5s with s = pr2 2 pr3 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' prk k ≥ 3 not divisible by 5, then either k = 2, or k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In the first case w5(α) = s 2 ≥ 3 2, and in the second case w5(α) = k� j=2 p rj j k ≥ 2k−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='3 k ≥ 2 ≥ 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' By Lemma 6, there are at least two cycles of length divisible by 5 (and not by 52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Suppose first there exactly two such cycles, α and α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' None of them can be of length 5 by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus w5(α) + w5(α′) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='3 2 = 3 = b(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Finally, if there are at least three cycles in g of length divisible by 5, then � α∈g w5(α) ≥ 3 = b(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The case pr = 7 is similar to the previous one, with the observation that for length l = 14, w7(α) = 12 7 ≥ 3 2 (Exception 10-14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' So, also in this case � α∈g w7(α) ≥ 3 = b(7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let pr = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If the length of a cycle α in g is l = 3, w3(α) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If l = 6, w3(α) = 2 (Exception 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If l = 12, w3(α) = 4 (Exception 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If l = 3s with s = pr2 2 pr3 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' prk k ≥ 5, then either k = 2, or k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In the first case w3(α) = s 2 ≥ 5 2, and in the second case w3(α) = k� j=2 p rj j k ≥ 2k−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='3 k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' By Lemma 6 there are at least two cycles in g of length divisible by 3 (and not by 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Suppose first there are exactly two such cycles α and α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' None of them can have length 3 by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then w3(α)+w3(α′) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='2 = 4 ≥ 3 = b(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Finally, if there are at least three cycles in g of length divisible by 3, then � α∈g w3(α) ≥ 3 = b(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Note that � α∈g w3(α) ≥ 4 unless there are exactly three cycles of length 3 and no other cycle of length divisible by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' We analyze now the case that 3 ∦ n and pr = 2 or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In the first situation, there is at least one cycle α of even length l (not divisible by 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If l = 2, w2(α) = 1 (General case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If l = 6, w2(α) = 3 (Exception 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If l = 10 or l = 14, then w2(α) = 1 (Exception 10-14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If l = 2s with s = pr2 2 pr3 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' prk k ̸= 1, 3, 5, 7 (odd), then w2(α) = k� j=2 p rj j k ≥ 3k−1 k ≥ 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus � α∈g w2(α) ≥ 1 = b(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' We consider the second situation, pr = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If α has length l = 4, then w4(α) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If l = 12, w4(α) = 3 (Exception 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If l = 4s with s = pr2 2 pr3 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' prk k ≥ 5 (odd), SMALLEST POSETS WITH GIVEN CYCLIC AUTOMORPHISM GROUP 11 then k = 2 or k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' For k = 2 we have w4(α) = s 2 ≥ 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' For k ≥ 3, w4(α) ≥ 3k−1 k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' By Lemma 6, g contains at least two cycles of lengths divisible by 4 (and not by 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Suppose first there are exactly two such cycles, α and α′, of lengths l, l′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If l = l′ = 4, then by Lemma 10, there exists a third and a fourth cycle β, β′ of lengths 2m and 2m′ for some odd m, m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The weights w4(β) that we obtain for each m are the halves of the weights that we obtained for 2 in cycles of the same length when 2 ∥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Namely, if m = 1, w4(β) = 1 2 (General case);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' if m = 3, w4(β) = 3 2 (Exception 6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' if m = 5, 7, w4(β) = 1 2 (Exception 10-14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' if m = pr2 2 pr3 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' prk k ̸= 1, 3, 5, 7 then w4(β) ≥ 3k−1 2k ≥ 3 4 (General case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The same happens with β′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus w4(α) + w4(α′) + w4(β) + w4(β′) ≥ 1 + 1 + 1 2 + 1 2 = 3 = b(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If instead l = 4 and l′ = 12, then w4(α) + w4(α′) = 1 + 3 = 4 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If l = 4 and l′ = 4s for some odd s ≥ 5, then w4(α) + w4(α′) ≥ 1 + 5 2 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If both l and l′ are greater than 4, then w4(α) + w4(α′) ≥ 5 2 + 5 2 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Finally, if there are at least three cycles of length divisible by 4, then � α∈g w4(α) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus, in any case � α∈g w4(α) ≥ 3 = b(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' It only remains to analyze the case 3 ∥ n and 2 ∥ n and the case 3 ∥ n and 4 ∥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If 3 ∥ n and 2 ∥ n, recall that we have already proved that � α∈g w3(α) ≥ 4 or there are exactly three cycles of length 3 and no other cycle of length divisible by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In the first case � α∈g (2w2(α) + 3w3(α)) ≥ � α∈g 3w3(α) ≥ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' In the second case, there exists a cycle β in g of even length m ̸= 6, so w2(β) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus � α∈g (2w2(α) + 3w3(α)) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='1 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='3 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The last case is 3 ∥ n and 4 ∥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Note that if there are no cycles of length 6 nor 12 in g, then the computation � α∈g w4(α) ≥ 3 remains valid as Exceptions 6 and 12 do not occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus � α∈g (3w3(α)+4w4(α)) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='3+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='3 = 21 > 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If there are at least two 12-cycles, then � α∈g (3w3(α) + 4w4(α)) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='4 = 24 > 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If there is no 12-cycle in g and � α∈g w4(α) < 3, then we must be in the case that there is a 6-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' This already implies � α∈g w3(α) ≥ 4, while the existence of two cycles of length divisible by 4 implies � α∈g w4(α) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus � α∈g (3w3(α) + 4w4(α)) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='4 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='2 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus we may assume g has a unique 12-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' By Lemma 6 there is another cycle of length divisible by 4, so � α∈g w4(α) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' On the other hand, � α∈g w3(α) ≥ 4 + 2 = 6, as the weight of 3 in a 12-cycle is 4 and by Lemmas 6 and 9 there are either two more cycles of lengths divisible by 3 or just one, but of length not 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thus � α∈g (3w3(α) + 4w4(α)) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='6 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='1 = 22 > 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' □ Corollary 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Let n = pr1 1 pr2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' prk k , where the pi are different primes and ri ≥ 1 for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Then the minimum number β(Zn) of points in a poset with cyclic automorphism group of order n is k� i=1 b(pri i )pri i − 1 if 3 ∥ n and 4 ∥ n, and k� i=1 b(pri i )pri i otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If P is a poset with Aut(P) ≃ Zn generated by g, then the number of points in P is at least � α∈g |α| = � α∈g � pr|n wpr(α)pr ≥ k� i=1 ( � α∈g wpri i (α))pri i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' If both 3 and 4 exactly divide n, 12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' BARMAK AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' BARRETO by Theorem 11 this is � pri i ̸=3,4 ( � α∈g wpri i (α))pri i + � α∈g (3w3(α) + 4w4(α)) ≥ � pri i ̸=3,4 b(pri i )pri i + 3b(3) + 4b(4) − 1 = k� i=1 b(pri i )pri i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Otherwise, the bound is one more than this number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The bound is attained by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' □ References [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Arlinghaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The structure of minimal graphs with given abelian automorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Thesis, Wayne State University, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' [2] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Arlinghaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' The classification of minimal graphs with given abelian automorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' 57(1985), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' 330, viii+86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' [3] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Babai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' On the minimum order of graphs with given group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Canad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' 17(1974), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' 4, 467-470.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' [4] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Babai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Finite digraphs with given regular automorphism groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Hungar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' 11(1980), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' 4, 257-270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Barreto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Sobre los posets m´as chicos con grupo de automorfismos abeliano dado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Tesis de Licen- ciatura, Universidad de Buenos Aires, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' [6] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Birkhoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Sobre los grupos de automorfismos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Argentina 11(1946), 155-157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' [7] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Frucht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Herstellung von Graphen mit vorgegebener abstrakter Gruppe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Compositio Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' 6(1939), 239-250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Frucht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Graphs of degree 3 with given abstract group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Canad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' 1(1949) 305-378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Frucht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' On the construction of partially ordered systems with a given group of automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' 72(1950), 195-199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Meriwether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Smallest graphs with a given cyclic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' 1963, unpublished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' (1967) [11] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Sabidussi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' On the minimum order of graphs with given automorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Monatsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' 63(1959), 124-127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Universidad de Buenos Aires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Facultad de Ciencias Exactas y Naturales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Departamento de Matem´atica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Buenos Aires, Argentina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' CONICET-Universidad de Buenos Aires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Instituto de Investigaciones Matem´aticas Luis A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Santal´o (IMAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Buenos Aires, Argentina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content=' Email address: jbarmak@dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='uba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='ar Email address: abarreto@dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='uba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} +page_content='ar' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFAT4oBgHgl3EQf0B4_/content/2301.08701v1.pdf'} diff --git a/R9AzT4oBgHgl3EQf0f4J/content/2301.01783v1.pdf b/R9AzT4oBgHgl3EQf0f4J/content/2301.01783v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..29e20bf6cefeec572b19401191c5ce310580f108 --- /dev/null +++ b/R9AzT4oBgHgl3EQf0f4J/content/2301.01783v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2008b431e44624e56663b383554130aed4265eeb1c5be1f458e373bcfd537e62 +size 1083680 diff --git a/R9AzT4oBgHgl3EQf0f4J/vector_store/index.pkl b/R9AzT4oBgHgl3EQf0f4J/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..cf9e0126527f3a7965f77e22efc05b8e65c5978e --- /dev/null +++ b/R9AzT4oBgHgl3EQf0f4J/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ad6ef2d36d245dbcec0dbbd58adaf21392a9ac327ce07cb0a56a3c9d829bfd55 +size 168624 diff --git a/SdAyT4oBgHgl3EQfVPfZ/vector_store/index.faiss b/SdAyT4oBgHgl3EQfVPfZ/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..79c12ba975fac23473df24f9ab2bd95c6a6d7941 --- /dev/null +++ b/SdAyT4oBgHgl3EQfVPfZ/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6f800df66326703c007cf0cba23a7074b1ec523dd205164250a3c358d4abd633 +size 6422573 diff --git a/U9E0T4oBgHgl3EQfVQCg/content/tmp_files/2301.02262v1.pdf.txt b/U9E0T4oBgHgl3EQfVQCg/content/tmp_files/2301.02262v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bfd97ac1dbf8ed57515fb906a8ed0a265ea94b26 --- /dev/null +++ b/U9E0T4oBgHgl3EQfVQCg/content/tmp_files/2301.02262v1.pdf.txt @@ -0,0 +1,681 @@ +SINGING VOICE SYNTHESIS BASED ON FRAME-LEVEL SEQUENCE-TO-SEQUENCE +MODELS CONSIDERING VOCAL TIMING DEVIATION +Miku Nishihara, Yukiya Hono, Kei Hashimoto, Yoshihiko Nankaku, and Keiichi Tokuda +Department of Computer Science, Nagoya Institute of Technology, Nagoya, Japan +ABSTRACT +This paper proposes singing voice synthesis (SVS) based on frame- +level sequence-to-sequence models considering vocal timing devia- +tion. In SVS, it is essential to synchronize the timing of singing with +temporal structures represented by scores, taking into account that +there are differences between actual vocal timing and note start tim- +ing. In many SVS systems including our previous work, phoneme- +level score features are converted into frame-level ones on the basis +of phoneme boundaries obtained by external aligners to take into +account vocal timing deviations. Therefore, the sound quality is af- +fected by the aligner accuracy in this system. To alleviate this prob- +lem, we introduce an attention mechanism with frame-level features. +In the proposed system, the attention mechanism absorbs alignment +errors in phoneme boundaries. Additionally, we evaluate the system +with pseudo-phoneme-boundaries defined by heuristic rules based +on musical scores when there is no aligner. The experimental results +show the effectiveness of the proposed system. +Index Terms— Singing voice synthesis, frame-level sequence- +to-sequence models, attention mechanism, vocal timing deviation, +pseudo-phoneme-boundaries +1. INTRODUCTION +Statistical parametric singing voice synthesis (SVS) [1–4] has been +developed along with the spread of machine learning. This method +statistically models not only acoustic features but also the temporal +structures of singing voices because such a structure is highly depen- +dent on the tempo and note durations of the musical piece. Although +the musical score information is input into the system, the actual vo- +cal timing is usually different from the note start timing described +in the musical score. That is why modeling the temporal structure +synchronized with the musical score has become an important task +in SVS. +SVS systems based on deep neural networks (DNNs) [2–4] that +can generate natural singing have become commonly used. +The +systems use DNNs to model the mapping function from score fea- +tures, such as notes and phonemes, to acoustic features extracted +from singing voices. Generally, the lengths of phoneme-level score +features used as input and that of frame-level acoustic features used +as output are different. In our previous work, Sinsy [2], phoneme- +level score feature sequences are converted to frame-level ones on +the basis of phoneme boundaries obtained by using pre-trained hid- +den semi-Markov models (HSMMs) [5] as an aligner for training +DNNs. During synthesis, the time-lag model and phoneme duration +model determine the phoneme boundaries of the singing voice by +taking into account vocal timing deviations related to the note start +time given by the musical score. +This work was supported by JSPS KAKENHI Grant Number +JP22H03614, +CASIO SCIENCE PROMOTION FOUNDATION, and +FOUNDATION OF PUBLIC INTEREST OF TATEMATSU. +Text-to-speech (TTS) synthesis systems based on a sequence- +to-sequence (seq2seq) model [6–14] have been reported to gener- +ate natural speech. SVS systems based on a seq2seq model have +also been proposed [15–21] following seq2seq TTS. However, it is +difficult for seq2seq models to figure out the relationships of fea- +ture sequences with significantly different sequence lengths, such as +a phoneme-level score feature sequence and a frame-level acoustic +one all at once. Furthermore, for SVS, it is difficult to predict acous- +tic features even for the same phoneme or the same singer because +the temporal structure of the singing voice depends on the tempo +and note duration of the song. Therefore, in this paper, we focus on +the framework used in many SVS systems including our previous +work where phoneme-level score feature sequences are converted to +frame-level ones in advance on the basis of phoneme boundaries. +We have developed an SVS system called Sinsy [2]. In this sys- +tem, acoustic features and temporal structures of singing voices are +modeled by independent DNNs. This system can synthesize natural +singing voices in accordance with the input musical scores without +complicated operations like manual parameter adjustment. However, +in this system, the quality of the synthesized singing voice signifi- +cantly depends on the accuracy of phoneme boundaries, i.e. align- +ments between phonemes and acoustic features, used in training. Al- +though phoneme boundaries used in training are usually estimated +by external aligners such as pre-trained HSMMs, obtained phoneme +boundaries often include alignment errors, which affect subsequent +acoustic model training. +In this paper, to solve this problem, we introduce an attention +mechanism into the acoustic model of Sinsy. The attention mech- +anism captures the temporal structure of the training data and can +absorb alignment errors in phoneme boundaries due to the aligner. +In addition, the attention mechanism enables high flexibility in +weighting when converting features, which is expected to improve +the sound quality. We also introduce pseudo-phoneme-boundaries +which are boundaries determined by heuristic rules. Since no ex- +ternal model is used to estimate the pseudo-phoneme-boundaries, +there is no need to use forced alignment. We compare these methods +in experiments and describe the relationship among the alignment +accuracy in the proposed system. +2. DNN-BASED SINGING VOICE SYNTHESIS +Generally in TTS synthesis systems, a change of duration from the +original speech is acceptable as long as it is not much different from +the actual one. However, if we synthesize singing voices regardless +of the temporal structure, it may cause unusual sounds due to the +unacceptable deviation of the beat between the singing voice and the +accompaniment. Therefore, it is essential to synchronize the timing +of singing with temporal structures, such as tempo and note duration, +represented in scores. +Figure 1 shows the outline of the conventional DNN-based SVS +arXiv:2301.02262v1 [eess.AS] 5 Jan 2023 + +!" +" +!# +$% +&#" +! +" +! +$ +&#" +" +# +% $ +!" +" +!# +$% +&#" +$' +!" +" +!# +$% +43 +&'()%*% ++",#-.() +/.*%01#2 +34405#6%+ +/.*.)278(+%1 +34405#6%+ +3",#-.()78(+%1 +&'()%*%01%9%1 +:;(,%7<%#-",% +<,#*%01%9%1 +:;(,%7<%#-",% +34405#6%+ +=;("6-.;78(+%1 +<,#*%01%9%1 +=;("6-.;7<%#-",% +&'()%*%01%9%1 +:;(,%7<%#-",% +Fig. 1. Outline of the DNN-based SVS system. +system [2]. +This system models acoustic features and temporal +structures independently. Acoustic features represent spectrum and +fundamental frequency, and score features represent lyrics, note +duration, and pitch described in musical scores. +In model train- +ing, phoneme boundaries are first estimated by aligners such as +HSMMs. Then, time-lag models, phoneme duration models, and +acoustic models are trained by using DNNs, respectively. In syn- +thesis, phoneme-level score feature sequences without information +of each phoneme durations are first converted to phoneme-level +score feature ones by the time-lag models. Then, frame-level feature +sequences are obtained on the basis of the phoneme-level ones and +phoneme boundaries determined by the phoneme duration models. +Finally, frame-level acoustic features are generated by inputting the +obtained frame-level score features into the acoustic model. DNN +for acoustic models represent mapping functions from frame-level +score features to frame-level acoustic ones. +Therefore, accurate +phoneme boundaries are essential for training appropriate acoustic +models and generating singing voices that synchronize the musical +scores. +There are also cases when vocal timing deviations occur because +of intentional singing expressions or unique habits of the singer. Be- +cause of this, the start position of the note determined from the mu- +sical score and the actual vocal timing are not always the same. This +deviation in vocal timing is defined as the difference between the +start time of a note and that of the first phoneme of the note in this +paper. The note start timings and phoneme duration obtained from +the scores deviate from the actual singing timing, which must be +taken into account to match synthesized singing voice to the tempo +of the song in the case of SVS. +The deviation estimated by the time-lag model is used for the es- +timation of note boundaries, and the phoneme duration model is used +for the estimation of phoneme boundaries based on them. Therefore, +appropriate modeling of vocal timing deviations is one of the essen- +tial factors to generate a natural singing voice. +The sequence of note duration L and the sequence of deviations +predicted by the DNN-based time-lag model ˆg are represented as +follows: +L = (L1, L2, . . . , LN), +(1) +ˆg = (ˆg1, ˆg2, . . . , ˆgN), +(2) +where N is the number of notes included in a song. Note that it is +always ˆg1 = 0 because there is no timing deviations in the first note. +yu +u +ya +ke +pau +y +u y +k +u +a +e k +pau +Phoneme-level +Score Feature +Heuristic Upsampling +yu +u +ya +ke +43 +Frame-level +Acoustic Feature +Attention-based DNN +Frame-level +Score Feature +y +u y +k +u +a +e +k +pau +Fig. 2. Outline of seq2seq SVS system based on pseudo-phoneme- +boundaries. +The n-th adjusted note duration ˆLn is obtained by +ˆLn = +� +Ln − ˆgn + ˆgn+1 +(n < N), +Ln − ˆgn +(n = N). +(3) +In this paper, the duration distribution of the k-th phoneme in the +n-th note is represented by the Gaussian distribution with mean µnk +and variance σ2 +nk. The phoneme duration ˆdnk is calculated under the +constraint of the note duration ˆLn, which is obtained by taking into +account the deviation in the vocal timing, as follows +ˆdnk = µnk + ρnσ2 +nk, +(4) +ρn = +ˆLn − �Kn +k=1 µnk +�Kn +k=1 σ2 +nk +, +(5) +where Kn is the number of phonemes in the n-th note. Frame-level +score features can be appropriately obtained by considering the vocal +timing. +3. PROPOSED SYSTEM +In the conventional SVS systems, Sinsy [2], the alignment accuracy +is easily affected so the alignment errors might be propagated. To ad- +dress this problem, we adapt frame-level seq2seq models based on +an attention mechanism to our conventional SVS system to absorb +the alignment errors. The attention mechanism can absorb deviations +between the actual vocal timing and estimated phoneme boundaries. +We also propose a method using pseudo-phoneme-boundaries that +does not require forced alignment for estimating phoneme bound- +aries. Figure 2 shows an outline of this system. +3.1. Model architecture +Figure 3 shows the model architecture of SVS system based on +frame-level seq2seq models. The proposed system is based on the +Tacotron 2 [6]. Unlike Tacotron2, the attention mechanism in this +paper is driven frame by frame. The encoder uses as input the score +features that have already been converted from phoneme-level to +frame-level ones on the basis of previously determined phoneme +boundaries. The decoder outputs frame-level acoustic features by +considering the deviation in vocal timing using the attention mecha- +nism. +3.2. Embedding note features into the decoder +In the proposed method, the vocal timing of the singing voice needs +to be adjusted to the appropriate position on the basis of the musical + +!"#$$%&%'( +!")'(*+&,(+&%'( +-'&,".,+&/0, +12'(,3,45,6,5 +7*'0,".,+&/0, +.0+3,45,6,5 +7*'0,".,+&/0, +.0+3,45,6,5 +-'&,".,+&/0, +10,$%*&,$" +#*'/8&%*".,+&/0, +9:8+3:5%(; <+8,$"'(" +=8&%3+&%'( +9:8+3:5%(; +>%(,+0"10'?,*&%'( +-'&," +1%&*2 +>%(,+0"10'?,*&%'( +>'*+&%'("7,(8%&%6," +#&&,(&%'( +<%$%0,*&%'(+5">7@A +B")'(6">+C,08 +D"9(%$%0,*&%'(+5" +>7@A">+C,08 +D">+C,0"10,4-,& +E")'(6">+C,0 +1'8&4-,& +>%(,+0"10'?,*&%'( +#&&,(&%'("F,%;2& +F,%;2&,$"7/3 +1%&*2"-'03+5%G+&%'( +!"#$%&' +(&#$%&' +Fig. 3. Model architecture of seq2seq SVS system that is driven +frame by frame. +score by the attention mechanism. For this purpose, the decoder in- +put is a vector concatenated from the output of the decoder Pre-Net +and the extracted features of the target notes from the score features. +The frame-level note features are obtained by upsampling the note +features in the phoneme-level score features in accordance with the +note duration given by the musical score and concatenating it with +the frame positioning information within the note. Preliminary ex- +periments confirmed that the note feature information used as an ad- +ditional query enables the attention mechanism to adjust estimated +phoneme boundaries into appropriate phoneme boundaries. +3.3. Pitch normalization +Pitch normalization is performed as in the conventional SVS sys- +tem [2], where the difference between the note pitch in the musical +score and the logarithmic fundamental frequency (F0) of the singing +voice is modeled by a DNN. A note pitch sequence is obtained in +accordance with the input musical score taking into account the ac- +tual vocal timing. Pitch normalization requires a frame-level note +pitch sequence, which is obtained by attention weighting of the note +pitch sequence generated on the basis of the pre-estimated frame- +level phoneme boundaries at each decoder step. +3.4. Training criteria +The alignment obtained by the attentional mechanism of the pro- +posed method might follow the diagonal except near phoneme +boundaries because the score feature sequences are based on the es- +timated phoneme boundaries. Therefore, we use a guided attention +loss [22] in addition to the summed mean squared error (MSE) from +before and after the post-net. +3.5. Heuristic upsampling using the pseudo-phoneme-boundaries +We propose an aligner-free method using a heuristic rule that reflects +our singing tendencies as appropriate phoneme boundaries are given +by the attention mechanism. As the heuristic rule for setting pseudo- +phoneme-boundaries, this paper defines them in accordance with the +rule of shifting the consonant of the first phoneme to the front un- +der the constraint of maximum consonant duration Dc +max. The k-th +phoneme duration in the n-th note dnk is given by +dnk = +� +� +� +dc +n +if consonant +Ln − Cn · dc +n +Kn − Cn +otherwise +(6) +where Cn is the number of consonants in the n-th note, and dc +n is +dc +n = min(Dc +max, Ln/Kn) +(7) +In this paper, Dc +max is set to 30. As previously described, there is +a deviation between the start timing of notes on the score and the +actual vocal timing. Since it is difficult to express the pitch of a note +when it is a consonant, it tends to be pronounced a little earlier, so the +vowel is assigned to the start position of the note. The same approach +is used in the time-lag model used in the conventional SVS system +[2]. Preliminary experiments also indicated that shifting consonants +forward improves the naturalness of the synthesized singing voice. +In addition, the duration of consonants is not significantly affected +by the note duration and is known to be shorter than that of vowels. +To take into account these tendencies, we adopted this heuristic rule, +which shifts the first consonant of a note to the previous note position +and sets the maximum consonant duration. +3.6. Relation to prior work +In Sinsy [2], the score features are converted from phoneme-level +to frame-level ones on the basis of phoneme boundaries in advance +to enable the DNN to train the correspondence between phoneme- +level score features and frame-level acoustic ones. During the train- +ing stage, the phoneme boundaries are obtained by forced alignment +used the trained HSMMs, and during the synthesis stage, the bound- +aries are estimated explicitly from the vocal timing and phoneme +duration models in accordance with Eq. (4). In the proposed sys- +tem, the score features input to the seq2seq model are converted in +advance from phoneme-level to frame-level ones on the basis of the +phoneme boundaries like Sinsy. Furthermore, the seq2seq model +models the relationship between the score and acoustic features. +Bytesing [15] is a seq2seq SVS system using score features con- +verted into frame-level based on a duration model, which is similar +to the proposed method. However, the synthesized singing voice +does not follow the timing of the score because it does not take into +account vocal timing deviations. Therefore, the synthesized voices +must be manually adjusted to mix with the musical accompaniment. +In contrast, the proposed system can synthesize voices that are syn- +chronized with the temporal structure of the score because it takes +into account the vocal timing deviation when estimating phoneme +boundaries and is absorbed by the attention mechanism. +On the +other hand, strategies that reduce the dependence on alignment have +also been proposed. The authors of [16] produces frame-level score +features with phoneme boundaries estimated by a simple duration +model on the basis of average phoneme durations and absorbs vo- +cal timing deviations by using a seq2seq model with self-attention. +This requires manual correction to construct an appropriate duration +model. Incidentally, [17] does not use any external model but uses +heuristic rules to generate frame-level score features like our pro- +posed method. When multiple phonemes are assigned to a single +note, [17] assumes insufficiently, whereas we expect our system to +maintain naturalness for the variable number of consonants and long +tones since it constrains the maximum consonant duration. +4. EXPERIMENTS +4.1. Experimental conditions +We conducted experiments using 70 Japanese children’s songs per- +formed by a single female singer: 60 songs were used for train- +ing, and the rest were used for evaluation. Singing voice signals +were sampled at 48 kHz and each sample was quantized by 16 bits. +The acoustic feature consisted of 50-dimensional mel-cepstral coef- +ficients extracted by WORLD [23], a continuous log F0 value, 25- +dimensional analysis aperiodicity measures, 1-dimensional vibrato + +component, and a voiced/unvoiced binary flag. The difference be- +tween the original log F0 and the median-smoothed one was used +as the vibrato component. The acoustic features were extracted with +a 5-ms frame shift. The frame-level score feature for the input of +the encoder was a 272-dimensional feature vector consisting of the +context of the current phoneme and note information, frame position +in the current phoneme, and the phoneme duration. The frame-level +note feature for the input of the decoder was a 92-dimensional fea- +ture vector consisting of the context of the current note, the frame +position in the current note, and the note duration. The reduction +factor was set to 3 for all methods. All methods were combined +with the pre-trained PeriodNet [24], a non-AR neural vocoder with +a parallel structure, to generate waveforms from predicted acoustic +features. Five-state, left-to-right, no-skip HSMMs were used to ob- +tain the alignment of the score and acoustic features. The following +four systems were compared. +fal w/o att In the training stage, phoneme boundaries obtained by +forced alignment using trained HSMMs are used, while in +the synthesis stage, they were explicitly estimated using the +time-lag model and phoneme duration model in accordance +with Eq. (4). This system architecture is similar to the con- +ventional SVS system [2] that uses multiple DNNs, except +that the acoustic model structure has been modified on the +basis of Tacotron 2-like encoder-decoder model without the +attention mechanism. +fal w/ att The method to define phoneme boundaries is the same as +that of fal w/o att and the attention mechanism is used in this +system. +model w/ att In both the training and synthesis stages, the bound- +aries were explicitly estimated using the time-lag model and +phoneme duration model in accordance with Eq. (4). +pseudo w/ att In both the training and synthesis stages, the first +consonants in each note were deliberately shifted and then the +pseudo-phoneme boundaries were calculated in accordance +with Eq. (6). +The systems, fal w/ att, model w/ att, and pseudo w/ att, are based +on frame-level seq2seq models. The difference between them is the +phoneme boundaries for extracting frame-level score features. As +comparison methods, model w/o att and pseudo w/o att were also +considered. However, we excluded them from the experiments be- +cause they generated unnatural singing voice samples such as miss- +ing phonemes due to failing to absorb the mismatch of phoneme +boundaries without attention mechanisms. +4.2. Subjective evaluation +We performed two mean opinion score (MOS) tests to evaluate the +naturalness of the synthesized singing voices. Each of the 14 na- +tive Japanese-speaking participants evaluated 15 phrases randomly +selected from the test songs. In the first experiment, we asked the +participants to evaluate the quality of the synthesized singing voices, +focusing on the naturalness of the sound quality and pitch. In the sec- +ond experiment, a click sound generated on the basis of the tempo +of the score was mixed with the synthesized singing voices and the +naturalness of the vocal timing was evaluated. +The MOS results are plotted in Fig. 4.1 Compared with fal w/o +att, fal w/ att greatly improved the naturalness score. These results +suggest that the use of an attention mechanism improves sound qual- +ity compared with the conventional DNN-based SVS method. The +1Audio samples: https://www.sp.nitech.ac.jp/˜mkring/ +demo/ICASSP2023/ +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +fal w/o att +fal w/ att +model w/ att +pseudo w/ att +Mean Opinion Score +95% confidence intervals +3.63 +3.91 +3.61 +3.68 +(a) The score for the sound quality. +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +fal w/o att +fal w/ att +model w/ att +pseudo w/ att +Mean Opinion Score +95% confidence intervals +4.28 +4.31 +3.93 +3.99 +(b) The score for the vocal timing. +Fig. 4. Mean opinion scores. +scores for vocal timing of fal w/ att and fal w/o att were almost the +same. +In fal w/ att, the time-lag and phoneme duration models used +to estimate phoneme boundaries during synthesis were the same as +those used in model w/ att to estimate boundaries during both train- +ing and synthesis. However, for fal w/ att, the phoneme boundaries +are obtained from the forced alignment obtained from the actual +singing voice data during training. In addition, model w/ att gen- +erated voices that had a tendency to be delayed and unstable regard- +ing the vocal timing. When we checked the phoneme boundaries +estimated by the time-lag and phoneme duration models, the start +timing of each phoneme was later and more fluctuating than other +methods. The results comparing fal w/ att and model w/ att also in- +dicate the amount of vocal timing deviation and the alignment errors +that can be absorbed by the attention mechanism are limited because +the phoneme boundaries estimated by the time-lag and phoneme du- +ration models for training data were less accurate than the forced +alignment. Therefore, the accuracy of phoneme boundary estima- +tion during training has a strong effect on naturalness. +pseudo w/ att and model w/ att showed similar scores in both +tests. +In addition, these two methods showed synthesized sound +quality comparable to that with the aligner, fal w/o att. This re- +sult indicates that a natural singing voice can be generated without +any external models, and that appropriate vocal timing modeling can +be achieved from pseudo-phoneme-boundaries based on the musical +score. The phoneme boundaries used in pseudo w/ att and model w/ +att are not consistent with the actual vocal timing of the singing data. +Such deviations are appropriately modeled by the attention mecha- +nism in our proposed system. Although the timing is not as good as +the conventional method, the results are sufficiently practical. +5. CONCLUSION +In this paper, we introduced frame-level seq2seq models consid- +ering vocal timing deviation to our conventional DNN-based SVS +system, Sinsy. We also proposed a method using pseudo-phoneme- +boundaries that do not require external aligners. Experimental re- +sults show that the proposed method improves the naturalness of +the synthesized singing voice from the conventional method. In ad- +dition, the naturalness score of the proposed method with pseudo- +phoneme-boundaries was comparable to those of the conventional +method with high-accuracy aligners. Future work includes to com- +pare our method with existing ones and to improve the accuracy of +vocal timing estimation by the attention mechanism. + +6. REFERENCES +[1] Keiichiro Oura, Ayami Mase, Tomohiko Yamada, Satoru +Muto, Yoshihiko Nankaku, and Keiichi Tokuda, “Recent de- +velopment of the HMM-based singing voice synthesis system– +Sinsy,” in Seventh ISCA Workshop on Speech Synthesis, 2010, +pp. 211–216. +[2] Yukiya Hono, Kei Hashimoto, Keiichiro Oura, Yoshihiko +Nankaku, and Keiichi Tokuda, “Sinsy: A deep neural network- +based singing voice synthesis system,” IEEE/ACM Transac- +tions on Audio, Speech, and Language Processing, vol. 29, pp. +2803–2815, 2021. +[3] Yukiya Hono, Kei Hashimoto, Keiichiro Oura, Yoshihiko +Nankaku, and Keiichi Tokuda, “Singing voice synthesis based +on generative adversarial networks,” in Proc. ICASSP. IEEE, +2019, pp. 6955–6959. +[4] Masanari Nishimura, Kei Hashimoto, Keiichiro Oura, Yoshi- +hiko Nankaku, and Keiichi Tokuda, “Singing voice synthesis +based on deep neural networks,” in Proc. Interspeech, 2016, +pp. 2478–2482. +[5] Heiga +Zen, +Keiichi +Tokuda, +Takashi +Masuko, +Takao +Kobayasih, and Tadashi Kitamura, +“A hidden semi-Markov +model-based speech synthesis system,” IEICE transactions on +information and systems, vol. 90, no. 5, pp. 825–834, 2007. +[6] Jonathan Shen, Ruoming Pang, Ron J Weiss, Mike Schuster, +Navdeep Jaitly, Zongheng Yang, Zhifeng Chen, Yu Zhang, +Yuxuan Wang, Rj Skerrv-Ryan, Rif A Saurous, Yannis +Agiomyrgiannakis, and Yonghui Wu, “Natural TTS synthe- +sis by conditioning WaveNet on mel spectrogram predictions,” +in Proc. ICASSP, 2018, pp. 4779–4783. +[7] Yi Ren, Yangjun Ruan, Xu Tan, Tao Qin, Sheng Zhao, Zhou +Zhao, and Tie-Yan Liu, “Fastspeech: Fast, robust and control- +lable text to speech,” Advances in Neural Information Process- +ing Systems, vol. 32, 2019. +[8] Yi Ren, Chenxu Hu, Xu Tan, Tao Qin, Sheng Zhao, Zhou Zhao, +and Tie-Yan Liu, “FastSpeech 2: Fast and high-quality end-to- +end text to speech,” arXiv preprint arXiv:2006.04558, 2020. +[9] Naihan Li, Shujie Liu, Yanqing Liu, Sheng Zhao, and Ming +Liu, “Neural speech synthesis with transformer network,” in +Proc. AAAI Conference on Artificial Intelligence, 2019, vol. 33, +pp. 6706–6713. +[10] Wei Ping, Kainan Peng, Andrew Gibiansky, Sercan O Arik, +Ajay Kannan, Sharan Narang, Jonathan Raiman, and John +Miller, “Deep voice 3: Scaling text-to-speech with convolu- +tional sequence learning,” arXiv preprint arXiv:1710.07654, +2017. +[11] Chengzhu Yu, Heng Lu, Na Hu, Meng Yu, Chao Weng, Kun +Xu, Peng Liu, Deyi Tuo, Shiyin Kang, Guangzhi Lei, et al., +“DurIAN: Duration informed attention network for speech +synthesis,” pp. 2027–2031, 2020. +[12] Jaehyeon Kim, Jungil Kong, and Juhee Son, +“Conditional +variational autoencoder with adversarial learning for end-to- +end text-to-speech,” in International Conference on Machine +Learning. PMLR, 2021, pp. 5530–5540. +[13] Yin-Ping Cho, Yu Tsao, Hsin-Min Wang, +and Yi-Wen +Liu, +“Mandarin singing voice synthesis with denoising +diffusion probabilistic wasserstein gan,” +arXiv preprint +arXiv:2209.10446, 2022. +[14] Dan Lim, Sunghee Jung, and Eesung Kim, +“JETS: Jointly +training FastSpeech2 and HiFi-Gan for end to end text to +speech,” arXiv preprint arXiv:2203.16852, 2022. +[15] Yu Gu, Xiang Yin, Yonghui Rao, Yuan Wan, Benlai Tang, +Yang Zhang, Jitong Chen, Yuxuan Wang, and Zejun Ma, +“Bytesing: A chinese singing voice synthesis system using du- +ration allocated encoder-decoder acoustic models and wavernn +vocoders,” in Proc. ISCSLP, 2021, pp. 1–5. +[16] Merlijn Blaauw and Jordi Bonada, +“Sequence-to-sequence +singing synthesis using the feed-forward transformer,” in Proc. +ICASSP, 2020, pp. 7229–7233. +[17] Juheon Lee, Hyeong-Seok Choi, Chang-Bin Jeon, Junghyun +Koo, and Kyogu Lee, “Adversarially trained end-to-end korean +singing voice synthesis system,” Proc. Interspeech, pp. 2588– +2592, 2019. +[18] Peiling Lu, Jie Wu, Jian Luan, Xu Tan, and Li Zhou, “Xiaoic- +esing: A high-quality and integrated singing voice synthesis +system,” in Proc. Interspeech, 2020, pp. 1306–1310. +[19] Jiawei Chen, Xu Tan, Jian Luan, Tao Qin, and Tie-Yan Liu, +“Hifisinger: Towards high-fidelity neural singing voice synthe- +sis,” arXiv preprint arXiv:2009.01776, 2020. +[20] Jie Wu and Jian Luan, +“Adversarially trained multi-singer +sequence-to-sequence singing synthesizer,” +in Proc. Inter- +speech, 2020, pp. 1296–1300. +[21] Jiatong Shi, Shuai Guo, Nan Huo, Yuekai Zhang, and Qin Jin, +“Sequence-to-sequence singing voice synthesis with percep- +tual entropy loss,” in Proc.ICASSP, 2021, pp. 76–80. +[22] Hideyuki Tachibana, Katsuya Uenoyama, and Shunsuke Ai- +hara, +“Efficiently trainable text-to-speech system based on +deep convolutional networks with guided attention,” in Proc. +ICASSP. IEEE, 2018, pp. 4784–4788. +[23] Masanori Morise, Fumiya Yokomori, and Kenji Ozawa, +“WORLD: a vocoder-based high-quality speech synthesis sys- +tem for real-time applications,” IEICE transactions on infor- +mation and systems, vol. 99, no. 7, pp. 1877–1884, 2016. +[24] Yukiya Hono, Shinji Takaki, Kei Hashimoto, Keiichiro Oura, +Yoshihiko Nankaku, and Keiichi Tokuda, +“PeriodNet: A +non-autoregressive waveform generation model with a struc- +ture separating periodic and aperiodic components,” in Proc. +ICASSP. IEEE, 2021, pp. 6049–6053. + diff --git a/U9E0T4oBgHgl3EQfVQCg/content/tmp_files/load_file.txt b/U9E0T4oBgHgl3EQfVQCg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..17103ef732e38a5713328c1b560d580bbd2cb55b --- /dev/null +++ b/U9E0T4oBgHgl3EQfVQCg/content/tmp_files/load_file.txt @@ -0,0 +1,331 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf,len=330 +page_content='SINGING VOICE SYNTHESIS BASED ON FRAME-LEVEL SEQUENCE-TO-SEQUENCE MODELS CONSIDERING VOCAL TIMING DEVIATION Miku Nishihara, Yukiya Hono, Kei Hashimoto, Yoshihiko Nankaku, and Keiichi Tokuda Department of Computer Science, Nagoya Institute of Technology, Nagoya, Japan ABSTRACT This paper proposes singing voice synthesis (SVS) based on frame- level sequence-to-sequence models considering vocal timing devia- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' In SVS, it is essential to synchronize the timing of singing with temporal structures represented by scores, taking into account that there are differences between actual vocal timing and note start tim- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' In many SVS systems including our previous work, phoneme- level score features are converted into frame-level ones on the basis of phoneme boundaries obtained by external aligners to take into account vocal timing deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Therefore, the sound quality is af- fected by the aligner accuracy in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' To alleviate this prob- lem, we introduce an attention mechanism with frame-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' In the proposed system, the attention mechanism absorbs alignment errors in phoneme boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Additionally, we evaluate the system with pseudo-phoneme-boundaries defined by heuristic rules based on musical scores when there is no aligner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The experimental results show the effectiveness of the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Index Terms— Singing voice synthesis, frame-level sequence- to-sequence models, attention mechanism, vocal timing deviation, pseudo-phoneme-boundaries 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' INTRODUCTION Statistical parametric singing voice synthesis (SVS) [1–4] has been developed along with the spread of machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' This method statistically models not only acoustic features but also the temporal structures of singing voices because such a structure is highly depen- dent on the tempo and note durations of the musical piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Although the musical score information is input into the system, the actual vo- cal timing is usually different from the note start timing described in the musical score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' That is why modeling the temporal structure synchronized with the musical score has become an important task in SVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' SVS systems based on deep neural networks (DNNs) [2–4] that can generate natural singing have become commonly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The systems use DNNs to model the mapping function from score fea- tures, such as notes and phonemes, to acoustic features extracted from singing voices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Generally, the lengths of phoneme-level score features used as input and that of frame-level acoustic features used as output are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' In our previous work, Sinsy [2], phoneme- level score feature sequences are converted to frame-level ones on the basis of phoneme boundaries obtained by using pre-trained hid- den semi-Markov models (HSMMs) [5] as an aligner for training DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' During synthesis, the time-lag model and phoneme duration model determine the phoneme boundaries of the singing voice by taking into account vocal timing deviations related to the note start time given by the musical score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' This work was supported by JSPS KAKENHI Grant Number JP22H03614, CASIO SCIENCE PROMOTION FOUNDATION, and FOUNDATION OF PUBLIC INTEREST OF TATEMATSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Text-to-speech (TTS) synthesis systems based on a sequence- to-sequence (seq2seq) model [6–14] have been reported to gener- ate natural speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' SVS systems based on a seq2seq model have also been proposed [15–21] following seq2seq TTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' However, it is difficult for seq2seq models to figure out the relationships of fea- ture sequences with significantly different sequence lengths, such as a phoneme-level score feature sequence and a frame-level acoustic one all at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Furthermore, for SVS, it is difficult to predict acous- tic features even for the same phoneme or the same singer because the temporal structure of the singing voice depends on the tempo and note duration of the song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Therefore, in this paper, we focus on the framework used in many SVS systems including our previous work where phoneme-level score feature sequences are converted to frame-level ones in advance on the basis of phoneme boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' We have developed an SVS system called Sinsy [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' In this sys- tem, acoustic features and temporal structures of singing voices are modeled by independent DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' This system can synthesize natural singing voices in accordance with the input musical scores without complicated operations like manual parameter adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' However, in this system, the quality of the synthesized singing voice signifi- cantly depends on the accuracy of phoneme boundaries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' align- ments between phonemes and acoustic features, used in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Al- though phoneme boundaries used in training are usually estimated by external aligners such as pre-trained HSMMs, obtained phoneme boundaries often include alignment errors, which affect subsequent acoustic model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' In this paper, to solve this problem, we introduce an attention mechanism into the acoustic model of Sinsy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The attention mech- anism captures the temporal structure of the training data and can absorb alignment errors in phoneme boundaries due to the aligner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' In addition, the attention mechanism enables high flexibility in weighting when converting features, which is expected to improve the sound quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' We also introduce pseudo-phoneme-boundaries which are boundaries determined by heuristic rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Since no ex- ternal model is used to estimate the pseudo-phoneme-boundaries, there is no need to use forced alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' We compare these methods in experiments and describe the relationship among the alignment accuracy in the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' DNN-BASED SINGING VOICE SYNTHESIS Generally in TTS synthesis systems, a change of duration from the original speech is acceptable as long as it is not much different from the actual one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' However, if we synthesize singing voices regardless of the temporal structure, it may cause unusual sounds due to the unacceptable deviation of the beat between the singing voice and the accompaniment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Therefore, it is essential to synchronize the timing of singing with temporal structures, such as tempo and note duration, represented in scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Figure 1 shows the outline of the conventional DNN-based SVS arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='02262v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='AS] 5 Jan 2023 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='" " !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='# $% &#" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' " !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' $ &#" " # % $ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='" " !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='# $% &#" $\' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='" " !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='# $% 43 &\'()%*% +",#-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' () /.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' *%01#2 34405#6%+ /.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' )278(+%1 34405#6%+ 3",#-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=" ()78(+%1 &'()%*%01%9%1 :;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='(,%7<%#-",% <,#*%01%9%1 :;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='(,%7<%#-",% 34405#6%+ =;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='("6-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='78(+%1 <,#*%01%9%1 =;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='("6-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='7<%#-",% &\'()%*%01%9%1 :;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='(,%7<%#-",% Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Outline of the DNN-based SVS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' system [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' This system models acoustic features and temporal structures independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Acoustic features represent spectrum and fundamental frequency, and score features represent lyrics, note duration, and pitch described in musical scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' In model train- ing, phoneme boundaries are first estimated by aligners such as HSMMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Then, time-lag models, phoneme duration models, and acoustic models are trained by using DNNs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' In syn- thesis, phoneme-level score feature sequences without information of each phoneme durations are first converted to phoneme-level score feature ones by the time-lag models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Then, frame-level feature sequences are obtained on the basis of the phoneme-level ones and phoneme boundaries determined by the phoneme duration models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Finally, frame-level acoustic features are generated by inputting the obtained frame-level score features into the acoustic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' DNN for acoustic models represent mapping functions from frame-level score features to frame-level acoustic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Therefore, accurate phoneme boundaries are essential for training appropriate acoustic models and generating singing voices that synchronize the musical scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' There are also cases when vocal timing deviations occur because of intentional singing expressions or unique habits of the singer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Be- cause of this, the start position of the note determined from the mu- sical score and the actual vocal timing are not always the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' This deviation in vocal timing is defined as the difference between the start time of a note and that of the first phoneme of the note in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The note start timings and phoneme duration obtained from the scores deviate from the actual singing timing, which must be taken into account to match synthesized singing voice to the tempo of the song in the case of SVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The deviation estimated by the time-lag model is used for the es- timation of note boundaries, and the phoneme duration model is used for the estimation of phoneme boundaries based on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Therefore, appropriate modeling of vocal timing deviations is one of the essen- tial factors to generate a natural singing voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The sequence of note duration L and the sequence of deviations predicted by the DNN-based time-lag model ˆg are represented as follows: L = (L1, L2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' , LN), (1) ˆg = (ˆg1, ˆg2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' , ˆgN), (2) where N is the number of notes included in a song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Note that it is always ˆg1 = 0 because there is no timing deviations in the first note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' yu u ya ke pau y u y k u a e k pau Phoneme-level Score Feature Heuristic Upsampling yu u ya ke 43 Frame-level Acoustic Feature Attention-based DNN Frame-level Score Feature y u y k u a e k pau Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Outline of seq2seq SVS system based on pseudo-phoneme- boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The n-th adjusted note duration ˆLn is obtained by ˆLn = � Ln − ˆgn + ˆgn+1 (n < N), Ln − ˆgn (n = N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' (3) In this paper, the duration distribution of the k-th phoneme in the n-th note is represented by the Gaussian distribution with mean µnk and variance σ2 nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The phoneme duration ˆdnk is calculated under the constraint of the note duration ˆLn, which is obtained by taking into account the deviation in the vocal timing, as follows ˆdnk = µnk + ρnσ2 nk, (4) ρn = ˆLn − �Kn k=1 µnk �Kn k=1 σ2 nk , (5) where Kn is the number of phonemes in the n-th note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Frame-level score features can be appropriately obtained by considering the vocal timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' PROPOSED SYSTEM In the conventional SVS systems, Sinsy [2], the alignment accuracy is easily affected so the alignment errors might be propagated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' To ad- dress this problem, we adapt frame-level seq2seq models based on an attention mechanism to our conventional SVS system to absorb the alignment errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The attention mechanism can absorb deviations between the actual vocal timing and estimated phoneme boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' We also propose a method using pseudo-phoneme-boundaries that does not require forced alignment for estimating phoneme bound- aries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Figure 2 shows an outline of this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Model architecture Figure 3 shows the model architecture of SVS system based on frame-level seq2seq models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The proposed system is based on the Tacotron 2 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Unlike Tacotron2, the attention mechanism in this paper is driven frame by frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The encoder uses as input the score features that have already been converted from phoneme-level to frame-level ones on the basis of previously determined phoneme boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The decoder outputs frame-level acoustic features by considering the deviation in vocal timing using the attention mecha- nism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Embedding note features into the decoder In the proposed method, the vocal timing of the singing voice needs to be adjusted to the appropriate position on the basis of the musical !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' "#$$%&%\'( !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' ")\'(*+&,(+&%\'( \'&,".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=',+&/0, 12\'(,3,45,6,5 7*\'0,".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=',+&/0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='0+3,45,6,5 7*\'0,".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=',+&/0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='0+3,45,6,5 \'&,".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=',+&/0, 10,$%*&,$" #*\'/8&%*".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=',+&/0, 9:8+3:5%(;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' <+8,$"\'(" =8&%3+&%\'( 9:8+3:5%(;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' >%(,+0"10\'?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=',*&%\'( \'&," 1%&*2 >%(,+0"10\'?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=',*&%\'( >\'*+&%\'("7,(8%&%6," #&&,(&%\'( <%$%0,*&%\'(+5">7@A B")\'(6">+C,08 D"9(%$%0,*&%\'(+5" >7@A">+C,08 D">+C,0"10,4-,& E")\'(6">+C,0 1\'8&4-,& >%(,+0"10\'?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=',*&%\'( #&&,(&%\'("F,%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='2& F,%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='2&,$"7/3 1%&*2"-\'03+5%G+&%\'( !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' "#$%&\' (&#$%&\' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Model architecture of seq2seq SVS system that is driven frame by frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' score by the attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' For this purpose, the decoder in- put is a vector concatenated from the output of the decoder Pre-Net and the extracted features of the target notes from the score features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The frame-level note features are obtained by upsampling the note features in the phoneme-level score features in accordance with the note duration given by the musical score and concatenating it with the frame positioning information within the note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Preliminary ex- periments confirmed that the note feature information used as an ad- ditional query enables the attention mechanism to adjust estimated phoneme boundaries into appropriate phoneme boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Pitch normalization Pitch normalization is performed as in the conventional SVS sys- tem [2], where the difference between the note pitch in the musical score and the logarithmic fundamental frequency (F0) of the singing voice is modeled by a DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' A note pitch sequence is obtained in accordance with the input musical score taking into account the ac- tual vocal timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Pitch normalization requires a frame-level note pitch sequence, which is obtained by attention weighting of the note pitch sequence generated on the basis of the pre-estimated frame- level phoneme boundaries at each decoder step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Training criteria The alignment obtained by the attentional mechanism of the pro- posed method might follow the diagonal except near phoneme boundaries because the score feature sequences are based on the es- timated phoneme boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Therefore, we use a guided attention loss [22] in addition to the summed mean squared error (MSE) from before and after the post-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Heuristic upsampling using the pseudo-phoneme-boundaries We propose an aligner-free method using a heuristic rule that reflects our singing tendencies as appropriate phoneme boundaries are given by the attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' As the heuristic rule for setting pseudo- phoneme-boundaries, this paper defines them in accordance with the rule of shifting the consonant of the first phoneme to the front un- der the constraint of maximum consonant duration Dc max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The k-th phoneme duration in the n-th note dnk is given by dnk = � � � dc n if consonant Ln − Cn · dc n Kn − Cn otherwise (6) where Cn is the number of consonants in the n-th note, and dc n is dc n = min(Dc max, Ln/Kn) (7) In this paper, Dc max is set to 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' As previously described, there is a deviation between the start timing of notes on the score and the actual vocal timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Since it is difficult to express the pitch of a note when it is a consonant, it tends to be pronounced a little earlier, so the vowel is assigned to the start position of the note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The same approach is used in the time-lag model used in the conventional SVS system [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Preliminary experiments also indicated that shifting consonants forward improves the naturalness of the synthesized singing voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' In addition, the duration of consonants is not significantly affected by the note duration and is known to be shorter than that of vowels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' To take into account these tendencies, we adopted this heuristic rule, which shifts the first consonant of a note to the previous note position and sets the maximum consonant duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Relation to prior work In Sinsy [2], the score features are converted from phoneme-level to frame-level ones on the basis of phoneme boundaries in advance to enable the DNN to train the correspondence between phoneme- level score features and frame-level acoustic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' During the train- ing stage, the phoneme boundaries are obtained by forced alignment used the trained HSMMs, and during the synthesis stage, the bound- aries are estimated explicitly from the vocal timing and phoneme duration models in accordance with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' In the proposed sys- tem, the score features input to the seq2seq model are converted in advance from phoneme-level to frame-level ones on the basis of the phoneme boundaries like Sinsy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Furthermore, the seq2seq model models the relationship between the score and acoustic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Bytesing [15] is a seq2seq SVS system using score features con- verted into frame-level based on a duration model, which is similar to the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' However, the synthesized singing voice does not follow the timing of the score because it does not take into account vocal timing deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Therefore, the synthesized voices must be manually adjusted to mix with the musical accompaniment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' In contrast, the proposed system can synthesize voices that are syn- chronized with the temporal structure of the score because it takes into account the vocal timing deviation when estimating phoneme boundaries and is absorbed by the attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' On the other hand, strategies that reduce the dependence on alignment have also been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The authors of [16] produces frame-level score features with phoneme boundaries estimated by a simple duration model on the basis of average phoneme durations and absorbs vo- cal timing deviations by using a seq2seq model with self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' This requires manual correction to construct an appropriate duration model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Incidentally, [17] does not use any external model but uses heuristic rules to generate frame-level score features like our pro- posed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' When multiple phonemes are assigned to a single note, [17] assumes insufficiently, whereas we expect our system to maintain naturalness for the variable number of consonants and long tones since it constrains the maximum consonant duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' EXPERIMENTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Experimental conditions We conducted experiments using 70 Japanese children’s songs per- formed by a single female singer: 60 songs were used for train- ing, and the rest were used for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Singing voice signals were sampled at 48 kHz and each sample was quantized by 16 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The acoustic feature consisted of 50-dimensional mel-cepstral coef- ficients extracted by WORLD [23], a continuous log F0 value, 25- dimensional analysis aperiodicity measures, 1-dimensional vibrato component, and a voiced/unvoiced binary flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The difference be- tween the original log F0 and the median-smoothed one was used as the vibrato component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The acoustic features were extracted with a 5-ms frame shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The frame-level score feature for the input of the encoder was a 272-dimensional feature vector consisting of the context of the current phoneme and note information, frame position in the current phoneme, and the phoneme duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The frame-level note feature for the input of the decoder was a 92-dimensional fea- ture vector consisting of the context of the current note, the frame position in the current note, and the note duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The reduction factor was set to 3 for all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' All methods were combined with the pre-trained PeriodNet [24], a non-AR neural vocoder with a parallel structure, to generate waveforms from predicted acoustic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Five-state, left-to-right, no-skip HSMMs were used to ob- tain the alignment of the score and acoustic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The following four systems were compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' fal w/o att In the training stage, phoneme boundaries obtained by forced alignment using trained HSMMs are used, while in the synthesis stage, they were explicitly estimated using the time-lag model and phoneme duration model in accordance with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' This system architecture is similar to the con- ventional SVS system [2] that uses multiple DNNs, except that the acoustic model structure has been modified on the basis of Tacotron 2-like encoder-decoder model without the attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' fal w/ att The method to define phoneme boundaries is the same as that of fal w/o att and the attention mechanism is used in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' model w/ att In both the training and synthesis stages, the bound- aries were explicitly estimated using the time-lag model and phoneme duration model in accordance with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' pseudo w/ att In both the training and synthesis stages, the first consonants in each note were deliberately shifted and then the pseudo-phoneme boundaries were calculated in accordance with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The systems, fal w/ att, model w/ att, and pseudo w/ att, are based on frame-level seq2seq models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The difference between them is the phoneme boundaries for extracting frame-level score features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' As comparison methods, model w/o att and pseudo w/o att were also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' However, we excluded them from the experiments be- cause they generated unnatural singing voice samples such as miss- ing phonemes due to failing to absorb the mismatch of phoneme boundaries without attention mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Subjective evaluation We performed two mean opinion score (MOS) tests to evaluate the naturalness of the synthesized singing voices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Each of the 14 na- tive Japanese-speaking participants evaluated 15 phrases randomly selected from the test songs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' In the first experiment, we asked the participants to evaluate the quality of the synthesized singing voices, focusing on the naturalness of the sound quality and pitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' In the sec- ond experiment, a click sound generated on the basis of the tempo of the score was mixed with the synthesized singing voices and the naturalness of the vocal timing was evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The MOS results are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='1 Compared with fal w/o att, fal w/ att greatly improved the naturalness score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' These results suggest that the use of an attention mechanism improves sound qual- ity compared with the conventional DNN-based SVS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The 1Audio samples: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='nitech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='jp/˜mkring/ demo/ICASSP2023/ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='0 fal w/o att fal w/ att model w/ att pseudo w/ att Mean Opinion Score 95% confidence intervals 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='63 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='91 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='61 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='68 (a) The score for the sound quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='0 fal w/o att fal w/ att model w/ att pseudo w/ att Mean Opinion Score 95% confidence intervals 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='93 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='99 (b) The score for the vocal timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Mean opinion scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' scores for vocal timing of fal w/ att and fal w/o att were almost the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' In fal w/ att, the time-lag and phoneme duration models used to estimate phoneme boundaries during synthesis were the same as those used in model w/ att to estimate boundaries during both train- ing and synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' However, for fal w/ att, the phoneme boundaries are obtained from the forced alignment obtained from the actual singing voice data during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' In addition, model w/ att gen- erated voices that had a tendency to be delayed and unstable regard- ing the vocal timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' When we checked the phoneme boundaries estimated by the time-lag and phoneme duration models, the start timing of each phoneme was later and more fluctuating than other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The results comparing fal w/ att and model w/ att also in- dicate the amount of vocal timing deviation and the alignment errors that can be absorbed by the attention mechanism are limited because the phoneme boundaries estimated by the time-lag and phoneme du- ration models for training data were less accurate than the forced alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Therefore, the accuracy of phoneme boundary estima- tion during training has a strong effect on naturalness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' pseudo w/ att and model w/ att showed similar scores in both tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' In addition, these two methods showed synthesized sound quality comparable to that with the aligner, fal w/o att.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' This re- sult indicates that a natural singing voice can be generated without any external models, and that appropriate vocal timing modeling can be achieved from pseudo-phoneme-boundaries based on the musical score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' The phoneme boundaries used in pseudo w/ att and model w/ att are not consistent with the actual vocal timing of the singing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Such deviations are appropriately modeled by the attention mecha- nism in our proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Although the timing is not as good as the conventional method, the results are sufficiently practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' CONCLUSION In this paper, we introduced frame-level seq2seq models consid- ering vocal timing deviation to our conventional DNN-based SVS system, Sinsy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' We also proposed a method using pseudo-phoneme- boundaries that do not require external aligners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Experimental re- sults show that the proposed method improves the naturalness of the synthesized singing voice from the conventional method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' In ad- dition, the naturalness score of the proposed method with pseudo- phoneme-boundaries was comparable to those of the conventional method with high-accuracy aligners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Future work includes to com- pare our method with existing ones and to improve the accuracy of vocal timing estimation by the attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' REFERENCES [1] Keiichiro Oura, Ayami Mase, Tomohiko Yamada, Satoru Muto, Yoshihiko Nankaku, and Keiichi Tokuda, “Recent de- velopment of the HMM-based singing voice synthesis system– Sinsy,” in Seventh ISCA Workshop on Speech Synthesis, 2010, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 211–216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [2] Yukiya Hono, Kei Hashimoto, Keiichiro Oura, Yoshihiko Nankaku, and Keiichi Tokuda, “Sinsy: A deep neural network- based singing voice synthesis system,” IEEE/ACM Transac- tions on Audio, Speech, and Language Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 29, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 2803–2815, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [3] Yukiya Hono, Kei Hashimoto, Keiichiro Oura, Yoshihiko Nankaku, and Keiichi Tokuda, “Singing voice synthesis based on generative adversarial networks,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' ICASSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' IEEE, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 6955–6959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [4] Masanari Nishimura, Kei Hashimoto, Keiichiro Oura, Yoshi- hiko Nankaku, and Keiichi Tokuda, “Singing voice synthesis based on deep neural networks,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Interspeech, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 2478–2482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [5] Heiga Zen, Keiichi Tokuda, Takashi Masuko, Takao Kobayasih, and Tadashi Kitamura, “A hidden semi-Markov model-based speech synthesis system,” IEICE transactions on information and systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 90, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 825–834, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [6] Jonathan Shen, Ruoming Pang, Ron J Weiss, Mike Schuster, Navdeep Jaitly, Zongheng Yang, Zhifeng Chen, Yu Zhang, Yuxuan Wang, Rj Skerrv-Ryan, Rif A Saurous, Yannis Agiomyrgiannakis, and Yonghui Wu, “Natural TTS synthe- sis by conditioning WaveNet on mel spectrogram predictions,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' ICASSP, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 4779–4783.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [7] Yi Ren, Yangjun Ruan, Xu Tan, Tao Qin, Sheng Zhao, Zhou Zhao, and Tie-Yan Liu, “Fastspeech: Fast, robust and control- lable text to speech,” Advances in Neural Information Process- ing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [8] Yi Ren, Chenxu Hu, Xu Tan, Tao Qin, Sheng Zhao, Zhou Zhao, and Tie-Yan Liu, “FastSpeech 2: Fast and high-quality end-to- end text to speech,” arXiv preprint arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='04558, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [9] Naihan Li, Shujie Liu, Yanqing Liu, Sheng Zhao, and Ming Liu, “Neural speech synthesis with transformer network,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' AAAI Conference on Artificial Intelligence, 2019, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 6706–6713.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [10] Wei Ping, Kainan Peng, Andrew Gibiansky, Sercan O Arik, Ajay Kannan, Sharan Narang, Jonathan Raiman, and John Miller, “Deep voice 3: Scaling text-to-speech with convolu- tional sequence learning,” arXiv preprint arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='07654, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [11] Chengzhu Yu, Heng Lu, Na Hu, Meng Yu, Chao Weng, Kun Xu, Peng Liu, Deyi Tuo, Shiyin Kang, Guangzhi Lei, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=', “DurIAN: Duration informed attention network for speech synthesis,” pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 2027–2031, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [12] Jaehyeon Kim, Jungil Kong, and Juhee Son, “Conditional variational autoencoder with adversarial learning for end-to- end text-to-speech,” in International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' PMLR, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 5530–5540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [13] Yin-Ping Cho, Yu Tsao, Hsin-Min Wang, and Yi-Wen Liu, “Mandarin singing voice synthesis with denoising diffusion probabilistic wasserstein gan,” arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='10446, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [14] Dan Lim, Sunghee Jung, and Eesung Kim, “JETS: Jointly training FastSpeech2 and HiFi-Gan for end to end text to speech,” arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='16852, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [15] Yu Gu, Xiang Yin, Yonghui Rao, Yuan Wan, Benlai Tang, Yang Zhang, Jitong Chen, Yuxuan Wang, and Zejun Ma, “Bytesing: A chinese singing voice synthesis system using du- ration allocated encoder-decoder acoustic models and wavernn vocoders,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' ISCSLP, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [16] Merlijn Blaauw and Jordi Bonada, “Sequence-to-sequence singing synthesis using the feed-forward transformer,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' ICASSP, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 7229–7233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [17] Juheon Lee, Hyeong-Seok Choi, Chang-Bin Jeon, Junghyun Koo, and Kyogu Lee, “Adversarially trained end-to-end korean singing voice synthesis system,” Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Interspeech, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 2588– 2592, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [18] Peiling Lu, Jie Wu, Jian Luan, Xu Tan, and Li Zhou, “Xiaoic- esing: A high-quality and integrated singing voice synthesis system,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Interspeech, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 1306–1310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [19] Jiawei Chen, Xu Tan, Jian Luan, Tao Qin, and Tie-Yan Liu, “Hifisinger: Towards high-fidelity neural singing voice synthe- sis,” arXiv preprint arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='01776, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [20] Jie Wu and Jian Luan, “Adversarially trained multi-singer sequence-to-sequence singing synthesizer,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' Inter- speech, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 1296–1300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [21] Jiatong Shi, Shuai Guo, Nan Huo, Yuekai Zhang, and Qin Jin, “Sequence-to-sequence singing voice synthesis with percep- tual entropy loss,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content='ICASSP, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 76–80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [22] Hideyuki Tachibana, Katsuya Uenoyama, and Shunsuke Ai- hara, “Efficiently trainable text-to-speech system based on deep convolutional networks with guided attention,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' ICASSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' IEEE, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 4784–4788.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [23] Masanori Morise, Fumiya Yokomori, and Kenji Ozawa, “WORLD: a vocoder-based high-quality speech synthesis sys- tem for real-time applications,” IEICE transactions on infor- mation and systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 99, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 1877–1884, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' [24] Yukiya Hono, Shinji Takaki, Kei Hashimoto, Keiichiro Oura, Yoshihiko Nankaku, and Keiichi Tokuda, “PeriodNet: A non-autoregressive waveform generation model with a struc- ture separating periodic and aperiodic components,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' ICASSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' IEEE, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} +page_content=' 6049–6053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQfVQCg/content/2301.02262v1.pdf'} diff --git a/V9FQT4oBgHgl3EQfbTbA/content/tmp_files/2301.13323v1.pdf.txt b/V9FQT4oBgHgl3EQfbTbA/content/tmp_files/2301.13323v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7c33427f96d3d8978bd22d5c75dbd2107db0fa98 --- /dev/null +++ b/V9FQT4oBgHgl3EQfbTbA/content/tmp_files/2301.13323v1.pdf.txt @@ -0,0 +1,6991 @@ +Published as a conference paper at ICLR 2023 +FAIRNESS AND ACCURACY UNDER DOMAIN GENER- +ALIZATION +Thai-Hoang Pham, Xueru Zhang, Ping Zhang +The Ohio State University, Columbus, OH 43210, USA +{pham.375,zhang.12807,zhang.10631}@osu.edu +ABSTRACT +As machine learning (ML) algorithms are increasingly used in high-stakes applica- +tions, concerns have arisen that they may be biased against certain social groups. +Although many approaches have been proposed to make ML models fair, they +typically rely on the assumption that data distributions in training and deployment +are identical. Unfortunately, this is commonly violated in practice and a model that +is fair during training may lead to an unexpected outcome during its deployment. +Although the problem of designing robust ML models under dataset shifts has +been widely studied, most existing works focus only on the transfer of accuracy. +In this paper, we study the transfer of both fairness and accuracy under domain +generalization where the data at test time may be sampled from never-before-seen +domains. We first develop theoretical bounds on the unfairness and expected loss +at deployment, and then derive sufficient conditions under which fairness and +accuracy can be perfectly transferred via invariant representation learning. Guided +by this, we design a learning algorithm such that fair ML models learned with +training data still have high fairness and accuracy when deployment environments +change. Experiments on real-world data validate the proposed algorithm. Model +implementation is available at https://github.com/pth1993/FATDM. +1 +INTRODUCTION +Machine learning (ML) algorithms trained with real-world data may have inherent bias and exhibit +discrimination against certain social groups. To address the unfairness in ML, existing studies have +proposed many fairness notions and developed approaches to learning models that satisfy these +fairness notions. However, these works are based on an implicit assumption that the data distributions +in training and deployment are the same, so that the fair models learned from training data can +be deployed to make fair decisions on testing data. Unfortunately, this assumption is commonly +violated in real-world applications such as healthcare e.g., it was shown that most US patient data +for training ML models are from CA, MA, and NY, with almost no representation from the other 47 +states (Kaushal et al., 2020). Because of the distribution shifts between training and deployment, a +model that is accurate and fair during training may behave in an unexpected way and induce poor +performance during deployment. Therefore, it is critical to account for distribution shifts and learn +fair models that are robust to potential changes in deployment environments. +The problem of learning models under distribution shifts has been extensively studied in the literature +and is typically referred to as domain adaptation/generalization, where the goal is to learn models +on source domain(s) that can be generalized to a different (but related) target domain. Specifically, +domain adaptation requires access to (unlabeled) data from the target domain at training time, and +the learned model can only be used at a specific target domain. In contrast, domain generalization +considers a more general setting when the target domain data are inaccessible during training; instead +it assumes there exists a set of source domains based on which the learned model can be generalized +to an unseen, novel target domain. For both problems, most studies focus only on the generalization +of accuracy across domains without considering fairness, e.g., by theoretically examining the relations +between accuracy at target and source domains (Mansour et al., 2008; 2009; Hoffman et al., 2018; +Zhao et al., 2018; Phung et al., 2021; Deshmukh et al., 2019; Muandet et al., 2013; Blanchard +et al., 2021; Albuquerque et al., 2019; Ye et al., 2021; Sicilia et al., 2021; Shui et al., 2022) or/and +developing practical methods (Albuquerque et al., 2019; Zhao et al., 2020; Li et al., 2018a; Sun & +1 +arXiv:2301.13323v1 [cs.LG] 30 Jan 2023 + +Published as a conference paper at ICLR 2023 +Saenko, 2016; Ganin et al., 2016; Ilse et al., 2020; Nguyen et al., 2021). To the best of our knowledge, +only Chen et al. (2022); Singh et al. (2021); Coston et al. (2019); Rezaei et al. (2021); Oneto et al. +(2019); Madras et al. (2018); Schumann et al. (2019); Yoon et al. (2020) considered the transfer +of fairness across domains. However, all of them focused on domain adaptation, and many also +imposed rather strong assumptions on distributional shifts (e.g., covariate shifts (Singh et al., 2021; +Coston et al., 2019; Rezaei et al., 2021), demographic shift (Giguere et al., 2022), prior probability +shift (Biswas & Mukherjee, 2021)) that may be violated in practice. Among them, most focused +on empirically examining how fairness properties are affected under distributional shifts, whereas +theoretical understandings are less studied (Schumann et al., 2019; Yoon et al., 2020). Details and +more related works are in Appendix A. +White +Asian +Black +Others +Hispanic or Latino +... +Training Data +White +Asian +Black +Others +Hispanic or Latino +(Fair) ML +CA +NY +MT +Source Domain 1 +Source Domain N +OH +FL +TX +... +Model +Unseen +Target Domain +Figure 1: An example of domain gener- +alization in healthcare: (fair) ML model +trained with patient data in CA, NY, etc., +can be deployed in other states by main- +taining high accuracy/fairness. +In this paper, we study the transfer of both fairness and +accuracy in domain generalization via invariant represen- +tation learning, where the data in target domain is unknown +and inaccessible during training. A motivating example +is shown in Figure 1. Specifically, we first establish a new +theoretical framework that develops interpretable bounds +on accuracy/fairness at a target domain under domain gen- +eralization, and then identify sufficient conditions under +which fairness/accuracy can be perfectly transferred to an +unseen target domain. Importantly, our theoretical bounds +are fundamentally different from the existing bounds, com- +pared to which ours are better connected with practical +algorithmic design, i.e., our bounds are aligned with the ob- +jective of adversarial learning-based algorithms, a method +that is widely used in domain generalization. +Inspired by the theoretical findings, we propose Fairness +and Accuracy Transfer by Density Matching (FATDM), a +two-stage learning framework such that the representations and fair model learned with source +domain data can be well-generalized to an unseen target domain. Last, we conduct the experiments +on real-world data; the empirical results show that fair ML models trained with our method still +attain a high accuracy and fairness when deployment environments differ from the training. Our main +contributions and findings are summarized as follows: +• We consider the transfer of both accuracy and fairness in domain generalization. To the best of our +knowledge, this is the first work studying domain generalization with fairness consideration. +• We develop upper bounds for expected loss (Thm. 1) and unfairness (Thm. 3) in target domains. +Notably, our bounds are significantly different from the existing bounds as discussed in Appendix +A. We also develop a lower bound for expected loss (Thm. 2); it indicates an inherent tradeoff of +the existing methods which learn marginal invariant representations for domain generalization. +• We identify sufficient conditions under which fairness and accuracy can be perfectly transferred +from source domains to target domains using invariant representation learning (Thm. 4). +• We propose a two-stage training framework (i.e., based on Thm. 5) that learns models in source +domains (Sec. 4), which can generalize both accuracy and fairness to target domain. +• We conduct experiments on real-world data to validate the effectiveness of the proposed method. +2 +PROBLEM FORMULATION +Notations. Let X, A, and Y denote the space of features, sensitive attribute (distinguishing different +groups, e.g., race/gender), and label, respectively. Let Z be the representation space induced from +X by representation mapping g : X → Z. We use X, A, Y , Z to denote random variables that +take values in X, A, Y, Z and x, a, y, z the realizations. A domain D is specified by distribution +PD : X × A × Y → [0, 1] and labeling function fD : X → Y∆, where ∆ is a probability simplex +over Y. Similarly, let hD : Z → Y∆ be a labeling function from representation space for domain D. +Note that fD, hD, g are stochastic functions and fD = hD ◦ g.1 For simplicity, we use P V +D (or P V |U +D +) +to denote the induced marginal (or conditional) distributions of variable V (given U) in domain D. +1The deterministic labeling function is a special case when it follows Dirac delta distribution in ∆. +2 + +shutterstock.com · 554075098O +00 +O +O +Oshutterstock.com · 1380896705画 +Personal health +recordsPublished as a conference paper at ICLR 2023 +Error metric. Consider hypothesis �f = �h ◦ g : X → Y∆, where �h : Z → Y∆ is the hypothesis +directly used in representation space. Denote �f(x)y as the element on y-th dimension which predicts +the probability that label Y = y given X = x. Then the expected error of �f in domain D is defined as +ϵAcc +D ( �f) = ED[L( �f(X), Y )] for some loss function L : Y∆ × Y → R+ (e.g., 0-1 loss, cross-entropy +loss). Similarly, define the expected error of �h in representation space as ϵAcc +D (�h) = ED[L(�h(Z), Y )]. +Note that most existing works (Albuquerque et al., 2019; Zhao et al., 2018) focus on optimizing +ϵAcc +D (�h), while our goal is to attain high accuracy in input space, i.e., low ϵAcc +D ( �f). +Unfairness metric. We focus on group fairness notions (Makhlouf et al., 2021) that require certain +statistical measures to be equalized across different groups; many of them can be formulated as +(conditional) independence statements between random variables �f(X), A, Y , e.g., demographic +parity ( �f(X) ⊥ A: the likelihood of a positive outcome is the same across different groups) (Dwork +et al., 2012) , equalized odds ( �f(X) ⊥ A|Y : true positive rate (TPR) and false positive rate (FPR) +are the same across different groups), equal opportunity ( �f(X) ⊥ A|Y = 1 when Y = {0, 1}: TPR +is the same across different groups) (Hardt et al., 2016). In the paper, we will present the results +under equalized odds (EO) fairness with binary Y = {0, 1} and A = {0, 1}, while all the results (e.g., +methods, analysis) can be generalized to multi-class, multi-protected attributes, and other fairness +notions. Given hypothesis �f = �h ◦ g : X → Y∆, the violation of EO in domain D can be measured +as ϵEO +D ( �f) = � +y∈Y D +� +P +� +f(X)1|Y =y,A=0 +D +||P +� +f(X)1|Y =y,A=1 +D +� +for some distance metric D(·||·). +Problem setup. Consider a problem of domain generalization where a learning algorithm has access +to data {(xk, ak, yk, dk)}m +k=1 sampled from a set of N source domains {DS +i }i∈[N], where dk is the +domain label and [N] = {1, · · · , N}. Our goal is to learn a representation mapping g : X → Z and a +fair model �h : Z → Y∆ trained on source domains such that the model �f = �h ◦ g can be generalized +to an unseen target domain DT in terms of both accuracy and fairness. Specifically, we investigate +under what conditions and by what algorithms we can guarantee that attaining high accuracy and +fairness at source domains {DS +i }N +i=1 implies small ϵAcc +DT ( �f) and ϵEO +DT ( �f) at unknown target domain. +3 +THEORETICAL RESULTS +In this section, we present the results on the transfer of accuracy/fairness under domain generalization +via domain-invariant learning (proofs are shown in Appendix E). We first examine that for any model +�h : Z → Y∆ and any representation mapping g : X → Z, how the accuracy/fairness attained at +source domains {DS +i }N +i=1 can be affected when �f = �h ◦ g is deployed at any target domain DT . +Specifically, we can bound the error and unfairness at any target domain based on source domains. +Before presenting the results, we first introduce the discrepancy measure used for measuring the +dissimilarity between domains. +Discrepancy measure. We adopt Jensen-Shannon (JS) distance (Endres & Schindelin, 2003) to +measure the dissimilarity between two distributions. Formally, JS distance between distributions P +and P ′ is defined as +dJS(P, P ′) := +� +DJS(P||P ′), +where DJS(P||P ′) := 1 +2DKL(P|| P +P ′ +2 +) + 1 +2DKL(P ′|| P +P ′ +2 +) is JS divergence defined based on +Kullback–Leibler (KL) divergence DKL(·||·). Note that unlike KL divergence, JS divergence is +symmetric and bounded: 0 ≤ DJS(P||P ′) ≤ 1. +While different discrepancy measures such as H and H∆H divergences (Ben-David et al., 2010) (i.e., +definitions are given in Appendix A) were used in prior works, we particularly consider JS distance +because (1) it is aligned with training objective for discriminator in generative adversarial networks +(GAN) (Goodfellow et al., 2014), and many existing methods (Ganin et al., 2016; Albuquerque et al., +2019) for invariant representation learning are built based on GAN framework; (2) H and H∆H +divergences are limited to settings where the labeling functions fD are deterministic (Ben-David +et al., 2010; Albuquerque et al., 2019; Zhao et al., 2018). In contrast, our bounds admit the stochastic +labeling functions. The limitations of other discrepancy measures and existing bounds are discussed +in detail in Appendix A. +3 + +Published as a conference paper at ICLR 2023 +Theorem 1 (Upper bound: accuracy) For any hypothesis �h : Z → Y∆, any representation map- +ping g : X → Z, and any loss function L : Y∆ × Y → R+ that is upper bounded by C, the expected +error of �f = �h ◦ g : X → Y∆ at any unseen target domain DT is upper bounded:2 +ϵAcc +DT +� +�f +� +≤ 1 +N +N +� +i=1 +ϵAcc +DS +i +� +�f +� +� +�� +� +term (i) ++ +√ +2C min +i∈[N]dJS +� +P X,Y +DT , P X,Y +DS +i +� +� +�� +� +term (ii) ++ +√ +2C max +i,j∈[N]dJS +� +P Z,Y +DS +i , P Z,Y +DS +j +� +� +�� +� +term (iii) +(1) +The upper bound in Eq. (1) are interpretable and have three terms: term (i) is the averaged error of +source domains in input space; term (ii) is the discrepancy between the target domain and the source +domains in input space; term (iii) is the discrepancy between the source domains in representation +space.3 It provides guidance on learning the proper representation mapping g : X → Z: to ensure +small error at target domain ϵAcc +DT ( �f), we shall learn representations such that the upper bound of +ϵAcc +DT ( �f) is minimized. Because term (ii) depends on the unknown target domain DT and it’s evaluated +in input space X × Y, it is fixed and is out of control during training, we can only focus on term +(i) and term (iii), i.e., learn representations Z such that errors at source domains ϵAcc +DS +i ( �f) and the +discrepancy between source domains in the representation space dJS(P Z,Y +DS +i , P Z,Y +DS +j ) are minimized. +Corollary 1.1 ∀i, j, JS distance between P Z,Y +DS +i +and P Z,Y +DS +j +in Eq. (1) can be decomposed: +dJS +� +P Z,Y +DS +i , P Z,Y +DS +j +� += dJS +� +P Y +DS +i , P Y +DS +j +� ++ +� +2Ey∼P Y +DS +i,j +� +dJS +� +P Z|Y +DS +i , P Z|Y +DS +j +�2� +where P Y +DS +i,j = 1 +2 +� +P Y +DS +i + P Y +DS +j +� +. +Our algorithm in Sec. 4 is designed based on above decomposition: because P Y +DS +i solely depends +on source domain DS +i , we learn representations by minimizing dJS(P Z|Y +DS +i , P Z|Y +DS +j ), ∀i, j. Combining +Thm. 1 and Corollary 1.1, to ensure high accuracy at unseen target domain DT , we learn the +representation mapping g and model �h such that P Z|Y +DS +i +is invariant across source domains, and +meanwhile �f = �h ◦ g attains high accuracy at source domains. +Note that unlike our method, many existing works (Phung et al., 2021; Albuquerque et al., 2019; +Ganin et al., 2016) suggest that to ensure high accuracy in domain generalization, representation +mapping g should be learned such that P Z +DS +i is same across domains, i.e., small dJS(P Z +DS +i , P Z +DT ). +However, we show that the domain-invariant P Z +DS +i may adversely increase the error at target domain, +as indicated in the Thm. 2 below. +Theorem 2 (Lower bound: accuracy) Suppose L( �f(x), y) = � +ˆy∈Y �f(x)ˆyL(ˆy, y) where function +L : Y × Y → R+ is lower bounded by c when ˆy ̸= y, and is 0 when ˆy = y. If dJS(P Y +DS +i , P Y +DT ) ≥ +dJS(P Z +DS +i , P Z +DT ), the expected error of �f at source and target domains is lower bounded: +1 +N +N +� +i=1 +ϵAcc +DS +i ( �f) + ϵAcc +DT ( �f) ≥ +c +4|Y|N +N +� +i=1 +� +dJS(P Y +DS +i , P Y +DT ) − dJS(P Z +DS +i , P Z +DT ) +�4 +. +(2) +The above lower bound shows an inherent trade-off of approaches that minimize dJS(P Z +DS +i , P Z +DT ) +when learning the representations. Specifically, with the domain-invariant P Z +DS +i , the right hand side +of Eq. (2) may increase, resulting in an increased error at target domain ϵAcc +DT ( �f). +2The condition on the bounded loss is mild and can be satisfied by many loss functions. For example, cross- +entropy loss can be bounded by modifying the softmax output from +� +p1, p2, · · · , p|Y| +� to +� +ˆp1, ˆp2, · · · , ˆp|Y| +�, +where ˆpi = pi(1 − exp(−C)|Y|) + exp(−C), ∀i ∈ |Y|. +3In fact, a tighter upper bound for the loss at target domain can be established using strong data processing +inequality (Polyanskiy & Wu, 2017), as detailed in Appendix D +4 + +Published as a conference paper at ICLR 2023 +Similar to the loss, the unfairness at target domain can also be upper bounded, as presented in Thm. 3. +Theorem 3 (Upper bound: fairness) Consider a special case where the unfairness measure is +defined as the distance between means of two distributions: +ϵEO +D ( �f) = � +y∈{0,1} +���ED +� +�f(X)1|Y = y, A = 0 +� +− ED +� +�f(X)1|Y = y, A = 1 +���� , +then the unfairness at any unseen target domain DT is upper bounded: +ϵEO +DT +� +�f +� +≤ +1 +N +N +� +i=1 +ϵEO +DS +i +� +�f +� ++ +√ +2 min +i∈[N] +� +y∈{0,1} +� +a∈{0,1} +dJS +� +P X|Y =y,A=a +DT +, P X|Y =y,A=a +DS +i +� ++ +√ +2 max +i,j∈[N] +� +y∈{0,1} +� +a∈{0,1} +dJS +� +P Z|Y =y,A=a +DS +i +, P Z|Y =y,A=a +DS +j +� +Similar to Thm. 1, the upper bound in Thm. 3 also has three terms and the second term is out of +control during training because it depends on the unseen target domain and is defined in input space. +Therefore, to maintain fairness at target domain DT , we learn the representation mapping g and +model �h such that P Z|Y,A +DS +i +is invariant across source domains, and meanwhile �f = �h ◦ g attains high +fairness at source domains. +The results above characterize the relations between accuracy/fairness at any target and source +domains under any representation mapping g and model �h. Next, we identify conditions under which +the accuracy/fairness attained at sources can be perfectly transferred to a target domain. +Theorem 4 (Sufficient condition for perfect transfer) Consider N source domains {DS +i }N +i=1 and +an unseen target domain DT . Define set Λ = {Dt : Dt = �N +i=1 πiDS +i , {πi} ∈ ∆N−1}. +1. (Transfer of fairness) ∀DT ∈ Λ, if g is the mapping under which P Z|Y,A +DS +i +is the same across all +source domains, then ϵEO +DS +i (�h) = ϵEO +DT (�h) = ϵEO +DS +i ( �f) = ϵEO +DT ( �f), ∀i. +2. (Transfer of accuracy) ∀DT ∈ Λ, if P Y +DS +i is the same and if g is the mapping under which P Z|Y +DS +i +is the same across all source domains, then ϵAcc +DS +i (�h) = ϵAcc +DT (�h) = ϵAcc +DS +i ( �f) = ϵAcc +DT ( �f), ∀i. +Thm. 4 indicates the possibility of attaining the perfect transfer of accuracy/fairness and examples of +such representation mappings are provided. Note that these results are consistent with Thm. 1 and +Thm. 3, which also suggest learning domain-invariant representations P Z|Y +DS +i +and P Z|Y,A +DS +i +. +4 +PROPOSED ALGORITHM +Table 1: Usages of terms in Eq. (3) to guarantee +the fairness and accuracy in target domain. +Loss terms +Usages +Lcls +Mimimize ϵAcc +DS +i +Lfair +Mimimize ϵEO +DS +i +Linv +Minimize dJS +� +P Z|Y +DS +i , P Z|Y +DS +j +� +and dJS +� +P Z|Y,A +DS +i +, P Z|Y,A +DS +j +� +Lcls + Lfair + Linv +Mimimize ϵAcc +DT and ϵEO +DT +The accuracy and fairness upper bounds in Sec. 3 +shed light on designing robust ML model that can +preserve high accuracy and fairness on unseen tar- +get domains. Specifically, the model consists of +representation mapping g : X → Z and classi- +fier �h : Z → Y such that (1) the prediction errors +and unfairness of �f = �h ◦ g on source domains +are minimized; and (2) the discrepancy of learned +conditional representations (i.e, P Z|Y +DS +i +and P Z|Y,A +DS +i +) +among source domains is minimized. That is, +min +g,�h +Lcls(g,�h) + ωLfair(g,�h) + γLinv(g) +(3) +where Lcls, Lfair, and Linv are expected losses that penalize incorrect classification, unfairness, +and discrepancy among source domains. Hyper-parameters ω > 0 and γ > 0 control the accuracy- +fairness trade-off and accuracy-invariant representation trade-off, respectively. The usages of these +three losses are summarized in Table 1. +5 + +Published as a conference paper at ICLR 2023 +Adversarial learning framework (Goodfellow et al., 2014). Linv in Eq. (3) can be optimized +directly with adversarial learning. This is because the training objective of the discriminator in GAN +is aligned with our goal of minimizing JS distance between P Z|Y +DS +i +(or P Z|Y,A +DS +i +) among source domains, +as mentioned in Sec. 3. Specifically, define a set of discriminators K = {ky : y ∈ Y} ∪ {ky,a : y ∈ +Y, a ∈ A}; each discriminator ky (resp. ky,a) aims to distinguish whether a sample with label y +(resp. label y and sensitive attribute a) comes from a particular domain (i.e., maximize Linv). The +representation mapping g should be learned to increase the error of discriminators (i.e., minimize +Linv). Therefore, the model and discriminators can be trained simultaneously by playing a two-player +minimax game (i.e., ming maxK Linv(g)). Combine with the objective of minimizing prediction +error and unfairness (i.e., ming,�h Lcls(g,�h) + ωLfair(g,�h)), the overall learning objective is: +min +g,�h +max +K +Lcls(g,�h) + ωLfair(g,�h) + γLinv(g) +(4) +However, the above adversarial learning framework for learning domain-invariant representation may +not work well when |Y × A| is large: as the label space and sensitive attribute space get larger, the +number of discriminators to be learned increases and the training can be highly unstable. A naive +solution to tackling this issue is to use one discriminator ∀y ∈ Y, a ∈ A. However, this would result +in the reduced mutual information between representations and label/sensitive attribute, which may +hurt the accuracy. We thus propose another approach to learn the domain-invariant representations. + +source domain +source domain +Figure 2: 1D illustration of domain-invariant rep- +resentation. To transfer accuracy and fairness to +target domains, we need to find representation z +such that P z|y +DS +i and P z|y,a +DS +i +are domain-invariant. +Proposed solution to learning invariant rep- +resentations. For any domain D, we have: +P Z|y +D += +� +X +P Z,x|y +D +dx = +� +X +P Z|xP x|y +D dx +P Z|y,a +D += +� +X +P Z,x|y,a +D +dx = +� +X +P Z|xP x|y,a +D +dx +where P Z|x is domain-independent so we drop +D in subscript. +Given any two source do- +mains DS +i and DS +j , in general P X|y +DS +i +̸= P X|y +DS +j +and P X|y,a +DS +i +̸= +P X|y,a +DS +j +so that it is non- +trivial to achieve domain-invariant representa- +tions P Z|y +DS +i += P Z|y +DS +j and P Z|y,a +DS +i += P Z|y,a +DS +j +. How- +ever, if there exist invertible functions my +i,j : X → X and my,a +i,j : X → X that can match the density +functions of X from DS +i to DS +j such that P X|y +DS +i += P +my +i,j(X)|y +DS +j +and P X|y,a +DS +i += P +my,a +i,j (X)|y,a +DS +j +, and +if we can find the representation Z such that P Z|x +DS +i += P +Z|my +i,j(x) +DS +i +and P Z|x +DS +i += P +Z|my,a +i,j (x) +DS +i +, then +∀y ∈ Y, a ∈ A, we have: +P Z|y +DS +i += +� +X +P Z|xP x|y +DS +i dx = +� +X +P Z|x′P x′|y +DS +j dx′ = P Z|y +DS +j +P Z|y,a +DS +i += +� +X +P Z|xP X|y,a +DS +i +dx = +� +X +P Z|x′′P x′′|y,a +DS +j +dx′′ = P Z|y,a +DS +j +where x′ = my +i,j(x) and x′′ = my,a +i,j (x). This observation suggests that to minimize the discrepancy +of representation distributions among source domains, we can first find the density mapping functions +my +i,j and my,a +i,j , ∀y, a, i, j, and then minimize the discrepancies between P Z|x, P Z|x′, and P Z|x′′, +∀x. This is formally shown in Thm. 5 below. +Theorem 5 If there exist invertible mappings my +i,j and my,a +i,j such that P X|y +DS +i += P +my +i,j(X)|y +DS +j +and +P X|y,a +DS +i += P +my,a +i,j (X)|y,a +DS +j +, ∀y, a, i, j, and if the representation mapping are in the form of g := P Z|x = +N(µ(x), σ2Id), where µ(x) is the function of x and d is the dimension of the representation space +Z, then minimizing dJS +� +P Z|y +DS +i , P Z|y +DS +j +� +and dJS +� +P Z|y,a +DS +i +, P Z|y,a +DS +j +� +can be reduced to minimizing +��µ(x) − µ +� +my +i,j(x) +��� +2 and +��µ(x) − µ +� +my,a +i,j (x) +��� +2, respectively. +6 + +sourcedomain2 +source domainj +Density +-5 +U +10 +15source domain +source domainj +Density +5 +U +10 +15source domain +source domainj +Density +-2 +0 +2 +8 +ry.asource domain +source domainj +Density +0Published as a conference paper at ICLR 2023 +Based on Thm. 5, we propose a two-stage learning approach FATDM, as stated below. +Remark 1 (Fairness and Accuracy Transfer by Density Matching (FATDM)) Given +the +exis- +tence of density matching functions my +i,j and my,a +i,j , and representation mapping g := N(µ(x), σ2Id), +domain-invariant representations can be learned via a two-stage process: (i) finding these mapping +functions my +i,j and my,a +i,j ; (ii) minimizing the mean squared errors between µ(x) and µ +� +my +i,j(x) +� +, +and µ(x) and µ +� +my,a +i,j (x) +� +, ∀i, j ∈ [N], x ∈ X, y ∈ Y, a ∈ A. +Stage 1: learning mapping functions my +i,j and my,a +i,j across source domains. Many approaches can +be leveraged to estimate my +i,j and my,a +i,j from data. In our study, we adopt StarGAN (Choi et al., 2018) +and CycleGAN (Zhu et al., 2017) as examples; both frameworks are widely used in multi-domain +image-to-image translation and can be leveraged. In our algorithm, we independently train two +translation models DensityMatchY and DensityMatchY,A using StarGAN or CycleGAN, with each +used for learning {my +i,j}y∈Y,i,j∈[N] and {my,a +i,j }y∈Y,a∈A,i,j∈[N], respectively. +Stage 1: Finding +Stage 2: Enforcing +source domain +source domain +Figure 3: FATDM: two-stage training +Specifically, DensityMatchY (or DensityMatchY,A) con- +sists of a generator G : X × [N] × [N] → X and a discrim- +inator D : X → [N] × {0, 1}. The generator takes in real +image x and a pair of domain labels i, j as input and generates +a fake image; the discriminator aims to predict the domain +label of the image generated by the generator and distinguish +whether it is fake or real. G and D are learned simultaneously +by solving the minimax game, and their loss functions are +specified in Appendix B. When the training is completed, +we obtain two optimal generators from DensityMatchY and +DensityMatchY,A, denoted as GY and GY,A. We shall use +GY (·, i, j) (resp. GY,A(·, i, j)) directly as the density map- +ping function {my +i,j(·)}y∈Y (resp. {my,a +i,j (·)}y∈Y,a∈A). +Stage 2: learning domain-invariant representation. Given +GY and GY,A learned in stage 1, we are ready to learn the invariant representation Z by finding +g : X → Z such that g := N(µ(x), σ2Id) and minimizes the following: +Linv = Ed,d′,d′′∼{DS +i }i∈[N] [Lmse(µ(X), µ(X′)) + Lmse(µ(X), µ(X′′))] +(5) +where d, d′, d′′ are domain labels sampled from source domains, X is features sampled from domain +d, X′ = GY (X, d, d′), X′′ = GY,A(X, d, d′′), Lmse is mean squared error. The pseudo-code of our +proposed model (FATDM) is in Algorithm 1. The detailed architecture of FATDM is in Appendix B. +Algorithm 1: Fairness and Accuracy Transfer by Density Matching (FATDM) +Input: Training dataset Dtrain from N source domains {DS +i }N +i=1 +Output: representation mapping g, classifier �h, density matching functions GY , GY,A +1 Procedure Density_Matching(Dtrain) +/* Procedure for training GY +is similar but not presented +*/ +2 +while training DensityMatchY,A is not end do +3 +Sample y ∼ Y, a ∼ A and data batch B = {xk, dk|ak = a, yk = y}|B| +k=1 from Dtrain ; +4 +Update GY,A based on the objectives of the minimax game (Appendix B). +5 Procedure Invariant_Representation_Learning(Dtrain, GY , GY,A) +6 +while training FATDM is not end do +7 +Sample data batch B = {xk, ak, yk, dk}|B| +k=1 from Dtrain ; +8 +Sample lists of domain labels {d′ +i}|B| +k=1 and {d′′ +i }|B| +k=1; +9 +Generate sets of artificial images {x′ +k}|B| +k=1 and {x′′ +k}|B| +k=1 by GY and GY,A ; +10 +Update g, ˆh by optimizing Eq. (3) with Linv defined in Eq. (5). +Remark 2 (Summary of theoretical results and proposed algorithm) Thm. 1 and Thm. 3 suggest +a way to ensure high accuracy and fairness in target domain: by minimizing the source error ϵAcc +Ds +i (i.e., +Lcls in Eq. (3)), the source unfairness ϵEO +Ds +i (i.e.,Lfair in Eq. (3)), and the discrepancies between source +domains dJS +� +P Z|Y =y +DS +i +, P Z|Y =y +DS +j +� +and dJS +� +P Z|Y =y,A=a +DS +i +, P Z|Y =y,A=a +DS +j +� +(i.e., Linv in Eq. (3)). The +7 + +Density +-2 +U +2 +4sourcedomain2 +source domainj +Density +-5 +U +10 +15Published as a conference paper at ICLR 2023 +common way to optimize Eq. (3) using adversarial learning (Eq. (4)) is not stable when |Y × A| is +large. Thm. 5 states that instead of using adversarial learning, Eq. (3) can be optimized via 2-stage +learning: (i) find mappings my +i,j and my,a +i,j ( Density_Matching in Alg. 1) and (ii) minimize +Eq. (3) with Linv defined in Eq. (5) ( Invariant_Representation_Learning in Alg. 1). +5 +EXPERIMENTS +We conduct experiments on MIMIC-CXR database (Johnson et al., 2019), which includes 377,110 +chest X-ray images associated with 227,827 imaging studies about 14 diseases performed at the Beth +Israel Deaconess Medical Center. Importantly, these images are linked with MIMIC-IV database +(Johnson et al., 2021) which includes patients’ information such as age, and race; these can serve as +sensitive attributes for measuring the unfairness. Based on MIMIC-CXR and MIMIC-IV data, we +construct two datasets on two diseases: +• Cardiomegaly disease: we first extract all images related to Cardiomegaly disease, and the corre- +sponding labels (i.e., positive/negative) and sensitive attributes (i.e., male/female); then we partition +the data into four domain-specific datasets based on age (i.e., [18, 40), [40, 60), [60, 80), [80, 100)). +We consider age as domain label because it captures the real scenario that there are distribution +shifts across patients with different ages. +• Edema disease: we extract all images related to Edema disease, and corresponding labels (i.e., pos- +itive/negative) and sensitive attributes (i.e., age with ranges [18, 40), [40, 60), [60, 80), [80, 100)). +Unlike Cardiomegaly data, we construct the dataset for each domain by first sampling images from +Edema data followed by θ degree counter-clockwise rotation, where θ ∈ {0◦, 15◦, 30◦, 45◦, 60◦}. +We consider rotation degree as domain label to model the scenario where there is rotational +misalignment among images collected from different devices. +Next, we focus on Cardiomegaly disease and the results for Edema disease are shown in Appendix C. +Baselines. We compare our method (i.e., FATDM-StarGAN and FATDM-CycleGAN) with exist- +ing methods for domain generalization, including empirical risk minimization, domain invariant +representation learning, and distributionally robust optimization, as detailed below. +• Empirical risk minimization (ERM): The baseline that considers all source domains as one domain. +• Domain invariant representation learning: Method that aims to achieve the invariant across source +domains. We experiment with G2DM (Albuquerque et al., 2019), DANN (Ganin et al., 2016), CDANN +(Li et al., 2018c), CORAL (Sun & Saenko, 2016), IRM (Arjovsky et al., 2019). These models focus +on accuracy transfer by enforcing the invariance of distributions P Z +DS +i or P Z|Y +DS +i . +• Distributionally robust optimization: Method that learns a model at worst-case distribution to hope +it can generalize well on test data. We experiment with GroupDRO (Sagawa et al., 2019) that +minimizes the worst-case training loss over a set of pre-defined groups through regularization. +• ATDM: A variant of FATDM-StarGAN that solely focuses on accuracy transfer. That is, we only +enforce the invariance of P Z|Y +DS +i +during learning which is similar to Nguyen et al. (2021). +The implementations of these models except G2DM are adapted from DomainBed framework (Gulra- +jani & Lopez-Paz, 2020). For G2DM, we use the author-provided implementation. For all models, we +use ResNet18 (He et al., 2016) as the backbone module of representation mapping g : X → Z; and +fairness constraint Lfair is enforced as a regularization term added to the original objective functions. +Experiment setup. We follow leave-one-out domain setting in which 3 domains are used for training +and the remaining domain serves as the unseen target domain and is used for evaluation. Several +metrics are considered to measure the unfairness and error of each model in target domain, including: +• Error: cross-entropy loss (CE), misclassification rate (MR), AUROC := 1−AUROC, AUPR := +1−AUPR, F1 := 1−F1, where AUROC, AUPR, F1 are area under receiver operating characteristic +curve, area under precision-recall curve, F1 score, respectively. +• Unfairness: we consider both equalized odds and equal opportunity fairness notion, and adopt +mean distance (MD) and earth mover’s distance (EMD) as distance metric D(·||·). +Fairness and accuracy on target domains. We first compare our method with baselines in terms of +the optimal trade-off (Pareto frontier) between accuracy and fairness on target domains under different +metric pairs. Figure 4 shows the error-unfairness curves (as ω varies from 0 (no fairness constraint) to +10 (strong fairness constraint)), with AUROC and MR as error metric, and equalized odds (measured +under distance metrics MD and EMD) as fairness notion; the results for other error metrics are similar +8 + +Published as a conference paper at ICLR 2023 +Figure 4: Fairness-accuracy trade-off (Pareto frontier) of FATDM-StarGAN, FATDM-CycleGAN, +and baseline methods: error-unfairness curves are constructed by varying ω ∈ [0, 10] and the values +of error and unfairness are normalized to [0, 1]. Lower-left points indicate the model has a better +fairness-accuracy trade-off (Pareto optimality). +Figure 5: Prediction performances (AUROC, AUPR, Accuracy, F1) of FATDM-StarGAN on Car- +diomegaly disease data when varying hyper-parameter γ at different levels of fairnsess constraint +ω. +and shown in Appendix C. Our observations are as follows: (1) As expected, there is a trade-off +between fairness and accuracy: for all methods, increasing ω improves fairness but reduces accuracy. +(2) Among all methods, the Pareto frontiers of FATDM-StarGAN and FATDM-CycleGAN are the +bottom leftmost, implying that our method attains a better fairness-accuracy trade-off than baselines. +(3) Although fairness constraint is imposed during training for all methods, the fairness attained at +source domains cannot be well-generalized to the target domain under other methods. These results +validate our theorems and show that enforcing the domain-invariant P Z|Y +DS +i +and P Z|Y,A +DS +i +when learning +representations ensures the transfer of both accuracy and fairness. It is worth-noting that under this +dataset, the domain-invariant P Z|Y +DS +i +(accuracy transfer) does not imply the domain-invariant P Z|Y,A +DS +i +(fairness transfer). This is because domain DS +i (i.e., age) is correlated with label Y (i.e., has a disease) +and sensitive attribute A (i.e., gender), making the distribution P Y,A +DS +i +different across domains. +Impact of density mapping model. To investigate whether the performance gain of our method +is due to the use of any specific density mapping model, we adopt StarGAN and CycleGAN +architectures to learn density mapping functions in our method and compare their performances. +Figure 4 shows that FATDM-StarGAN and CycleGAN achieve similar fairness-accuracy trade-off +at the target domains and both of them outperform the baselines. This result shows that our method is +not limited to any specific density mapping model and is broadly applicable to other architectures. +Impact of invariant representation constraints. We also examine the impact of Linv on the perfor- +mance of FATDM-StarGAN at target domains, where we vary the hyper-parameter γ ∈ [0, 5e2] at +different levels of fairness (i.e., fix ω = 1, 5, 10) and examine how the prediction performances (i.e., +AUROC, AUPR, accuracy and F1) could change. Figure 5 shows that enforcing domain-invariant +constraint Linv helps transfer the performance from source to target domain, and γ that attains the +highest accuracy at target domain can be different for different levels of fairness. The results also +indicate the fairness-accuracy trade-off, i.e., for any γ, enforcing stronger fairness constraints (large +ω) could hurt prediction performances. +6 +CONCLUSION +In this paper, we theoretically and empirically demonstrate how to achieve fair and accurate predic- +tions in unknown testing environments. To the best of our knowledge, our work provides the first +theoretical analysis to understand the efficiency of invariant representation learning in transferring +both fairness and accuracy under domain generalization. In particular, we first propose the upper +bounds of prediction error and unfairness in terms of JS-distance, then design the two-stage learning +9 + +--+- ERM +G2DM +DANN +CDANN +CORAL +GroupDRO +IRM +ATDM +FATDM-StarGAN +FATDM-CycleGAN +1.0 +1.0 +0.8 +(MD) +(MD) +0.8 +0.6 +Unfairness ( +Unfairness +Unfairness ( +Unfairness ( + 0. +0.2 +2 +0.0 +0.0 +0.0 + 0'0 +0.0 +0.2 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Error AUROC +Error (MR) +Error AUROC +Error +(MR)0.84 +0.79 +0.915 +0.83- +0.78 +0.82 +0.910 +0.77 +C 0.82 +JRO +R 0.905 +0.76 +~0.80 +0.81 +0.75 +0.895 +0.78 +0.80 +0.74 +0.890- +0.73 +0.79 +0.76 +0.885 +1e1 +1el +1e22e23e24e2 5e2 +0 le-1 le0 lel 1e2 2e2 3e2 4e2 5e2Published as a conference paper at ICLR 2023 +method that minimizes these upper bounds by learning domain-invariant representations. Experiments +on the real-world clinical data demonstrate the effectiveness of our study. +REPRODUCIBILITY STATEMENT +The original chest X-ray images and the corresponding metadata can be downloaded from +PhysioNet (https://physionet.org/content/mimic-cxr-jpg/2.0.0/; +https: +//physionet.org/content/mimiciv/2.0/). Codes for data processing and proposed +algorithms are in supplementary materials. Technical details of the proposed algorithms and experi- +mental settings are in Appendix B. Additional experimental results are in Appendix C. Lemmas used +in proofs of the theorems in the main paper are in Appendix D. Complete proofs of the theorems in +the main paper and the corresponding lemmas are in Appendix E. +ACKNOWLEDGEMENTS +This work was funded in part by the National Science Foundation under award number IIS-2145625, +by the National Institutes of Health under award number UL1TR002733, and by The Ohio State +University President’s Research Excellence Accelerator Grant. +REFERENCES +Alekh Agarwal, Alina Beygelzimer, Miroslav Dudík, John Langford, and Hanna Wallach. +A +reductions approach to fair classification. In International Conference on Machine Learning, pp. +60–69. PMLR, 2018. +Isabela Albuquerque, João Monteiro, Mohammad Darvishi, Tiago H Falk, and Ioannis Mitliagkas. +Generalizing to unseen domains via distribution matching. arXiv preprint arXiv:1911.00804, 2019. +Martin Arjovsky, Léon Bottou, Ishaan Gulrajani, and David Lopez-Paz. Invariant risk minimization. +arXiv preprint arXiv:1907.02893, 2019. +Yahav Bechavod, Christopher Jung, and Steven Z Wu. Metric-free individual fairness in online +learning. Advances in neural information processing systems, 33:11214–11225, 2020. +Shai Ben-David, John Blitzer, Koby Crammer, Alex Kulesza, Fernando Pereira, and Jennifer Wortman +Vaughan. A theory of learning from different domains. Machine learning, 79(1):151–175, 2010. +Asia J Biega, Krishna P Gummadi, and Gerhard Weikum. Equity of attention: Amortizing individual +fairness in rankings. In The 41st international acm sigir conference on research & development in +information retrieval, pp. 405–414, 2018. +Arpita Biswas and Suvam Mukherjee. Ensuring fairness under prior probability shifts. In Proceedings +of the 2021 AAAI/ACM Conference on AI, Ethics, and Society, pp. 414–424, 2021. +Gilles Blanchard, Aniket Anand Deshmukh, Urun Dogan, Gyemin Lee, and Clayton Scott. Domain +generalization by marginal transfer learning. Journal of Machine Learning Research, 22:1–55, +2021. +Fabio M Carlucci, Antonio D’Innocente, Silvia Bucci, Barbara Caputo, and Tatiana Tommasi. +Domain generalization by solving jigsaw puzzles. In Proceedings of the IEEE/CVF Conference on +Computer Vision and Pattern Recognition, pp. 2229–2238, 2019. +Yatong Chen, Reilly Raab, Jialu Wang, and Yang Liu. Fairness transferability subject to bounded +distribution shift. arXiv preprint arXiv:2206.00129, 2022. +Yunjey Choi, Minje Choi, Munyoung Kim, Jung-Woo Ha, Sunghun Kim, and Jaegul Choo. Star- +gan: Unified generative adversarial networks for multi-domain image-to-image translation. In +Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 8789–8797, +2018. +10 + +Published as a conference paper at ICLR 2023 +Amanda Coston, Karthikeyan Natesan Ramamurthy, Dennis Wei, Kush R Varshney, Skyler Speakman, +Zairah Mustahsan, and Supriyo Chakraborty. Fair transfer learning with missing protected attributes. +In Proceedings of the 2019 AAAI/ACM Conference on AI, Ethics, and Society, pp. 91–98, 2019. +Aniket Anand Deshmukh, Yunwen Lei, Srinagesh Sharma, Urun Dogan, James W Cutler, and +Clayton Scott. A generalization error bound for multi-class domain generalization. arXiv preprint +arXiv:1905.10392, 2019. +Luc Devroye, Abbas Mehrabian, and Tommy Reddad. The total variation distance between high- +dimensional gaussians. arXiv preprint arXiv:1810.08693, 2018. +Cynthia Dwork, Moritz Hardt, Toniann Pitassi, Omer Reingold, and Richard Zemel. Fairness through +awareness. In Proceedings of the 3rd innovations in theoretical computer science conference, pp. +214–226, 2012. +D.M. Endres and J.E. Schindelin. A new metric for probability distributions. IEEE Transactions on +Information Theory, 49(7):1858–1860, 2003. doi: 10.1109/TIT.2003.813506. +Yaroslav Ganin and Victor Lempitsky. Unsupervised domain adaptation by backpropagation. In +International conference on machine learning, pp. 1180–1189. PMLR, 2015. +Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François +Laviolette, Mario Marchand, and Victor Lempitsky. Domain-adversarial training of neural networks. +The journal of machine learning research, 17(1):2096–2030, 2016. +Stephen Giguere, Blossom Metevier, Bruno Castro da Silva, Yuriy Brun, Philip S Thomas, and Scott +Niekum. Fairness guarantees under demographic shift. In International Conference on Learning +Representations, 2022. +Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, +Aaron Courville, and Yoshua Bengio. Generative adversarial nets. Advances in neural information +processing systems, 27, 2014. +Ishaan Gulrajani and David Lopez-Paz. In search of lost domain generalization. arXiv preprint +arXiv:2007.01434, 2020. +Swati Gupta and Vijay Kamble. Individual fairness in hindsight. Journal of Machine Learning +Research, 22(144):1–35, 2021. +Moritz Hardt, Eric Price, and Nati Srebro. Equality of opportunity in supervised learning. Advances +in neural information processing systems, 29, 2016. +Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image +recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, +pp. 770–778, 2016. +Judy Hoffman, Mehryar Mohri, and Ningshan Zhang. Algorithms and theory for multiple-source +adaptation. Advances in Neural Information Processing Systems, 31, 2018. +Weihua Hu, Gang Niu, Issei Sato, and Masashi Sugiyama. Does distributionally robust supervised +learning give robust classifiers? In International Conference on Machine Learning, pp. 2029–2037. +PMLR, 2018. +Zeyi Huang, Haohan Wang, Eric P Xing, and Dong Huang. Self-challenging improves cross-domain +generalization. In European Conference on Computer Vision, pp. 124–140. Springer, 2020. +Maximilian Ilse, Jakub M Tomczak, Christos Louizos, and Max Welling. Diva: Domain invariant +variational autoencoders. In Medical Imaging with Deep Learning, pp. 322–348. PMLR, 2020. +Seogkyu Jeon, Kibeom Hong, Pilhyeon Lee, Jewook Lee, and Hyeran Byun. Feature stylization and +domain-aware contrastive learning for domain generalization. In Proceedings of the 29th ACM +International Conference on Multimedia, pp. 22–31, 2021. +Alistair Johnson, Lucas Bulgarelli, Tom Pollard, Steven Horng, Leo Anthony Celi, and Mark Roger. +Mimic-iv. https://doi.org/10.13026/s6n6-xd98, 2021. +11 + +Published as a conference paper at ICLR 2023 +Alistair EW Johnson, Tom J Pollard, Nathaniel R Greenbaum, Matthew P Lungren, Chih-ying Deng, +Yifan Peng, Zhiyong Lu, Roger G Mark, Seth J Berkowitz, and Steven Horng. Mimic-cxr-jpg, a +large publicly available database of labeled chest radiographs. arXiv preprint arXiv:1901.07042, +2019. +Faisal Kamiran and Toon Calders. Data preprocessing techniques for classification without discrimi- +nation. Knowledge and information systems, 33(1):1–33, 2012. +Amit Kaushal, Russ Altman, and Curt Langlotz. Geographic distribution of us cohorts used to train +deep learning algorithms. Jama, 324(12):1212–1213, 2020. +Daehee Kim, Youngjun Yoo, Seunghyun Park, Jinkyu Kim, and Jaekoo Lee. Selfreg: Self-supervised +contrastive regularization for domain generalization. In Proceedings of the IEEE/CVF International +Conference on Computer Vision, pp. 9619–9628, 2021. +Pang Wei Koh, Shiori Sagawa, Henrik Marklund, Sang Michael Xie, Marvin Zhang, Akshay Bal- +subramani, Weihua Hu, Michihiro Yasunaga, Richard Lanas Phillips, Irena Gao, et al. Wilds: A +benchmark of in-the-wild distribution shifts. In International Conference on Machine Learning, +pp. 5637–5664. PMLR, 2021. +David Krueger, Ethan Caballero, Joern-Henrik Jacobsen, Amy Zhang, Jonathan Binas, Dinghuai +Zhang, Remi Le Priol, and Aaron Courville. Out-of-distribution generalization via risk extrapola- +tion (rex). In International Conference on Machine Learning, pp. 5815–5826. PMLR, 2021. +Haoliang Li, Sinno Jialin Pan, Shiqi Wang, and Alex C Kot. Domain generalization with adversarial +feature learning. In Proceedings of the IEEE conference on computer vision and pattern recognition, +pp. 5400–5409, 2018a. +Ya Li, Mingming Gong, Xinmei Tian, Tongliang Liu, and Dacheng Tao. Domain generalization +via conditional invariant representations. In Proceedings of the AAAI conference on artificial +intelligence, volume 32, 2018b. +Ya Li, Xinmei Tian, Mingming Gong, Yajing Liu, Tongliang Liu, Kun Zhang, and Dacheng Tao. +Deep domain generalization via conditional invariant adversarial networks. In Proceedings of the +European Conference on Computer Vision (ECCV), pp. 624–639, 2018c. +Zheren Li, Zhiming Cui, Sheng Wang, Yuji Qi, Xi Ouyang, Qitian Chen, Yuezhi Yang, Zhong Xue, +Dinggang Shen, and Jie-Zhi Cheng. Domain generalization for mammography detection via +multi-style and multi-view contrastive learning. In International Conference on Medical Image +Computing and Computer-Assisted Intervention, pp. 98–108. Springer, 2021. +Evan Z Liu, Behzad Haghgoo, Annie S Chen, Aditi Raghunathan, Pang Wei Koh, Shiori Sagawa, +Percy Liang, and Chelsea Finn. Just train twice: Improving group robustness without training +group information. In International Conference on Machine Learning, pp. 6781–6792. PMLR, +2021. +David Madras, Elliot Creager, Toniann Pitassi, and Richard Zemel. Learning adversarially fair and +transferable representations. In International Conference on Machine Learning, pp. 3384–3393. +PMLR, 2018. +Karima Makhlouf, Sami Zhioua, and Catuscia Palamidessi. Machine learning fairness notions: +Bridging the gap with real-world applications. Information Processing & Management, 58(5): +102642, 2021. +Yishay Mansour, Mehryar Mohri, and Afshin Rostamizadeh. Domain adaptation with multiple +sources. Advances in neural information processing systems, 21, 2008. +Yishay Mansour, Mehryar Mohri, and Afshin Rostamizadeh. Multiple source adaptation and the rényi +divergence. In Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence, +pp. 367–374, 2009. +Takeru Miyato, Toshiki Kataoka, Masanori Koyama, and Yuichi Yoshida. Spectral normalization for +generative adversarial networks. arXiv preprint arXiv:1802.05957, 2018. +12 + +Published as a conference paper at ICLR 2023 +Krikamol Muandet, David Balduzzi, and Bernhard Schölkopf. Domain generalization via invariant +feature representation. In International Conference on Machine Learning, pp. 10–18. PMLR, +2013. +A Tuan Nguyen, Toan Tran, Yarin Gal, and Atilim Gunes Baydin. Domain invariant representation +learning with domain density transformations. Advances in Neural Information Processing Systems, +34, 2021. +Luca Oneto, Michele Donini, Andreas Maurer, and Massimiliano Pontil. Learning fair and transfer- +able representations. arXiv preprint arXiv:1906.10673, 2019. +Trung Phung, Trung Le, Tung-Long Vuong, Toan Tran, Anh Tran, Hung Bui, and Dinh Phung. On +learning domain-invariant representations for transfer learning with multiple sources. Advances in +Neural Information Processing Systems, 34, 2021. +Yury Polyanskiy and Yihong Wu. Lecture notes on information theory. Lecture Notes for ECE563 +(UIUC) and, 6(2012-2016):7, 2014. +Yury Polyanskiy and Yihong Wu. Strong data-processing inequalities for channels and bayesian +networks. In Convexity and Concentration, pp. 211–249. Springer, 2017. +Fengchun Qiao, Long Zhao, and Xi Peng. Learning to learn single domain generalization. In +Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. +12556–12565, 2020. +Alexandre Rame, Corentin Dancette, and Matthieu Cord. Fishr: Invariant gradient variances for +out-of-distribution generalization. arXiv preprint arXiv:2109.02934, 2021. +Ashkan Rezaei, Anqi Liu, Omid Memarrast, and Brian D Ziebart. Robust fairness under covariate +shift. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pp. 9419–9427, +2021. +Shiori Sagawa, Pang Wei Koh, Tatsunori B Hashimoto, and Percy Liang. Distributionally robust +neural networks. In International Conference on Learning Representations, 2019. +Candice Schumann, Xuezhi Wang, Alex Beutel, Jilin Chen, Hai Qian, and Ed H Chi. Transfer of +machine learning fairness across domains. arXiv preprint arXiv:1906.09688, 2019. +Shiv Shankar, Vihari Piratla, Soumen Chakrabarti, Siddhartha Chaudhuri, Preethi Jyothi, and Sunita +Sarawagi. Generalizing across domains via cross-gradient training. In International Conference +on Learning Representations, 2018. +Yuge Shi, Jeffrey Seely, Philip HS Torr, N Siddharth, Awni Hannun, Nicolas Usunier, and Gabriel +Synnaeve. Gradient matching for domain generalization. arXiv preprint arXiv:2104.09937, 2021. +Changjian Shui, Boyu Wang, and Christian Gagné. On the benefits of representation regularization in +invariance based domain generalization. Machine Learning, pp. 1–21, 2022. +Anthony Sicilia, Xingchen Zhao, and Seong Jae Hwang. Domain adversarial neural networks for +domain generalization: When it works and how to improve. arXiv preprint arXiv:2102.03924, +2021. +Harvineet Singh, Rina Singh, Vishwali Mhasawade, and Rumi Chunara. Fairness violations and +mitigation under covariate shift. +In Proceedings of the 2021 ACM Conference on Fairness, +Accountability, and Transparency, pp. 3–13, 2021. +Baochen Sun and Kate Saenko. Deep coral: Correlation alignment for deep domain adaptation. In +European conference on computer vision, pp. 443–450. Springer, 2016. +Chris Xing Tian, Haoliang Li, Xiaofei Xie, Yang Liu, and Shiqi Wang. Neuron coverage-guided +domain generalization. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2022. +Riccardo Volpi, Hongseok Namkoong, Ozan Sener, John C Duchi, Vittorio Murino, and Silvio +Savarese. Generalizing to unseen domains via adversarial data augmentation. Advances in neural +information processing systems, 31, 2018. +13 + +Published as a conference paper at ICLR 2023 +Jingge Wang, Yang Li, Liyan Xie, and Yao Xie. Class-conditioned domain generalization via +wasserstein distributional robust optimization. arXiv preprint arXiv:2109.03676, 2021. +Haotian Ye, Chuanlong Xie, Tianle Cai, Ruichen Li, Zhenguo Li, and Liwei Wang. Towards a +theoretical framework of out-of-distribution generalization. Advances in Neural Information +Processing Systems, 34, 2021. +Taeho Yoon, Jaewook Lee, and Woojin Lee. Joint transfer of model knowledge and fairness over +domains using wasserstein distance. IEEE Access, 8:123783–123798, 2020. +Muhammad Bilal Zafar, Isabel Valera, Manuel Gomez-Rodriguez, and Krishna P Gummadi. Fairness +constraints: A flexible approach for fair classification. The Journal of Machine Learning Research, +20(1):2737–2778, 2019. +Rich Zemel, Yu Wu, Kevin Swersky, Toni Pitassi, and Cynthia Dwork. Learning fair representations. +In International conference on machine learning, pp. 325–333. PMLR, 2013. +Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical +risk minimization. In International Conference on Learning Representations, 2018. +Xueru Zhang, Mohammadmahdi Khaliligarekani, Cem Tekin, et al. Group retention when using +machine learning in sequential decision making: the interplay between user dynamics and fairness. +Advances in Neural Information Processing Systems, 32, 2019. +Xueru Zhang, Ruibo Tu, Yang Liu, Mingyan Liu, Hedvig Kjellstrom, Kun Zhang, and Cheng Zhang. +How do fair decisions fare in long-term qualification? Advances in Neural Information Processing +Systems, 33:18457–18469, 2020. +Han Zhao, Shanghang Zhang, Guanhang Wu, José MF Moura, Joao P Costeira, and Geoffrey J Gordon. +Adversarial multiple source domain adaptation. Advances in neural information processing systems, +31, 2018. +Shanshan Zhao, Mingming Gong, Tongliang Liu, Huan Fu, and Dacheng Tao. Domain generalization +via entropy regularization. Advances in Neural Information Processing Systems, 33:16096–16107, +2020. +Kaiyang Zhou, Yongxin Yang, Timothy Hospedales, and Tao Xiang. Learning to generate novel +domains for domain generalization. In European conference on computer vision, pp. 561–578. +Springer, 2020. +Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to-image translation +using cycle-consistent adversarial networks. In Proceedings of the IEEE international conference +on computer vision, pp. 2223–2232, 2017. +14 + +Published as a conference paper at ICLR 2023 +A +RELATED WORKS +Domain generalization/Domain adaptation: In many real scenarios of machine learning, data in +training phase is sampled from one or many source domains, while in the testing phase, data is +sampled from an unseen target domain. Many works have been proposed to design robust ML models +that can achieve good performances in deployment environment depending on whether they can +access to the target data (domain adaptation) or not (domain generalization). However, most of these +models focus only on transfering accuracy from source to target domains and can be categorized +into five main approaches: (1) data manipulation (Volpi et al., 2018; Qiao et al., 2020; Zhou et al., +2020; Zhang et al., 2018; Shankar et al., 2018); (2) domain-invariant representation learning (Li et al., +2018b;a; Ganin & Lempitsky, 2015; Ganin et al., 2016; Phung et al., 2021; Nguyen et al., 2021); (3) +distributional robustness (Krueger et al., 2021; Liu et al., 2021; Koh et al., 2021; Wang et al., 2021; +Sagawa et al., 2019; Hu et al., 2018), (4) gradient operation (Huang et al., 2020; Shi et al., 2021; +Rame et al., 2021; Tian et al., 2022), and (5) self-supervised learning (Carlucci et al., 2019; Kim +et al., 2021; Jeon et al., 2021; Li et al., 2021). +Fairness in Machine Learning: Many fairness notions have been proposed to measure the unfairness +in ML model, and they can be roughly classified into two classes: Individual fairness considers +the equity at the individual-level and it requires that similar individuals should be treated similarly +(Biega et al., 2018; Bechavod et al., 2020; Gupta & Kamble, 2021; Dwork et al., 2012). Group +fairness attains a certain balance in the group-level, where the entire population is first partitioned +into multiple groups and certain statistical measures are equalized across different groups (Hardt +et al., 2016; Zhang et al., 2019; 2020). Various approaches have also been developed to satisfy +these fairness notions, they roughly fall into three categories: (1) Pre-processing: modifying training +dataset to remove bias before learning an ML model (Kamiran & Calders, 2012; Zemel et al., 2013). +(2) In-processing: attain fairness during the training process by imposing certain fairness constraint +or modifying loss function. (Zafar et al., 2019; Agarwal et al., 2018) (3) Post-processing: altering +the output of an existing algorithm to satisfy a fairness constraint after training (Hardt et al., 2016). +However, most of these methods assume the data distributions at training and testing are the same. In +contrast, we study fairness problem under domain generalization in this paper. +Fairness under Domain Adaptation: There are some studies proposed to achieve good fairness +when the testing environment changes but all of them focused on the domain adaptation setting. The +most common adaptation setup is learning under the assumption of covariate shift. For example, +Singh et al. (2021) leveraged a feature selection method in a causal graph describing data to mitigate +fairness violation under covariate shift of distribution in testing data. Coston et al. (2019) proposed +the weighting methods that can give fair prediction under covariate shift between source and target +distribution when access to the sensitive attributes is prohibited. Rezaei et al. (2021) sought fair +decisions by optimizing a worst-case testing performance. Besides convariate shift, there are some +works proposed to handle other types of distribution shift including demographic shift and prior +probability shift. Instead of learning fair model directly, Oneto et al. (2019) and Madras et al. (2018) +find fair representation that can generalize to the new tasks. Aside from empirical studies, Schumann +et al. (2019) and Yoon et al. (2020) developed theoretical frameworks to examine fairness transfer +in domain adaptation setting and then offered modeling approaches to achieve good fairness in the +target domain. +Comparison with existing bounds in the literature: We compare our bounds with most commons +bound in the fields of domain adaptation and domain generalization as follows. +Accuracy bounds in domain adaptation. +• Bounds in Ben-David et al. (2010): +ϵAcc +DT +� +�f +� +≤ ϵAcc +DS +� +�f +� ++ DT V +� +P X +DS ∥ P X +DT +� ++ +min +D∈{DS,DT }ED [|fDS(X) − fDT (X)|] +This bound is for binary classification problem under domain adaptation. The classification error +in target domain is bounded by the error in source domain, the total variation distance of feature +distribution between source and target domain, and the misalignment of the labeling function +between source and target domain. The limitation of this bound is that (1) it’s only applicable to +settings with zero-one loss function and deterministic labeling function; (2) estimating the total +variation distance is hard in practice and it doesn’t relate the feature and representation spaces. +15 + +Published as a conference paper at ICLR 2023 +This paper also provides another accuracy bound based on H∆H divergence:. +ϵAcc +DT +� +�f +� +≤ ϵAcc +DS +� +�f +� ++ DH∆H +� +P X +DS ∥ P X +DT +� ++ inf +� +f +� +ϵAcc +DT +� +�f +� ++ ϵAcc +DS +� +�f +�� +where DH∆H +� +P X +DS ∥ P X +DT +� += sup +� +f1, � +f1 +���PDS +� +�f1(X) ̸= �f2(X) +� +− PDT +� +�f1(X) ̸= �f2(X) +���� is the +H∆H divergence. However, it has the same limitations as total variation distance mentioned above. +Accuracy bounds in domain generalization. +• Bounds in Albuquerque et al. (2019): +ϵAcc +DT +� +�f +� +≤ +N +� +i=1 +πiϵAcc +DS +i +� +�f +� ++ max +j,k∈[N]DH +� +P X +DS +j ∥ P X +DS +k +� ++ DH +� +P X +DS +∗ ∥ P X +DT +� ++ +min +D∈{DS +∗ ,DT }ED +���fDS +∗ (X) − fDT (X) +��� +where DH +� +P X +DS ∥ P X +DT +� += sup +� +f +���PDS +� +�f(X) = 1 +� +− PDT +� +�f(X) = 1 +���� is the H divergence, +P X +DS +∗ = arg min +π +DH +��N +i=1 πiP X +DS +i ∥ P X +DT +� +is the mixture of source domains that is closest to +target domain with respect to H divergence. In this bound, the classification error in target domain +is bounded by the convex combination of errors in source domains, the H divergence between +source domains, the H divergence between target domain and its nearest mixture of source domains, +and the misalignment of the labeling function between mixture source domains and target domain. +Because this bound is constructed based on H divergence, it also has the limitations for the bounds +in domain adaptation (Ben-David et al., 2010) as we mentioned. This bound can be transformed to +the representation space Z by replacing X by Z in its formula. Then, this bound suggests enforcing +invariant constraint of marginal distribution of representation Z across source domains, which has +inherent trade-off as shown in Thm. 2. Because the target domain is unknown during training, the +mixing weights {πi}N +i=1 are not useful for algorithmic design. +• Bounds in Phung et al. (2021): +ϵAcc +DT +� +�f +� +≤ +N +� +i=1 +πiϵAcc +DS +i +� +�f +� ++ Cmax +i∈[N]EDS +i +����� +����fDT (X)y − fDS +i (X)y +��� +�|Y| +y=1 +���� +1 +� ++ +N +� +i=1 +N +� +j=1 +C�2πj +N +d1/2 +� +P Z +DT , P Z +DS +i +� ++ +N +� +i=1 +N +� +j=1 +C�2πj +N +d1/2 +� +P Z +DS +i , P Z +DS +j +� +where d1/2 +� +P X +DS +i , P X +DS +j +� += +� +D1/2 +� +P X +DS +i ∥ P X +DS +j +� +is Hellinger distance defined based on +Hellinger divergence D1/2 +� +P X +DS +i ∥ P X +DS +j +� += 2 +� +X +�� +P X +DS +i − +� +P X +DS +j +�2 +dX. This bound re- +lates the feature and representation spaces that the classification error of target domain defined in +feature space is bounded by classification errors of source domains defined in feature space, the +misalignment of labeling function between target and source domains, and the Hellinger distances +between source and target domains and between source domains of marginal distribution of rep- +resentation Z. While this bound is not limited to zero-one loss and the labeling function can be +stochastic, it suggests the alignment of marginal distribution of representation Z across source +domains for generalization. Moreover, estimating Hellinger distance can be hard in practice. +The mismatch between existing bounds and adversarial learning approach for domain generalization. +All existing bounds mentioned above suggest minimizing the distances between representation +distributions across source domains with respect to some discrepancy measures such as H divergence, +total variation distance, and Hellinger distance. Based on these bounds, adversarial learning-based +models are often proposed to minimize these distances. However, there is a misalignment between +the objectives of adversarial learning and the bounds which results in the gap between theoretical +findings and practical algorithms. +16 + +Published as a conference paper at ICLR 2023 +In particular, it has been shown that the objective of the minimax game between the representation +mapping and the discriminator is equivalent to minimizing the JS divergence between representation +distributions across source domains (Goodfellow et al., 2014). However, minimizing JS divergence +does not guarantee the minimization of common distances used in the existing bounds. The details +are as follows. +• H divergence: We show that JS divergence is not the upper bound of H divergence. Con- +sider an example with two distributions P(X) and Q(X) where +� +P(X) = 0 +w.p 1/3 +P(X) = 1 +w.p 2/3 and +� +Q(X) = 0 +w.p 1/3 +Q(X) = 1 +w.p 2/3. By definition, DH(P ∥ Q) ∼ 0.33 > DJS(P ∥ Q) ∼ 0.08. +• Total variation distance: We have DJS(P ∥ Q) ≤ DT V (P ∥ Q) ∀P, Q where DJS and DT V are +JS divergence and total variation distance, respectively. Then, minimizing JS divergence does not +guarantee the minimization of total variation distance. +• Hellinger distance: We have DJS(P ∥ Q) ≤ +√ +2d1/2(P, Q) +∀P, Q where d1/2 is Hellinger +distance and total variation distance, respectively. Then, minimizing JS divergence does not +guarantee the minimization of Hellinger distance. +Different from the existing bounds, our bounds are based on JS divergence/distance. Then they align +with the adversarial learning approach for domain generalization in general, and with our proposed +method FATDM in particular. +Advantages of our proposed bounds in domain generalization. +In summary, our proposed bounds has several advantages in terms of the following: +• Most existing bounds (Ben-David et al., 2010; Albuquerque et al., 2019) do not relates feature +and representation spaces so it is not clear how performance in input space is affected by the +representations. In contrast, our bounds connect the representation and input spaces; this further +guides us to find representations that lead to good performances in input space. +• Most prior studies adopt H divergence to measure the dissimilarity between domains, which is +limited to deterministic labeling functions and zero-one loss (Ben-David et al., 2010; Albuquerque +et al., 2019). In contrast, our bound is more general and is applicable to settings where domains are +specified by stochastic labeling functions and general loss functions. +• Distant metrics (i.e., total variation distance, H divergence, Hellinger divergence, etc.) used in +existing bounds (Ben-David et al., 2010; Albuquerque et al., 2019; Phung et al., 2021) are hard +to compute in practice. In contrast, our bounds use JS divergence which is aligned with training +objective for discriminator in adversarial learning Goodfellow et al. (2014). +• Existing bounds for domain generalization only imply the alignment of marginal distribution of +feature across source domains (Albuquerque et al., 2019; Phung et al., 2021). As shown in Thm. 2, +methods that learn invariance of marginal distribution have an inherent trade-off and may increase +the lower bound of expected loss. In contrast, our bounds suggest the alignment of label-conditional +distribution of feature across source domains which has been verified to be more effective in +empirical studies (Li et al., 2018b;c; Zhao et al., 2020; Nguyen et al., 2021). +• Regarding the fairness, our work is the first that bounds the unfairness in domain generalization. In +particular, our bounds suggest enforcing the invariant constraint of feature distribution given label +and sensitive attribute across source domains to transfer fairness to the unseen target domain. +B +DETAILS OF ALGORITHM FATDM +FATDM consists of density mapping functions my +i,j and my,a +i,j , ∀y ∈ Y, a ∈ A, i, j ∈ [N] (learned +by two DensityMatch models), feature mapping function g (ResNet18 model), and the clas- +sifier �h. In our study, we experiment with two different DensityMatch architectures: Star- +GAN (i.e., in FATDM-StarGAN) and CycleGAN (in FATDM-CycleGAN). We show the details +of FATDM-StarGAN below. For FATDM-CycleGAN, the only difference is we used CycleGAN +as DensityMatch instead of StarGAN. The details of CycleGAN were presented in the original +paper (Zhu et al., 2017). +17 + +Published as a conference paper at ICLR 2023 +For FATDM-StarGAN, each DensityMatchY (or DensityMatchY,A) consists of a generator +G : X × [N] × [N] → X and a discriminator D : X → [N] × {0, 1}. The generator takes in real +image x and a pair of domain labels i, j as input and generates a fake image; the discriminator aims +to predict the domain label of the image generated by the generator and distinguish whether it is fake +or real. G and D are learned simultaneously by solving the following optimizations: +Discriminator’s objective: min LStarGAN +D +:= −LStarGAN +adv ++ λclsLStarGAN +cls(real) +Generator’s objective: min LStarGAN +G +:= LStarGAN +adv ++ λclsLStarGAN +cls(fake) + λrecLStarGAN +rec +(6) +where LStarGAN +adv +is the adversarial loss, LStarGAN +cls(fake) , LStarGAN +cls(real) are domain classification loss with respect +to fake and real images respectively, LStarGAN +rec +is the reconstruction loss. The specific formulations +of these loss functions are in Choi et al. (2018). λcls and λrec are hyper-parameters that control the +relative importance of domain classification and reconstruction losses, respectively, compared to the +adversarial loss. +In our experiments, input images are resized to (256, 256) and normalized into the range [−1, 1]. The +dimension of representation space Z is set to 512. ω (hyper-parameter that controls accuracy-fairness +trade-off) varies from 0 to 10 with step sizes 0.0002 for ω ∈ [0, 0.002], 0.002 for ω ∈ [0.002, 0.1] +and 0.2 for ω ∈ [0.2, 10], and γ (hyper-parameter that controls accuracy-invariance trade-off) is set +to 0.1 (after hyper-parameter tuning). Models (FATDM and baselines) are implemented by PyTorch +library version 1.11 and is trained on multiple computer nodes (each model instance is trained on a +single node which has 4 CPUs, 8GB of memory, and a single GPU (P100 or V100)). One domain’s +data is used for testing and the other domains’ data is used for training (10% of training data is +used for validation). Each model is trained with 10 epoches and the results are from the epoch +with best performance on the validation set. Figure 6 visualizes the two-stage training process of +FATDM-StarGAN. The detailed architectures of FATDM-StarGAN are shown in Tables 2-5. We +have also provided all code for these models in supplemental material. +18 + +Published as a conference paper at ICLR 2023 +Figure 6: Two-stage training of FATDM-StarGAN. For stage 1, we only show the training process +for DensityMatchY,A (training process for DensityMatchY is similar.) +Table 2: Architecture of StarGAN generators GY and GY,A - Density mapping functions my +i,j and +my,a +i,j ∀y ∈ Y, a ∈ A, i, j ∈ [N]. This architecture is similar to the one in the original paper Choi +et al. (2018) except for the first convolution layer where number of input channels is 1 (for grayscale +images) and input shape is (h, w, 1 + 2nc). (h, w) is the size of input images, IN is instance +batchnorm, and ReLU is Rectified Linear Unit. N: number of output channels, K: kernel size, S: +stride szie, P: padding size are convolution and deconvolution layers’ hyper-parameters. +Part +Input → Output Shape +Layer Information +Down-sampling +(h, w, 1 + 2nc) → (h, w, 64) +CONV-(N64, K7x7, S1, P3), IN, ReLU +(h, w, 64) → +� h +2 , w +2 , 128 +� +CONV-(N128, K4x4, S2, P1), IN, ReLU +� h +2 , w +2 , 128 +� +→ +� h +4 , w +4 , 256 +� +CONV-(N256, K4x4, S2, P1), IN, ReLU +Bottleneck +� h +4 , w +4 , 256 +� +→ +� h +4 , w +4 , 256 +� +Residual Block: CONV-(N256, K3x3, S1, P1), IN, ReLU +� h +4 , w +4 , 256 +� +→ +� h +4 , w +4 , 256 +� +Residual Block: CONV-(N256, K3x3, S1, P1), IN, ReLU +� h +4 , w +4 , 256 +� +→ +� h +4 , w +4 , 256 +� +Residual Block: CONV-(N256, K3x3, S1, P1), IN, ReLU +� h +4 , w +4 , 256 +� +→ +� h +4 , w +4 , 256 +� +Residual Block: CONV-(N256, K3x3, S1, P1), IN, ReLU +� h +4 , w +4 , 256 +� +→ +� h +4 , w +4 , 256 +� +Residual Block: CONV-(N256, K3x3, S1, P1), IN, ReLU +� h +4 , w +4 , 256 +� +→ +� h +4 , w +4 , 256 +� +Residual Block: CONV-(N256, K3x3, S1, P1), IN, ReLU +Up-sampling +� h +4 , w +4 , 256 +� +→ +� h +2 , w +2 , 128 +� +DECONV-(N128, K4x4, S2, P1), IN, ReLU +� h +2 , w +2 , 128 +� +→ (h, w, 64) +DECONV-(N64, K4x4, S2, P1), IN, ReLU +(h, w, 64) → (h, w, 3) +CONV-(N3, K7x7, S1, P3), IN, ReLU +19 + +Stage 1 +Source +Target +Domain Label +Domain Label +Sample +Target +Source +Fake Image +Domain Label +DY,A +Domain Label +yEy,aEA +Sample +batch +DY,A +Reconstructed +GY,A +CStarGAN + data +Image +Lady +Source +Target +Real Image +Domain Label +Domain Label +Real Image +Fake Image +Real Image +Source +Target +Source +Target +Batch of images with Y = y, A = a +Domain Label +DomainLabel +DomainLabel +DomainLabel +Sample data for training +Discriminator Training +Generator Training +Stage 2 +Linv +GY +Fake +FakeImage +Classification +Lds +Representation +Label +Sensitive +Real Image +Attribute +Target +Domain Label +Source +Target +Real Image +Real +(Generated) +9 +h +Classification +Source +Domain Label +Domain Label +Representation +Label +Domain Label +Batch of images +Fake Image +Fake +GY,A +Sensitive +Lfair +Representation +Attribute +Sample data for training +Stage 2 TrainingPublished as a conference paper at ICLR 2023 +Table 3: Architecture of StarGAN discriminators. This architecture is similar to the one in the original +paper Choi et al. (2018) except for the first convolution layer where number of input channels is 1 +(for grayscale images). (h, w) is the size of input images, nd is the number of domains, and Leaky +ReLU is Leaky Rectified Linear Unit. N: number of output channels, K: kernel size, S: stride szie, P: +padding size are convolution layers’ hyper-parameters. +Layer +Input → Output Shape +Layer Information +Input Layer +(h, w, 1) → +� h +2 , w +2 , 64 +� +CONV-(N64, K4x4, S2, P1), Leaky ReLU +Hidden Layer +� h +2 , w +2 , 64 +� +→ +� h +4 , w +4 , 128 +� +CONV-(N128, K4x4, S2, P1), Leaky ReLU +Hidden Layer +� h +4 , w +4 , 128 +� +→ +� h +8 , w +8 , 256 +� +CONV-(N256, K4x4, S2, P1), Leaky ReLU +Hidden Layer +� h +8 , w +8 , 256 +� +→ +� h +16, w +16, 512 +� +CONV-(N512, K4x4, S2, P1), Leaky ReLU +Hidden Layer +� h +16, w +16, 512 +� +→ +� h +32, w +32, 1024 +� +CONV-(N1024, K4x4, S2, P1), Leaky ReLU +Hidden Layer +� h +32, w +32, 1024 +� +→ +� h +64, w +64, 2048 +� +CONV-(N2048, K4x4, S2, P1), Leaky ReLU +Output Layer (Dsrc) +� h +64, w +64, 2048 +� +→ +� h +64, w +64, 1 +� +CONV-(N1, K3x3, S1, P1) +Output Layer (Dcls) +� h +64, w +64, 2048 +� +→ (1, 1, nd) +CONV-(N(nd), K h +64 × w +64, S1, P0) +Table 4: Architecture of feature mapping g. This architecture is similar to ResNet18 model He et al. +(2016) except for the first convolution layer where number of input channels is 1 (for grayscale +images) and the last layer where output dimension is nz - dimension of representation space Z. (h, w) +is the size of input images, BN is batchnorm, MaxPool is max pooling, AvePool is average pooling, +and ReLU is Rectified Linear Unit. N: number of output channels, K: kernel size, S: stride szie, P: +padding size are convolution layers’ hyper-parameters. +Part +Input → Output Shape +Layer Information +Input +(h, w, 1) → +� h +2 , w +2 , 64 +� +CONV-(N64, K7x7, S2, P3), BN, ReLU, MaxPool +Bottleneck +� h +2 , w +2 , 64 +� +→ +� h +4 , w +4 , 64 +� +Residual Block: CONV-(N64, K3x3, S1, P1), BN, ReLU, +CONV-(N64, K3x3, S1, P1), BN +� h +4 , w +4 , 64 +� +→ +� h +8 , w +8 , 128 +� +Residual Block: CONV-(N128, K3x3, S1, P1), BN, ReLU, +CONV-(N128, K3x3, S1, P1), BN +� h +8 , w +8 , 128 +� +→ +� h +16, w +16, 256 +� +Residual Block: CONV-(N256, K3x3, S1, P1), BN, ReLU, +CONV-(N256, K3x3, S1, P1), BN +� h +16, w +16, 256 +� +→ (1, 1, 512) +Residual Block: CONV-(N512, K3x3, S1, P1), IN, ReLU, +CONV-(N512, K3x3, S1, P1), BN, AvgPool +Output +(1, 1, 512) → nz +LINEAR-(512, nz) +Table 5: Architecture of classifier �h. nz is the dimension of representation space Z. +Layer +Input → Output Shape +Layer Information +Hidden Layer +nz → nz +2 +LINEAR- +� +nz, nz +2 +� +, ReLU +Hidden Layer +nz +2 → nz +4 +LINEAR- +� nz +2 , nz +4 +� +, ReLU +Output Layer +nz +4 → 1 +LINEAR- +� nz +4 , 1 +� +, Sigmoid +20 + +Published as a conference paper at ICLR 2023 +C +ADDITIONAL EXPERIMENTS +Experimental results with all unfairness and error metrics. +In this section, we provide more +experimental results about fairness and accuracy under domain generalization. In particular, we +investigate fairness-accuracy trade-off on the two clinical image datasets including Cardiomegaly and +Edema diseases with respect to different fairness criteria (i.e., Equalized Odds, Equal Opportunity), +and unfairness (i.e., MD and EMD) and error (i.e., CE, MR, AUROC, AUPR, F1) measures. Figure 7 +(Cardiomegaly disease - Equalized Odds), Figure 8 (Cardiomegaly disease - Equal Opportunity), +Figure 9 (Edema disease - Equalized Odds), and Figure 10 (Edema disease - Equal Opportunity) +show the unfairness-error curves of our models as well as baselines for these two datasets. As we can +see, our model outperforms other baselines in terms of fairness-accuracy trade-off. The curve of our +model is the bottom-leftmost compared to other baselines in all measures showing the clear benefit of +(1) enforcing conditional invariant constraints for accuracy and fairness transfer and (2) using the +two-stage training process to stabilize training compared to adversarial learning approach. We also +quantify our observations by calculating the areas under these unfairness-error curves, in which the +smaller area indicates the better accuracy-fairness trade-off. As shown in Tables 6 and 7, our model +has the smallest areas under the curve and achieves significantly better fairness-accuracy trade-off for +both equalized odd and equal opportunity compared to other methods. +Impact of the number of source domains. +Our work focuses on transferring fairness and accuracy +under domain generalization when the target domain data are inaccessible during training. Instead, +it relies on a set of source domains to generalize to an unseen, novel target domain. We investigate +the relationship between the fairness-accuracy trade-off on the target domain and the number of +source domains during training. In particular, we evaluate the performances of FATDM and ERM on +Edema dataset with different numbers of source domains. Similar to the previous experiment, we first +construct the dataset for each domain by rotating images with θ degree, where θ ∈ {0◦, 15◦, 30◦} +when the number of domain is 3, θ ∈ {0◦, 15◦, 30◦, 45◦} when the number of domain is 4, and +θ ∈ {0◦, 15◦, 30◦, 45◦, 60◦} when the number of domain is 5. The number of images per domain +is adapted to ensure the training set size is fixed for the three cases. We follow the leave-one-out +domain setting in which one domain serves as the unseen target domain for evaluation while the rest +domains are for training; the average results across target domains are reported. +Figure 11 shows error-unfairness curves of FATDM and ERM when training with 2, 3, and 4 source +domains. We observe that training with more source domains does not always help the model achieve +better fairness-accuracy trade-off on unseen target domains. In particular, the performances of both +FATDM and ERM are the best when training with 2 source domains and the worst when training with +3 source domains. We conjecture the reason that adding more source domains may help reduce the +discrepancy between source and target domains (term (ii) in Thm. 1 and Thm. 3), but it may make it +more difficult to minimize the source error and unfairness (term (i) in Thm. 1 and Thm. 3) and to +learn invariant representation across the source domains (term (iii) in Thm. 1 and Thm. 3). Thus, our +suggestion in practice is to conduct an ablation study to find the optimal number of source domains. +Simultaneous and sequential training comparison. +In all experiments we conducted so far, +the fairness constraint Lfair is optimized simultaneously with the prediction error Lacc and the +domain-invariant constraint Linv for all methods. To investigate whether FATDM still attains a +better accuracy-fairness trade-off when the processes of invariant representation learning and fair +model training are decoupled, we conduct another set of experiments where models (FATDM (i.e., +FATDM-StarGAN) and baselines G2DM, DANN, CDANN) are learned in a sequential matter: for +each model, we first learn the representation mapping g by optimizing Linv and Lacc; using the +representations generated by the fixed g, we then learn the fair classifier by optimizing Lacc and Lfair. +The models trained based on the above procedure are named FATDM-seq, G2DM-seq, DANN-seq, +and CDANN-seq; and their corresponding error-unfairness curves are shown in Figure 12. The +results show that FATDM-seq still attains the best accuracy-fairness trade-off at target domain +compared to G2DM-seq, DANN-seq, CDANN-seq. Our method is effective no matter whether +Lfair and Linv are optimized simultaneously or sequentially. +The reason that our method consistently outperforms the baselines for both settings is that the +invariant-representation learning in baseline methods only guarantees the transfer of accuracy but +not fairness. Even though a fairness regularizer is imposed to ensure the model is fair at source +21 + +Published as a conference paper at ICLR 2023 +domains (no matter whether invariant representations and fair classifier are trained simultaneously or +sequentially), this fairness cannot be preserved at the target domain due to the potential distributional +shifts. The key to ensuring the transfer of fairness is to learn representations such that P(Z|Y, A) is +domain-invariant; this must be done during the representation learning process. From Thm 3, we +can see that unfairness at target domain ϵEO +DT can still blow up if P Z|Y,A is different across domains, +regardless of how fair the model is at source domains (i.e., small ϵEO +DS +i ). +Figure 7: Error-unfairness curves with respect to equalized odds of FATDM and baselines on Car- +diomegaly disease dataset. +22 + +-- ERM +-●- G2DM +O- DANN +-- CDANN +-- CORAL +GroupDRO +IRM +●- ATDM +-- FATDM-StarGAN +-- FATDM-CycleGAN +1.0 +.0 +0.8 +(EMD) +(MD) +Unfairness +Unfairness +0.2 +0.2 + 0'0 +0.0 +0.0 +0.2 +10 +0.0 +0.2 +0.8 +Error AUROC +Error AUROC +(MD) +(EMD) +Unfairness +Unfairness ( +0.2 +0.0 +0.0 +0.0 +0.2 +0.4 +1.0 +0.0 +0.2 +0. +0.8 +Error AUPR +Error AUPR +1.0 +(EMD) +(MD) +Unfairness +Unfairness +0.4 +0.2 +0.2 +0.0 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +Error (CE) +Error (CE) +1.0 +(MD) +(EMD) +J.8 +Unfairness +Unfairness ( +0.4 +0.4 + 0.2 +0.0 +0'0 +0.2 +0.4 +0.8 +1.0 +0.0 +0.2 +0.6 +0.8 +Error (MR) +Error (MR) +(MD) +(EMD) +Unfairness ( +J.6 +Unfairness ( +0.4 +0.2 +0.2 +0.0 +0.0 +0.2 +0.4 +8'0 +L.0 +0.0 +0.2 +0.4 +0.6 +0.8 +Error (F)Published as a conference paper at ICLR 2023 +Figure 8: Error-unfairness curves with respect to equal opportunity of FATDM and baselines on +Cardiomegaly disease dataset. +23 + +-- ERM +G2DM +O- DANN +-CDANN +CORAL +GroupDRO +IRM +-●-ATDM +-- FATDM-StarGAN +-- FATDM-CycleGAN +1.0 +(MD) +(EMD) +Unfairness +Unfairness ( +0.2 +0.2 +0'0 +0.0 +0.0 +0.2 +0.8 +1.0 +0.0 +0.2 +0. +Error AUROC +Error AUROC +(EMD) +(MD) +0.8 +Unfairness ( +Unfairness ( +0.4 +中 +0.2 +0.0 +0.0 +0.0 +0.2 +0.4 +0.8 +1.0 +0.0 +0.2 +0.4 +0.8 +Error AUPR +Error AUPR +1.0 +(MD) +(EMD) +Unfairness +Unfairness +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0'0 +0.2 +0.4 + 0.6 +0.8 +Error (CE) +Error (CE) +(EMD) +(MD) +Unfairness +Unfairness ( +02 +0.0 +0.0 +0.0 +0.2 +0.4 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +Error (MR) +Error (MR) +Unfairness (EMD) +(MD) +: +Unfairness ( +1 s +0.2 +0.2 +0.0 +0.0 +0.0 +0.2 +0.4 +0.8 +0.2 +0.4 +0.6 +0.8 +Error (F))Published as a conference paper at ICLR 2023 +Figure 9: Error-unfairness curves with respect to equalized odds of FATDM and baselines on Edema +disease dataset. +24 + +-- ERM +●-G2DM +O- DANN +CDANN +GroupDRO +O- CORAL +-O- FATDM-StarGAN +1.0 +(MD) +(EMD) +0.8 +Unfairness +Unfairness ( +0.4 +0.2 +0.2 +0.0 +0.0 - +0.0 +0.2 +0.8 +. +0.0 +0.2 +0.8 +1.0 +Error AUROC +Error AUROC +(EMD) +(MD) +0.8 +0. 6 +Unfairness ( +Unfairness ( +0.4 +0.4 +0.2 +0.2 +0.0 +0'0 +0.0 +0.2 +0.4 +0.8 +1.0 +0.0 +0.2 +0.8 +Error AUPR +Error AUPR +1.0 +1.0 +(MD) +(EMD) +LE + 0.6 +Unfairness +Unfairness ( +0.4 +0.4 +0.2 +0.2 +F 0'0 +0.0 - +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +Error (CE) +Error (CE) +ti! +(MD) +(EMD) +0.8 +0.6 +Unfairness +Unfairness ( +0.4 +0.4 +0.2 - +0.0 +0.0 +0.0 +0.2 +0.6 +0.8 +1.0 +0.0 +0.4 +0.2 +0.4 +0.8 +1.0 +Error (MR) +Error (MR) +1.0 +1.0 +Unfairness (EMD) +(MD) +Unfairness +0.4 +0.2 +0.2 +0.0 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0Published as a conference paper at ICLR 2023 +Figure 10: Error-unfairness curves with respect to equal opportunity of FATDM and baselines on +Edema disease dataset. +25 + +-O- ERM +●-G2DM +- DANN +CDANN +GroupDRO +CORAL +-O- FATDM-StarGAN +1.0 +Unfairness (EMD) +(MD) +Unfairness ( +0.4 +11 +0.2 +0.2 +F 0°0 +0.0 - +0.0 +0.2 +0.8 +0.0 +0.2 +0.8 +1.0 +Error AUROC +Error AUROC +Unfairness (MD) +1 +Unfairness (EMD) +0.8 +0. 6 +0.f +0.4 +0.4 +11 +0.2 +02 +0.0 +0.0 - +0.0 +0.4 +0. +1.0 +0.0 +0.2 +0.8 +Error AUPR +Error AUPR +1.0 +1.0 +(MD) +Unfairness (EMD) + 0.6 +Unfairness +0.4 +0.4 +0.2 +0.2 +F 0'0 +0.0 +0.2 +0.6 +0.8 +1.0 +0.0 +0.4 +0.2 +0.4 +0.6 +0.8 +1.0 +Error (CE) +Error (CE) +1.0 +(MD) +(EMD) +Unfairness ( +0.6 +0.6 +Unfairness ( +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 + 0.0 +0.2 +0.4 +0.6 +0.8 +1.0 + 0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Error (MR) +Error (MR) +1.0 +Unfairness (EMD) +(MD) +0.8 +Unfairness +0.4 +0.2 +0.0 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.4 +0.6 +0.8 +1.0 +Error (Fl)Published as a conference paper at ICLR 2023 +Table 6: Area under the error-unfairness curves (Cardiomegaly disease dataset). +Error - Unfairness +Method +ERM +G2DM +DANN +CDANN +CORAL +GroupDRO +IRM +FATDM +Equalized Odds +AUROC - MD +0.5575 +0.6093 +0.7571 +0.7224 +0.7239 +0.7039 +0.6784 +0.0935 +AUPRC - MD +0.5463 +0.6301 +0.7730 +0.6883 +0.7300 +0.7152 +0.6967 +0.0291 +CE - MD +0.2861 +0.2601 +0.4622 +0.4232 +0.4424 +0.3148 +0.3370 +0.2152 +MR - MD +0.6312 +0.4906 +0.6795 +0.6667 +0.6683 +0.6382 +0.5721 +0.2439 +F1 - MD +0.5901 +0.4150 +0.6507 +0.6547 +0.5745 +0.5360 +0.5025 +0.3365 +AUROC - EMD +0.7326 +0.7106 +0.8342 +0.7931 +0.8075 +0.7845 +0.7991 +0.1099 +AUPRC - EMD +0.6901 +0.7146 +0.8308 +0.7577 +0.7918 +0.7806 +0.7945 +0.0437 +CE - EMD +0.5158 +0.4443 +0.6143 +0.5788 +0.5873 +0.4911 +0.5274 +0.3384 +MR - EMD +0.7056 +0.5795 +0.7137 +0.6979 +0.6902 +0.6571 +0.6483 +0.2045 +F1 - EMD +0.6866 +0.5328 +0.7279 +0.7019 +0.6515 +0.6027 +0.6120 +0.2888 +Equal +Opportunity +AUROC - MD +0.5128 +0.6001 +0.6999 +0.6686 +0.5935 +0.6288 +0.5910 +0.0750 +AUPRC - MD +0.5419 +0.6718 +0.7086 +0.7189 +0.6423 +0.6761 +0.6435 +0.0262 +CE - MD +0.3690 +0.4272 +0.5094 +0.4492 +0.3780 +0.2737 +0.3582 +0.2754 +MR - MD +0.3203 +0.5068 +0.5252 +0.5512 +0.4897 +0.4368 +0.4173 +0.1778 +F1 - MD +0.2134 +0.4570 +0.4608 +0.5207 +0.4017 +0.3561 +0.3510 +0.2737 +AUROC - EMD +0.6119 +0.7184 +0.7649 +0.7517 +0.6720 +0.7068 +0.6780 +0.0947 +AUPRC - EMD +0.6321 +0.7684 +0.7718 +0.7877 +0.6912 +0.7335 +0.7200 +0.0448 +CE - EMD +0.5092 +0.6093 +0.6264 +0.6141 +0.4737 +0.4340 +0.4917 +0.3070 +MR - EMD +0.4619 +0.6420 +0.6325 +0.6532 +0.5790 +0.5515 +0.5298 +0.1918 +F1 - EMD +0.3876 +0.6122 +0.5942 +0.6496 +0.5101 +0.4898 +0.4889 +0.3108 +Table 7: Area under the error-unfairness curves (Edema disease dataset). +Error - Unfairness +Method +ERM +G2DM +DANN +CDANN +CORAL +GroupDRO +FATDM +Equalized Odds +AUROC - MD +0.3395 +0.2765 +0.2972 +0.2548 +0.3642 +0.3627 +0.0633 +AUPRC - MD +0.2865 +0.2446 +0.2561 +0.2304 +0.3052 +0.2998 +0.0771 +CE - MD +0.1096 +0.1266 +0.1243 +0.1192 +0.1269 +0.1179 +0.0341 +MR - MD +0.3929 +0.3525 +0.3509 +0.3302 +0.4303 +0.4240 +0.0656 +F1 - MD +0.4213 +0.3219 +0.3527 +0.4178 +0.4283 +0.4189 +0.1369 +AUROC - EMD +0.4277 +0.3813 +0.3637 +0.3419 +0.4419 +0.4394 +0.2729 +AUPRC - EMD +0.3868 +0.3588 +0.3285 +0.3245 +0.3958 +0.3921 +0.3041 +CE - EMD +0.2366 +0.2401 +0.2348 +0.2334 +0.2447 +0.2339 +0.1827 +MR - MD +0.4592 +0.4435 +0.4017 +0.3904 +0.4942 +0.4792 +0.2802 +F1 - MD +0.5186 +0.4642 +0.4132 +0.4827 +0.5180 +0.5029 +0.3855 +Equal +Opportunity +AUROC - MD +0.2488 +0.2139 +0.2085 +0.1806 +0.2696 +0.2625 +0.0218 +AUPRC - MD +0.2606 +0.2381 +0.2297 +0.2035 +0.2937 +0.2874 +0.0168 +CE - MD +0.1540 +0.1839 +0.1689 +0.1572 +0.1487 +0.1446 +0.0234 +MR - MD +0.2967 +0.2652 +0.2620 +0.2516 +0.3101 +0.2999 +0.0468 +F1 - MD +0.2848 +0.2195 +0.2534 +0.2613 +0.2973 +0.2975 +0.0502 +AUROC - EMD +0.2736 +0.2472 +0.2449 +0.2155 +0.2897 +0.2841 +0.1121 +AUPRC - EMD +0.2653 +0.2451 +0.2429 +0.2176 +0.2852 +0.2812 +0.0912 +CE - EMD +0.2083 +0.2318 +0.2355 +0.2147 +0.2055 +0.2003 +0.1159 +MR - MD +0.3409 +0.3162 +0.3258 +0.3026 +0.3442 +0.3388 +0.1872 +F1 - MD +0.3237 +0.2756 +0.3031 +0.3008 +0.3215 +0.3271 +0.1779 +26 + +Published as a conference paper at ICLR 2023 +Figure 11: Error-unfairness curves with respect to equalized odds of FATDM and ERM on Edema +disease dataset when training with different numbers of source domains. Names in the figure legend +are in the form of X-Y where X is the model and Y is the number of source domains (e.g., ERM-2 +means training ERM on two source domains.) +27 + +ERM-2 +ERM-3 +ERM-4 +FATDM-2 +FATDM-3 +FATDM-4 +1.0 +(MD) +(EMD) +3'0 +U. +0.6 +Unfairness +Unfairness ( +0.4 +0.2 +0'0 +0.0 +0.2 +0.0 +0.2 +1.0 +Error AUROC +Error AUROC +.( +(EMD) +(MD) +0.6 +Unfairness +Unfairness ( +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.8 +1.0 +Error AUPR +Error AUPR +(EMD) +(MD) +Unfairness +Unfairness +0.4 +0.2 +0.2 +0.0 +0.0 +0.0 +0.0 +0.2 +0.8 +1.0 +0.4 +0.8 +0.4 +Error (CE) +Error (CE) +1.0 +(EMD) + (MD) +0.8 +Unfairness +0. +0.6 +Unfairness ( +0.4 +0.4 + 0.2 +F 0°0 +0'0 +0.0 +0.2 +1.0 +0.0 +0.2 +0.8 +1.0 +Error (MR) +Error (MR) +. +(EMD) +(MD) +0. + 0.6 +Unfairness +Unfairness ( +0.4 +0.2 +0.2 + 0.0 - +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +0.0 +0.2 +0.4 +. +0.8 +1.0 +Error (Fl) +Error (F)Published as a conference paper at ICLR 2023 +(a) Equalized Odds +(b) Equal Opportunity +Figure 12: Fairness-accuracy trade-off (Pareto frontier) of models trained with simultaneous and +sequential (i.e., models with ‘-seq’ suffix) approaches, and FATDM-CycleGAN (i.e., use CycleGAN +instead of StarGAN as density mapping functions) on Cardiomegaly disease dataset: error-unfairness +curves are constructed by varying ω ∈ [0, 10] and the values of error and unfairness are normalized +to [0, 1]. Lower-left points and the smaller area under the curve indicate the model has a better +fairness-accuracy trade-off (Pareto optimality). +D +ADDITIONAL RESULTS & LEMMAS +D.1 +TIGHTER UPPER BOUND FOR ACCURACY +Corollary 5.1 We can replace term (ii) in Thm. 1 with the following term to attain a tighter upper +bound for accuracy: +√ +2C min +i∈[N] +� +dJS +� +P Y +DT , P Y +DS +i +� ++ +� +2ηT V Ez∼PDT +i (z) +� +dJS +� +P X|Y +DT , P X|Y +DT +i +�2�� +. +where ηT V = +sup +P X +Di̸=P X +Dj +DT V +� +P Z +Di,P Z +Dj +� +DT V +� +P X +Di,P X +Dj +� ≤ 1 is called Dobrushin’s coefficient (Polyanskiy & Wu, +2017). +This result suggests that we can further optimize term (ii) in Thm. 1 by minimizing ηT V . It has +been shown in Shui et al. (2022) that ηT V can be controlled by Lipschitz constant of the feature +mapping g : X → Z when g follows Gaussian distribution. The Lipschitz constant of g, in turn, can +be upper bounded by the Frobenius norm of Jacobian matrix with respect to g (Miyato et al., 2018). +However, in practice, we found that computing Jacobian matrix of g is computationally expensive +when dimension of representation Z is large, and optimizing it together with invariant constraints +does not improve the performances of models in our experiments. +D.2 +LEMMAS FOR PROVING THEOREM 1 +Lemma 6 Let X be the random variable in domains Di and Dj, and E be an event that P X +Dj ≥ P X +Di, +then we have: � +E +���P X +Dj − P X +Di +��� dX = +� +E +���P X +Dj − P X +Di +��� dX = 1 +2 +� ���P X +Dj − P X +Di +��� dX +where E is the complement of event E. +28 + +ii +G2DM +G2DM-seq +DANN-seq +CDANN +- CDANN-seq +FATDM +FATDM-seq +1.0 +1.0 +1.0 +0.8 +0.8 +Unfairness (EMD) +8'0 +Unfairness (EMD) +(MD) +(MD) +.8 +0.6 +Unfairness ( +Unfairness ( +0.2 +0.0 +0.0 +0.0 +0.0 +0.0 +0.2 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Error AUROC +Error (MR) +Error AUROC +Error (MR)G2DM +G2DM-seq +DANN-seq +CDANN +i +CDANN-seq +FATDM +FATDM-seq +1.0 +1.0 +1.0 +1.0 +0.8 - +0.8 +(EMD) +8'0 +(MD) +(EMD) +(MD) +Unfairness +Unfairness ( +Unfairness ( +Unfairness +0.4 +2.4 +0.2 + 0.2 +1.2 +0.0 +0.0 +0.0 +0.0 - +0.0 +0.2 +0.8 +1.0 +0.0 +0.2 +0.4 +0.8 +1.0 +0.0 +0.2 +0.4 +0.8 +1.0 +0.0 +0.2 +0.4 +0.8 +1.0 +Error AUROC +Error (MR) +Error AUROC +Error (MR)Published as a conference paper at ICLR 2023 +Lemma 7 Let X be the random variable in domains Di and Dj, let f : X → R+ be a non-negative +function bounded by C, then we have: +EDj[f(X)] − EDi[f(X)] ≤ C +√ +2 +� +min +� +DKL +� +P X +Di ∥ P X +Dj +� +, DKL +� +P X +Dj ∥ P X +Di +�� +where DKL(· ∥ ·) is the KL-divergence between two distributions. +Lemma 8 Suppose loss function L is upper bounded by C and consider a classifier �f : X → Y. the +expected classification error of �f in domain Dj can be upper bounded by its error in domain Di: +ϵAcc +Dj +� +�f +� +≤ ϵAcc +Di +� +�f +� ++ +√ +2CdJS +� +P X,Y +Dj , P X,Y +Di +� +where X, Y are random variables denoting feature and label in domains Di and Dj. +Lemma 9 Consider two distributions P X +Di and P X +Dj over X. Let P Z +Di and P Z +Dj be the induced +distributions over Z by mapping function g : X → Z, then we have: +dJS(P X +Di, P X +Dj) ≥ dJS(P Z +Di, P Z +Dj) +Lemma 10 +(Phung et al., 2021) Consider domain D with joint distribution P X,Y +D +and labeling +function fD : X → Y∆ from feature space to label space. Given mapping function g : X → Z +from feature to representation space, we define labeling function hD : Z → Y∆ from representation +space to label space as hD(Z)Y = fD(X)Y ◦ g−1(Z) = +� +g−1(Z) fD(X)Y P X +D dX +� +g−1(Z) P X +D dX +. Similarly, let +�f be the hypothesis from feature space, then the corresponding hypothesis �h from representation +space under the mapping function g is computed as �h(Z)Y = +� +g−1(Z) � +f(X)Y P X +D dX +� +g−1(Z) P X +D dX +. Let ϵAcc +D ( �f) = +ED +� +L( �f(X), Y ) +� +and ϵAcc +D (�h) = ED +� +L(�h(Z), Y ) +� +be expected errors defined with respect to feature +space and representation space, respectively. We have: +ϵAcc +D +� +�f +� += ϵAcc +D +� +�h +� +D.3 +LEMMAS FOR PROVING COROLLARY 1.1 +Lemma 11 Consider two random variables X, Y . Let P X,Y +Di +, P X,Y +Dj +be two joint distributions defined +in domains Di and Dj, respectively. Then, JS-divergence DJS +� +P X,Y +Di +∥ P X,Y +Dj +� +and KL-divergence +DKL +� +P X,Y +Di +∥ P X,Y +Dj +� +can be decomposed as follows: +DKL +� +P X,Y +Di +∥ P X,Y +Dj +� += DKL +� +P Y +Di ∥ P Y +Dj +� ++ EDi +� +DKL +� +P X|Y +Di +∥ P X|Y +Dj +�� +DJS +� +P X,Y +Di +∥ P X,Y +Dj +� +≤ DJS +� +P Y +Di ∥ P Y +Dj +� ++ EDi +� +DJS +� +P X|Y +Di +∥ P X|Y +Dj +�� ++ EDj +� +DJS +� +P X|Y +Di +∥ P X|Y +Dj +�� +D.4 +LEMMAS FOR PROVING THEOREM 2 +Lemma 12 Under Assumption in Theorem 2, the following holds for any domain D: +� +ϵAcc +D ( �f) = +� +ED[L( �f(X), Y )] ≥ +� +2c +|Y|dJS(P Y +D , P +�Y +D )2, ∀ �f +where �Y is the prediction made by randomized predictor �f. +29 + +Published as a conference paper at ICLR 2023 +D.5 +LEMMAS FOR PROVING THEOREM 3 +Definition 13 Given domain Di with binary random variable A denoting the sensitive attribute, the +unfairness measures that evaluate the violation of equalized odd (EO) and equal opportunity (EP) +criteria between sensitive groups of this domain are defined as follows. +ϵEO +Di +� +�f +� += +���R0,0 +Di +� +�f +� +− R0,1 +Di +� +�f +���� + +���R1,0 +Di +� +�f +� +− R1,1 +Di +� +�f +���� +ϵEP +Di +� +�f +� += +���R1,0 +Di +� +�f +� +− R1,1 +Di +� +�f +���� +where Ry,a +Di +� +�f +� += EDi +� +�f(X)1|Y = y, A = a +� +. +Lemma 14 Given two domains Di and Dj, under Definition 13, Ry,a +Dj +� +�f +� +can be bounded by +Ry,a +Di +� +�f +� +as follows. +Ry,a +Dj +� +�f +� +≤ Ry,a +Di +� +�f +� ++ +√ +2dJS +� +P X|Y =y,A=a +Dj +, P X|Y =y,A=a +Di +� +∀y, a ∈ {0, 1} +Lemma 15 Given two domains Di and Dj, under Definition 13, the unfairness in domain Dj can +be upper bounded by the unfairness measure in domain Di as follows. +ϵEO +Dj +� +�f +� +≤ ϵEO +Di +� +�f +� ++ +√ +2 +� +y=0,1 +� +a=0,1 +dJS +� +P X|Y =y,A=a +Dj +, P X|Y =y,A=a +Di +� +ϵEP +Dj +� +�f +� +≤ ϵEP +Di +� +�f +� ++ +√ +2 +� +a=0,1 +dJS +� +P X|Y =1,A=a +Dj +, P X|Y =1,A=a +Di +� +Lemma 16 Consider domain D with distribution P X,Y +D +and labeling function fD : X → Y∆. Given +mapping function g : X → Z from feature to representation space, we define labeling function +hD : Z → Y∆ from representation space to label space as hD(Z)Y = fD(X)Y ◦ g−1(Z) = +� +g−1(Z) fD(X)Y P X +D dX +� +g−1(Z) P X +D dX +. Similarly, let �f be the hypothesis from feature space, then the corresponding +hypothesis �h from representation space under the mapping function g is computed as �h(Z)Y = +� +g−1(Z) � +f(X)Y P X +D dX +� +g−1(Z) P X +D dX +. Under Definition 13, we have: +ϵEO +D +� +�f +� += ϵEO +D +� +�h +� +ϵEP +D +� +�f +� += ϵEP +D +� +�h +� +D.6 +LEMMAS FOR PROVING THEOREM 5 +Lemma 17 Consider two domains Di and Dj, if there exist invertible mappings my +i,j and my,a +i,j such +that P X|y +Di += P +my +i,j(X)|y +Dj +and P X|y,a +Di += P +my,a +i,j (X)|y,a +Dj +, ∀y ∈ Y, a ∈ A, then DJS +� +P Z|y +Di +∥ P Z|y +Dj +� +and DJS +� +P Z|y,a +Di +∥ P Z|y,a +Dj +� +can be upper bounded by +� +x P x|y +Di DJS +� +P Z|x ∥ P Z|my +i,j(x)� +dx and +� +x P x|y,a +Di +DJS +� +P Z|x ∥ P Z|my,a +i,j (x)� +dx, respectively. +E +PROOFS +E.1 +PROOFS OF THEOREMS +Proof of Theorem 1. +First, we get the upper bound based on the representation space Z. Then, we +relate it with the feature space X. Let DS +∗ ∈ {DS +i }N +i=1 be the source domain that’s nearest to the +30 + +Published as a conference paper at ICLR 2023 +target domain DT . According to Lemma 8, we have upper bound of the expected classification error +for the target domain based on each of the source domain as follows. +ϵAcc +DT +� +�h +� +≤ ϵAcc +DS +i +� +�h +� ++ +√ +2CdJS +� +P Z,Y +DT , P Z,Y +DS +i +� +∀i ∈ [N] +Taking average of upper bounds based on all source domains, we have: +ϵAcc +DT +� +�h +� +≤ 1 +N +N +� +i=1 +ϵAcc +DS +i +� +�h +� ++ +√ +2C +N +N +� +i=1 +dJS +� +P Z,Y +DT , P Z,Y +DS +i +� +(1) +≤ 1 +N +N +� +i=1 +ϵAcc +DS +i +� +�h +� ++ +√ +2C +N +N +� +i=1 +dJS +� +P Z,Y +DT , P Z,Y +DS +∗ +� ++ +√ +2C +N +N +� +i=1 +dJS +� +P Z,Y +DS +∗ , P Z,Y +DS +i +� +(2) +≤ 1 +N +N +� +i=1 +ϵAcc +DS +i +� +�h +� ++ +√ +2C min +i∈[N]dJS +� +P Z,Y +DT , P Z,Y +DS +i +� ++ +√ +2C max +i,j∈[N]dJS +� +P Z,Y +DS +i , P Z,Y +DS +j +� +(7) +Here we have +(1) +≤ by using triangle inequality for JS-distance: dJS(P, R) ≤ dJS(P, Q) + dJS(Q, R) +with P, Q, and R = PDT , PDS +∗ and PDS +i , respectively. We have +(2) +≤ because DS +∗ ∈ {DS +i }N +i=1 then +dJS +� +P Z,Y +DS +∗ , P Z,Y +DS +i +� +≤ max +i,j∈[N]dJS +� +P Z,Y +DS +i , P Z,Y +DS +j +� +. Similarly, we can obtain the upper bound based +on the feature space X as follows. +ϵAcc +DT +� +�f +� +≤ 1 +N +N +� +i=1 +ϵAcc +DS +i +� +�f +� ++ +√ +2C min +i∈[N]dJS +� +P X,Y +DT , P X,Y +DS +i +� ++ +√ +2C max +i,j∈[N]dJS +� +P X,Y +DS +i +, P X,Y +DS +j +� +(8) +However, the bounds in Eq. (7) and Eq. (8) are based on either feature space or representation space, +which is not readily to use for practical algorithmic design because the actual objective is to minimize +ϵAcc +DT +� +�f +� +in feature space by controlling Z in representation space. According to Lemmas 9 and 10, +we can derive the bound that relates feature and representation spaces as follows. +ϵAcc +DT +� +�f +� += ϵAcc +DT +� +�h +� +≤ 1 +N +N +� +i=1 +ϵAcc +DS +i +� +�h +� ++ +√ +2C min +i∈[N]dJS +� +P Z,Y +DT , P Z,Y +DS +i +� ++ +√ +2C max +i,j∈[N]dJS +� +P Z,Y +DS +i , P Z,Y +DS +j +� +≤ 1 +N +N +� +i=1 +ϵAcc +DS +i +� +�f +� ++ +√ +2C min +i∈[N]dJS +� +P X,Y +DT , P X,Y +DS +i +� ++ +√ +2C max +i,j∈[N]dJS +� +P Z,Y +DS +i , P Z,Y +DS +j +� +(9) +Proof of Corollary 1.1. +dJS +� +P Z,Y +DS +i , P Z,Y +DS +j +� += +� +DJS +� +P Z,Y +DS +i +∥ P Z,Y +DS +j +� +(1) +≤ +� +DJS +� +P Y +DS +i ∥ P Y +DS +j +� ++ 2Ez∼PDS +i,j (z) +� +DJS +� +P Z|Y +DS +i +∥ P Z|Y +DS +j +�� +(2) +≤ dJS +� +P Y +DS +i , P Y +DS +j +� ++ +� +2Ez∼PDS +i,j (z) +� +dJS +� +P Z|Y +DS +i , P Z|Y +DS +j +�2� +Here we have +(1) +≤ by using Lemma 11 to decompose the JS-divergence of the joint distributions and +(2) +≤ by using inequality +√ +a + b ≤ √a + +√ +b. +31 + +Published as a conference paper at ICLR 2023 +This new upper bound, combined with Thm. 1 suggests learning representation Z such that P Z|Y +DS +i +is invariant across source domains, or in another word, Z ⊥ D | Y . This result is consistent with +Thm. 4: when the target domain DT is the mixture of source domains {DS +i }N +i=1, and when P Y +DS +i and +P Z|Y +DS +i +are invariant across source domains, we have dJS +� +P Z,Y +DT , P Z,Y +DS +i +� += dJS +� +P Z,Y +DS +i , P Z,Y +DS +j +� += 0, +implying ϵAcc +DT +� +�f +� +≤ 1 +N +�N +i=1 ϵAcc +DS +i +� +�f +� += ϵAcc +DS +i +� +�f +� +∀i ∈ [N]. +Proof of Corollary 5.1 (tighter upper bound for accuracy). +The bound in Eq. (9) is constructed +using Lemma 9. Indeed, we can make this bound tighter using the strong data processing inequality +for JS-divergence (Polyanskiy & Wu, 2017), as stated below. +DJS +� +P Z +Di ∥ P Z +Dj +� +≤ ηJSDJS +� +P X +Di ∥ P X +Dj +� +≤ ηT V DJS +� +P X +Di ∥ P X +Dj +� +where Z is random variable induced from random variable X, and P X +Di and P X +Di are two distribution +over X, and ηJS = +sup +P X +Di̸=P X +Dj +DJS +� +P Z +Di,P Z +Dj +� +DJS +� +P X +Di,P X +Dj +� ≤ ηT V = +sup +P X +Di̸=P X +Dj +DT V +� +P Z +Di,P Z +Dj +� +DT V +� +P X +Di,P X +Dj +� ≤ 1, DT V is the +total variation distance. ηT V is called the Dobrushin’s coefficient (Polyanskiy & Wu, 2017). +Apply Lemma 11 and this inequality to the second term in the right hand side of Eq. (7) (similar to +the proof of Corollary 1.1), we have: +√ +2C min +i∈[N]dJS +� +P Z,Y +DT , P Z,Y +DS +i +� +≤ +√ +2C min +i∈[N] +� +dJS +� +P Y +DT , P Y +DS +i +� ++ +� +2Ez∼PDT +i (z) +� +dJS +� +P Z|Y +DT , P Z|Y +DT +i +�2�� +≤ +√ +2C min +i∈[N] +� +dJS +� +P Y +DT , P Y +DS +i +� ++ +� +2ηT V Ez∼PDT +i (z) +� +dJS +� +P X|Y +DT , P X|Y +DT +i +�2�� +(10) +Proof of Theorem 2. +Consider a source domain DS +i and target domain DT . Because JS-distance +dJS(·, ·) is a distance metric, we have triangle inequality: +dJS(P Y +DS +i , P Y +DT ) ≤ dJS(P Y +DS +i , P +�Y +DS +i ) + dJS(P +�Y +DS +i , P +�Y +DT ) + dJS(P +�Y +DT , P Y +DT ) +Since X +g +−→ Z +�h +−→ �Y , we have dJS(P �Y +DS +i , P �Y +DT ) ≤ dJS(P Z +DS +i , P Z +DT ). Using Lemma 12, the +following holds when dJS(P Y +DS +i , P Y +DT ) ≥ dJS(P Z +DS +i , P Z +DT ) +� +dJS(P Y +DS +i , P Y +DT ) − dJS(P Z +DS +i , P Z +DT ) +�2 +≤ +� +dJS(P Y +DS +i , P +�Y +DS +i ) + dJS(P +�Y +DT , P Y +DT ) +�2 +≤ +2 +� +dJS(P Y +DS +i , P +�Y +DS +i )2 + dJS(P +�Y +DT , P Y +DT )2� +≤ +2 +� +2c +|Y| +�� +ϵAcc +DS +i ( �f) + +� +ϵAcc +DT ( �f) +� +≤ +� +4|Y| +c +� +ϵAcc +DS +i ( �f) + ϵAcc +DT ( �f) +� +The last inequality is by AM-GM inequality. +Therefore, when dJS(P Y +DS +i , P Y +DT ) ≥ dJS(P Z +DS +i , P Z +DT ), we have +ϵAcc +DS +i ( �f) + ϵAcc +DT ( �f) ≥ +c +4|Y| +� +dJS(P Y +DS +i , P Y +DT ) − dJS(P Z +DS +i , P Z +DT ) +�4 +The above holds for any source domain DS +i . Average over all N source domains, we have +1 +N +N +� +i=1 +ϵAcc +DS +i ( �f) + ϵAcc +DT ( �f) ≥ +c +4|Y|N +N +� +i=1 +� +dJS(P Y +DS +i , P Y +DT ) − dJS(P Z +DS +i , P Z +DT ) +�4 +32 + +Published as a conference paper at ICLR 2023 +Proof of Theorem 3. +The proof is based on Lemmas 15 and 16 and similar to the proof of Thm. 1. +Let DS +∗ ∈ {DS +i }N +i=1 be the source domain nearest to the target domain DT . According to Lemma 15, +we have upper bound of the unfairness measured with respect to the representation space for the +target domain based on each of the source domain. For equal opportunity (EP), we have: +ϵEP +DT +� +�h +� +≤ ϵEP +DS +i +� +�h +� ++ +√ +2 +� +a∈{0,1} +dJS +� +P Z|Y =1,A=a +DT +, P Z|Y =1,A=a +DS +i +� +Taking average of upper bounds based on all source domains, we have: +ϵEP +DT +� +�h +� +≤ 1 +N +N +� +i=1 +ϵEP +DS +i +� +�h +� ++ +√ +2 +N +N +� +i=1 +� +a∈{0,1} +dJS +� +P Z|Y =1,A=a +DT +, P Z|Y =1,A=a +DS +i +� +≤ 1 +N +N +� +i=1 +ϵEP +DS +i +� +�h +� ++ +√ +2 +N +N +� +i=1 +� +a∈{0,1} +dJS +� +P Z|Y =1,A=a +DT +, P Z|Y =1,A=a +DS +∗ +� ++ +√ +2 +N +N +� +i=1 +� +a∈{0,1} +dJS +� +P Z|Y =1,A=a +DS +∗ +, P Z|Y =1,A=a +DS +i +� +≤ 1 +N +N +� +i=1 +ϵEP +DS +i +� +�h +� ++ +√ +2 min +i∈[N] +� +a∈{0,1} +dJS +� +P Z|Y =1,A=a +DT +, P Z|Y =1,A=a +DS +i +� ++ +√ +2 max +i,j∈[N] +� +a∈{0,1} +dJS +� +P Z|Y =1,A=a +DS +i +, P Z|Y =1,A=a +DS +j +� +According to Lemmas 9 and 16. we can relate this bound to the feature space as follows. +ϵEP +DT +� +�f +� += ϵEP +DT +� +�h +� +≤ 1 +N +N +� +i=1 +ϵEP +DS +i +� +�h +� ++ +√ +2 min +i∈[N] +� +a∈{0,1} +dJS +� +P Z|Y =1,A=a +DT +, P Z|Y =1,A=a +DS +i +� ++ +√ +2 max +i,j∈[N] +� +a∈{0,1} +dJS +� +P Z|Y =1,A=a +DS +i +, P Z|Y =1,A=a +DS +j +� +≤ 1 +N +N +� +i=1 +ϵEP +DS +i +� +�f +� ++ +√ +2 min +i∈[N] +� +a∈{0,1} +dJS +� +P X|Y =1,A=a +DT +, P X|Y =1,A=a +DS +i +� ++ +√ +2 max +i,j∈[N] +� +a∈{0,1} +dJS +� +P Z|Y =1,A=a +DS +i +, P Z|Y =1,A=a +DS +j +� +Similarly, we got the upper bound for unfairness measure with respect to equalized odds as follows. +ϵEO +DT +� +�f +� +≤ 1 +N +N +� +i=1 +ϵEO +DS +i +� +�f +� ++ +√ +2 min +i∈[N] +� +y∈{0,1} +� +a∈{0,1} +dJS +� +P X|Y =y,A=a +DT +, P X|Y =y,A=a +DS +i +� ++ +√ +2 max +i,j∈[N] +� +y∈{0,1} +� +a∈{0,1} +dJS +� +P Z|Y =y,A=a +DS +i +, P Z|Y =y,A=a +DS +j +� +(11) +Proof of Theorem 4. +Consider two source domains, DS +i and DS +j , if P Y +DS +i = P Y +DS +j , we can learn the +mapping function g = Pθ (Z|X) such that P Z|Y +DS +i += P Z|Y +DS +j . Note that this mapping function always +exists. In particular, the trivial solution for Z that satisfies P Z|Y +DS +i += P Z|Y +DS +j +is making Z ⊥ Y, D (e.g., +33 + +Published as a conference paper at ICLR 2023 +Pθ (Z|X) = N (0, I)). Then we have: +ϵAcc +DS +i +� +�h +� += Ez∼P Z +DS +i +,y∼hDS +i (z) +� +L +� +�h (Z) , Y +�� += Ey∼P Y +DS +i +,z∼P Z|Y +DS +i +� +L +� +�h (Z) , Y +�� += Ey∼P Y +DS +j +,z∼P Z|Y +DS +j +� +L +� +�h (Z) , Y +�� += Ez∼P Z +DS +j +,y∼hDS +j (z) +� +L +� +�h (Z) , Y +�� += ϵAcc +DS +j +� +�h +� +For unseen target domain DT in Λ, we have: +ϵAcc +DT +� +�h +� += EDT +� +L +� +�h (Z) , Y +�� += +� +Z×Y +L +� +�h (Z) , Y +� +P Y,Z +DT dY dZ += +� +Z×Y +L +� +�h (Z) , Y +� N +� +i=1 +πiP Y,Z +DS +i +dY dZ += +N +� +i=1 +πi +� +Z×Y +L +� +�h (Z) , Y +� +P Y,Z +DS +i +dY dZ += +N +� +i=1 +πiEDS +i +� +L +� +�h (Z) , Y +�� += EDS +i +� +L +� +�h (Z) , Y +�� +∀i ∈ [N] += ϵAcc +DS +i +� +�h +� +∀i ∈ [N] +By Lemma 10, we have ϵAcc +DT +� +�h +� += ϵAcc +DS +i +� +�h +� += ϵAcc +DT +� +�f +� += ϵAcc +DS +i +� +�f +� +. +For fairness, we only give the proof for equalized odds (EO), we can easily get the similar derivation +for equal opportunity. For any Z that satisfies P Z|Y =y,A=a +DS +i += P Z|Y =y,A=a +DS +j +∀y, a ∈ {0, 1}, we have: +ϵEO +DS +i +� +�h +� += +� +y∈{0,1} +D +� +P +�h(Z)1|Y =y,A=0 +DS +i +∥ P +�h(Z)1|Y =y,A=1 +DS +i +� += +� +y∈{0,1} +D +� +P +�h(Z)1|Y =y,A=0 +DS +j +∥ P +�h(Z)1|Y =y,A=1 +DS +j +� += ϵEO +DS +j +� +�h +� +For unseen target domain DT in Λ, we have: +ϵEO +DT +� +�h +� += +� +y∈{0,1} +D +� +P +�h(Z)1|Y =y,A=0 +DT +∥ P +�h(Z)1|Y =y,A=1 +DT +� += +� +y∈{0,1} +D +� N +� +i=1 +πiP +�h(Z)1|Y =y,A=0 +DS +i +∥ +N +� +i=1 +πiP +�h(Z)1|Y =y,A=1 +DS +i +� += +� +y∈{0,1} +D +� +P +�h(Z)1|Y =y,A=0 +DS +i +∥ P +�h(Z)1|Y =y,A=1 +DS +i +� +∀i ∈ [N] += ϵEO +DS +i +� +�h +� +∀i ∈ [N] +34 + +Published as a conference paper at ICLR 2023 +Similar to the proof of accuracy, Z that satisfies P Z|Y =y,A=a +DS +i += P Z|Y =y,A=a +DS +j +∀y, a ∈ {0, 1}, i, j ∈ +[N] always exists. The trivial solution for is Z that satisfies Z ⊥ Y, A, D. +By Lemma 16, we have ϵEO +DT +� +�h +� += ϵEO +DS +i +� +�h +� += ϵEO +DT +� +�f +� += ϵEO +DS +i +� +�f +� +. +For equal opportunity (EP), Z only need to satisfy the condition for positive label, i.e., P Z|Y =1,A=a +DS +i += +P Z|Y =1,A=a +DS +j +∀a ∈ {0, 1}, i, j ∈ [N]. +Proof of Theorem 5. +According to Lemma 17, we have: +DJS +� +P Z|y +Di +∥ P Z|y +Dj +� +≤ +� +x +P x|y +Dj DJS +� +P Z|x +Di +∥ P +Z|my +i,j(x) +Di +� +dx +(12) +Then, +minimizing +DJS +� +P Z|y +Di +∥ P Z|y +Dj +� +can +be +achieved +by +minimizing +DJS +� +P Z|x ∥ P Z|my +i,j(x)� +∀x ∈ X. +We can upper bound DJS +� +P Z|x ∥ P Z|my +i,j(x)� +as +follows +DJS +� +P Z|x ∥ P Z|my +i,j(x)� +≤ DT V +� +P Z|x ∥ P Z|my +i,j(x)� +≤ +√ +2 d1/2 +� +P Z|x, P Z|my +i,j(x)� +(1) += +√ +2 d1/2 +� +N +� +µ(x); σ2Id +� +, N +� +µ +� +my +i,j(x) +� +; σ2Id +�� +(13) +where DT V and d1/2 are total variation distance and Hellinger distance between two distributions, +respectively. We have +(1) += because of our choice for representation mapping g(x) := P Z|x = +N +� +µ(x); σ2Id +� +. According to Devroye et al. (2018), the Hellinger distance between two multivariate +normal distributions over Rd has a closed form as follows +d1/2 (N (µ1; Σ1) , N (µ2; Σ2)) += +� +� +� +�1 − det (Σ1)1/4 det (Σ2)1/4 +det +� Σ1+Σ2 +2 +�1/2 +exp +� +−1 +8 (µ1 − µ2)T +�Σ1 + Σ2 +2 +�−1 +(µ1 + µ2) +� +(14) +where µ1, µ2, Σ1, Σ2 are mean vectors and covariance matrices of the two normal distributions. In +Eq. (14), let µ1 = µ(x), µ2 = µ +� +my +i,j(x) +� +, Σ1 = Σ2 = σ2Id, then we have: +d1/2 +� +N +� +µ(x); σ2Id +� +, N +� +µ +� +my +i,j(x) +� +; σ2Id +�� += +� +1 − exp +� +− +1 +8dσ2 +� +µ (x) − µ +� +my +i,j(x) +��T � +µ (x) − µ +� +my +i,j(x) +��� += +� +1 − exp +� +− +1 +8dσ2 +��µ (x) − µ +� +my +i,j(x) +���2 +2 +� +(15) +From Eq. (15), we can see that Helinger distance between two representation distributions P Z|x +and P Z|my +i,j(x) is the function of their means µ (x) and µ +� +my +i,j(x) +� +. Combining this with Eq. +(12) and Eq. (13), we conclude that minimizing dJS +� +P Z|y +DS +i , P Z|y +DS +j +� +can be reduced to minimizing +��µ(x) − µ +� +my +i,j(x) +��� +2 which can be implemented as the mean square error between µ(x) and +µ +� +my +i,j(x) +� +in practice. Proof for dJS +� +P Z|y,a +DS +i +, P Z|y,a +DS +j +� +is derived in the similar way. +35 + +Published as a conference paper at ICLR 2023 +E.2 +PROOFS OF LEMMAS +Proof of Lemma 6. +We have: +� +E +���P X +Dj − P X +Di +��� dX = +� +E +� +P X +Dj − P X +Di +� +dX += +� +E∪E +� +P X +Dj − P X +Di +� +dX − +� +E +� +P X +Dj − P X +Di +� +dX += +� +E +� +P X +Di − P X +Dj +� +dX += +� +E +���P X +Dj − P X +Di +��� dX += 1 +2 +� ���P X +Dj − P X +Di +��� dX +Proof of Lemma 7. +We have: +EDj [f(X)] = +� +X +f(X)P X +DjdX = +� +X +f(X)P X +DidX + +� +X +f(X) +� +P X +Dj − P X +Di +� +dX += EDi [f(X)] + +� +X +f(X) +� +P X +Dj − P X +Di +� +dX += EDi [f(X)] + +� +E +f(X) +� +P X +Dj − P X +Di +� +dX + +� +E +f(X) +� +P X +Dj − P X +Di +� +dX +(1) +≤ EDi [f(X)] + +� +E +f(X) +� +P X +Dj − P X +Di +� +dX +(2) +≤ EDi [f(X)] + C +� +E +� +P X +Dj − P X +Di +� +dX += EDi [f(X)] + C +� +E +���P X +Dj − P X +Di +��� dX +(3) +≤ EDi [f(X)] + C +2 +� ���P X +Dj − P X +Di +��� dX +(4) +≤ EDi [f(X)] + C +2 +� +2 min +� +DKL +� +P X +Di ∥ P X +Dj +� +, DKL +� +P X +Dj ∥ P X +Di +�� += EDi [f(X)] + C +√ +2 +� +min +� +DKL +� +P X +Di ∥ P X +Dj +� +, DKL +� +P X +Dj ∥ P X +Di +�� +where E is the event that P X +Dj ≥ P X +Di and E is the complement of E. We have +(1) +≤ because +� +E f(X) +� +P X +Dj − P X +Di +� +dX ≤ 0; +(2) +≤ because f(X) is non-negative function and is bounded by +C; +(3) +≤ by using Lemma 6; +(4) +≤ by using Pinsker’s inequality between total variation norm and KL- +divergence. +Proof of Lemma 8. +Applying Lemma 7 and replacing X by (X, Y ), f by loss function L, Di by +Di,j, we have: +ϵAcc +Dj +� +�f +� +− EDi,j +� +L( �f(X), Y ) +� += EDj +� +L( �f(X), Y ) +� +− EDi,j +� +L( �f(X), Y ) +� +≤ C +√ +2 +� +min +� +DKL +� +P X,Y +Dj +∥ P X,Y +Di,j +� +, DKL +� +P X,Y +Di,j ∥ P X,Y +Dj +�� +≤ C +√ +2 +� +DKL +� +P X,Y +Dj +∥ P X,Y +Di,j +� +(16) +36 + +Published as a conference paper at ICLR 2023 +Applying Lemma 7 again and replacing X by (X, Y ), f by loss function L, Dj by Di,j, we have: +EDi,j +� +L( �f(X), Y ) +� +− ϵAcc +Di +� +�f +� += EDi,j +� +L( �f(X), Y ) +� +− EDi +� +L( �f(X), Y ) +� +≤ C +√ +2 +� +min +� +DKL +� +P X,Y +Di +∥ P X,Y +Di,j +� +, DKL +� +P X,Y +Di,j ∥ P X,Y +Di +�� +≤ C +√ +2 +� +DKL +� +P X,Y +Di +∥ P X,Y +Di,j +� +(17) +Adding Eq. (16) to Eq. (17), we have: +ϵAcc +Dj +� +�f +� +− ϵAcc +Di +� +�f +� +≤ C +√ +2 +�� +DKL +� +P X,Y +Di +∥ P X,Y +Di,j +� ++ +� +DKL +� +P X,Y +Dj +∥ P X,Y +Di,j +�� +(1) +≤ C +√ +2 +� +2 +� +DKL +� +P X,Y +Di +∥ P X,Y +Di,j +� ++ DKL +� +P X,Y +Dj +∥ P X,Y +Di,j +�� += C +√ +2 +� +4DJS +� +P X,Y +Di +∥ P X,Y +Dj +� += +√ +2CdJS +� +P X,Y +Di +, P X,Y +Dj +� +Here we have +(1) +≤ by using Cauchy–Schwarz inequality. +Proof of Lemma 9. +Note that the JS-divergence DJS +� +P X +Di ∥ P X +Dj +� +can be understood as the +mutual information between a random variable X associated with the mixture distribution P X +Di,j = +1 +2 +� +P X +Di + P X +Dj +� +and the equiprobable binary random variable T used to switch between P X +Di and +P X +Dj to create the mixture distribution P X +Di,j. In particular, we have: +DJS +� +P X +Di ∥ P X +Dj +� += 1 +2 +� +DKL +� +P X +Di ∥ P X +Di,j +� ++ DJS +� +P X +Dj ∥ P X +Di,j +�� += 1 +2 +� � +log P X +Di − log P X +Di,j +� +P X +DidX ++ 1 +2 +� � +log P X +Dj − log P X +Di,j +� +P X +DjdX += +�1 +2 +� +log +� +P X +Di +� +P X +Didx + 1 +2 +� +log +� +P X +Dj +� +P X +DjdX +� +− +� +log +� +P X +Di,j +� +P X +Di,jdX += −H(X|T) + H(X) += I(X; T) +where H(X) is the entropy of X, H(X|T) is the entropy of X conditioned on T, and I(X; T) is +the mutual information between X and T. Similarly, we also have DJS((P Z +Di ∥ P Z +Dj)) = I(Z; T). +Because Z is induced from X by the mapping function h then we have Z ⊥ T | X and the Markov +chain T → X → Z. According to data processing inequality for mutual information (Polyanskiy & +Wu, 2014), we have I(X; T) ≥ I(Z; T) which implies DJS((P X +Di ∥ P X +Dj)) ≥ DJS((P Z +Di ∥ P Z +Dj)). +Taking square root on both sides, we have dJS(P X +Di, P X +Dj) ≥ dJS(P Z +Di, P Z +Dj). +37 + +Published as a conference paper at ICLR 2023 +Proof of Lemma 10. +We have: +ϵAcc +D +� +�h +� += Ez∼P Z +D ,y∼hD(z) +� +L +� +�h (Z) , Y +�� += +|Y| +� +y=1 +Ez∼P Z +D +� +L +� +�h (Z) , y +� +hD(Z)y +� += +|Y| +� +y=1 +� +Z +L +� +�h (Z) , y +� +hD(Z)yP Z +DdZ += +|Y| +� +y=1 +� +Z +L +� +�h (Z) , y +� � +g−1(Z) fD(X)yP X +D dX +� +g−1(Z) P X +D dX +� +g−1(Z) +P X +D dXdZ += +|Y| +� +y=1 +� +Z +L +� +�h (Z) , y +� � +g−1(Z) +fD(X)yP X +D dXdZ += +|Y| +� +y=1 +� +Z +� +g−1(Z) +L +� +�h (g(X)) , y +� +fD(X)yP X +D dXdZ += +|Y| +� +y=1 +� +Z +� +X +1 +� +X ∈ g−1(Z) +� +L +� +�h (Z) , y +� +fD(X)yP X +D dXdZ += +|Y| +� +y=1 +� +X +� +Z +1 (Z = g(X)) L +� +�h (Z) , y +� +fD(X)yP X +D dXdZ += +|Y| +� +y=1 +� +X +L +� +�h (g(X)) , y +� +fD(X)yP X +D dXdZ += +|Y| +� +y=1 +� +X +L +� +�f (X) , y +� +fD(X)yP X +D dX += ϵAcc +D +� +�f +� +Proof of Lemma 11. +We show the decomposition for KL-divergence first and then use the result to +derive the decomposition for JS-divergence. We have: +DKL +� +P X,Y +Di +∥ P X,Y +Dj +� += EDi +� +log P X,Y +Di +− log P X,Y +Dj +� += EDi +� +log P Y +Di + log P X|Y +Di +� +− EDi +� +log P Y +Dj + log P X|Y +Dj +� += EDi +� +log P Y +Di − log P Y +Dj +� ++ EDi +� +log P X|Y +Di +− log P X|Y +Dj +� += EDi +� +log P Y +Di − log P Y +Dj +� ++ Ey∼P Y +Di +� +Ex∼P X|y +Di +� +log P X|Y +Di +− log P X|Y +Dj +�� += DKL +� +P Y +Di ∥ P Y +Dj +� ++ EDi +� +DKL +� +P X|Y +Di +∥ P X|Y +Dj +�� +38 + +Published as a conference paper at ICLR 2023 +DJS +� +P X,Y +Di +∥ P X,Y +Dj +� += 1 +2 +� +DKL +� +P X,Y +Di +∥ P X,Y +Di,j +�� ++ 1 +2 +� +DKL +� +P X,Y +Dj +∥ P X,Y +Di,j +�� += 1 +2 +� +DKL +� +P Y +Di ∥ P Y +Di,j +�� ++ 1 +2 +� +EDi +� +DKL +� +P X|Y +Di +∥ P X|Y +Di,j +��� ++ 1 +2 +� +DKL +� +P Y +Dj ∥ P Y +Di,j +�� ++ 1 +2 +� +EDj +� +DKL +� +P X|Y +Dj +∥ P X|Y +Di,j +��� += DJS +� +P Y +Di ∥ P Y +Dj +� ++ 1 +2 +� +EDi +� +DKL +� +P X|Y +Di +∥ P X|Y +Di,j +��� ++ 1 +2 +� +EDj +� +DKL +� +P X|Y +Dj +∥ P X|Y +Di,j +��� +≤ DJS +� +P Y +Di ∥ P Y +Dj +� ++ 1 +2 +� +EDi +� +DKL +� +P X|Y +Di +∥ P X|Y +Di,j +��� ++ 1 +2 +� +EDi +� +DKL +� +P X|Y +Dj +∥ P X|Y +Di,j +��� ++ 1 +2 +� +EDj +� +DKL +� +P X|Y +Dj +∥ P X|Y +Di,j +��� ++ 1 +2 +� +EDj +� +DKL +� +P X|Y +Di +∥ P X|Y +Di,j +��� += DJS +� +P Y +Di ∥ P Y +Dj +� ++ EDi +� +DJS +� +P X|Y +Di +∥ P X|Y +Dj +�� ++ EDj +� +DJS +� +P X|Y +Di +∥ P X|Y +Dj +�� +Proof of Lemma 12. +ED +� +L( �f(X), Y ) +� += +ED +� +�� +�y∈Y +�f(X)�yL(�y, Y ) +� +� +(1) +≥ +c EX +� +�� +�y∈Y +�f(X)�y Pr(Y ̸= �y|X) +� +� +(2) += +c EX +� +1 − �f(X)T f(X) +� +(3) +≥ +c +2 EX +���� �f(X) − f(X) +��� +2 +2 +� +(4) +≥ +c +2 +1 +|Y|EX +����� �f(X) − f(X) +��� +1 +�2� +(5) +≥ +c +2 +1 +|Y| +� ���EX +� +�f(X) − f(X) +���� +1 +�2 += +c +2 +1 +|Y| +���P +�Y +D − P Y +D +��� +2 +1 +(6) +≥ +2c +|Y|DJS +� +P Y +D ∥ P +�Y +D +�2 += +2c +|Y| · dJS +� +P Y +D , P +�Y +D +�4 +Here we have +(1) +≥ is because of the assumption that L(�y, y) is lower bounded by c when �y ̸= y; +(2) += is because �f(X)T 1 = || �f(X)||1 = 1; +(3) +≥ is because || �f(X)||2 ≤ || �f(X)||1 = 1; +(4) +≥ is because +|| �f(X)||2 ≥ +1 +√ +|Y||| �f(X)||1; +(5) +≥ is by using Jensen’s inequality; +(6) +≥ is by using JS-divergence lower +bound of total variation distance. +39 + +Published as a conference paper at ICLR 2023 +Proof of Lemma 14. +Similar to the proof in Lemma 8, we apply Lemma 7 for Ry,a +Di and Ry,a +Dj and +note that �f(X)y is bounded by 1. Then ∀y, a ∈ {0, 1}, we have: +Ry,a +Dj − EDi,j +� +�f(X)y|Y = y, A = a +� += EDj +� +�f(X)y|Y = y, A = a +� +− EDi,j +� +�f(X)y|Y = y, A = a +� +≤ +1 +√ +2 +� +min +� +DKL +� +P X|Y =y,A=a +Di +∥ P X|Y =y,A=a +Di,j +� +, DKL +� +P X|Y =y,A=a +Di,j +∥ P X|Y =y,A=a +Di +�� +≤ +1 +√ +2 +� +DKL +� +P X|Y =y,A=a +Dj +∥ P X|Y =y,A=a +Di,j +� +(18) +EDi,j +� +�f(X)y|Y = y, A = a +� +− Ry,a +Di += EDi,j +� +�f(X)y|Y = y, A = a +� +− EDi +� +�f(X)y|Y = y, A = a +� +≤ +1 +√ +2 +� +min +� +DKL +� +P X|Y =y,A=a +Dj +∥ P X|Y =y,A=a +Di,j +� +, DKL +� +P X|Y =y,A=a +Di,j +∥ P X|Y =y,A=a +Dj +�� +≤ +1 +√ +2 +� +DKL +� +P X|Y =y,A=a +Di +∥ P X|Y =y,A=a +Di,j +� +(19) +Adding Eq. (18) to Eq. (19), we have: +Ry,a +Dj − Ry,a +Di +≤ +1 +√ +2 +�� +DKL +� +P X|Y =y,A=a +Di +∥ P X|Y =y,A=a +Di,j +� ++ +� +DKL +� +P X|Y =y,A=a +Dj +∥ P X|Y =y,A=a +Di,j +�� +≤ +√ +2dJS +� +P X|Y =y,A=a +Dj +, P X|Y =y,A=a +Di,j +� +Proof of Lemma 15. +We give the proof for unfairness measure w.r.t. to equal opportunity first and +then use this result to derive the proof for unfairness measure w.r.t. to equalized odd. Without loss of +generality, assign group indices 1, 0 be such that R1,0 +Dj +� +�f +� +≥ R1,1 +Dj +� +�f +� +. Then we have: +ϵEP +Dj +� +�f +� += +���R1,0 +Dj +� +�f +� +− R1,1 +Dj +� +�f +���� += R1,0 +Dj +� +�f +� +− R1,1 +Dj +� +�f +� += R1,0 +Dj +� +�f +� +− EDj +� +�f(X)1|Y = 1, A = 1 +� += R1,0 +Dj +� +�f +� ++ EDj +� +1 − �f(X)1|Y = 1, A = 1 +� +− 1 += R1,0 +Dj +� +�f +� ++ R1,1 +Dj +� +1 − �f +� +− 1 +where 1 is vector with all 1’s. By Lemma 14, we have: +R1,0 +Dj +� +�f +� +≤ R1,0 +Di +� +�f +� ++ +√ +2dJS +� +P X|Y =1,A=0 +Dj +, P X|Y =1,A=0 +Di +� +R1,1 +Dj +� +1 − �f +� +≤ R1,1 +Di +� +1 − �f +� ++ +√ +2dJS +� +P X|Y =1,A=1 +Dj +, P X|Y =1,A=1 +Di +� +Sum above two inequalities and add −1 at both sides, we have, +ϵEP +Dj +� +�f +� += R1,0 +Dj +� +�f +� ++ R1,1 +Dj +� +1 − �f +� +− 1 +≤ R1,0 +Di +� +�f +� ++ R1,1 +Di +� +1 − �f +� +− 1 + +√ +2 +� +a=0,1 +dJS +� +P X|Y =1,A=a +Dj +, P X|Y =1,A=a +Di +� +≤ ϵEP +Di +� +�f +� ++ +√ +2 +� +a=0,1 +dJS +� +P X|Y =1,A=a +Dj +, P X|Y =1,A=a +Di +� +(20) +40 + +Published as a conference paper at ICLR 2023 +Similarly, we have: +���R0,0 +Dj +� +�f +� +− R0,1 +Dj +� +�f +���� ≤ +���R0,0 +Di +� +�f +� +− R0,1 +Di +� +�f +���� + +√ +2 +� +a=0,1 +dJS +� +P X|Y =0,A=a +Dj +, P X|Y =0,A=a +Di +� +(21) +Sum both Eq. (20) and Eq. (21), we have: +ϵEO +Dj +� +�f +� +≤ ϵEO +Di +� +�f +� ++ +√ +2 +� +y=0,1 +� +a=0,1 +dJS +� +P X|Y =y,A=a +Dj +, P X|Y =y,A=a +Di +� +Proof of Lemma 16. +Similar to the proof of Lemma 10, Ry,a +Di +� +�f +� += Ry,a +Di +� +�h +� +∀y, a ∈ {0, 1}. +Then, we have: +ϵEO +Di +� +�f +� += +���R0,0 +Di +� +�f +� +− R0,1 +Di +� +�f +���� + +���R1,0 +Di +� +�f +� +− R1,1 +Di +� +�f +���� += +���R0,0 +Di +� +�h +� +− R0,1 +Di +� +�h +���� + +���R1,0 +Di +� +�h +� +− R1,1 +Di +� +�h +���� += ϵEO +Di +� +�h +� +ϵEP +Di +� +�f +� += +���R1,0 +Di +� +�f +� +− R1,1 +Di +� +�f +���� += +���R1,0 +Di +� +�h +� +− R1,1 +Di +� +�h +���� += ϵEP +Di +� +�h +� +Proof of Lemma 17. +∀y ∈ Y, we have: +DJS +� +P Z|y +i +∥ P Z|y +j +� (1) += DJS +�� +X +P Z|xP x|y +i +dx ∥ +� +X +P Z|my +i,j(x)P +my +i,j(x)|y +j +dmy +i,j(x) +� +(2) += DJS +�� +X +P Z|xP x|y +i +dx ∥ +� +X +P Z|my +i,j(x)P +my +i,j(x)|y +j +dx +� +(3) += DJS +�� +X +P Z|xP x|y +i +dx ∥ +� +X +P Z|my +i,j(x)P x|y +i +dx +� +(4) +≤ +� +X +P x|y +i +DJS +� +P Z|x ∥ P Z|my +i,j(x)� +dx +Here we have +(1) += is because of law of total probability and Z ⊥ Y |X; +(2) += is because my +i,j is invertible +function; +(3) += is because P x|y +i += P +my +i,j(x)|y +j +∀x ∈ X; +(4) +≤ is because of joint complexity of JS +divergence. By similar derivation, ∀y ∈ Y, a ∈ A, we have: +DJS +� +P Z|y,a +i +∥ P Z|y,a +j +� +≤ +� +X +P x|y,a +i +DJS +� +P Z|x ∥ P Z|my,a +i,j (x)� +dx +41 + diff --git a/V9FQT4oBgHgl3EQfbTbA/content/tmp_files/load_file.txt b/V9FQT4oBgHgl3EQfbTbA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8621b387a6ab3881310604c1560749336df68bc7 --- /dev/null +++ b/V9FQT4oBgHgl3EQfbTbA/content/tmp_files/load_file.txt @@ -0,0 +1,2694 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf,len=2693 +page_content='Published as a conference paper at ICLR 2023 FAIRNESS AND ACCURACY UNDER DOMAIN GENER- ALIZATION Thai-Hoang Pham, Xueru Zhang, Ping Zhang The Ohio State University, Columbus, OH 43210, USA {pham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='375,zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='12807,zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='10631}@osu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='edu ABSTRACT As machine learning (ML) algorithms are increasingly used in high-stakes applica- tions, concerns have arisen that they may be biased against certain social groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Although many approaches have been proposed to make ML models fair, they typically rely on the assumption that data distributions in training and deployment are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Unfortunately, this is commonly violated in practice and a model that is fair during training may lead to an unexpected outcome during its deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Although the problem of designing robust ML models under dataset shifts has been widely studied, most existing works focus only on the transfer of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In this paper, we study the transfer of both fairness and accuracy under domain generalization where the data at test time may be sampled from never-before-seen domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We first develop theoretical bounds on the unfairness and expected loss at deployment, and then derive sufficient conditions under which fairness and accuracy can be perfectly transferred via invariant representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Guided by this, we design a learning algorithm such that fair ML models learned with training data still have high fairness and accuracy when deployment environments change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Experiments on real-world data validate the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Model implementation is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='com/pth1993/FATDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1 INTRODUCTION Machine learning (ML) algorithms trained with real-world data may have inherent bias and exhibit discrimination against certain social groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' To address the unfairness in ML, existing studies have proposed many fairness notions and developed approaches to learning models that satisfy these fairness notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' However, these works are based on an implicit assumption that the data distributions in training and deployment are the same, so that the fair models learned from training data can be deployed to make fair decisions on testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Unfortunately, this assumption is commonly violated in real-world applications such as healthcare e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', it was shown that most US patient data for training ML models are from CA, MA, and NY, with almost no representation from the other 47 states (Kaushal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Because of the distribution shifts between training and deployment, a model that is accurate and fair during training may behave in an unexpected way and induce poor performance during deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Therefore, it is critical to account for distribution shifts and learn fair models that are robust to potential changes in deployment environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The problem of learning models under distribution shifts has been extensively studied in the literature and is typically referred to as domain adaptation/generalization, where the goal is to learn models on source domain(s) that can be generalized to a different (but related) target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Specifically, domain adaptation requires access to (unlabeled) data from the target domain at training time, and the learned model can only be used at a specific target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In contrast, domain generalization considers a more general setting when the target domain data are inaccessible during training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' instead it assumes there exists a set of source domains based on which the learned model can be generalized to an unseen, novel target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' For both problems, most studies focus only on the generalization of accuracy across domains without considering fairness, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', by theoretically examining the relations between accuracy at target and source domains (Mansour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Hoffman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Phung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Deshmukh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Muandet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Blanchard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Albuquerque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Sicilia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Shui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2022) or/and developing practical methods (Albuquerque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Sun & 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='13323v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='LG] 30 Jan 2023 Published as a conference paper at ICLR 2023 Saenko, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Ganin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Ilse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' To the best of our knowledge, only Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Coston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Rezaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Oneto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Madras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Schumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2020) considered the transfer of fairness across domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' However, all of them focused on domain adaptation, and many also imposed rather strong assumptions on distributional shifts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', covariate shifts (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Coston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Rezaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021), demographic shift (Giguere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2022), prior probability shift (Biswas & Mukherjee, 2021)) that may be violated in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Among them, most focused on empirically examining how fairness properties are affected under distributional shifts, whereas theoretical understandings are less studied (Schumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Details and more related works are in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' White Asian Black Others Hispanic or Latino .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Training Data White Asian Black Others Hispanic or Latino (Fair) ML CA NY MT Source Domain 1 Source Domain N OH FL TX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Model Unseen Target Domain Figure 1: An example of domain gener- alization in healthcare: (fair) ML model trained with patient data in CA, NY, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', can be deployed in other states by main- taining high accuracy/fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In this paper, we study the transfer of both fairness and accuracy in domain generalization via invariant represen- tation learning, where the data in target domain is unknown and inaccessible during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' A motivating example is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Specifically, we first establish a new theoretical framework that develops interpretable bounds on accuracy/fairness at a target domain under domain gen- eralization, and then identify sufficient conditions under which fairness/accuracy can be perfectly transferred to an unseen target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Importantly, our theoretical bounds are fundamentally different from the existing bounds, com- pared to which ours are better connected with practical algorithmic design, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', our bounds are aligned with the ob- jective of adversarial learning-based algorithms, a method that is widely used in domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Inspired by the theoretical findings, we propose Fairness and Accuracy Transfer by Density Matching (FATDM), a two-stage learning framework such that the representations and fair model learned with source domain data can be well-generalized to an unseen target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Last, we conduct the experiments on real-world data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' the empirical results show that fair ML models trained with our method still attain a high accuracy and fairness when deployment environments differ from the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Our main contributions and findings are summarized as follows: We consider the transfer of both accuracy and fairness in domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' To the best of our knowledge, this is the first work studying domain generalization with fairness consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We develop upper bounds for expected loss (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1) and unfairness (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 3) in target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Notably, our bounds are significantly different from the existing bounds as discussed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We also develop a lower bound for expected loss (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' it indicates an inherent tradeoff of the existing methods which learn marginal invariant representations for domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We identify sufficient conditions under which fairness and accuracy can be perfectly transferred from source domains to target domains using invariant representation learning (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We propose a two-stage training framework (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', based on Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 5) that learns models in source domains (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 4), which can generalize both accuracy and fairness to target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We conduct experiments on real-world data to validate the effectiveness of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 2 PROBLEM FORMULATION Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Let X, A, and Y denote the space of features, sensitive attribute (distinguishing different groups, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', race/gender), and label, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Let Z be the representation space induced from X by representation mapping g : X → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We use X, A, Y , Z to denote random variables that take values in X, A, Y, Z and x, a, y, z the realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' A domain D is specified by distribution PD : X × A × Y → [0, 1] and labeling function fD : X → Y∆, where ∆ is a probability simplex over Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Similarly, let hD : Z → Y∆ be a labeling function from representation space for domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Note that fD, hD, g are stochastic functions and fD = hD ◦ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1 For simplicity, we use P V D (or P V |U D ) to denote the induced marginal (or conditional) distributions of variable V (given U) in domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1The deterministic labeling function is a special case when it follows Dirac delta distribution in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 2 shutterstock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='com · 554075098O 00 O O Oshutterstock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='com · 1380896705画 Personal health recordsPublished as a conference paper at ICLR 2023 Error metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Consider hypothesis �f = �h ◦ g : X → Y∆, where �h : Z → Y∆ is the hypothesis directly used in representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Denote �f(x)y as the element on y-th dimension which predicts the probability that label Y = y given X = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Then the expected error of �f in domain D is defined as ϵAcc D ( �f) = ED[L( �f(X), Y )] for some loss function L : Y∆ × Y → R+ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 0-1 loss, cross-entropy loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Similarly, define the expected error of �h in representation space as ϵAcc D (�h) = ED[L(�h(Z), Y )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Note that most existing works (Albuquerque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2018) focus on optimizing ϵAcc D (�h), while our goal is to attain high accuracy in input space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', low ϵAcc D ( �f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Unfairness metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We focus on group fairness notions (Makhlouf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021) that require certain statistical measures to be equalized across different groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' many of them can be formulated as (conditional) independence statements between random variables �f(X), A, Y , e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', demographic parity ( �f(X) ⊥ A: the likelihood of a positive outcome is the same across different groups) (Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2012) , equalized odds ( �f(X) ⊥ A|Y : true positive rate (TPR) and false positive rate (FPR) are the same across different groups), equal opportunity ( �f(X) ⊥ A|Y = 1 when Y = {0, 1}: TPR is the same across different groups) (Hardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In the paper, we will present the results under equalized odds (EO) fairness with binary Y = {0, 1} and A = {0, 1}, while all the results (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', methods, analysis) can be generalized to multi-class, multi-protected attributes, and other fairness notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Given hypothesis �f = �h ◦ g : X → Y∆, the violation of EO in domain D can be measured as ϵEO D ( �f) = � y∈Y D � P � f(X)1|Y =y,A=0 D ||P � f(X)1|Y =y,A=1 D � for some distance metric D(·||·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Problem setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Consider a problem of domain generalization where a learning algorithm has access to data {(xk, ak, yk, dk)}m k=1 sampled from a set of N source domains {DS i }i∈[N], where dk is the domain label and [N] = {1, · · · , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Our goal is to learn a representation mapping g : X → Z and a fair model �h : Z → Y∆ trained on source domains such that the model �f = �h ◦ g can be generalized to an unseen target domain DT in terms of both accuracy and fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Specifically, we investigate under what conditions and by what algorithms we can guarantee that attaining high accuracy and fairness at source domains {DS i }N i=1 implies small ϵAcc DT ( �f) and ϵEO DT ( �f) at unknown target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 3 THEORETICAL RESULTS In this section, we present the results on the transfer of accuracy/fairness under domain generalization via domain-invariant learning (proofs are shown in Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We first examine that for any model �h : Z → Y∆ and any representation mapping g : X → Z, how the accuracy/fairness attained at source domains {DS i }N i=1 can be affected when �f = �h ◦ g is deployed at any target domain DT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Specifically, we can bound the error and unfairness at any target domain based on source domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Before presenting the results, we first introduce the discrepancy measure used for measuring the dissimilarity between domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Discrepancy measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We adopt Jensen-Shannon (JS) distance (Endres & Schindelin, 2003) to measure the dissimilarity between two distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Formally, JS distance between distributions P and P ′ is defined as dJS(P, P ′) := � DJS(P||P ′), where DJS(P||P ′) := 1 2DKL(P|| P +P ′ 2 ) + 1 2DKL(P ′|| P +P ′ 2 ) is JS divergence defined based on Kullback–Leibler (KL) divergence DKL(·||·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Note that unlike KL divergence, JS divergence is symmetric and bounded: 0 ≤ DJS(P||P ′) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' While different discrepancy measures such as H and H∆H divergences (Ben-David et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2010) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', definitions are given in Appendix A) were used in prior works, we particularly consider JS distance because (1) it is aligned with training objective for discriminator in generative adversarial networks (GAN) (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2014), and many existing methods (Ganin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Albuquerque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019) for invariant representation learning are built based on GAN framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2) H and H∆H divergences are limited to settings where the labeling functions fD are deterministic (Ben-David et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Albuquerque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In contrast, our bounds admit the stochastic labeling functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The limitations of other discrepancy measures and existing bounds are discussed in detail in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 3 Published as a conference paper at ICLR 2023 Theorem 1 (Upper bound: accuracy) For any hypothesis �h : Z → Y∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' any representation map- ping g : X → Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' and any loss function L : Y∆ × Y → R+ that is upper bounded by C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' the expected error of �f = �h ◦ g : X → Y∆ at any unseen target domain DT is upper bounded:2 ϵAcc DT � �f � ≤ 1 N N � i=1 ϵAcc DS i � �f � � �� � term (i) + √ 2C min i∈[N]dJS � P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y DT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y DS i � � �� � term (ii) + √ 2C max i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j∈[N]dJS � P Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y DS i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y DS j � � �� � term (iii) (1) The upper bound in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (1) are interpretable and have three terms: term (i) is the averaged error of source domains in input space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' term (ii) is the discrepancy between the target domain and the source domains in input space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' term (iii) is the discrepancy between the source domains in representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3 It provides guidance on learning the proper representation mapping g : X → Z: to ensure small error at target domain ϵAcc DT ( �f), we shall learn representations such that the upper bound of ϵAcc DT ( �f) is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Because term (ii) depends on the unknown target domain DT and it’s evaluated in input space X × Y, it is fixed and is out of control during training, we can only focus on term (i) and term (iii), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', learn representations Z such that errors at source domains ϵAcc DS i ( �f) and the discrepancy between source domains in the representation space dJS(P Z,Y DS i , P Z,Y DS j ) are minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1 ∀i, j, JS distance between P Z,Y DS i and P Z,Y DS j in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (1) can be decomposed: dJS � P Z,Y DS i , P Z,Y DS j � = dJS � P Y DS i , P Y DS j � + � 2Ey∼P Y DS i,j � dJS � P Z|Y DS i , P Z|Y DS j �2� where P Y DS i,j = 1 2 � P Y DS i + P Y DS j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Our algorithm in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 4 is designed based on above decomposition: because P Y DS i solely depends on source domain DS i , we learn representations by minimizing dJS(P Z|Y DS i , P Z|Y DS j ), ∀i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Combining Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1, to ensure high accuracy at unseen target domain DT , we learn the representation mapping g and model �h such that P Z|Y DS i is invariant across source domains, and meanwhile �f = �h ◦ g attains high accuracy at source domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Note that unlike our method, many existing works (Phung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Albuquerque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Ganin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2016) suggest that to ensure high accuracy in domain generalization, representation mapping g should be learned such that P Z DS i is same across domains, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', small dJS(P Z DS i , P Z DT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' However, we show that the domain-invariant P Z DS i may adversely increase the error at target domain, as indicated in the Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Theorem 2 (Lower bound: accuracy) Suppose L( �f(x), y) = � ˆy∈Y �f(x)ˆyL(ˆy, y) where function L : Y × Y → R+ is lower bounded by c when ˆy ̸= y, and is 0 when ˆy = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' If dJS(P Y DS i , P Y DT ) ≥ dJS(P Z DS i , P Z DT ), the expected error of �f at source and target domains is lower bounded: 1 N N � i=1 ϵAcc DS i ( �f) + ϵAcc DT ( �f) ≥ c 4|Y|N N � i=1 � dJS(P Y DS i , P Y DT ) − dJS(P Z DS i , P Z DT ) �4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2) The above lower bound shows an inherent trade-off of approaches that minimize dJS(P Z DS i , P Z DT ) when learning the representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Specifically, with the domain-invariant P Z DS i , the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2) may increase, resulting in an increased error at target domain ϵAcc DT ( �f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 2The condition on the bounded loss is mild and can be satisfied by many loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' For example, cross- entropy loss can be bounded by modifying the softmax output from � p1, p2, · · · , p|Y| � to � ˆp1, ˆp2, · · · , ˆp|Y| �, where ˆpi = pi(1 − exp(−C)|Y|) + exp(−C), ∀i ∈ |Y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 3In fact, a tighter upper bound for the loss at target domain can be established using strong data processing inequality (Polyanskiy & Wu, 2017), as detailed in Appendix D 4 Published as a conference paper at ICLR 2023 Similar to the loss, the unfairness at target domain can also be upper bounded, as presented in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Theorem 3 (Upper bound: fairness) Consider a special case where the unfairness measure is defined as the distance between means of two distributions: ϵEO D ( �f) = � y∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} ���ED � �f(X)1|Y = y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' A = 0 � − ED � �f(X)1|Y = y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' A = 1 ���� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' then the unfairness at any unseen target domain DT is upper bounded: ϵEO DT � �f � ≤ 1 N N � i=1 ϵEO DS i � �f � + √ 2 min i∈[N] � y∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} � a∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} dJS � P X|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P X|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS i � + √ 2 max i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j∈[N] � y∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} � a∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} dJS � P Z|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS j � Similar to Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1, the upper bound in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 3 also has three terms and the second term is out of control during training because it depends on the unseen target domain and is defined in input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Therefore, to maintain fairness at target domain DT , we learn the representation mapping g and model �h such that P Z|Y,A DS i is invariant across source domains, and meanwhile �f = �h ◦ g attains high fairness at source domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The results above characterize the relations between accuracy/fairness at any target and source domains under any representation mapping g and model �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Next, we identify conditions under which the accuracy/fairness attained at sources can be perfectly transferred to a target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Theorem 4 (Sufficient condition for perfect transfer) Consider N source domains {DS i }N i=1 and an unseen target domain DT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Define set Λ = {Dt : Dt = �N i=1 πiDS i , {πi} ∈ ∆N−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (Transfer of fairness) ∀DT ∈ Λ, if g is the mapping under which P Z|Y,A DS i is the same across all source domains, then ϵEO DS i (�h) = ϵEO DT (�h) = ϵEO DS i ( �f) = ϵEO DT ( �f), ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (Transfer of accuracy) ∀DT ∈ Λ, if P Y DS i is the same and if g is the mapping under which P Z|Y DS i is the same across all source domains, then ϵAcc DS i (�h) = ϵAcc DT (�h) = ϵAcc DS i ( �f) = ϵAcc DT ( �f), ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 4 indicates the possibility of attaining the perfect transfer of accuracy/fairness and examples of such representation mappings are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Note that these results are consistent with Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1 and Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 3, which also suggest learning domain-invariant representations P Z|Y DS i and P Z|Y,A DS i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 4 PROPOSED ALGORITHM Table 1: Usages of terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (3) to guarantee the fairness and accuracy in target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Loss terms Usages Lcls Mimimize ϵAcc DS i Lfair Mimimize ϵEO DS i Linv Minimize dJS � P Z|Y DS i , P Z|Y DS j � and dJS � P Z|Y,A DS i , P Z|Y,A DS j � Lcls + Lfair + Linv Mimimize ϵAcc DT and ϵEO DT The accuracy and fairness upper bounds in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 3 shed light on designing robust ML model that can preserve high accuracy and fairness on unseen tar- get domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Specifically, the model consists of representation mapping g : X → Z and classi- fier �h : Z → Y such that (1) the prediction errors and unfairness of �f = �h ◦ g on source domains are minimized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' and (2) the discrepancy of learned conditional representations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e, P Z|Y DS i and P Z|Y,A DS i ) among source domains is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' That is, min g,�h Lcls(g,�h) + ωLfair(g,�h) + γLinv(g) (3) where Lcls, Lfair, and Linv are expected losses that penalize incorrect classification, unfairness, and discrepancy among source domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Hyper-parameters ω > 0 and γ > 0 control the accuracy- fairness trade-off and accuracy-invariant representation trade-off, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The usages of these three losses are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 5 Published as a conference paper at ICLR 2023 Adversarial learning framework (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Linv in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (3) can be optimized directly with adversarial learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' This is because the training objective of the discriminator in GAN is aligned with our goal of minimizing JS distance between P Z|Y DS i (or P Z|Y,A DS i ) among source domains, as mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Specifically, define a set of discriminators K = {ky : y ∈ Y} ∪ {ky,a : y ∈ Y, a ∈ A};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' each discriminator ky (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ky,a) aims to distinguish whether a sample with label y (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' label y and sensitive attribute a) comes from a particular domain (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', maximize Linv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The representation mapping g should be learned to increase the error of discriminators (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', minimize Linv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Therefore, the model and discriminators can be trained simultaneously by playing a two-player minimax game (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', ming maxK Linv(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Combine with the objective of minimizing prediction error and unfairness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', ming,�h Lcls(g,�h) + ωLfair(g,�h)), the overall learning objective is: min g,�h max K Lcls(g,�h) + ωLfair(g,�h) + γLinv(g) (4) However, the above adversarial learning framework for learning domain-invariant representation may not work well when |Y × A| is large: as the label space and sensitive attribute space get larger, the number of discriminators to be learned increases and the training can be highly unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' A naive solution to tackling this issue is to use one discriminator ∀y ∈ Y, a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' However, this would result in the reduced mutual information between representations and label/sensitive attribute, which may hurt the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We thus propose another approach to learn the domain-invariant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' source domain source domain Figure 2: 1D illustration of domain-invariant rep- resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' To transfer accuracy and fairness to target domains, we need to find representation z such that P z|y DS i and P z|y,a DS i are domain-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Proposed solution to learning invariant rep- resentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' For any domain D, we have: P Z|y D = � X P Z,x|y D dx = � X P Z|xP x|y D dx P Z|y,a D = � X P Z,x|y,a D dx = � X P Z|xP x|y,a D dx where P Z|x is domain-independent so we drop D in subscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Given any two source do- mains DS i and DS j , in general P X|y DS i ̸= P X|y DS j and P X|y,a DS i ̸= P X|y,a DS j so that it is non- trivial to achieve domain-invariant representa- tions P Z|y DS i = P Z|y DS j and P Z|y,a DS i = P Z|y,a DS j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' How- ever,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' if there exist invertible functions my i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j : X → X and my,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j : X → X that can match the density functions of X from DS i to DS j such that P X|y DS i = P my i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j(X)|y DS j and P X|y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a DS i = P my,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j (X)|y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a DS j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' and if we can find the representation Z such that P Z|x DS i = P Z|my i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j(x) DS i and P Z|x DS i = P Z|my,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j (x) DS i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' then ∀y ∈ Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' a ∈ A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' we have: P Z|y DS i = � X P Z|xP x|y DS i dx = � X P Z|x′P x′|y DS j dx′ = P Z|y DS j P Z|y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a DS i = � X P Z|xP X|y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a DS i dx = � X P Z|x′′P x′′|y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a DS j dx′′ = P Z|y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a DS j where x′ = my i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j(x) and x′′ = my,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' This observation suggests that to minimize the discrepancy of representation distributions among source domains, we can first find the density mapping functions my i,j and my,a i,j , ∀y, a, i, j, and then minimize the discrepancies between P Z|x, P Z|x′, and P Z|x′′, ∀x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' This is formally shown in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Theorem 5 If there exist invertible mappings my i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j and my,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j such that P X|y DS i = P my i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j(X)|y DS j and P X|y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a DS i = P my,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j (X)|y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a DS j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ∀y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' and if the representation mapping are in the form of g := P Z|x = N(µ(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' σ2Id),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' where µ(x) is the function of x and d is the dimension of the representation space Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' then minimizing dJS � P Z|y DS i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z|y DS j � and dJS � P Z|y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a DS i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z|y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a DS j � can be reduced to minimizing ��µ(x) − µ � my i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j(x) ��� 2 and ��µ(x) − µ � my,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j (x) ��� 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 6 sourcedomain2 source domainj Density 5 U 10 15source domain source domainj Density 5 U 10 15source domain source domainj Density 2 0 2 8 ry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='asource domain source domainj Density 0Published as a conference paper at ICLR 2023 Based on Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 5, we propose a two-stage learning approach FATDM, as stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Remark 1 (Fairness and Accuracy Transfer by Density Matching (FATDM)) Given the exis- tence of density matching functions my i,j and my,a i,j , and representation mapping g := N(µ(x), σ2Id), domain-invariant representations can be learned via a two-stage process: (i) finding these mapping functions my i,j and my,a i,j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (ii) minimizing the mean squared errors between µ(x) and µ � my i,j(x) � , and µ(x) and µ � my,a i,j (x) � , ∀i, j ∈ [N], x ∈ X, y ∈ Y, a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Stage 1: learning mapping functions my i,j and my,a i,j across source domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Many approaches can be leveraged to estimate my i,j and my,a i,j from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In our study, we adopt StarGAN (Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2018) and CycleGAN (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2017) as examples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' both frameworks are widely used in multi-domain image-to-image translation and can be leveraged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In our algorithm, we independently train two translation models DensityMatchY and DensityMatchY,A using StarGAN or CycleGAN, with each used for learning {my i,j}y∈Y,i,j∈[N] and {my,a i,j }y∈Y,a∈A,i,j∈[N], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Stage 1: Finding Stage 2: Enforcing source domain source domain Figure 3: FATDM: two-stage training Specifically, DensityMatchY (or DensityMatchY,A) con- sists of a generator G : X × [N] × [N] → X and a discrim- inator D : X → [N] × {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The generator takes in real image x and a pair of domain labels i, j as input and generates a fake image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' the discriminator aims to predict the domain label of the image generated by the generator and distinguish whether it is fake or real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' G and D are learned simultaneously by solving the minimax game, and their loss functions are specified in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' When the training is completed, we obtain two optimal generators from DensityMatchY and DensityMatchY,A, denoted as GY and GY,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We shall use GY (·, i, j) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' GY,A(·, i, j)) directly as the density map- ping function {my i,j(·)}y∈Y (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' {my,a i,j (·)}y∈Y,a∈A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Stage 2: learning domain-invariant representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Given GY and GY,A learned in stage 1, we are ready to learn the invariant representation Z by finding g : X → Z such that g := N(µ(x), σ2Id) and minimizes the following: Linv = Ed,d′,d′′∼{DS i }i∈[N] [Lmse(µ(X), µ(X′)) + Lmse(µ(X), µ(X′′))] (5) where d, d′, d′′ are domain labels sampled from source domains, X is features sampled from domain d, X′ = GY (X, d, d′), X′′ = GY,A(X, d, d′′), Lmse is mean squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The pseudo-code of our proposed model (FATDM) is in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The detailed architecture of FATDM is in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Algorithm 1: Fairness and Accuracy Transfer by Density Matching (FATDM) Input: Training dataset Dtrain from N source domains {DS i }N i=1 Output: representation mapping g, classifier �h, density matching functions GY , GY,A 1 Procedure Density_Matching(Dtrain) /* Procedure for training GY is similar but not presented / 2 while training DensityMatchY,A is not end do 3 Sample y ∼ Y, a ∼ A and data batch B = {xk, dk|ak = a, yk = y}|B| k=1 from Dtrain ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 4 Update GY,A based on the objectives of the minimax game (Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 5 Procedure Invariant_Representation_Learning(Dtrain, GY , GY,A) 6 while training FATDM is not end do 7 Sample data batch B = {xk, ak, yk, dk}|B| k=1 from Dtrain ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 8 Sample lists of domain labels {d′ i}|B| k=1 and {d′′ i }|B| k=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 9 Generate sets of artificial images {x′ k}|B| k=1 and {x′′ k}|B| k=1 by GY and GY,A ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 10 Update g, ˆh by optimizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (3) with Linv defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Remark 2 (Summary of theoretical results and proposed algorithm) Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1 and Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 3 suggest a way to ensure high accuracy and fairness in target domain: by minimizing the source error ϵAcc Ds i (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', Lcls in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (3)), the source unfairness ϵEO Ds i (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=',Lfair in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (3)), and the discrepancies between source domains dJS � P Z|Y =y DS i , P Z|Y =y DS j � and dJS � P Z|Y =y,A=a DS i , P Z|Y =y,A=a DS j � (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', Linv in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The 7 Density 2 U 2 4sourcedomain2 source domainj Density 5 U 10 15Published as a conference paper at ICLR 2023 common way to optimize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (3) using adversarial learning (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (4)) is not stable when |Y × A| is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 5 states that instead of using adversarial learning, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (3) can be optimized via 2-stage learning: (i) find mappings my i,j and my,a i,j ( Density_Matching in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1) and (ii) minimize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (3) with Linv defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (5) ( Invariant_Representation_Learning in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 5 EXPERIMENTS We conduct experiments on MIMIC-CXR database (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019), which includes 377,110 chest X-ray images associated with 227,827 imaging studies about 14 diseases performed at the Beth Israel Deaconess Medical Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Importantly, these images are linked with MIMIC-IV database (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021) which includes patients’ information such as age, and race;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' these can serve as sensitive attributes for measuring the unfairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Based on MIMIC-CXR and MIMIC-IV data, we construct two datasets on two diseases: Cardiomegaly disease: we first extract all images related to Cardiomegaly disease, and the corre- sponding labels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', positive/negative) and sensitive attributes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', male/female);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' then we partition the data into four domain-specific datasets based on age (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', [18, 40), [40, 60), [60, 80), [80, 100)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We consider age as domain label because it captures the real scenario that there are distribution shifts across patients with different ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Edema disease: we extract all images related to Edema disease, and corresponding labels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', pos- itive/negative) and sensitive attributes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', age with ranges [18, 40), [40, 60), [60, 80), [80, 100)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Unlike Cardiomegaly data, we construct the dataset for each domain by first sampling images from Edema data followed by θ degree counter-clockwise rotation, where θ ∈ {0◦, 15◦, 30◦, 45◦, 60◦}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We consider rotation degree as domain label to model the scenario where there is rotational misalignment among images collected from different devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Next, we focus on Cardiomegaly disease and the results for Edema disease are shown in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We compare our method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', FATDM-StarGAN and FATDM-CycleGAN) with exist- ing methods for domain generalization, including empirical risk minimization, domain invariant representation learning, and distributionally robust optimization, as detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Empirical risk minimization (ERM): The baseline that considers all source domains as one domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Domain invariant representation learning: Method that aims to achieve the invariant across source domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We experiment with G2DM (Albuquerque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019), DANN (Ganin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2016), CDANN (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2018c), CORAL (Sun & Saenko, 2016), IRM (Arjovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' These models focus on accuracy transfer by enforcing the invariance of distributions P Z DS i or P Z|Y DS i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Distributionally robust optimization: Method that learns a model at worst-case distribution to hope it can generalize well on test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We experiment with GroupDRO (Sagawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019) that minimizes the worst-case training loss over a set of pre-defined groups through regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ATDM: A variant of FATDM-StarGAN that solely focuses on accuracy transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' That is, we only enforce the invariance of P Z|Y DS i during learning which is similar to Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The implementations of these models except G2DM are adapted from DomainBed framework (Gulra- jani & Lopez-Paz, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' For G2DM, we use the author-provided implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' For all models, we use ResNet18 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2016) as the backbone module of representation mapping g : X → Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' and fairness constraint Lfair is enforced as a regularization term added to the original objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Experiment setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We follow leave-one-out domain setting in which 3 domains are used for training and the remaining domain serves as the unseen target domain and is used for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Several metrics are considered to measure the unfairness and error of each model in target domain, including: Error: cross-entropy loss (CE), misclassification rate (MR), AUROC := 1−AUROC, AUPR := 1−AUPR, F1 := 1−F1, where AUROC, AUPR, F1 are area under receiver operating characteristic curve, area under precision-recall curve, F1 score, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Unfairness: we consider both equalized odds and equal opportunity fairness notion, and adopt mean distance (MD) and earth mover’s distance (EMD) as distance metric D(·||·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Fairness and accuracy on target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We first compare our method with baselines in terms of the optimal trade-off (Pareto frontier) between accuracy and fairness on target domains under different metric pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Figure 4 shows the error-unfairness curves (as ω varies from 0 (no fairness constraint) to 10 (strong fairness constraint)), with AUROC and MR as error metric, and equalized odds (measured under distance metrics MD and EMD) as fairness notion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' the results for other error metrics are similar 8 Published as a conference paper at ICLR 2023 Figure 4: Fairness-accuracy trade-off (Pareto frontier) of FATDM-StarGAN, FATDM-CycleGAN, and baseline methods: error-unfairness curves are constructed by varying ω ∈ [0, 10] and the values of error and unfairness are normalized to [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Lower-left points indicate the model has a better fairness-accuracy trade-off (Pareto optimality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Figure 5: Prediction performances (AUROC, AUPR, Accuracy, F1) of FATDM-StarGAN on Car- diomegaly disease data when varying hyper-parameter γ at different levels of fairnsess constraint ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' and shown in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Our observations are as follows: (1) As expected, there is a trade-off between fairness and accuracy: for all methods, increasing ω improves fairness but reduces accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2) Among all methods, the Pareto frontiers of FATDM-StarGAN and FATDM-CycleGAN are the bottom leftmost, implying that our method attains a better fairness-accuracy trade-off than baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (3) Although fairness constraint is imposed during training for all methods, the fairness attained at source domains cannot be well-generalized to the target domain under other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' These results validate our theorems and show that enforcing the domain-invariant P Z|Y DS i and P Z|Y,A DS i when learning representations ensures the transfer of both accuracy and fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' It is worth-noting that under this dataset, the domain-invariant P Z|Y DS i (accuracy transfer) does not imply the domain-invariant P Z|Y,A DS i (fairness transfer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' This is because domain DS i (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', age) is correlated with label Y (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', has a disease) and sensitive attribute A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', gender), making the distribution P Y,A DS i different across domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Impact of density mapping model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' To investigate whether the performance gain of our method is due to the use of any specific density mapping model, we adopt StarGAN and CycleGAN architectures to learn density mapping functions in our method and compare their performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Figure 4 shows that FATDM-StarGAN and CycleGAN achieve similar fairness-accuracy trade-off at the target domains and both of them outperform the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' This result shows that our method is not limited to any specific density mapping model and is broadly applicable to other architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Impact of invariant representation constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We also examine the impact of Linv on the perfor- mance of FATDM-StarGAN at target domains, where we vary the hyper-parameter γ ∈ [0, 5e2] at different levels of fairness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', fix ω = 1, 5, 10) and examine how the prediction performances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', AUROC, AUPR, accuracy and F1) could change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Figure 5 shows that enforcing domain-invariant constraint Linv helps transfer the performance from source to target domain, and γ that attains the highest accuracy at target domain can be different for different levels of fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The results also indicate the fairness-accuracy trade-off, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', for any γ, enforcing stronger fairness constraints (large ω) could hurt prediction performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 6 CONCLUSION In this paper, we theoretically and empirically demonstrate how to achieve fair and accurate predic- tions in unknown testing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' To the best of our knowledge, our work provides the first theoretical analysis to understand the efficiency of invariant representation learning in transferring both fairness and accuracy under domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In particular, we first propose the upper bounds of prediction error and unfairness in terms of JS-distance, then design the two-stage learning 9 --+- ERM G2DM DANN CDANN CORAL GroupDRO IRM ATDM FATDM-StarGAN FATDM-CycleGAN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 (MD) (MD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 Unfairness ( Unfairness Unfairness ( Unfairness ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content="0 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Error AUROC Error (MR) Error AUROC Error (MR)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='915 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='83- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='910 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='77 C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='82 JRO R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='905 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='76 ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='890- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='885 1e1 1el 1e22e23e24e2 5e2 0 le-1 le0 lel 1e2 2e2 3e2 4e2 5e2Published as a conference paper at ICLR 2023 method that minimizes these upper bounds by learning domain-invariant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Experiments on the real-world clinical data demonstrate the effectiveness of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' REPRODUCIBILITY STATEMENT The original chest X-ray images and the corresponding metadata can be downloaded from PhysioNet (https://physionet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='org/content/mimic-cxr-jpg/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0/;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' https: //physionet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='org/content/mimiciv/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Codes for data processing and proposed algorithms are in supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Technical details of the proposed algorithms and experi- mental settings are in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Additional experimental results are in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Lemmas used in proofs of the theorems in the main paper are in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Complete proofs of the theorems in the main paper and the corresponding lemmas are in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work was funded in part by the National Science Foundation under award number IIS-2145625, by the National Institutes of Health under award number UL1TR002733, and by The Ohio State University President’s Research Excellence Accelerator Grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' REFERENCES Alekh Agarwal, Alina Beygelzimer, Miroslav Dudík, John Langford, and Hanna Wallach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' A reductions approach to fair classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 60–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Isabela Albuquerque, João Monteiro, Mohammad Darvishi, Tiago H Falk, and Ioannis Mitliagkas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Generalizing to unseen domains via distribution matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' arXiv preprint arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='00804, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Martin Arjovsky, Léon Bottou, Ishaan Gulrajani, and David Lopez-Paz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Invariant risk minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' arXiv preprint arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='02893, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Yahav Bechavod, Christopher Jung, and Steven Z Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Metric-free individual fairness in online learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Advances in neural information processing systems, 33:11214–11225, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Shai Ben-David, John Blitzer, Koby Crammer, Alex Kulesza, Fernando Pereira, and Jennifer Wortman Vaughan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' A theory of learning from different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Machine learning, 79(1):151–175, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Asia J Biega, Krishna P Gummadi, and Gerhard Weikum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Equity of attention: Amortizing individual fairness in rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In The 41st international acm sigir conference on research & development in information retrieval, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 405–414, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Arpita Biswas and Suvam Mukherjee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Ensuring fairness under prior probability shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In Proceedings of the 2021 AAAI/ACM Conference on AI, Ethics, and Society, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 414–424, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Gilles Blanchard, Aniket Anand Deshmukh, Urun Dogan, Gyemin Lee, and Clayton Scott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Domain generalization by marginal transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Journal of Machine Learning Research, 22:1–55, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Fabio M Carlucci, Antonio D’Innocente, Silvia Bucci, Barbara Caputo, and Tatiana Tommasi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Domain generalization by solving jigsaw puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 2229–2238, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Yatong Chen, Reilly Raab, Jialu Wang, and Yang Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Fairness transferability subject to bounded distribution shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='00129, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Yunjey Choi, Minje Choi, Munyoung Kim, Jung-Woo Ha, Sunghun Kim, and Jaegul Choo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Star- gan: Unified generative adversarial networks for multi-domain image-to-image translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 8789–8797, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 10 Published as a conference paper at ICLR 2023 Amanda Coston, Karthikeyan Natesan Ramamurthy, Dennis Wei, Kush R Varshney, Skyler Speakman, Zairah Mustahsan, and Supriyo Chakraborty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Fair transfer learning with missing protected attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In Proceedings of the 2019 AAAI/ACM Conference on AI, Ethics, and Society, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 91–98, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Aniket Anand Deshmukh, Yunwen Lei, Srinagesh Sharma, Urun Dogan, James W Cutler, and Clayton Scott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' A generalization error bound for multi-class domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' arXiv preprint arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='10392, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Luc Devroye, Abbas Mehrabian, and Tommy Reddad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The total variation distance between high- dimensional gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' arXiv preprint arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='08693, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Cynthia Dwork, Moritz Hardt, Toniann Pitassi, Omer Reingold, and Richard Zemel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Fairness through awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In Proceedings of the 3rd innovations in theoretical computer science conference, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 214–226, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Endres and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Schindelin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' A new metric for probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' IEEE Transactions on Information Theory, 49(7):1858–1860, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1109/TIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='813506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Yaroslav Ganin and Victor Lempitsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Unsupervised domain adaptation by backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In International conference on machine learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1180–1189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' PMLR, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario Marchand, and Victor Lempitsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Domain-adversarial training of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The journal of machine learning research, 17(1):2096–2030, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Stephen Giguere, Blossom Metevier, Bruno Castro da Silva, Yuriy Brun, Philip S Thomas, and Scott Niekum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Fairness guarantees under demographic shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In International Conference on Learning Representations, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Generative adversarial nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Advances in neural information processing systems, 27, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Ishaan Gulrajani and David Lopez-Paz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In search of lost domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' arXiv preprint arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='01434, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Swati Gupta and Vijay Kamble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Individual fairness in hindsight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Journal of Machine Learning Research, 22(144):1–35, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Moritz Hardt, Eric Price, and Nati Srebro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Equality of opportunity in supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Advances in neural information processing systems, 29, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Deep residual learning for image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 770–778, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Judy Hoffman, Mehryar Mohri, and Ningshan Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Algorithms and theory for multiple-source adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 31, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Weihua Hu, Gang Niu, Issei Sato, and Masashi Sugiyama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Does distributionally robust supervised learning give robust classifiers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 2029–2037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Zeyi Huang, Haohan Wang, Eric P Xing, and Dong Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Self-challenging improves cross-domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In European Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 124–140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Maximilian Ilse, Jakub M Tomczak, Christos Louizos, and Max Welling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Diva: Domain invariant variational autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In Medical Imaging with Deep Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 322–348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Seogkyu Jeon, Kibeom Hong, Pilhyeon Lee, Jewook Lee, and Hyeran Byun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Feature stylization and domain-aware contrastive learning for domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In Proceedings of the 29th ACM International Conference on Multimedia, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 22–31, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Alistair Johnson, Lucas Bulgarelli, Tom Pollard, Steven Horng, Leo Anthony Celi, and Mark Roger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Mimic-iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='13026/s6n6-xd98, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 11 Published as a conference paper at ICLR 2023 Alistair EW Johnson, Tom J Pollard, Nathaniel R Greenbaum, Matthew P Lungren, Chih-ying Deng, Yifan Peng, Zhiyong Lu, Roger G Mark, Seth J Berkowitz, and Steven Horng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Mimic-cxr-jpg, a large publicly available database of labeled chest radiographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' arXiv preprint arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='07042, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Faisal Kamiran and Toon Calders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Data preprocessing techniques for classification without discrimi- nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Knowledge and information systems, 33(1):1–33, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Amit Kaushal, Russ Altman, and Curt Langlotz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Geographic distribution of us cohorts used to train deep learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Jama, 324(12):1212–1213, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Daehee Kim, Youngjun Yoo, Seunghyun Park, Jinkyu Kim, and Jaekoo Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Selfreg: Self-supervised contrastive regularization for domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 9619–9628, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Pang Wei Koh, Shiori Sagawa, Henrik Marklund, Sang Michael Xie, Marvin Zhang, Akshay Bal- subramani, Weihua Hu, Michihiro Yasunaga, Richard Lanas Phillips, Irena Gao, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Wilds: A benchmark of in-the-wild distribution shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 5637–5664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' David Krueger, Ethan Caballero, Joern-Henrik Jacobsen, Amy Zhang, Jonathan Binas, Dinghuai Zhang, Remi Le Priol, and Aaron Courville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Out-of-distribution generalization via risk extrapola- tion (rex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 5815–5826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Haoliang Li, Sinno Jialin Pan, Shiqi Wang, and Alex C Kot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Domain generalization with adversarial feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 5400–5409, 2018a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Ya Li, Mingming Gong, Xinmei Tian, Tongliang Liu, and Dacheng Tao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Domain generalization via conditional invariant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In Proceedings of the AAAI conference on artificial intelligence, volume 32, 2018b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Ya Li, Xinmei Tian, Mingming Gong, Yajing Liu, Tongliang Liu, Kun Zhang, and Dacheng Tao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Deep domain generalization via conditional invariant adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In Proceedings of the European Conference on Computer Vision (ECCV), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 624–639, 2018c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Zheren Li, Zhiming Cui, Sheng Wang, Yuji Qi, Xi Ouyang, Qitian Chen, Yuezhi Yang, Zhong Xue, Dinggang Shen, and Jie-Zhi Cheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Domain generalization for mammography detection via multi-style and multi-view contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In International Conference on Medical Image Computing and Computer-Assisted Intervention, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 98–108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Springer, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Evan Z Liu, Behzad Haghgoo, Annie S Chen, Aditi Raghunathan, Pang Wei Koh, Shiori Sagawa, Percy Liang, and Chelsea Finn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Just train twice: Improving group robustness without training group information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 6781–6792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' David Madras, Elliot Creager, Toniann Pitassi, and Richard Zemel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Learning adversarially fair and transferable representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 3384–3393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Karima Makhlouf, Sami Zhioua, and Catuscia Palamidessi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Machine learning fairness notions: Bridging the gap with real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Information Processing & Management, 58(5): 102642, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Yishay Mansour, Mehryar Mohri, and Afshin Rostamizadeh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Domain adaptation with multiple sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Advances in neural information processing systems, 21, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Yishay Mansour, Mehryar Mohri, and Afshin Rostamizadeh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Multiple source adaptation and the rényi divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 367–374, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Takeru Miyato, Toshiki Kataoka, Masanori Koyama, and Yuichi Yoshida.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Spectral normalization for generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' arXiv preprint arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='05957, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 12 Published as a conference paper at ICLR 2023 Krikamol Muandet, David Balduzzi, and Bernhard Schölkopf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Domain generalization via invariant feature representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 10–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' PMLR, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' A Tuan Nguyen, Toan Tran, Yarin Gal, and Atilim Gunes Baydin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Domain invariant representation learning with domain density transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Luca Oneto, Michele Donini, Andreas Maurer, and Massimiliano Pontil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Learning fair and transfer- able representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' arXiv preprint arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='10673, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Trung Phung, Trung Le, Tung-Long Vuong, Toan Tran, Anh Tran, Hung Bui, and Dinh Phung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' On learning domain-invariant representations for transfer learning with multiple sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Yury Polyanskiy and Yihong Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Lecture notes on information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Lecture Notes for ECE563 (UIUC) and, 6(2012-2016):7, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Yury Polyanskiy and Yihong Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Strong data-processing inequalities for channels and bayesian networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In Convexity and Concentration, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 211–249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Springer, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Fengchun Qiao, Long Zhao, and Xi Peng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Learning to learn single domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 12556–12565, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Alexandre Rame, Corentin Dancette, and Matthieu Cord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Fishr: Invariant gradient variances for out-of-distribution generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' arXiv preprint arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='02934, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Ashkan Rezaei, Anqi Liu, Omid Memarrast, and Brian D Ziebart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Robust fairness under covariate shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 9419–9427, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Shiori Sagawa, Pang Wei Koh, Tatsunori B Hashimoto, and Percy Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Distributionally robust neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In International Conference on Learning Representations, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Candice Schumann, Xuezhi Wang, Alex Beutel, Jilin Chen, Hai Qian, and Ed H Chi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Transfer of machine learning fairness across domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' arXiv preprint arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='09688, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Shiv Shankar, Vihari Piratla, Soumen Chakrabarti, Siddhartha Chaudhuri, Preethi Jyothi, and Sunita Sarawagi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Generalizing across domains via cross-gradient training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In International Conference on Learning Representations, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Yuge Shi, Jeffrey Seely, Philip HS Torr, N Siddharth, Awni Hannun, Nicolas Usunier, and Gabriel Synnaeve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Gradient matching for domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='09937, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Changjian Shui, Boyu Wang, and Christian Gagné.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' On the benefits of representation regularization in invariance based domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1–21, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Anthony Sicilia, Xingchen Zhao, and Seong Jae Hwang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Domain adversarial neural networks for domain generalization: When it works and how to improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' arXiv preprint arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='03924, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Harvineet Singh, Rina Singh, Vishwali Mhasawade, and Rumi Chunara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Fairness violations and mitigation under covariate shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 3–13, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Baochen Sun and Kate Saenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Deep coral: Correlation alignment for deep domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In European conference on computer vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 443–450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Springer, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Chris Xing Tian, Haoliang Li, Xiaofei Xie, Yang Liu, and Shiqi Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Neuron coverage-guided domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' IEEE Transactions on Pattern Analysis and Machine Intelligence, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Riccardo Volpi, Hongseok Namkoong, Ozan Sener, John C Duchi, Vittorio Murino, and Silvio Savarese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Generalizing to unseen domains via adversarial data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Advances in neural information processing systems, 31, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 13 Published as a conference paper at ICLR 2023 Jingge Wang, Yang Li, Liyan Xie, and Yao Xie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Class-conditioned domain generalization via wasserstein distributional robust optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' arXiv preprint arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='03676, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Haotian Ye, Chuanlong Xie, Tianle Cai, Ruichen Li, Zhenguo Li, and Liwei Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Towards a theoretical framework of out-of-distribution generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Taeho Yoon, Jaewook Lee, and Woojin Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Joint transfer of model knowledge and fairness over domains using wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' IEEE Access, 8:123783–123798, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Muhammad Bilal Zafar, Isabel Valera, Manuel Gomez-Rodriguez, and Krishna P Gummadi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Fairness constraints: A flexible approach for fair classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The Journal of Machine Learning Research, 20(1):2737–2778, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Rich Zemel, Yu Wu, Kevin Swersky, Toni Pitassi, and Cynthia Dwork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Learning fair representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In International conference on machine learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 325–333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' PMLR, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' mixup: Beyond empirical risk minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In International Conference on Learning Representations, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Xueru Zhang, Mohammadmahdi Khaliligarekani, Cem Tekin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Group retention when using machine learning in sequential decision making: the interplay between user dynamics and fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Xueru Zhang, Ruibo Tu, Yang Liu, Mingyan Liu, Hedvig Kjellstrom, Kun Zhang, and Cheng Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' How do fair decisions fare in long-term qualification?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 33:18457–18469, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Han Zhao, Shanghang Zhang, Guanhang Wu, José MF Moura, Joao P Costeira, and Geoffrey J Gordon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Adversarial multiple source domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Advances in neural information processing systems, 31, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Shanshan Zhao, Mingming Gong, Tongliang Liu, Huan Fu, and Dacheng Tao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Domain generalization via entropy regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 33:16096–16107, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Kaiyang Zhou, Yongxin Yang, Timothy Hospedales, and Tao Xiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Learning to generate novel domains for domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In European conference on computer vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 561–578.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Unpaired image-to-image translation using cycle-consistent adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In Proceedings of the IEEE international conference on computer vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 2223–2232, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 14 Published as a conference paper at ICLR 2023 A RELATED WORKS Domain generalization/Domain adaptation: In many real scenarios of machine learning, data in training phase is sampled from one or many source domains, while in the testing phase, data is sampled from an unseen target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Many works have been proposed to design robust ML models that can achieve good performances in deployment environment depending on whether they can access to the target data (domain adaptation) or not (domain generalization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' However, most of these models focus only on transfering accuracy from source to target domains and can be categorized into five main approaches: (1) data manipulation (Volpi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Shankar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2) domain-invariant representation learning (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Ganin & Lempitsky, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Ganin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Phung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (3) distributional robustness (Krueger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Koh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Sagawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2018), (4) gradient operation (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Rame et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2022), and (5) self-supervised learning (Carlucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Fairness in Machine Learning: Many fairness notions have been proposed to measure the unfairness in ML model, and they can be roughly classified into two classes: Individual fairness considers the equity at the individual-level and it requires that similar individuals should be treated similarly (Biega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Bechavod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Gupta & Kamble, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Group fairness attains a certain balance in the group-level, where the entire population is first partitioned into multiple groups and certain statistical measures are equalized across different groups (Hardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Various approaches have also been developed to satisfy these fairness notions, they roughly fall into three categories: (1) Pre-processing: modifying training dataset to remove bias before learning an ML model (Kamiran & Calders, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Zemel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2) In-processing: attain fairness during the training process by imposing certain fairness constraint or modifying loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (Zafar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2018) (3) Post-processing: altering the output of an existing algorithm to satisfy a fairness constraint after training (Hardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' However, most of these methods assume the data distributions at training and testing are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In contrast, we study fairness problem under domain generalization in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Fairness under Domain Adaptation: There are some studies proposed to achieve good fairness when the testing environment changes but all of them focused on the domain adaptation setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The most common adaptation setup is learning under the assumption of covariate shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' For example, Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2021) leveraged a feature selection method in a causal graph describing data to mitigate fairness violation under covariate shift of distribution in testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Coston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2019) proposed the weighting methods that can give fair prediction under covariate shift between source and target distribution when access to the sensitive attributes is prohibited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Rezaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2021) sought fair decisions by optimizing a worst-case testing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Besides convariate shift, there are some works proposed to handle other types of distribution shift including demographic shift and prior probability shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Instead of learning fair model directly, Oneto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2019) and Madras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2018) find fair representation that can generalize to the new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Aside from empirical studies, Schumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2019) and Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2020) developed theoretical frameworks to examine fairness transfer in domain adaptation setting and then offered modeling approaches to achieve good fairness in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Comparison with existing bounds in the literature: We compare our bounds with most commons bound in the fields of domain adaptation and domain generalization as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Accuracy bounds in domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Bounds in Ben-David et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2010): ϵAcc DT � �f � ≤ ϵAcc DS � �f � + DT V � P X DS ∥ P X DT � + min D∈{DS,DT }ED [|fDS(X) − fDT (X)|] This bound is for binary classification problem under domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The classification error in target domain is bounded by the error in source domain, the total variation distance of feature distribution between source and target domain, and the misalignment of the labeling function between source and target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The limitation of this bound is that (1) it’s only applicable to settings with zero-one loss function and deterministic labeling function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2) estimating the total variation distance is hard in practice and it doesn’t relate the feature and representation spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 15 Published as a conference paper at ICLR 2023 This paper also provides another accuracy bound based on H∆H divergence:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ϵAcc DT � �f � ≤ ϵAcc DS � �f � + DH∆H � P X DS ∥ P X DT � + inf � f � ϵAcc DT � �f � + ϵAcc DS � �f �� where DH∆H � P X DS ∥ P X DT � = sup � f1, � f1 ���PDS � �f1(X) ̸= �f2(X) � − PDT � �f1(X) ̸= �f2(X) ���� is the H∆H divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' However, it has the same limitations as total variation distance mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Accuracy bounds in domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Bounds in Albuquerque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2019): ϵAcc DT � �f � ≤ N � i=1 πiϵAcc DS i � �f � + max j,k∈[N]DH � P X DS j ∥ P X DS k � + DH � P X DS ∗ ∥ P X DT � + min D∈{DS ∗ ,DT }ED ���fDS ∗ (X) − fDT (X) ��� where DH � P X DS ∥ P X DT � = sup � f ���PDS � �f(X) = 1 � − PDT � �f(X) = 1 ���� is the H divergence, P X DS ∗ = arg min π DH ��N i=1 πiP X DS i ∥ P X DT � is the mixture of source domains that is closest to target domain with respect to H divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In this bound, the classification error in target domain is bounded by the convex combination of errors in source domains, the H divergence between source domains, the H divergence between target domain and its nearest mixture of source domains, and the misalignment of the labeling function between mixture source domains and target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Because this bound is constructed based on H divergence, it also has the limitations for the bounds in domain adaptation (Ben-David et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2010) as we mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' This bound can be transformed to the representation space Z by replacing X by Z in its formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Then, this bound suggests enforcing invariant constraint of marginal distribution of representation Z across source domains, which has inherent trade-off as shown in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Because the target domain is unknown during training, the mixing weights {πi}N i=1 are not useful for algorithmic design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Bounds in Phung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2021): ϵAcc DT � �f � ≤ N � i=1 πiϵAcc DS i � �f � + Cmax i∈[N]EDS i ����� ����fDT (X)y − fDS i (X)y ��� �|Y| y=1 ���� 1 � + N � i=1 N � j=1 C�2πj N d1/2 � P Z DT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z DS i � + N � i=1 N � j=1 C�2πj N d1/2 � P Z DS i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z DS j � where d1/2 � P X DS i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P X DS j � = � D1/2 � P X DS i ∥ P X DS j � is Hellinger distance defined based on Hellinger divergence D1/2 � P X DS i ∥ P X DS j � = 2 � X �� P X DS i − � P X DS j �2 dX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' This bound re- lates the feature and representation spaces that the classification error of target domain defined in feature space is bounded by classification errors of source domains defined in feature space, the misalignment of labeling function between target and source domains, and the Hellinger distances between source and target domains and between source domains of marginal distribution of rep- resentation Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' While this bound is not limited to zero-one loss and the labeling function can be stochastic, it suggests the alignment of marginal distribution of representation Z across source domains for generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Moreover, estimating Hellinger distance can be hard in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The mismatch between existing bounds and adversarial learning approach for domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' All existing bounds mentioned above suggest minimizing the distances between representation distributions across source domains with respect to some discrepancy measures such as H divergence, total variation distance, and Hellinger distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Based on these bounds, adversarial learning-based models are often proposed to minimize these distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' However, there is a misalignment between the objectives of adversarial learning and the bounds which results in the gap between theoretical findings and practical algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 16 Published as a conference paper at ICLR 2023 In particular, it has been shown that the objective of the minimax game between the representation mapping and the discriminator is equivalent to minimizing the JS divergence between representation distributions across source domains (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' However, minimizing JS divergence does not guarantee the minimization of common distances used in the existing bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The details are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' H divergence: We show that JS divergence is not the upper bound of H divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Con- sider an example with two distributions P(X) and Q(X) where � P(X) = 0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='p 1/3 P(X) = 1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='p 2/3 and � Q(X) = 0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='p 1/3 Q(X) = 1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='p 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' By definition, DH(P ∥ Q) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='33 > DJS(P ∥ Q) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Total variation distance: We have DJS(P ∥ Q) ≤ DT V (P ∥ Q) ∀P, Q where DJS and DT V are JS divergence and total variation distance, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Then, minimizing JS divergence does not guarantee the minimization of total variation distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Hellinger distance: We have DJS(P ∥ Q) ≤ √ 2d1/2(P, Q) ∀P, Q where d1/2 is Hellinger distance and total variation distance, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Then, minimizing JS divergence does not guarantee the minimization of Hellinger distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Different from the existing bounds, our bounds are based on JS divergence/distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Then they align with the adversarial learning approach for domain generalization in general, and with our proposed method FATDM in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Advantages of our proposed bounds in domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In summary, our proposed bounds has several advantages in terms of the following: Most existing bounds (Ben-David et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Albuquerque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019) do not relates feature and representation spaces so it is not clear how performance in input space is affected by the representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In contrast, our bounds connect the representation and input spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' this further guides us to find representations that lead to good performances in input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Most prior studies adopt H divergence to measure the dissimilarity between domains, which is limited to deterministic labeling functions and zero-one loss (Ben-David et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Albuquerque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In contrast, our bound is more general and is applicable to settings where domains are specified by stochastic labeling functions and general loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Distant metrics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', total variation distance, H divergence, Hellinger divergence, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=') used in existing bounds (Ben-David et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Albuquerque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Phung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021) are hard to compute in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In contrast, our bounds use JS divergence which is aligned with training objective for discriminator in adversarial learning Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Existing bounds for domain generalization only imply the alignment of marginal distribution of feature across source domains (Albuquerque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Phung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' As shown in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 2, methods that learn invariance of marginal distribution have an inherent trade-off and may increase the lower bound of expected loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In contrast, our bounds suggest the alignment of label-conditional distribution of feature across source domains which has been verified to be more effective in empirical studies (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Regarding the fairness, our work is the first that bounds the unfairness in domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In particular, our bounds suggest enforcing the invariant constraint of feature distribution given label and sensitive attribute across source domains to transfer fairness to the unseen target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' B DETAILS OF ALGORITHM FATDM FATDM consists of density mapping functions my i,j and my,a i,j , ∀y ∈ Y, a ∈ A, i, j ∈ [N] (learned by two DensityMatch models), feature mapping function g (ResNet18 model), and the clas- sifier �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In our study, we experiment with two different DensityMatch architectures: Star- GAN (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', in FATDM-StarGAN) and CycleGAN (in FATDM-CycleGAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We show the details of FATDM-StarGAN below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' For FATDM-CycleGAN, the only difference is we used CycleGAN as DensityMatch instead of StarGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The details of CycleGAN were presented in the original paper (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 17 Published as a conference paper at ICLR 2023 For FATDM-StarGAN, each DensityMatchY (or DensityMatchY,A) consists of a generator G : X × [N] × [N] → X and a discriminator D : X → [N] × {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The generator takes in real image x and a pair of domain labels i, j as input and generates a fake image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' the discriminator aims to predict the domain label of the image generated by the generator and distinguish whether it is fake or real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' G and D are learned simultaneously by solving the following optimizations: Discriminator’s objective: min LStarGAN D := −LStarGAN adv + λclsLStarGAN cls(real) Generator’s objective: min LStarGAN G := LStarGAN adv + λclsLStarGAN cls(fake) + λrecLStarGAN rec (6) where LStarGAN adv is the adversarial loss, LStarGAN cls(fake) , LStarGAN cls(real) are domain classification loss with respect to fake and real images respectively, LStarGAN rec is the reconstruction loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The specific formulations of these loss functions are in Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' λcls and λrec are hyper-parameters that control the relative importance of domain classification and reconstruction losses, respectively, compared to the adversarial loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In our experiments, input images are resized to (256, 256) and normalized into the range [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The dimension of representation space Z is set to 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ω (hyper-parameter that controls accuracy-fairness trade-off) varies from 0 to 10 with step sizes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0002 for ω ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='002], 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='002 for ω ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='002, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1] and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 for ω ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2, 10], and γ (hyper-parameter that controls accuracy-invariance trade-off) is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1 (after hyper-parameter tuning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Models (FATDM and baselines) are implemented by PyTorch library version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='11 and is trained on multiple computer nodes (each model instance is trained on a single node which has 4 CPUs, 8GB of memory, and a single GPU (P100 or V100)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' One domain’s data is used for testing and the other domains’ data is used for training (10% of training data is used for validation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Each model is trained with 10 epoches and the results are from the epoch with best performance on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Figure 6 visualizes the two-stage training process of FATDM-StarGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The detailed architectures of FATDM-StarGAN are shown in Tables 2-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We have also provided all code for these models in supplemental material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 18 Published as a conference paper at ICLR 2023 Figure 6: Two-stage training of FATDM-StarGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' For stage 1, we only show the training process for DensityMatchY,A (training process for DensityMatchY is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=') Table 2: Architecture of StarGAN generators GY and GY,A - Density mapping functions my i,j and my,a i,j ∀y ∈ Y, a ∈ A, i, j ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' This architecture is similar to the one in the original paper Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2018) except for the first convolution layer where number of input channels is 1 (for grayscale images) and input shape is (h, w, 1 + 2nc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (h, w) is the size of input images, IN is instance batchnorm, and ReLU is Rectified Linear Unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' N: number of output channels, K: kernel size, S: stride szie, P: padding size are convolution and deconvolution layers’ hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Part Input → Output Shape Layer Information Down-sampling (h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1 + 2nc) → (h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 64) CONV-(N64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K7x7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ReLU (h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 64) → � h 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 128 � CONV-(N128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K4x4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ReLU � h 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 128 � → � h 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 256 � CONV-(N256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K4x4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ReLU Bottleneck � h 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 256 � → � h 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 256 � Residual Block: CONV-(N256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K3x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ReLU � h 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 256 � → � h 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 256 � Residual Block: CONV-(N256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K3x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ReLU � h 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 256 � → � h 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 256 � Residual Block: CONV-(N256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K3x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ReLU � h 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 256 � → � h 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 256 � Residual Block: CONV-(N256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K3x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ReLU � h 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 256 � → � h 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 256 � Residual Block: CONV-(N256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K3x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ReLU � h 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 256 � → � h 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 256 � Residual Block: CONV-(N256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K3x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ReLU Up-sampling � h 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 256 � → � h 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 128 � DECONV-(N128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K4x4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ReLU � h 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 128 � → (h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 64) DECONV-(N64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K4x4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ReLU (h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 64) → (h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 3) CONV-(N3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K7x7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ReLU 19 Stage 1 Source Target Domain Label Domain Label Sample Target Source Fake Image Domain Label DY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A Domain Label yEy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='aEA Sample batch DY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A Reconstructed GY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A CStarGAN data Image Lady Source Target Real Image Domain Label Domain Label Real Image Fake Image Real Image Source Target Source Target Batch of images with Y = y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' A = a Domain Label DomainLabel DomainLabel DomainLabel Sample data for training Discriminator Training Generator Training Stage 2 Linv GY Fake FakeImage Classification Lds Representation Label Sensitive Real Image Attribute Target Domain Label Source Target Real Image Real (Generated) 9 h Classification Source Domain Label Domain Label Representation Label Domain Label Batch of images Fake Image Fake GY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A Sensitive Lfair Representation Attribute Sample data for training Stage 2 TrainingPublished as a conference paper at ICLR 2023 Table 3: Architecture of StarGAN discriminators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' This architecture is similar to the one in the original paper Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2018) except for the first convolution layer where number of input channels is 1 (for grayscale images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (h, w) is the size of input images, nd is the number of domains, and Leaky ReLU is Leaky Rectified Linear Unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' N: number of output channels, K: kernel size, S: stride szie, P: padding size are convolution layers’ hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Layer Input → Output Shape Layer Information Input Layer (h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1) → � h 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 64 � CONV-(N64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K4x4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Leaky ReLU Hidden Layer � h 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 64 � → � h 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 128 � CONV-(N128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K4x4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Leaky ReLU Hidden Layer � h 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 128 � → � h 8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 256 � CONV-(N256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K4x4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Leaky ReLU Hidden Layer � h 8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 256 � → � h 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 512 � CONV-(N512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K4x4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Leaky ReLU Hidden Layer � h 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 512 � → � h 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1024 � CONV-(N1024,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K4x4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Leaky ReLU Hidden Layer � h 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1024 � → � h 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 2048 � CONV-(N2048,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K4x4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Leaky ReLU Output Layer (Dsrc) � h 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 2048 � → � h 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1 � CONV-(N1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K3x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1) Output Layer (Dcls) � h 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 2048 � → (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' nd) CONV-(N(nd),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K h 64 × w 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P0) Table 4: Architecture of feature mapping g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' This architecture is similar to ResNet18 model He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2016) except for the first convolution layer where number of input channels is 1 (for grayscale images) and the last layer where output dimension is nz - dimension of representation space Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (h, w) is the size of input images, BN is batchnorm, MaxPool is max pooling, AvePool is average pooling, and ReLU is Rectified Linear Unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' N: number of output channels, K: kernel size, S: stride szie, P: padding size are convolution layers’ hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Part Input → Output Shape Layer Information Input (h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1) → � h 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 64 � CONV-(N64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K7x7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' BN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ReLU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' MaxPool Bottleneck � h 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 64 � → � h 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 64 � Residual Block: CONV-(N64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K3x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' BN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ReLU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' CONV-(N64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K3x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' BN � h 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 64 � → � h 8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 128 � Residual Block: CONV-(N128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K3x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' BN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ReLU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' CONV-(N128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K3x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' BN � h 8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 128 � → � h 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 256 � Residual Block: CONV-(N256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K3x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' BN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ReLU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' CONV-(N256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K3x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' BN � h 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' w 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 256 � → (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 512) Residual Block: CONV-(N512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K3x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' IN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ReLU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' CONV-(N512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' K3x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' BN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' AvgPool Output (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 512) → nz LINEAR-(512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' nz) Table 5: Architecture of classifier �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' nz is the dimension of representation space Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Layer Input → Output Shape Layer Information Hidden Layer nz → nz 2 LINEAR- � nz, nz 2 � , ReLU Hidden Layer nz 2 → nz 4 LINEAR- � nz 2 , nz 4 � , ReLU Output Layer nz 4 → 1 LINEAR- � nz 4 , 1 � , Sigmoid 20 Published as a conference paper at ICLR 2023 C ADDITIONAL EXPERIMENTS Experimental results with all unfairness and error metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In this section, we provide more experimental results about fairness and accuracy under domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In particular, we investigate fairness-accuracy trade-off on the two clinical image datasets including Cardiomegaly and Edema diseases with respect to different fairness criteria (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', Equalized Odds, Equal Opportunity), and unfairness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', MD and EMD) and error (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', CE, MR, AUROC, AUPR, F1) measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Figure 7 (Cardiomegaly disease - Equalized Odds), Figure 8 (Cardiomegaly disease - Equal Opportunity), Figure 9 (Edema disease - Equalized Odds), and Figure 10 (Edema disease - Equal Opportunity) show the unfairness-error curves of our models as well as baselines for these two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' As we can see, our model outperforms other baselines in terms of fairness-accuracy trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The curve of our model is the bottom-leftmost compared to other baselines in all measures showing the clear benefit of (1) enforcing conditional invariant constraints for accuracy and fairness transfer and (2) using the two-stage training process to stabilize training compared to adversarial learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We also quantify our observations by calculating the areas under these unfairness-error curves, in which the smaller area indicates the better accuracy-fairness trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' As shown in Tables 6 and 7, our model has the smallest areas under the curve and achieves significantly better fairness-accuracy trade-off for both equalized odd and equal opportunity compared to other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Impact of the number of source domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Our work focuses on transferring fairness and accuracy under domain generalization when the target domain data are inaccessible during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Instead, it relies on a set of source domains to generalize to an unseen, novel target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We investigate the relationship between the fairness-accuracy trade-off on the target domain and the number of source domains during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In particular, we evaluate the performances of FATDM and ERM on Edema dataset with different numbers of source domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Similar to the previous experiment, we first construct the dataset for each domain by rotating images with θ degree, where θ ∈ {0◦, 15◦, 30◦} when the number of domain is 3, θ ∈ {0◦, 15◦, 30◦, 45◦} when the number of domain is 4, and θ ∈ {0◦, 15◦, 30◦, 45◦, 60◦} when the number of domain is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The number of images per domain is adapted to ensure the training set size is fixed for the three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We follow the leave-one-out domain setting in which one domain serves as the unseen target domain for evaluation while the rest domains are for training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' the average results across target domains are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Figure 11 shows error-unfairness curves of FATDM and ERM when training with 2, 3, and 4 source domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We observe that training with more source domains does not always help the model achieve better fairness-accuracy trade-off on unseen target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In particular, the performances of both FATDM and ERM are the best when training with 2 source domains and the worst when training with 3 source domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We conjecture the reason that adding more source domains may help reduce the discrepancy between source and target domains (term (ii) in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1 and Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 3), but it may make it more difficult to minimize the source error and unfairness (term (i) in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1 and Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 3) and to learn invariant representation across the source domains (term (iii) in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1 and Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Thus, our suggestion in practice is to conduct an ablation study to find the optimal number of source domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Simultaneous and sequential training comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In all experiments we conducted so far, the fairness constraint Lfair is optimized simultaneously with the prediction error Lacc and the domain-invariant constraint Linv for all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' To investigate whether FATDM still attains a better accuracy-fairness trade-off when the processes of invariant representation learning and fair model training are decoupled, we conduct another set of experiments where models (FATDM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', FATDM-StarGAN) and baselines G2DM, DANN, CDANN) are learned in a sequential matter: for each model, we first learn the representation mapping g by optimizing Linv and Lacc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' using the representations generated by the fixed g, we then learn the fair classifier by optimizing Lacc and Lfair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The models trained based on the above procedure are named FATDM-seq, G2DM-seq, DANN-seq, and CDANN-seq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' and their corresponding error-unfairness curves are shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The results show that FATDM-seq still attains the best accuracy-fairness trade-off at target domain compared to G2DM-seq, DANN-seq, CDANN-seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Our method is effective no matter whether Lfair and Linv are optimized simultaneously or sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The reason that our method consistently outperforms the baselines for both settings is that the invariant-representation learning in baseline methods only guarantees the transfer of accuracy but not fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Even though a fairness regularizer is imposed to ensure the model is fair at source 21 Published as a conference paper at ICLR 2023 domains (no matter whether invariant representations and fair classifier are trained simultaneously or sequentially), this fairness cannot be preserved at the target domain due to the potential distributional shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The key to ensuring the transfer of fairness is to learn representations such that P(Z|Y, A) is domain-invariant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' this must be done during the representation learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' From Thm 3, we can see that unfairness at target domain ϵEO DT can still blow up if P Z|Y,A is different across domains, regardless of how fair the model is at source domains (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', small ϵEO DS i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Figure 7: Error-unfairness curves with respect to equalized odds of FATDM and baselines on Car- diomegaly disease dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 22 -- ERM -●- G2DM O- DANN -- CDANN -- CORAL GroupDRO IRM ●- ATDM -- FATDM-StarGAN -- FATDM-CycleGAN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 (EMD) (MD) Unfairness Unfairness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content="2 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 Error AUROC Error AUROC (MD) (EMD) Unfairness Unfairness ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 Error AUPR Error AUPR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 (EMD) (MD) Unfairness Unfairness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 Error (CE) Error (CE) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 (MD) (EMD) J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 Unfairness Unfairness ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content="0 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 Error (MR) Error (MR) (MD) (EMD) Unfairness ( J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 Unfairness ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content="4 8'0 L." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 Error (F)Published as a conference paper at ICLR 2023 Figure 8: Error-unfairness curves with respect to equal opportunity of FATDM and baselines on Cardiomegaly disease dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 23 -- ERM G2DM O- DANN CDANN CORAL GroupDRO IRM -●-ATDM -- FATDM-StarGAN -- FATDM-CycleGAN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 (MD) (EMD) Unfairness Unfairness ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content="2 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Error AUROC Error AUROC (EMD) (MD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 Unfairness ( Unfairness ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 中 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 Error AUPR Error AUPR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 (MD) (EMD) Unfairness Unfairness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content="0 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 Error (CE) Error (CE) (EMD) (MD) Unfairness Unfairness ( 02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 Error (MR) Error (MR) Unfairness (EMD) (MD) : Unfairness ( 1 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 Error (F))Published as a conference paper at ICLR 2023 Figure 9: Error-unfairness curves with respect to equalized odds of FATDM and baselines on Edema disease dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 24 -- ERM ●-G2DM O- DANN CDANN GroupDRO O- CORAL O- FATDM-StarGAN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 (MD) (EMD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 Unfairness Unfairness ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Error AUROC Error AUROC (EMD) (MD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 6 Unfairness ( Unfairness ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content="0 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 Error AUPR Error AUPR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 (MD) (EMD) LE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 Unfairness Unfairness ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content="2 F 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 Error (CE) Error (CE) ti!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (MD) (EMD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 Unfairness Unfairness ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Error (MR) Error (MR) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Unfairness (EMD) (MD) Unfairness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0Published as a conference paper at ICLR 2023 Figure 10: Error-unfairness curves with respect to equal opportunity of FATDM and baselines on Edema disease dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 25 O- ERM ●-G2DM DANN CDANN GroupDRO CORAL O- FATDM-StarGAN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Unfairness (EMD) (MD) Unfairness ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 F 0°0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Error AUROC Error AUROC Unfairness (MD) 1 Unfairness (EMD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 Error AUPR Error AUPR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 (MD) Unfairness (EMD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 Unfairness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content="2 F 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Error (CE) Error (CE) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 (MD) (EMD) Unfairness ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 Unfairness ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Error (MR) Error (MR) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Unfairness (EMD) (MD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 Unfairness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Error (Fl)Published as a conference paper at ICLR 2023 Table 6: Area under the error-unfairness curves (Cardiomegaly disease dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Error - Unfairness Method ERM G2DM DANN CDANN CORAL GroupDRO IRM FATDM Equalized Odds AUROC - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7571 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7239 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6784 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0935 AUPRC - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5463 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7730 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6883 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0291 CE - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2861 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2601 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4622 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4232 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4424 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3370 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2152 MR - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6312 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6795 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6683 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6382 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5721 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2439 F1 - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5901 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6507 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6547 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5360 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3365 AUROC - EMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7326 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7931 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7991 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1099 AUPRC - EMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6901 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7146 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8308 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7577 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7918 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7806 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0437 CE - EMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4443 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6143 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5788 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5873 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4911 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5274 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3384 MR - EMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5795 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6571 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6483 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2045 F1 - EMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6866 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7279 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6515 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2888 Equal Opportunity AUROC - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6686 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6288 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5910 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0750 AUPRC - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5419 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6718 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7189 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6761 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6435 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0262 CE - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3690 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4272 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4492 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3780 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3582 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2754 MR - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4897 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4368 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4173 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1778 F1 - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4570 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4608 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3561 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3510 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2737 AUROC - EMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7184 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7649 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7517 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6720 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6780 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0947 AUPRC - EMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7718 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7877 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7335 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='7200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0448 CE - EMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4340 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4917 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3070 MR - EMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4619 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6420 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6532 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5790 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5515 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5298 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1918 F1 - EMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3876 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4898 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4889 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3108 Table 7: Area under the error-unfairness curves (Edema disease dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Error - Unfairness Method ERM G2DM DANN CDANN CORAL GroupDRO FATDM Equalized Odds AUROC - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3395 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2765 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2972 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2548 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3642 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0633 AUPRC - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2865 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2446 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2561 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2304 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0771 CE - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1243 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0341 MR - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3929 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3509 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3302 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4303 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0656 F1 - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4213 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3527 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4178 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4189 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1369 AUROC - EMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4277 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3813 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3637 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3419 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4419 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4394 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2729 AUPRC - EMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3868 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3588 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3285 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3245 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3041 CE - EMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2366 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2401 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2348 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2334 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2447 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2339 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1827 MR - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4592 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4435 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3904 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4792 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2802 F1 - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5186 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4642 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4827 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3855 Equal Opportunity AUROC - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2488 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1806 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2696 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0218 AUPRC - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2606 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2381 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2297 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2937 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2874 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0168 CE - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1540 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1839 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1689 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1572 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1487 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1446 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0234 MR - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2652 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2620 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2516 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0468 F1 - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2848 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2534 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0502 AUROC - EMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2736 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2472 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2897 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2841 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1121 AUPRC - EMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2653 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2451 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2429 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2812 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0912 CE - EMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2318 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2147 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1159 MR - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3409 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3162 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3442 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3388 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1872 F1 - MD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3237 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2756 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3271 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1779 26 Published as a conference paper at ICLR 2023 Figure 11: Error-unfairness curves with respect to equalized odds of FATDM and ERM on Edema disease dataset when training with different numbers of source domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Names in the figure legend are in the form of X-Y where X is the model and Y is the number of source domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', ERM-2 means training ERM on two source domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=') 27 ERM-2 ERM-3 ERM-4 FATDM-2 FATDM-3 FATDM-4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content="0 (MD) (EMD) 3'0 U." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 Unfairness Unfairness ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content="2 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Error AUROC Error AUROC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ( (EMD) (MD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 Unfairness Unfairness ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Error AUPR Error AUPR (EMD) (MD) Unfairness Unfairness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 Error (CE) Error (CE) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 (EMD) (MD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 Unfairness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 Unfairness ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content="2 F 0°0 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Error (MR) Error (MR) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (EMD) (MD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 Unfairness Unfairness ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Error (Fl) Error (F)Published as a conference paper at ICLR 2023 (a) Equalized Odds (b) Equal Opportunity Figure 12: Fairness-accuracy trade-off (Pareto frontier) of models trained with simultaneous and sequential (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', models with ‘-seq’ suffix) approaches, and FATDM-CycleGAN (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', use CycleGAN instead of StarGAN as density mapping functions) on Cardiomegaly disease dataset: error-unfairness curves are constructed by varying ω ∈ [0, 10] and the values of error and unfairness are normalized to [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Lower-left points and the smaller area under the curve indicate the model has a better fairness-accuracy trade-off (Pareto optimality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' D ADDITIONAL RESULTS & LEMMAS D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1 TIGHTER UPPER BOUND FOR ACCURACY Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1 We can replace term (ii) in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1 with the following term to attain a tighter upper bound for accuracy: √ 2C min i∈[N] � dJS � P Y DT , P Y DS i � + � 2ηT V Ez∼PDT i (z) � dJS � P X|Y DT , P X|Y DT i �2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' where ηT V = sup P X Di̸=P X Dj DT V � P Z Di,P Z Dj � DT V � P X Di,P X Dj � ≤ 1 is called Dobrushin’s coefficient (Polyanskiy & Wu, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' This result suggests that we can further optimize term (ii) in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1 by minimizing ηT V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' It has been shown in Shui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2022) that ηT V can be controlled by Lipschitz constant of the feature mapping g : X → Z when g follows Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The Lipschitz constant of g, in turn, can be upper bounded by the Frobenius norm of Jacobian matrix with respect to g (Miyato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' However, in practice, we found that computing Jacobian matrix of g is computationally expensive when dimension of representation Z is large, and optimizing it together with invariant constraints does not improve the performances of models in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 LEMMAS FOR PROVING THEOREM 1 Lemma 6 Let X be the random variable in domains Di and Dj, and E be an event that P X Dj ≥ P X Di, then we have: � E ���P X Dj − P X Di ��� dX = � E ���P X Dj − P X Di ��� dX = 1 2 � ���P X Dj − P X Di ��� dX where E is the complement of event E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 28 ii G2DM G2DM-seq DANN-seq CDANN CDANN-seq FATDM FATDM-seq 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content="8 Unfairness (EMD) 8'0 Unfairness (EMD) (MD) (MD) ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 Unfairness ( Unfairness ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Error AUROC Error (MR) Error AUROC Error (MR)G2DM G2DM-seq DANN-seq CDANN i CDANN-seq FATDM FATDM-seq 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content="8 (EMD) 8'0 (MD) (EMD) (MD) Unfairness Unfairness ( Unfairness ( Unfairness 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Error AUROC Error (MR) Error AUROC Error (MR)Published as a conference paper at ICLR 2023 Lemma 7 Let X be the random variable in domains Di and Dj, let f : X → R+ be a non-negative function bounded by C, then we have: EDj[f(X)] − EDi[f(X)] ≤ C √ 2 � min � DKL � P X Di ∥ P X Dj � , DKL � P X Dj ∥ P X Di �� where DKL(· ∥ ·) is the KL-divergence between two distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Lemma 8 Suppose loss function L is upper bounded by C and consider a classifier �f : X → Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' the expected classification error of �f in domain Dj can be upper bounded by its error in domain Di: ϵAcc Dj � �f � ≤ ϵAcc Di � �f � + √ 2CdJS � P X,Y Dj , P X,Y Di � where X, Y are random variables denoting feature and label in domains Di and Dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Lemma 9 Consider two distributions P X Di and P X Dj over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Let P Z Di and P Z Dj be the induced distributions over Z by mapping function g : X → Z, then we have: dJS(P X Di, P X Dj) ≥ dJS(P Z Di, P Z Dj) Lemma 10 (Phung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 2021) Consider domain D with joint distribution P X,Y D and labeling function fD : X → Y∆ from feature space to label space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Given mapping function g : X → Z from feature to representation space, we define labeling function hD : Z → Y∆ from representation space to label space as hD(Z)Y = fD(X)Y ◦ g−1(Z) = � g−1(Z) fD(X)Y P X D dX � g−1(Z) P X D dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Similarly, let �f be the hypothesis from feature space, then the corresponding hypothesis �h from representation space under the mapping function g is computed as �h(Z)Y = � g−1(Z) � f(X)Y P X D dX � g−1(Z) P X D dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Let ϵAcc D ( �f) = ED � L( �f(X), Y ) � and ϵAcc D (�h) = ED � L(�h(Z), Y ) � be expected errors defined with respect to feature space and representation space, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We have: ϵAcc D � �f � = ϵAcc D � �h � D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='3 LEMMAS FOR PROVING COROLLARY 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1 Lemma 11 Consider two random variables X, Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Let P X,Y Di , P X,Y Dj be two joint distributions defined in domains Di and Dj, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Then, JS-divergence DJS � P X,Y Di ∥ P X,Y Dj � and KL-divergence DKL � P X,Y Di ∥ P X,Y Dj � can be decomposed as follows: DKL � P X,Y Di ∥ P X,Y Dj � = DKL � P Y Di ∥ P Y Dj � + EDi � DKL � P X|Y Di ∥ P X|Y Dj �� DJS � P X,Y Di ∥ P X,Y Dj � ≤ DJS � P Y Di ∥ P Y Dj � + EDi � DJS � P X|Y Di ∥ P X|Y Dj �� + EDj � DJS � P X|Y Di ∥ P X|Y Dj �� D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='4 LEMMAS FOR PROVING THEOREM 2 Lemma 12 Under Assumption in Theorem 2, the following holds for any domain D: � ϵAcc D ( �f) = � ED[L( �f(X), Y )] ≥ � 2c |Y|dJS(P Y D , P �Y D )2, ∀ �f where �Y is the prediction made by randomized predictor �f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 29 Published as a conference paper at ICLR 2023 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='5 LEMMAS FOR PROVING THEOREM 3 Definition 13 Given domain Di with binary random variable A denoting the sensitive attribute, the unfairness measures that evaluate the violation of equalized odd (EO) and equal opportunity (EP) criteria between sensitive groups of this domain are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ϵEO Di � �f � = ���R0,0 Di � �f � − R0,1 Di � �f ���� + ���R1,0 Di � �f � − R1,1 Di � �f ���� ϵEP Di � �f � = ���R1,0 Di � �f � − R1,1 Di � �f ���� where Ry,a Di � �f � = EDi � �f(X)1|Y = y, A = a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Lemma 14 Given two domains Di and Dj, under Definition 13, Ry,a Dj � �f � can be bounded by Ry,a Di � �f � as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Ry,a Dj � �f � ≤ Ry,a Di � �f � + √ 2dJS � P X|Y =y,A=a Dj , P X|Y =y,A=a Di � ∀y, a ∈ {0, 1} Lemma 15 Given two domains Di and Dj, under Definition 13, the unfairness in domain Dj can be upper bounded by the unfairness measure in domain Di as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ϵEO Dj � �f � ≤ ϵEO Di � �f � + √ 2 � y=0,1 � a=0,1 dJS � P X|Y =y,A=a Dj , P X|Y =y,A=a Di � ϵEP Dj � �f � ≤ ϵEP Di � �f � + √ 2 � a=0,1 dJS � P X|Y =1,A=a Dj , P X|Y =1,A=a Di � Lemma 16 Consider domain D with distribution P X,Y D and labeling function fD : X → Y∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Given mapping function g : X → Z from feature to representation space, we define labeling function hD : Z → Y∆ from representation space to label space as hD(Z)Y = fD(X)Y ◦ g−1(Z) = � g−1(Z) fD(X)Y P X D dX � g−1(Z) P X D dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Similarly, let �f be the hypothesis from feature space, then the corresponding hypothesis �h from representation space under the mapping function g is computed as �h(Z)Y = � g−1(Z) � f(X)Y P X D dX � g−1(Z) P X D dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Under Definition 13, we have: ϵEO D � �f � = ϵEO D � �h � ϵEP D � �f � = ϵEP D � �h � D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='6 LEMMAS FOR PROVING THEOREM 5 Lemma 17 Consider two domains Di and Dj, if there exist invertible mappings my i,j and my,a i,j such that P X|y Di = P my i,j(X)|y Dj and P X|y,a Di = P my,a i,j (X)|y,a Dj , ∀y ∈ Y, a ∈ A, then DJS � P Z|y Di ∥ P Z|y Dj � and DJS � P Z|y,a Di ∥ P Z|y,a Dj � can be upper bounded by � x P x|y Di DJS � P Z|x ∥ P Z|my i,j(x)� dx and � x P x|y,a Di DJS � P Z|x ∥ P Z|my,a i,j (x)� dx, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' E PROOFS E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1 PROOFS OF THEOREMS Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' First, we get the upper bound based on the representation space Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Then, we relate it with the feature space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Let DS ∗ ∈ {DS i }N i=1 be the source domain that’s nearest to the 30 Published as a conference paper at ICLR 2023 target domain DT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' According to Lemma 8, we have upper bound of the expected classification error for the target domain based on each of the source domain as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ϵAcc DT � �h � ≤ ϵAcc DS i � �h � + √ 2CdJS � P Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y DT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y DS i � ∀i ∈ [N] Taking average of upper bounds based on all source domains,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' we have: ϵAcc DT � �h � ≤ 1 N N � i=1 ϵAcc DS i � �h � + √ 2C N N � i=1 dJS � P Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y DT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y DS i � (1) ≤ 1 N N � i=1 ϵAcc DS i � �h � + √ 2C N N � i=1 dJS � P Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y DT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y DS ∗ � + √ 2C N N � i=1 dJS � P Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y DS ∗ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y DS i � (2) ≤ 1 N N � i=1 ϵAcc DS i � �h � + √ 2C min i∈[N]dJS � P Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y DT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y DS i � + √ 2C max i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j∈[N]dJS � P Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y DS i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y DS j � (7) Here we have (1) ≤ by using triangle inequality for JS-distance: dJS(P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' R) ≤ dJS(P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Q) + dJS(Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' R) with P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' and R = PDT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' PDS ∗ and PDS i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We have (2) ≤ because DS ∗ ∈ {DS i }N i=1 then dJS � P Z,Y DS ∗ , P Z,Y DS i � ≤ max i,j∈[N]dJS � P Z,Y DS i , P Z,Y DS j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Similarly, we can obtain the upper bound based on the feature space X as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ϵAcc DT � �f � ≤ 1 N N � i=1 ϵAcc DS i � �f � + √ 2C min i∈[N]dJS � P X,Y DT , P X,Y DS i � + √ 2C max i,j∈[N]dJS � P X,Y DS i , P X,Y DS j � (8) However, the bounds in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (7) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (8) are based on either feature space or representation space, which is not readily to use for practical algorithmic design because the actual objective is to minimize ϵAcc DT � �f � in feature space by controlling Z in representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' According to Lemmas 9 and 10, we can derive the bound that relates feature and representation spaces as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ϵAcc DT � �f � = ϵAcc DT � �h � ≤ 1 N N � i=1 ϵAcc DS i � �h � + √ 2C min i∈[N]dJS � P Z,Y DT , P Z,Y DS i � + √ 2C max i,j∈[N]dJS � P Z,Y DS i , P Z,Y DS j � ≤ 1 N N � i=1 ϵAcc DS i � �f � + √ 2C min i∈[N]dJS � P X,Y DT , P X,Y DS i � + √ 2C max i,j∈[N]dJS � P Z,Y DS i , P Z,Y DS j � (9) Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' dJS � P Z,Y DS i , P Z,Y DS j � = � DJS � P Z,Y DS i ∥ P Z,Y DS j � (1) ≤ � DJS � P Y DS i ∥ P Y DS j � + 2Ez∼PDS i,j (z) � DJS � P Z|Y DS i ∥ P Z|Y DS j �� (2) ≤ dJS � P Y DS i , P Y DS j � + � 2Ez∼PDS i,j (z) � dJS � P Z|Y DS i , P Z|Y DS j �2� Here we have (1) ≤ by using Lemma 11 to decompose the JS-divergence of the joint distributions and (2) ≤ by using inequality √ a + b ≤ √a + √ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 31 Published as a conference paper at ICLR 2023 This new upper bound, combined with Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1 suggests learning representation Z such that P Z|Y DS i is invariant across source domains, or in another word, Z ⊥ D | Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' This result is consistent with Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 4: when the target domain DT is the mixture of source domains {DS i }N i=1, and when P Y DS i and P Z|Y DS i are invariant across source domains, we have dJS � P Z,Y DT , P Z,Y DS i � = dJS � P Z,Y DS i , P Z,Y DS j � = 0, implying ϵAcc DT � �f � ≤ 1 N �N i=1 ϵAcc DS i � �f � = ϵAcc DS i � �f � ∀i ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Proof of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1 (tighter upper bound for accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The bound in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (9) is constructed using Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Indeed, we can make this bound tighter using the strong data processing inequality for JS-divergence (Polyanskiy & Wu, 2017), as stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' DJS � P Z Di ∥ P Z Dj � ≤ ηJSDJS � P X Di ∥ P X Dj � ≤ ηT V DJS � P X Di ∥ P X Dj � where Z is random variable induced from random variable X, and P X Di and P X Di are two distribution over X, and ηJS = sup P X Di̸=P X Dj DJS � P Z Di,P Z Dj � DJS � P X Di,P X Dj � ≤ ηT V = sup P X Di̸=P X Dj DT V � P Z Di,P Z Dj � DT V � P X Di,P X Dj � ≤ 1, DT V is the total variation distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ηT V is called the Dobrushin’s coefficient (Polyanskiy & Wu, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Apply Lemma 11 and this inequality to the second term in the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (7) (similar to the proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1), we have: √ 2C min i∈[N]dJS � P Z,Y DT , P Z,Y DS i � ≤ √ 2C min i∈[N] � dJS � P Y DT , P Y DS i � + � 2Ez∼PDT i (z) � dJS � P Z|Y DT , P Z|Y DT i �2�� ≤ √ 2C min i∈[N] � dJS � P Y DT , P Y DS i � + � 2ηT V Ez∼PDT i (z) � dJS � P X|Y DT , P X|Y DT i �2�� (10) Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Consider a source domain DS i and target domain DT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Because JS-distance dJS(·, ·) is a distance metric, we have triangle inequality: dJS(P Y DS i , P Y DT ) ≤ dJS(P Y DS i , P �Y DS i ) + dJS(P �Y DS i , P �Y DT ) + dJS(P �Y DT , P Y DT ) Since X g −→ Z �h −→ �Y , we have dJS(P �Y DS i , P �Y DT ) ≤ dJS(P Z DS i , P Z DT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Using Lemma 12, the following holds when dJS(P Y DS i , P Y DT ) ≥ dJS(P Z DS i , P Z DT ) � dJS(P Y DS i , P Y DT ) − dJS(P Z DS i , P Z DT ) �2 ≤ � dJS(P Y DS i , P �Y DS i ) + dJS(P �Y DT , P Y DT ) �2 ≤ 2 � dJS(P Y DS i , P �Y DS i )2 + dJS(P �Y DT , P Y DT )2� ≤ 2 � 2c |Y| �� ϵAcc DS i ( �f) + � ϵAcc DT ( �f) � ≤ � 4|Y| c � ϵAcc DS i ( �f) + ϵAcc DT ( �f) � The last inequality is by AM-GM inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Therefore, when dJS(P Y DS i , P Y DT ) ≥ dJS(P Z DS i , P Z DT ), we have ϵAcc DS i ( �f) + ϵAcc DT ( �f) ≥ c 4|Y| � dJS(P Y DS i , P Y DT ) − dJS(P Z DS i , P Z DT ) �4 The above holds for any source domain DS i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Average over all N source domains, we have 1 N N � i=1 ϵAcc DS i ( �f) + ϵAcc DT ( �f) ≥ c 4|Y|N N � i=1 � dJS(P Y DS i , P Y DT ) − dJS(P Z DS i , P Z DT ) �4 32 Published as a conference paper at ICLR 2023 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The proof is based on Lemmas 15 and 16 and similar to the proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Let DS ∗ ∈ {DS i }N i=1 be the source domain nearest to the target domain DT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' According to Lemma 15, we have upper bound of the unfairness measured with respect to the representation space for the target domain based on each of the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' For equal opportunity (EP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' we have: ϵEP DT � �h � ≤ ϵEP DS i � �h � + √ 2 � a∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} dJS � P Z|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS i � Taking average of upper bounds based on all source domains,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' we have: ϵEP DT � �h � ≤ 1 N N � i=1 ϵEP DS i � �h � + √ 2 N N � i=1 � a∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} dJS � P Z|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS i � ≤ 1 N N � i=1 ϵEP DS i � �h � + √ 2 N N � i=1 � a∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} dJS � P Z|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS ∗ � + √ 2 N N � i=1 � a∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} dJS � P Z|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS ∗ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS i � ≤ 1 N N � i=1 ϵEP DS i � �h � + √ 2 min i∈[N] � a∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} dJS � P Z|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS i � + √ 2 max i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j∈[N] � a∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} dJS � P Z|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS j � According to Lemmas 9 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' we can relate this bound to the feature space as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ϵEP DT � �f � = ϵEP DT � �h � ≤ 1 N N � i=1 ϵEP DS i � �h � + √ 2 min i∈[N] � a∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} dJS � P Z|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS i � + √ 2 max i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j∈[N] � a∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} dJS � P Z|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS j � ≤ 1 N N � i=1 ϵEP DS i � �f � + √ 2 min i∈[N] � a∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} dJS � P X|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P X|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS i � + √ 2 max i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j∈[N] � a∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} dJS � P Z|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P Z|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS j � Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' we got the upper bound for unfairness measure with respect to equalized odds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ϵEO DT � �f � ≤ 1 N N � i=1 ϵEO DS i � �f � + √ 2 min i∈[N] � y∈{0,1} � a∈{0,1} dJS � P X|Y =y,A=a DT , P X|Y =y,A=a DS i � + √ 2 max i,j∈[N] � y∈{0,1} � a∈{0,1} dJS � P Z|Y =y,A=a DS i , P Z|Y =y,A=a DS j � (11) Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Consider two source domains, DS i and DS j , if P Y DS i = P Y DS j , we can learn the mapping function g = Pθ (Z|X) such that P Z|Y DS i = P Z|Y DS j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Note that this mapping function always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In particular, the trivial solution for Z that satisfies P Z|Y DS i = P Z|Y DS j is making Z ⊥ Y, D (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', 33 Published as a conference paper at ICLR 2023 Pθ (Z|X) = N (0, I)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Then we have: ϵAcc DS i � �h � = Ez∼P Z DS i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='y∼hDS i (z) � L � �h (Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y �� = Ey∼P Y DS i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='z∼P Z|Y DS i � L � �h (Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y �� = Ey∼P Y DS j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='z∼P Z|Y DS j � L � �h (Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y �� = Ez∼P Z DS j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='y∼hDS j (z) � L � �h (Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y �� = ϵAcc DS j � �h � For unseen target domain DT in Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' we have: ϵAcc DT � �h � = EDT � L � �h (Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y �� = � Z×Y L � �h (Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y � P Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Z DT dY dZ = � Z×Y L � �h (Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y � N � i=1 πiP Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Z DS i dY dZ = N � i=1 πi � Z×Y L � �h (Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y � P Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Z DS i dY dZ = N � i=1 πiEDS i � L � �h (Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y �� = EDS i � L � �h (Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y �� ∀i ∈ [N] = ϵAcc DS i � �h � ∀i ∈ [N] By Lemma 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' we have ϵAcc DT � �h � = ϵAcc DS i � �h � = ϵAcc DT � �f � = ϵAcc DS i � �f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' For fairness, we only give the proof for equalized odds (EO), we can easily get the similar derivation for equal opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' For any Z that satisfies P Z|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS i = P Z|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS j ∀y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' a ∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' we have: ϵEO DS i � �h � = � y∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} D � P �h(Z)1|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=0 DS i ∥ P �h(Z)1|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=1 DS i � = � y∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} D � P �h(Z)1|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=0 DS j ∥ P �h(Z)1|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=1 DS j � = ϵEO DS j � �h � For unseen target domain DT in Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' we have: ϵEO DT � �h � = � y∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} D � P �h(Z)1|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=0 DT ∥ P �h(Z)1|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=1 DT � = � y∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} D � N � i=1 πiP �h(Z)1|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=0 DS i ∥ N � i=1 πiP �h(Z)1|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=1 DS i � = � y∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1} D � P �h(Z)1|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=0 DS i ∥ P �h(Z)1|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=1 DS i � ∀i ∈ [N] = ϵEO DS i � �h � ∀i ∈ [N] 34 Published as a conference paper at ICLR 2023 Similar to the proof of accuracy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Z that satisfies P Z|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS i = P Z|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a DS j ∀y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' a ∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' j ∈ [N] always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' The trivial solution for is Z that satisfies Z ⊥ Y, A, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' By Lemma 16, we have ϵEO DT � �h � = ϵEO DS i � �h � = ϵEO DT � �f � = ϵEO DS i � �f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' For equal opportunity (EP), Z only need to satisfy the condition for positive label, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=', P Z|Y =1,A=a DS i = P Z|Y =1,A=a DS j ∀a ∈ {0, 1}, i, j ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' According to Lemma 17, we have: DJS � P Z|y Di ∥ P Z|y Dj � ≤ � x P x|y Dj DJS � P Z|x Di ∥ P Z|my i,j(x) Di � dx (12) Then, minimizing DJS � P Z|y Di ∥ P Z|y Dj � can be achieved by minimizing DJS � P Z|x ∥ P Z|my i,j(x)� ∀x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We can upper bound DJS � P Z|x ∥ P Z|my i,j(x)� as follows DJS � P Z|x ∥ P Z|my i,j(x)� ≤ DT V � P Z|x ∥ P Z|my i,j(x)� ≤ √ 2 d1/2 � P Z|x, P Z|my i,j(x)� (1) = √ 2 d1/2 � N � µ(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' σ2Id � , N � µ � my i,j(x) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' σ2Id �� (13) where DT V and d1/2 are total variation distance and Hellinger distance between two distributions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We have (1) = because of our choice for representation mapping g(x) := P Z|x = N � µ(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' σ2Id � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' According to Devroye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2018), the Hellinger distance between two multivariate normal distributions over Rd has a closed form as follows d1/2 (N (µ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Σ1) , N (µ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Σ2)) = � � � �1 − det (Σ1)1/4 det (Σ2)1/4 det � Σ1+Σ2 2 �1/2 exp � −1 8 (µ1 − µ2)T �Σ1 + Σ2 2 �−1 (µ1 + µ2) � (14) where µ1, µ2, Σ1, Σ2 are mean vectors and covariance matrices of the two normal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (14), let µ1 = µ(x), µ2 = µ � my i,j(x) � , Σ1 = Σ2 = σ2Id, then we have: d1/2 � N � µ(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' σ2Id � , N � µ � my i,j(x) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' σ2Id �� = � 1 − exp � − 1 8dσ2 � µ (x) − µ � my i,j(x) ��T � µ (x) − µ � my i,j(x) ��� = � 1 − exp � − 1 8dσ2 ��µ (x) − µ � my i,j(x) ���2 2 � (15) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (15), we can see that Helinger distance between two representation distributions P Z|x and P Z|my i,j(x) is the function of their means µ (x) and µ � my i,j(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Combining this with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (12) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (13), we conclude that minimizing dJS � P Z|y DS i , P Z|y DS j � can be reduced to minimizing ��µ(x) − µ � my i,j(x) ��� 2 which can be implemented as the mean square error between µ(x) and µ � my i,j(x) � in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Proof for dJS � P Z|y,a DS i , P Z|y,a DS j � is derived in the similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 35 Published as a conference paper at ICLR 2023 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 PROOFS OF LEMMAS Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We have: � E ���P X Dj − P X Di ��� dX = � E � P X Dj − P X Di � dX = � E∪E � P X Dj − P X Di � dX − � E � P X Dj − P X Di � dX = � E � P X Di − P X Dj � dX = � E ���P X Dj − P X Di ��� dX = 1 2 � ���P X Dj − P X Di ��� dX Proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='We have: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='EDj [f(X)] = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='f(X)P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='DjdX = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='f(X)P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='DidX + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='f(X) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Dj − P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Di ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='dX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='= EDi [f(X)] + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='f(X) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Dj − P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Di ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='dX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='= EDi [f(X)] + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='f(X) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Dj − P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Di ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='dX + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='f(X) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Dj − P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Di ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='dX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='≤ EDi [f(X)] + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='f(X) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Dj − P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Di ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='dX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='≤ EDi [f(X)] + C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Dj − P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Di ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='dX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='= EDi [f(X)] + C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='���P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Dj − P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Di ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='��� dX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='≤ EDi [f(X)] + C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ���P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Dj − P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Di ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='��� dX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='(4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='≤ EDi [f(X)] + C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='2 min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='DKL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Di ∥ P X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Dj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' DKL � P X Dj ∥ P X Di �� = EDi [f(X)] + C √ 2 � min � DKL � P X Di ∥ P X Dj � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' DKL � P X Dj ∥ P X Di �� where E is the event that P X Dj ≥ P X Di and E is the complement of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We have (1) ≤ because � E f(X) � P X Dj − P X Di � dX ≤ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2) ≤ because f(X) is non-negative function and is bounded by C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (3) ≤ by using Lemma 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (4) ≤ by using Pinsker’s inequality between total variation norm and KL- divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Proof of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Applying Lemma 7 and replacing X by (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' f by loss function L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Di by Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' we have: ϵAcc Dj � �f � − EDi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j � L( �f(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y ) � = EDj � L( �f(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y ) � − EDi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j � L( �f(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y ) � ≤ C √ 2 � min � DKL � P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Dj ∥ P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' DKL � P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j ∥ P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Dj �� ≤ C √ 2 � DKL � P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Dj ∥ P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j � (16) 36 Published as a conference paper at ICLR 2023 Applying Lemma 7 again and replacing X by (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' f by loss function L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Dj by Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' we have: EDi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j � L( �f(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y ) � − ϵAcc Di � �f � = EDi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j � L( �f(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y ) � − EDi � L( �f(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y ) � ≤ C √ 2 � min � DKL � P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Di ∥ P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' DKL � P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j ∥ P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Di �� ≤ C √ 2 � DKL � P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Di ∥ P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j � (17) Adding Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (16) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (17), we have: ϵAcc Dj � �f � − ϵAcc Di � �f � ≤ C √ 2 �� DKL � P X,Y Di ∥ P X,Y Di,j � + � DKL � P X,Y Dj ∥ P X,Y Di,j �� (1) ≤ C √ 2 � 2 � DKL � P X,Y Di ∥ P X,Y Di,j � + DKL � P X,Y Dj ∥ P X,Y Di,j �� = C √ 2 � 4DJS � P X,Y Di ∥ P X,Y Dj � = √ 2CdJS � P X,Y Di , P X,Y Dj � Here we have (1) ≤ by using Cauchy–Schwarz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Proof of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Note that the JS-divergence DJS � P X Di ∥ P X Dj � can be understood as the mutual information between a random variable X associated with the mixture distribution P X Di,j = 1 2 � P X Di + P X Dj � and the equiprobable binary random variable T used to switch between P X Di and P X Dj to create the mixture distribution P X Di,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' In particular, we have: DJS � P X Di ∥ P X Dj � = 1 2 � DKL � P X Di ∥ P X Di,j � + DJS � P X Dj ∥ P X Di,j �� = 1 2 � � log P X Di − log P X Di,j � P X DidX + 1 2 � � log P X Dj − log P X Di,j � P X DjdX = �1 2 � log � P X Di � P X Didx + 1 2 � log � P X Dj � P X DjdX � − � log � P X Di,j � P X Di,jdX = −H(X|T) + H(X) = I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' T) where H(X) is the entropy of X, H(X|T) is the entropy of X conditioned on T, and I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' T) is the mutual information between X and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Similarly, we also have DJS((P Z Di ∥ P Z Dj)) = I(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Because Z is induced from X by the mapping function h then we have Z ⊥ T | X and the Markov chain T → X → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' According to data processing inequality for mutual information (Polyanskiy & Wu, 2014), we have I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' T) ≥ I(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' T) which implies DJS((P X Di ∥ P X Dj)) ≥ DJS((P Z Di ∥ P Z Dj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Taking square root on both sides, we have dJS(P X Di, P X Dj) ≥ dJS(P Z Di, P Z Dj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 37 Published as a conference paper at ICLR 2023 Proof of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We have: ϵAcc D � �h � = Ez∼P Z D ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='y∼hD(z) � L � �h (Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y �� = |Y| � y=1 Ez∼P Z D � L � �h (Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' y � hD(Z)y � = |Y| � y=1 � Z L � �h (Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' y � hD(Z)yP Z DdZ = |Y| � y=1 � Z L � �h (Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' y � � g−1(Z) fD(X)yP X D dX � g−1(Z) P X D dX � g−1(Z) P X D dXdZ = |Y| � y=1 � Z L � �h (Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' y � � g−1(Z) fD(X)yP X D dXdZ = |Y| � y=1 � Z � g−1(Z) L � �h (g(X)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' y � fD(X)yP X D dXdZ = |Y| � y=1 � Z � X 1 � X ∈ g−1(Z) � L � �h (Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' y � fD(X)yP X D dXdZ = |Y| � y=1 � X � Z 1 (Z = g(X)) L � �h (Z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' y � fD(X)yP X D dXdZ = |Y| � y=1 � X L � �h (g(X)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' y � fD(X)yP X D dXdZ = |Y| � y=1 � X L � �f (X) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' y � fD(X)yP X D dX = ϵAcc D � �f � Proof of Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We show the decomposition for KL-divergence first and then use the result to derive the decomposition for JS-divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We have: DKL � P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Di ∥ P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Dj � = EDi � log P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Di − log P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Dj � = EDi � log P Y Di + log P X|Y Di � − EDi � log P Y Dj + log P X|Y Dj � = EDi � log P Y Di − log P Y Dj � + EDi � log P X|Y Di − log P X|Y Dj � = EDi � log P Y Di − log P Y Dj � + Ey∼P Y Di � Ex∼P X|y Di � log P X|Y Di − log P X|Y Dj �� = DKL � P Y Di ∥ P Y Dj � + EDi � DKL � P X|Y Di ∥ P X|Y Dj �� 38 Published as a conference paper at ICLR 2023 DJS � P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Di ∥ P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Dj � = 1 2 � DKL � P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Di ∥ P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j �� + 1 2 � DKL � P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Dj ∥ P X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='Y Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j �� = 1 2 � DKL � P Y Di ∥ P Y Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j �� + 1 2 � EDi � DKL � P X|Y Di ∥ P X|Y Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j ��� + 1 2 � DKL � P Y Dj ∥ P Y Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j �� + 1 2 � EDj � DKL � P X|Y Dj ∥ P X|Y Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j ��� = DJS � P Y Di ∥ P Y Dj � + 1 2 � EDi � DKL � P X|Y Di ∥ P X|Y Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j ��� + 1 2 � EDj � DKL � P X|Y Dj ∥ P X|Y Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j ��� ≤ DJS � P Y Di ∥ P Y Dj � + 1 2 � EDi � DKL � P X|Y Di ∥ P X|Y Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j ��� + 1 2 � EDi � DKL � P X|Y Dj ∥ P X|Y Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j ��� + 1 2 � EDj � DKL � P X|Y Dj ∥ P X|Y Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j ��� + 1 2 � EDj � DKL � P X|Y Di ∥ P X|Y Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j ��� = DJS � P Y Di ∥ P Y Dj � + EDi � DJS � P X|Y Di ∥ P X|Y Dj �� + EDj � DJS � P X|Y Di ∥ P X|Y Dj �� Proof of Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ED � L( �f(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y ) � = ED � �� �y∈Y �f(X)�yL(�y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Y ) � � (1) ≥ c EX � �� �y∈Y �f(X)�y Pr(Y ̸= �y|X) � � (2) = c EX � 1 − �f(X)T f(X) � (3) ≥ c 2 EX ���� �f(X) − f(X) ��� 2 2 � (4) ≥ c 2 1 |Y|EX ����� �f(X) − f(X) ��� 1 �2� (5) ≥ c 2 1 |Y| � ���EX � �f(X) − f(X) ���� 1 �2 = c 2 1 |Y| ���P �Y D − P Y D ��� 2 1 (6) ≥ 2c |Y|DJS � P Y D ∥ P �Y D �2 = 2c |Y| · dJS � P Y D ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P �Y D �4 Here we have (1) ≥ is because of the assumption that L(�y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' y) is lower bounded by c when �y ̸= y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2) = is because �f(X)T 1 = || �f(X)||1 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (3) ≥ is because || �f(X)||2 ≤ || �f(X)||1 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (4) ≥ is because || �f(X)||2 ≥ 1 √ |Y||| �f(X)||1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (5) ≥ is by using Jensen’s inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (6) ≥ is by using JS-divergence lower bound of total variation distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 39 Published as a conference paper at ICLR 2023 Proof of Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Similar to the proof in Lemma 8, we apply Lemma 7 for Ry,a Di and Ry,a Dj and note that �f(X)y is bounded by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Then ∀y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' a ∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' 1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' we have: Ry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a Dj − EDi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j � �f(X)y|Y = y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' A = a � = EDj � �f(X)y|Y = y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' A = a � − EDi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j � �f(X)y|Y = y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' A = a � ≤ 1 √ 2 � min � DKL � P X|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a Di ∥ P X|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' DKL � P X|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j ∥ P X|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a Di �� ≤ 1 √ 2 � DKL � P X|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a Dj ∥ P X|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j � (18) EDi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j � �f(X)y|Y = y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' A = a � − Ry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='a Di = EDi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j � �f(X)y|Y = y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' A = a � − EDi � �f(X)y|Y = y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' A = a � ≤ 1 √ 2 � min � DKL � P X|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a Dj ∥ P X|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' DKL � P X|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j ∥ P X|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a Dj �� ≤ 1 √ 2 � DKL � P X|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a Di ∥ P X|Y =y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='j � (19) Adding Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (18) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (19), we have: Ry,a Dj − Ry,a Di ≤ 1 √ 2 �� DKL � P X|Y =y,A=a Di ∥ P X|Y =y,A=a Di,j � + � DKL � P X|Y =y,A=a Dj ∥ P X|Y =y,A=a Di,j �� ≤ √ 2dJS � P X|Y =y,A=a Dj , P X|Y =y,A=a Di,j � Proof of Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' We give the proof for unfairness measure w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' to equal opportunity first and then use this result to derive the proof for unfairness measure w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' to equalized odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Without loss of generality, assign group indices 1, 0 be such that R1,0 Dj � �f � ≥ R1,1 Dj � �f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Then we have: ϵEP Dj � �f � = ���R1,0 Dj � �f � − R1,1 Dj � �f ���� = R1,0 Dj � �f � − R1,1 Dj � �f � = R1,0 Dj � �f � − EDj � �f(X)1|Y = 1, A = 1 � = R1,0 Dj � �f � + EDj � 1 − �f(X)1|Y = 1, A = 1 � − 1 = R1,0 Dj � �f � + R1,1 Dj � 1 − �f � − 1 where 1 is vector with all 1’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' By Lemma 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' we have: R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Dj � �f � ≤ R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Di � �f � + √ 2dJS � P X|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=0 Dj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P X|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=0 Di � R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1 Dj � 1 − �f � ≤ R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1 Di � 1 − �f � + √ 2dJS � P X|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=1 Dj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P X|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=1 Di � Sum above two inequalities and add −1 at both sides,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ϵEP Dj � �f � = R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Dj � �f � + R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1 Dj � 1 − �f � − 1 ≤ R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Di � �f � + R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1 Di � 1 − �f � − 1 + √ 2 � a=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1 dJS � P X|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a Dj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P X|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a Di � ≤ ϵEP Di � �f � + √ 2 � a=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1 dJS � P X|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a Dj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P X|Y =1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a Di � (20) 40 Published as a conference paper at ICLR 2023 Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' we have: ���R0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Dj � �f � − R0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1 Dj � �f ���� ≤ ���R0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='0 Di � �f � − R0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1 Di � �f ���� + √ 2 � a=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='1 dJS � P X|Y =0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a Dj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' P X|Y =0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content='A=a Di � (21) Sum both Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (20) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (21), we have: ϵEO Dj � �f � ≤ ϵEO Di � �f � + √ 2 � y=0,1 � a=0,1 dJS � P X|Y =y,A=a Dj , P X|Y =y,A=a Di � Proof of Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Similar to the proof of Lemma 10, Ry,a Di � �f � = Ry,a Di � �h � ∀y, a ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' Then, we have: ϵEO Di � �f � = ���R0,0 Di � �f � − R0,1 Di � �f ���� + ���R1,0 Di � �f � − R1,1 Di � �f ���� = ���R0,0 Di � �h � − R0,1 Di � �h ���� + ���R1,0 Di � �h � − R1,1 Di � �h ���� = ϵEO Di � �h � ϵEP Di � �f � = ���R1,0 Di � �f � − R1,1 Di � �f ���� = ���R1,0 Di � �h � − R1,1 Di � �h ���� = ϵEP Di � �h � Proof of Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' ∀y ∈ Y, we have: DJS � P Z|y i ∥ P Z|y j � (1) = DJS �� X P Z|xP x|y i dx ∥ � X P Z|my i,j(x)P my i,j(x)|y j dmy i,j(x) � (2) = DJS �� X P Z|xP x|y i dx ∥ � X P Z|my i,j(x)P my i,j(x)|y j dx � (3) = DJS �� X P Z|xP x|y i dx ∥ � X P Z|my i,j(x)P x|y i dx � (4) ≤ � X P x|y i DJS � P Z|x ∥ P Z|my i,j(x)� dx Here we have (1) = is because of law of total probability and Z ⊥ Y |X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (2) = is because my i,j is invertible function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (3) = is because P x|y i = P my i,j(x)|y j ∀x ∈ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' (4) ≤ is because of joint complexity of JS divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} +page_content=' By similar derivation, ∀y ∈ Y, a ∈ A, we have: DJS � P Z|y,a i ∥ P Z|y,a j � ≤ � X P x|y,a i DJS � P Z|x ∥ P Z|my,a i,j (x)� dx 41' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FQT4oBgHgl3EQfbTbA/content/2301.13323v1.pdf'} diff --git a/VNE3T4oBgHgl3EQf0QuB/content/tmp_files/2301.04736v1.pdf.txt b/VNE3T4oBgHgl3EQf0QuB/content/tmp_files/2301.04736v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7b4cdcf58ba0b5f39dfd04ba485fe7d49ce55f4e --- /dev/null +++ b/VNE3T4oBgHgl3EQf0QuB/content/tmp_files/2301.04736v1.pdf.txt @@ -0,0 +1,699 @@ +arXiv:2301.04736v1 [math.DS] 11 Jan 2023 +Twisted recurrence for dynamical systems with +exponential decay of correlations +Jiajie Zheng +Abstract +We study the set of points returning infinitely often to a sequence of +targets dependent on the starting points. With an assumption of decay +of correlations for L1 against bounded variations, we prove a generalized +quantitative recurrence result under Lipschitz twists. +1 +Introduction +Let (X, d) be a separable and compact metric space, and let (X, µ, T ) be a Borel +probability measure-preserving system. The Poincar´e Recurrence Theorem, see +e.g. [EW] states that almost all points in measurable dynamical systems return +close to themselves under a measure-preserving map; i.e., +lim inf +n→∞ d(T nx, x) = 0 +for almost every x ∈ X. +Boshernitzan quantified the speed of recurrence in [Bo]. +Namely, if the α- +dimensional Hausdorff measure of X is σ-finite for some α > 0, then +lim inf +n→∞ n1/αd(T nx, x) < ∞ +for almost every x ∈ X. +A natural generalization of the recurrence speed is to consider the following set +R(ψ) := {x ∈ X : d(T nx, x) < ψ(n) for infinitely many n} +given a function ψ : N → (0, ∞). Much has been done on the quantitative +recurrence theory since then; for example, see [BF, CWW, DFL, HLSW, KZ, +KKP, Pe]. +A topic closely related to recurrence theory is the so-called shrinking target +problem, which is concerned with determining the speed at which the orbit of +a µ-generic point accumulates near a fixed point y ∈ X. More precisely, for +ψ : N → (0, ∞) and a given point y ∈ X, one can define the set +A(ψ, y) := {x ∈ X : d(T nx, y) < ψ(n) for infinitely many n}. +There have been plenty of results concerning the zero-one laws for µ(A(ϕ, y)) +in specific systems; for example, see [CK, FMP, HNPV, KM, LLVZ, Ph]. +A more general setting called twisted recurrence, which can specialize into +shrinking target problem and quantitative recurrence, was introduced in [KZ, +LWW]. For ψ : N → (0, ∞) and a Borel measurable function f : X → X, one +can consider the set +R(ψ, f) := {x ∈ X : d(T nx, f(x)) < ψ(n) for infinitely many n}. +1 + +Clearly R(ψ, f) = R(ψ) if f is the identity function and R(ψ, f) = A(ψ, y) if f +is the constant function with value y. When f is Lipschitz, the zero-one laws +for µ(R(ψ, f)) were proved for some special systems in [KZ, LWW], including +classical dynamical systems like β-transformations, Gauss transformations and +left shift on Cantor sets. Specifically, in these cases +µ(R(ψ, f)) = +� +0 +if �∞ +n=1 ψ(n, x)δ < ∞ +1 +if �∞ +n=1 ψ(n, x)δ = ∞, +where δ is the Hausdorff dimension of the support of µ. +In this paper, we prove quantitative Lipschitz recurrent properties of dy- +namical systems with exponential decay of correlations. +To state the main +result of the paper, we need to adapt and modify the settings and assumptions +from [KKP]. For the rest of the paper, let X = [0, 1] and d be the standard +metric, and assume for any sequence of positive real number {Mn}n contained +in (0, 1), there exists a sequence of functions {rn : X → (0, 1)}n such that +rn(x) = inf{r : µ(B(x, r)) = Mn} for all x ∈ X. For a function f : X → X, the +f-twisted recurrence set we are interested is defined by +R({Mn}n, f) := {x ∈ X : T nx ∈ B(f(x), rn(x)) for infinitely many n}. +We say that µ is Ahlfors regular if there exist constants c, δ > 0 such that +1 +crs ≤ µ(B(x, r)) ≤ crs +∀x ∈ SuppX and balls B(x, r) ⊂ X +and that µ is upper Ahlfors regular if there exist constants c, δ > 0 such that +µ(B(x, r)) ≤ crs +∀x ∈ SuppX and balls B(x, r) ⊂ X. +(1) +If µ is Ahlfors regular, then R({Mn}n, f) = R(ψ, f) where ψ(n) = rn(x). In +general, these definitions are different. Before stating our main theorems, we +now specify the class of functions f which we deal with by our technique. f : +X → X is said to be Lipschitz if +sup +x,y∈X,x̸=y +d(f(x), f(y)) +d(x, y) +< ∞. +(2) +Definition 1. Let (X, µ, T ) be a measure-preserving system and p : N → R+ +be a sequence. We say that the correlations for the system decay as p for L1 +against bounded variation (BV), if +���� +� +(f ◦ T n) · g dµ − +� +f dµ +� +g dµ +���� ≤ ||f||L1 · ||g||BV · p(n) +for all n ∈ N and for all functions f with ||f||1 := +� +|f| dµ < ∞ and g with +||g||BV := supxi +� |g(xi+1) − g(xi)| + sup |g| < ∞. +2 + +Remark 1. Definition 1 is weaker than the uniform mixing condition found in +[KZ], as for any balls E, F ⊂ X, we can take f = χE and g = χF and we will +get +��µ(T −nE ∩ F) − µ(E)µ(F) +�� ≤ 3µ(E)p(n). +Remark 2. For a non-increasing function ψ : N → R>0, we know that there +exists some +α ∈ W(ψ) := {α ∈ [0, 1] : |qα−p| < ψ(q) for infinitely many natural numbers p, q}. +Then consider the system +T : [0, 1] → [0, 1], +x �→ x + α (mod1) +together with the Lebesgue measure. Then R(ψ, Id) = [0, 1], where Id : [0, 1] → +[0, 1] is the identity function. The convergence case of Theorem 1 would fail; +hence some mixing condition is need for a zero-one law for R(ψ, f) to hold. For +more details, see [KZ, §2]. +We first state the sufficient condition for R({Mn}n, f) = 0, which depends on +the convergence of �∞ +n=1 Mn. If there exists a countable partition of subinterval +{Xi}i∈I so that f|Xi is Lipschitz for all i ∈ I, then we say f is piecewise +Lipschitz. If there exists a countable partition of subinterval {Xi}i∈I so that +f|Xi is monotone for all i ∈ I, then we say f is piecewise monotone. +Theorem 1. Let p : N → R+ be a function and assume the correlations for +(X, µ, T ) decay as p for L1 against BV with �∞ +n=1 p(n) < ∞. Let {Mn}n be +a sequence contained in (0, 1). Suppose f : X → X is piecewise Lipschitz and +piecewise monotone. If �∞ +n=1 Mn < ∞, then µ(R({Mn}n, f)) = 0. +For the full measure part, +Theorem 2. Let p : N → R+, (X, µ, T ), {Mn}n and f : X → X be as in +Theorem 1. Additionally, suppose that +• There exist C > 0 and 0 < γ < 1 such that p(n) = Cγn. +• µ is upper Ahlfors regular. +• For any q > 0, +lim sup +N +N +� +n=⌊q log N⌋ +Mn = ∞ +(3) +Then µ(R({Mn}n, f)) = 1. +In [KKP], the authors proved Theorem 1 and Theorem 2 for f being the +identity map. We will apply some of the ideas and techniques used in the proof +in [KKP], but the proof is different due to the new setup. +We get an immediate corollary for Ahlfors regular systems. +3 + +Corollary 3. Let p(n) = Cγn for some C > 0 and 0 < γ < 1 and suppose the +correlations for (X, µ, T ) decays as p for L1 against BV. Suppose µ is δ-Ahlfors +regular. Assume f is piecewise Lipschitz and piecewise monotone. Then +1. If �∞ +n=1 ψ(n)δ < ∞, then R(ψ, f) is null. +2. If for any q > 0, +lim sup +N +N +� +n=⌊q log N⌋ +Mn = ∞, +(4) +then R(ψ, f) is full. +We shall remark that Corollary 3 is a generalization of the convergence part +of [KZ, Theorem 1.2] in the case X = [0, 1]. +Most notably, the expanding, +bounded distortion and conformality assumptions are omitted. Corollary 3 par- +tially generalizes the divergence case of [KZ, Theorem 1.2], with a stronger +summability assumption of the measures of the targets (4). +The structure of the paper is as follows. In §2, we reduce the proofs of The- +orem 1 and Theorem 2 to the case where f : X → X is Lipschitz and it only +changes monotonicity at finitely many points. In §3 we construct a sequence +of measurable sets whose limsup set is R({Mn}n, f), and we estimate the mea- +sure of each set and conclude Theorem 1. In §4, we study quasi-independence +properties of this sequence and prove Theorem 2. +Acknowledgements +The author would like to thank Tomas Persson for bringing this problem to +his attention and discussing possible generalizations. +The author is grateful +to Dmitry Kleinbock for his wonderful advice and guidance throughout this +project. +2 +Piecewise Lipschitz and piecewise monotone +twists +By the properties of R(ψ, f), to prove Theorem 1 and Theorem 2, it suffices to +show the statements for f Lipschitz and monotone. +Lemma 4. Theorem 1 and Theorem 2 hold for f Lipschitz and monotone. +We will prove Lemma 4 in §§3-4. Here we first conclude Theorem 1 and +Theorem 2 from this lemma. +Proofs of Theorem 1 and Theorem 2. Suppose there exists a countable collec- +tion of disjoint open intervals {Xi = (ai, bi)}i∈I so that � +i∈I Xi is full and f is +4 + +Lipschitz and monotone on each Xi. Then for each i ∈ I, f is bounded on Xi +and hence we can define +fi(x) = + + + + + +f(x) +if x ∈ (ai, bi) +limx→a− +i f(x) +if x ≤ ai +limx→b+ +i f(x) +if x ≥ bi. +Then fi is Lipschitz and monotone. By Lemma 4, Theorem 1 and Theorem 2 +hold for fi for each i ∈ I. When µ(R(ψ, fi)) = 0 for all i ∈ I, +µ(R(ψ, f)) = +� +i∈I +µ(R(ψ, f) ∩ Xi) = +� +i∈I +µ(R(ψ, fi) ∩ Xi) = 0; +when µ(R(ψ, fi)) = 1 for all i ∈ I, +µ(R(ψ, f)) = +� +i∈I +µ(R(ψ, f) ∩ Xi) = +� +i∈I +µ(R(ψ, fi) ∩ Xi) = +� +i∈I +µ(Xi) = 1, +so the theorems are proved once we prove Lemma 4 in the following sections. +3 +The convergence part +In this section we prove the convergence part of Lemma 4, thereby fixing p : +N → R+, (X, µ, T ), {Mn}n and f : X → X to be Lipschitz and monotone. Let +us define +Rn({Mn}n, f) := {x ∈ X : T nx ∈ B(f(x), rn(x))}. +Without ambiguity, we shall denote Rn({Mn}n, f) simply by Rn. +Clearly +R({Mn}n, f) = lim supn→∞ Rn. +We first prove a fact about the functions rn. +Lemma 5. For all n ∈ N, rn is 1-Lipschitz. +Proof. Let x, y ∈ [0, 1]. Without the loss of generality, suppose rn(x) ≤ rn(y). +Then +B(x, rn(x)) ⊂ B(y, rn(x) + d(x, y)), +so µ(B(y, rn(x) + d(x, y))) ≥ Mn and hence +rn(y) ≤ rn(x) + d(x, y), +i.e., |rn(y) − rn(x)| ≤ |x − y|. +For each n, we define Yn to be a subset of [0, 1]2 such that +Yn = {(x, y) : y ∈ B(f(x), rn(f(x)))}. +Then we have +5 + +Lemma 6. For each n ∈ N, Yn is an open subset of [0, 1]2. +Proof. Fix n ∈ N. We prove that Yn has closed complement in [0, 1]2. Let +{(xm, ym)}m be a Cauchy sequence in the complement of Yn and we denote +its limit in [0, 1]2 by (x, y). We show that (x, y) ̸∈ Yn. Let ε > 0. Since f is +continuous, there exists k ∈ N so that +|f(xk) − f(x)| < ε +L +and +|yk − y| < ε. +Then +|f(x) − y| ≥|f(xk) − yk| − |f(xk) − f(x)| − |yk − y| +≥rn(f(xk)) − 2ε +≥ +Lemma 5 +rn(f(x)) − 3ε. +Since ε is chosen arbitrarily, we must have |f(x) − y| ≥ rn(f(x)) as desired. +Now we are ready to estimate the measure of Rn for each n. +Lemma 7. For each n ∈ N, +|µ(Rn) − Mn| ≤ 3p(n). +Proof. Define Fn : [0, 1]2 → R to be the characteristic function of Yn. Since +Yn is open, we can approximate Fn by the following a sequence of uniformly +continuous functions {Fn,k}k, where +Fn,k(x, y) := +� +0 +if (x, y) ̸∈ Yn +min{1, kd((x, y), ∂Yn) +if (x, y) ∈ Yn +and ∂Yn denotes the boundary of Yn. Note that {Fn,k}k is increasing in k and +it converges pointwise to Fn, so by the monotone convergence theorem, for each +ε > 0, there exists some k so that +���� +� +Fn(x, T nx) dµ(x) − Fn,k(x, T nx) dµ(x) +���� < ε +and +���� +� +Fn − +� +Fn,k +���� < ε. +Since Fn,k is 2k-Lipschitz, we can choose a partition by intervals {Ih}m−1 +h=0 of +[0, 1] so that +�����Fn,k(x, y) − +m−1 +� +h=0 +Fn,k(x, yh)χIh(y) +����� < ε +6 + +for all x, y ∈ [0, 1], where yh is the middle point of Ih. Then consider the integral +� +m−1 +� +h=0 +Fn,k(x, yh)χIh(T nx) +(5) +For each summand in (5), apply the decay of correlations to get +���� +� +Fn,k(x, yh)χIh(T nx) dµ(x) − +� +Fn,k(x, yh) dµ(x) +� +χIh(x) dµ(x) +���� +≤ µ(Ih)||Fn,k(x, yh)||BV p(n) +Note that for each yh, +Fn,k(x, yh) = +� +1 +if d(f(x), yh) < rn(f(x)) +0 +else += χ{x:d(f(x),yh) 0 and 0 < γ < 1. Let +IN = +� +j : −2 +s logγ N ≤ j ≤ N +� +and +UN = +� +j∈IN +Rj. +Note that lim supn Rn = lim supN UN. +Let +SN = +� +j∈IN +µ(Rj) +and +σN = +� +j∈IN +Mj +By Lemma 7, we have +σN − c1 ≤ SN ≤ σN + c1 +9 + +where the constant c1 = Cγ−1. On the other hand, let +CN = +� +j,k∈IN +µ(Rj ∩ Rk) +and by Lemma 8, we have +CN =SN + 2 +� +j,k∈IN ,j>k +µ(Ej ∩ Ek) +≤SN + (1 + K1CN −1/s)σ2 +N + 2K +� +j,k∈IN ,j>k +(Mkγks/2 + Mj(γks/2 + γj−k) + γks/2) +where K = K2 + K3. Denote +DN := +� +j,k∈IN ,j>k +(Mkγks/2 + Mj(γks/2 + γj−k) + γks/2) +We show that RN is bounded, proceeding term by term. Each of the first two +and last sums is less than or equal to +N +� +j=− 2 +s logγ N +j−1 +� +k=− 2 +s logγ N +γks/2 ≤ +N +� +j=− 2 +s logγ N +c2e− log N ≤ c2 +for some constant c2 dependent on the bound in (3). For the third sum, +� +j,k∈IN ,j>k +Mjγj−k = +� +j=− 2 +s logγ N +Mj +j−1 +� +k=− 2 +s logγ N +γj−k ≤ c3σN +Hence +CN ≤SN + (1 + K1CN −1/2)σ2 +N + 2K(3c2 + c3σN) +≤σN + c1 + (1 + K1CN −1/2)σ2 +N + 2K(3c2 + c3σN) +Now we can use the Chung-Erd¨os inequality to conclude that +µ(UN) ≥ S2 +N +CN +≥ +(σN − c1)2 +(1 + K1CN −1/2)σ2 +N + c4σN + c5 +(6) +for some constants c4, c5; as we take lim supN→∞ in (6), we have that +lim sup +N +µ(UN) ≥ 1 +Hence we proved that lim supn Rn = lim supN UN = 1. +10 + +References +[BF] +S. Baker and M. Farmer, Quantitative recurrence properties for self- +conformal sets, Proc. Amer. Math. Soc. 149 (2021), no. 3, 1127–1138. +[Bo] +M. Boshernitzan, Quantitative recurrence results, Invent. Math. 113 +(1993), no. 3, 617–631. +[CK] +N. Chernov and D. Kleinbock, Dynamical Borel-Cantelli lemmas for +Gibbs measures, Israel J. Math. 122 (2001), 1–27. +[CWW] Y. Chang, M. Wu, and W. Wu, Quantitative recurrence properties and +homogeneous self-similar sets, Proc. Amer. Math. Soc. 147 (2019), +1453–1465. +[DFL] +D. Dolgopyat, B. Fayad and S. Liu, Multiple Borel Cantelli Lemma +in dynamics and MultiLog law for recurrence, +Preprint (2021), +arXiv:2103.08382. +[EW] +M. Einsiedler and T. Ward, Ergodic theory with a view towards number +theory, Graduate Texts in Mathematics, 259, Springer–Verlag London, +Ltd., London, 2011, xviii+481 pp. +[FMP] +L. Fern´andez, M. V. Mel´ıan, and D. Pestana, Quantitative mixing re- +sults and inner functions, Math. Ann. 337 (2007), no. 1, 233–251. +[HLSW] M. Hussain, B. Li, D. Simmons and B.-W. Wang, Dynamical Borel- +Cantelli lemma for recurrence theory, Ergodic Theory Dynam. Sys- +tems, DOI: https://doi.org/10.1017/etds.2021.23 (2021), 15 pp. +[HNPV] N. Haydn, M. Nicol, T. Persson and S. Vaienti, A note on Borel- +Cantelli lemmas for non-uniformly hyperbolic dynamical systems, Er- +godic Theory Dynam. Systems 33 (2013), no. 2, 475–498. +[KKP] +M. Kirsebom, P. Kunde, and T. Persson, On shrinking targets and +self-returning points, Preprint (2020), arXiv:2003.013613. +[KM] +D. Kleinbock and G. A. Margulis, Logarithm laws for flows on homo- +geneous spaces, Invent. Math. 138 (1999), no. 3, 451–494. +[KZ] +D. Kleinbock and J. Zheng, Dynamical Borel-Cantelli lemma for re- +currence under Lipschtiz twists, Preprint (2022), arXiv:2205.12366. +[Ku] +J. Kurzweil, On the metric theory of inhomogeneous Diophantine ap- +proximations, Studia Math. 15 (1955), 84–112. +[LLVZ] +B. Li, L. Liao, S. Velani and E. Zorin, The shrinking target problem +for matrix transformations of tori: revisiting the standard problem, +Preprint (2022), arXiv:2208.06112. +11 + +[LWW] +F. L¨u, B-W. Wang and J. Wu, Diophantine analysis of the expan- +sions of a fixed point under continuum many bases , Preprint (2021), +arXiv:arXiv:2103.00546. +[Pe] +T. Persson, A strong Borel–Cantelli lemma for recurrence, Preprint +(2022), arXiv:2202.07344. +[Ph] +W. Philipp, Some metrical theorems in number theory, Paci. Jour. +Math. 20 (1967), no. 1, 109–127. +12 + diff --git a/VNE3T4oBgHgl3EQf0QuB/content/tmp_files/load_file.txt b/VNE3T4oBgHgl3EQf0QuB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..721ed74332205b16333defde4c16743b5b71a6df --- /dev/null +++ b/VNE3T4oBgHgl3EQf0QuB/content/tmp_files/load_file.txt @@ -0,0 +1,270 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf,len=269 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content='04736v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content='DS] 11 Jan 2023 Twisted recurrence for dynamical systems with exponential decay of correlations Jiajie Zheng Abstract We study the set of points returning infinitely often to a sequence of targets dependent on the starting points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' With an assumption of decay of correlations for L1 against bounded variations, we prove a generalized quantitative recurrence result under Lipschitz twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' 1 Introduction Let (X, d) be a separable and compact metric space, and let (X, µ, T ) be a Borel probability measure-preserving system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' The Poincar´e Recurrence Theorem, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' [EW] states that almost all points in measurable dynamical systems return close to themselves under a measure-preserving map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=', lim inf n→∞ d(T nx, x) = 0 for almost every x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Boshernitzan quantified the speed of recurrence in [Bo].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Namely, if the α- dimensional Hausdorff measure of X is σ-finite for some α > 0, then lim inf n→∞ n1/αd(T nx, x) < ∞ for almost every x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' A natural generalization of the recurrence speed is to consider the following set R(ψ) := {x ∈ X : d(T nx, x) < ψ(n) for infinitely many n} given a function ψ : N → (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Much has been done on the quantitative recurrence theory since then;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' for example, see [BF, CWW, DFL, HLSW, KZ, KKP, Pe].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' A topic closely related to recurrence theory is the so-called shrinking target problem, which is concerned with determining the speed at which the orbit of a µ-generic point accumulates near a fixed point y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' More precisely, for ψ : N → (0, ∞) and a given point y ∈ X, one can define the set A(ψ, y) := {x ∈ X : d(T nx, y) < ψ(n) for infinitely many n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' There have been plenty of results concerning the zero-one laws for µ(A(ϕ, y)) in specific systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' for example, see [CK, FMP, HNPV, KM, LLVZ, Ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' A more general setting called twisted recurrence, which can specialize into shrinking target problem and quantitative recurrence, was introduced in [KZ, LWW].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' For ψ : N → (0, ∞) and a Borel measurable function f : X → X, one can consider the set R(ψ, f) := {x ∈ X : d(T nx, f(x)) < ψ(n) for infinitely many n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' 1 Clearly R(ψ, f) = R(ψ) if f is the identity function and R(ψ, f) = A(ψ, y) if f is the constant function with value y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' When f is Lipschitz, the zero-one laws for µ(R(ψ, f)) were proved for some special systems in [KZ, LWW], including classical dynamical systems like β-transformations, Gauss transformations and left shift on Cantor sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Specifically, in these cases µ(R(ψ, f)) = � 0 if �∞ n=1 ψ(n, x)δ < ∞ 1 if �∞ n=1 ψ(n, x)δ = ∞, where δ is the Hausdorff dimension of the support of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' In this paper, we prove quantitative Lipschitz recurrent properties of dy- namical systems with exponential decay of correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' To state the main result of the paper, we need to adapt and modify the settings and assumptions from [KKP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' For the rest of the paper, let X = [0, 1] and d be the standard metric, and assume for any sequence of positive real number {Mn}n contained in (0, 1), there exists a sequence of functions {rn : X → (0, 1)}n such that rn(x) = inf{r : µ(B(x, r)) = Mn} for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' For a function f : X → X, the f-twisted recurrence set we are interested is defined by R({Mn}n, f) := {x ∈ X : T nx ∈ B(f(x), rn(x)) for infinitely many n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' We say that µ is Ahlfors regular if there exist constants c, δ > 0 such that 1 crs ≤ µ(B(x, r)) ≤ crs ∀x ∈ SuppX and balls B(x, r) ⊂ X and that µ is upper Ahlfors regular if there exist constants c, δ > 0 such that µ(B(x, r)) ≤ crs ∀x ∈ SuppX and balls B(x, r) ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' (1) If µ is Ahlfors regular, then R({Mn}n, f) = R(ψ, f) where ψ(n) = rn(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' In general, these definitions are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Before stating our main theorems, we now specify the class of functions f which we deal with by our technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' f : X → X is said to be Lipschitz if sup x,y∈X,x̸=y d(f(x), f(y)) d(x, y) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' (2) Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Let (X, µ, T ) be a measure-preserving system and p : N → R+ be a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' We say that the correlations for the system decay as p for L1 against bounded variation (BV), if ���� � (f ◦ T n) · g dµ − � f dµ � g dµ ���� ≤ ||f||L1 · ||g||BV · p(n) for all n ∈ N and for all functions f with ||f||1 := � |f| dµ < ∞ and g with ||g||BV := supxi � |g(xi+1) − g(xi)| + sup |g| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' 2 Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Definition 1 is weaker than the uniform mixing condition found in [KZ], as for any balls E, F ⊂ X, we can take f = χE and g = χF and we will get ��µ(T −nE ∩ F) − µ(E)µ(F) �� ≤ 3µ(E)p(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' For a non-increasing function ψ : N → R>0, we know that there exists some α ∈ W(ψ) := {α ∈ [0, 1] : |qα−p| < ψ(q) for infinitely many natural numbers p, q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Then consider the system T : [0, 1] → [0, 1], x �→ x + α (mod1) together with the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Then R(ψ, Id) = [0, 1], where Id : [0, 1] → [0, 1] is the identity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' The convergence case of Theorem 1 would fail;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' hence some mixing condition is need for a zero-one law for R(ψ, f) to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' For more details, see [KZ, §2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' We first state the sufficient condition for R({Mn}n, f) = 0, which depends on the convergence of �∞ n=1 Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' If there exists a countable partition of subinterval {Xi}i∈I so that f|Xi is Lipschitz for all i ∈ I, then we say f is piecewise Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' If there exists a countable partition of subinterval {Xi}i∈I so that f|Xi is monotone for all i ∈ I, then we say f is piecewise monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Let p : N → R+ be a function and assume the correlations for (X, µ, T ) decay as p for L1 against BV with �∞ n=1 p(n) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Let {Mn}n be a sequence contained in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Suppose f : X → X is piecewise Lipschitz and piecewise monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' If �∞ n=1 Mn < ∞, then µ(R({Mn}n, f)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' For the full measure part, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Let p : N → R+, (X, µ, T ), {Mn}n and f : X → X be as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Additionally, suppose that There exist C > 0 and 0 < γ < 1 such that p(n) = Cγn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' µ is upper Ahlfors regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' For any q > 0, lim sup N N � n=⌊q log N⌋ Mn = ∞ (3) Then µ(R({Mn}n, f)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' In [KKP], the authors proved Theorem 1 and Theorem 2 for f being the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' We will apply some of the ideas and techniques used in the proof in [KKP], but the proof is different due to the new setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' We get an immediate corollary for Ahlfors regular systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' 3 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Let p(n) = Cγn for some C > 0 and 0 < γ < 1 and suppose the correlations for (X, µ, T ) decays as p for L1 against BV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Suppose µ is δ-Ahlfors regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Assume f is piecewise Lipschitz and piecewise monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' If �∞ n=1 ψ(n)δ < ∞, then R(ψ, f) is null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' If for any q > 0, lim sup N N � n=⌊q log N⌋ Mn = ∞, (4) then R(ψ, f) is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' We shall remark that Corollary 3 is a generalization of the convergence part of [KZ, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content='2] in the case X = [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Most notably, the expanding, bounded distortion and conformality assumptions are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Corollary 3 par- tially generalizes the divergence case of [KZ, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content='2], with a stronger summability assumption of the measures of the targets (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' In §2, we reduce the proofs of The- orem 1 and Theorem 2 to the case where f : X → X is Lipschitz and it only changes monotonicity at finitely many points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' In §3 we construct a sequence of measurable sets whose limsup set is R({Mn}n, f), and we estimate the mea- sure of each set and conclude Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' In §4, we study quasi-independence properties of this sequence and prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Acknowledgements The author would like to thank Tomas Persson for bringing this problem to his attention and discussing possible generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' The author is grateful to Dmitry Kleinbock for his wonderful advice and guidance throughout this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' 2 Piecewise Lipschitz and piecewise monotone twists By the properties of R(ψ, f), to prove Theorem 1 and Theorem 2, it suffices to show the statements for f Lipschitz and monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Theorem 1 and Theorem 2 hold for f Lipschitz and monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' We will prove Lemma 4 in §§3-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Here we first conclude Theorem 1 and Theorem 2 from this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Proofs of Theorem 1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Suppose there exists a countable collec- tion of disjoint open intervals {Xi = (ai, bi)}i∈I so that � i∈I Xi is full and f is 4 Lipschitz and monotone on each Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Then for each i ∈ I, f is bounded on Xi and hence we can define fi(x) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 f(x) if x ∈ (ai, bi) limx→a− i f(x) if x ≤ ai limx→b+ i f(x) if x ≥ bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Then fi is Lipschitz and monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' By Lemma 4, Theorem 1 and Theorem 2 hold for fi for each i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' When µ(R(ψ, fi)) = 0 for all i ∈ I, µ(R(ψ, f)) = � i∈I µ(R(ψ, f) ∩ Xi) = � i∈I µ(R(ψ, fi) ∩ Xi) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' when µ(R(ψ, fi)) = 1 for all i ∈ I, µ(R(ψ, f)) = � i∈I µ(R(ψ, f) ∩ Xi) = � i∈I µ(R(ψ, fi) ∩ Xi) = � i∈I µ(Xi) = 1, so the theorems are proved once we prove Lemma 4 in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' 3 The convergence part In this section we prove the convergence part of Lemma 4, thereby fixing p : N → R+, (X, µ, T ), {Mn}n and f : X → X to be Lipschitz and monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Let us define Rn({Mn}n, f) := {x ∈ X : T nx ∈ B(f(x), rn(x))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Without ambiguity, we shall denote Rn({Mn}n, f) simply by Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Clearly R({Mn}n, f) = lim supn→∞ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' We first prove a fact about the functions rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' For all n ∈ N, rn is 1-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Let x, y ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Without the loss of generality, suppose rn(x) ≤ rn(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Then B(x, rn(x)) ⊂ B(y, rn(x) + d(x, y)), so µ(B(y, rn(x) + d(x, y))) ≥ Mn and hence rn(y) ≤ rn(x) + d(x, y), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=', |rn(y) − rn(x)| ≤ |x − y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' For each n, we define Yn to be a subset of [0, 1]2 such that Yn = {(x, y) : y ∈ B(f(x), rn(f(x)))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Then we have 5 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' For each n ∈ N, Yn is an open subset of [0, 1]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Fix n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' We prove that Yn has closed complement in [0, 1]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Let {(xm, ym)}m be a Cauchy sequence in the complement of Yn and we denote its limit in [0, 1]2 by (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' We show that (x, y) ̸∈ Yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Since f is continuous, there exists k ∈ N so that |f(xk) − f(x)| < ε L and |yk − y| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Then |f(x) − y| ≥|f(xk) − yk| − |f(xk) − f(x)| − |yk − y| ≥rn(f(xk)) − 2ε ≥ Lemma 5 rn(f(x)) − 3ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Since ε is chosen arbitrarily, we must have |f(x) − y| ≥ rn(f(x)) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Now we are ready to estimate the measure of Rn for each n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' For each n ∈ N, |µ(Rn) − Mn| ≤ 3p(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Define Fn : [0, 1]2 → R to be the characteristic function of Yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Since Yn is open, we can approximate Fn by the following a sequence of uniformly continuous functions {Fn,k}k, where Fn,k(x, y) := � 0 if (x, y) ̸∈ Yn min{1, kd((x, y), ∂Yn) if (x, y) ∈ Yn and ∂Yn denotes the boundary of Yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Note that {Fn,k}k is increasing in k and it converges pointwise to Fn, so by the monotone convergence theorem, for each ε > 0, there exists some k so that ���� � Fn(x, T nx) dµ(x) − Fn,k(x, T nx) dµ(x) ���� < ε and ���� � Fn − � Fn,k ���� < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Since Fn,k is 2k-Lipschitz, we can choose a partition by intervals {Ih}m−1 h=0 of [0, 1] so that �����Fn,k(x, y) − m−1 � h=0 Fn,k(x, yh)χIh(y) ����� < ε 6 for all x, y ∈ [0, 1], where yh is the middle point of Ih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE3T4oBgHgl3EQf0QuB/content/2301.04736v1.pdf'} +page_content=' Then consider the integral � m−1 � h=0 Fn,k(x, yh)χIh(T nx) (5) For each summand in (5), apply the decay of correlations to get ���� � Fn,k(x, yh)χIh(T nx) dµ(x) − � Fn,k(x, yh) dµ(x) � χIh(x) dµ(x) ���� ≤ µ(Ih)||Fn,k(x, yh)||BV p(n) Note that for each yh, Fn,k(x, yh) = � 1 if d(f(x), yh) < rn(f(x)) 0 else = χ{x:d(f(x),yh) 4mm macroscopic sym- +metry breaking (species No. 198 of Ref. 19). +When disre- +garding the small angles associated with the elastic match- +ing of domain states and adopting to the pseudocubic crys- +tallographic notation, we can state that the polarization falls +along one of the six {001} directions and that 180 degree and +90-degree ferroelectric domain wall can be formed. The 90- +degree walls are particularly attractive for domain engineering +because they are also ferroelastic and can be induced by the +frustrative poling that favors more than one ferroelectric do- +main state. +The existence of charged 90-degree domain walls always +require almost perfect screening of the huge bound polar- +ization charge at such a wall. +The usual BaTiO3 crystals +are slightly n-doped semiconductors, so that the head-to-head +charged domain walls can be compensated by the naturally +present excess electrons6. +The concentration of the com- +pensation charge is so strikingly high that even the sponta- +neous formation of the two-dimensional electron gas at this +90-degree head-to-head domain wall have been observed in +BaTiO3 crystal1,20. +FIG. 1: +Charged domain walls in (111)-oriented BaTiO3 crystal +plates. Panels (a), (b) and (c) shows (01¯1), (10¯1) and (1¯10) sys- +tems of charged domain walls. These ex-situ optical micrographs +are taken in transmission mode, the viewing direction is that of the +formerly applied [111]-directed poling field. Red arrows stand for +the experimentally inferred polarization in the primary domain states +favored by the poling electric field, while the the polarizer-analyzer +orientation is indicated by the crossed directors. Panel (d) shows how +the mechanically compatible neutral (NDW) and charged (CDW) pri- +mary ferroelastic domain walls can be terminated on the (111) sur- +face. +It is worth noting that head-to-head and tail-to-tail ferroe- +lastic domains walls form parallel planes alternating in the +crystal (see Fig. 1). Consequently, the positively charged car- +riers, e.g. oxygen vacancies or other ionized mobile defects or +electron holes, also need to be present in the crystal in order +to form a stable domain pattern with multiple domain walls. +The sufficient amount of mobile positive charge carriers have +been secured using two alternative strategies. In the first ap- +arXiv:2301.03409v1 [cond-mat.mtrl-sci] 9 Jan 2023 + +a +b +(011) +[010] +(101) +[001] +[100] +[001] +7 +200μm +C +d +(011) +(011) +[010] +(110) +[oT] +(110) +(101) +[100] +(101) +(110) +CDW +NDW +[112] +[111]2 +proach, crystals containing a sufficient amount of oxygen va- +cancies were used9. In the second approach, the poling was +performed under the UV light illumination1, suggesting that +both photoexcited electrons and holes participated in screen- +ing of the polarization bound charge10. +Strictly speaking, the process of charged domain wall cre- +ation in BaTiO3 requires not only the presence of charge carri- +ers but also their efficient spatial redistribution. For example, +it has been recently emphasized that formation of the charged +domain walls in the polarization switching process is much +more understandable when such are conductive and connected +to the outer electrode21. However, the exact mechanism of the +90-degree charge domain pattern formation process was not +fully understood yet. In particular, there is no obvious reason +for an organized charge carrier separation before some sort +of bound charge modulation is formed in the crystal. Like- +wise, the idea that the macroscopic domain patterns could +be initially nucleated without the full compensation and only +then stabilized by the carriers of both signs appears unrealistic +when considering plausible charge carrier drift velocities and +the micron-sized distances between the charged domain walls. +In this work, we draw the attention to the fact that the 90- +degree charged domain patterns were so far created in bulk +crystals of BaTiO3 by the process of frustrative electric poling +in the vicinity of phase transitions9,10. With this in mind, we +have been revisiting the problem of the charged domain walls +creation by careful in-situ optical investigations of the early +stage of their formation near the cubic to tetragonal phase +transition9. We have realized that the charged domain walls +are actually originating from the charged superdomain walls, +separating nanotwinned martensitic plates, here also called su- +perdomains, that are formed in the process of the nucleation +of the ferroelectric phase itself. The optical observations are +complemented by an in-detph theoretical analysis of these su- +perdomain structures and walls. +The paper is organized as follows. In Section II., we present +the main experimental observation that are documented in +Figs. 1, 2 and 3. Then, in Section III. we expose the result of +the theoretical analysis. The part III.A summarizes the known +results of the Wechsler-Lieberman-Read (WLR) martensite +theory of phase fronts in BaTiO3. Among others, we clar- +ify there what exactly we mean by superdomains. The rest +of the Section III. describes various properties of the super- +domain walls and superdomain precipitates and the expected +consequences for the adopted geometry of the frustrative pol- +ing with [111] oriented field as we have derived them from +the mechanical compatibility conditions. The main results are +given in Tables I. and II. and in the summarizing Figure 4. +In Section IV., the theoretical background is used to analyze +and interpret the principal results of the Section II. Next, we +show that the so far derived conclusions are in agreement with +some additional observations and several proposals for further +investigation are discussed. Finally, the paper is concluded +by considerations about the role of possibly conductive phase +fronts. +II. +PRINCIPAL OBSERVATIONS +The main results of the present experiments are summa- +rized in Figs. 1-3. The samples used in all investigations were +(111)-oriented platelets. Formation of straight charged do- +main walls was most conveniently achieved in slightly off- +stoichiometric BaTiO3 single crystals with conductivity of the +order of 0.1-1 S/m. Similarly as is the Ref. 9, these charged +walls were induced by frustrative poling method across the cu- +bic to tetragonal phase transition. More precisely, the [111]- +oriented dc electric field of about 1 kV/mm field was applied +several K above the phase transition and then the sample was +cooled down through the transition while the field was kept +on. After cooling the sample down to the ambient temper- +ature and switching off the field, the contacts were removed +and the resulting domain patterns were thoroughly investi- +gated ex-situ. We have confirmed that most of the charged +domain walls grown in this way were stable at least over sev- +eral months. +Typically, such samples have shown an array of parallel +head-to-head and tail-to-tail domain walls separated by dis- +tances of about 100 microns (see Fig. 1). Interpretation of the +results shown in Fig. 1 is rather straightforward. Polarization +vectors were identified by the orientation of the optical indi- +catrix and the light extinction at the domain wall in the non- +polarized light following the procedure described in Ref. 9. +Distinction between the head-to-head and tail-to-tail could be +easily made based on the observation of the dark and light +line at the domain wall location9. Comparison with Fig. 1d +confirms that crystallographic orientations of these charged +domain walls agree with the three theoretically expected ori- +entations of primary charged domain walls, as discussed for +example in Refs. 9,22. +Repeating the same poling procedure using thin transpar- +ent electrodes enabled us to observe also the transient domain +structures formed at the early stage of the charge domain wall +formation. This experiment, essential for the present paper, +is reported in Fig. 2. It was carried out with a similar sam- +ple as that of Fig. 1. The precipitation started at temperature +Tstart which was about 402 K. Interestingly, we have observed +that the transformation started by a simultaneous nucleation +of many thin, lentil-shaped precipitates, appearing within the +bulk of the optically homogeneous and isotropic paraelectric +cubic phase. The boundaries of the precipitates perform back +and forth jerky motion, typical for the dynamics of ferroelastic +domain walls. +The image of Fig. 2a shows these precipitates at a tempera- +ture of about about Tstart −0.5 K. The cross section of the pre- +cipitate with the surface appears as a needle-shaped leaf. The +individual needles appear to be organized in clusters, suggest- +ing a systematic preference or a direct interaction among the +parallel precipitates. Fig. 2b shows the same area at a later +stage, corresponding to the temperature of about Tstart − 1 K. +One can see that precipitates formed compact fan-like domain +patterns delimited by two zig-zag boundaries (see Fig. 2b and +its schematic interpretation in Fig. 2e). Originally, three com- +peting systems of parallel needle-shaped leaves were coexist- +ing in the sample, each corresponding to of one of the charged + +3 +FIG. 2: Optical micrographs showing the charged domain wall formation process in (111)-oriented crystal plate under the electric field applied +along the viewing direction. Panel (a) suggests nucleation of the ferroelectric precipitates in the form of thin platelets embedded in the optically +isotropic paralectric matrix. In the viewing direction the platelet precipitates appear as needle-shaped leafs oriented similarly as the traces of +the charged domain walls in Fig. 1. Panel (b) reveals that the needle-shaped leafs have two blades, each of them decorated with a fine stripe +pattern, and that the precipitates become organized in a pattern with common zigzag phase fronts. Panel (c) shows that the fine stripe pattern +is fading out and the that stable charged domain walls are formed at the central lines of the needle-shape leaves. The panels (d,e,f) suggest +schematic representation of the corresponding observations of panels (a,b,c). Black spot in the center of images (a,b,c) is due to an electric +contact on the top electrode; polarizer-analyzer orientation is indicated in upper right corner of panel (c). +domain wall orientations shown in Fig. 1. Progressively, only +one of the three systems of precipitates extended over the +whole crystal (Fig. 2c). +A closer look to one of the thicker precipitates shown in +Fig. 2b allows to discern a central line, dividing each needle- +shaped leaf into two adjacent blades. Let us note that for the +given orientation of the crossed polarizers, one type of the +blades is systematically darker, while the other is barely seen. +The relevant detail is thus enlarged in Fig. 3a and schematized +in Fig. 3b. There one can also see that each of the adjacent +blade is decorated inside with a differently oriented fine-stripe +pattern. From the analysis of this fine-stripe pattern and the +overall geometry of the precipitates detailed in the next sec- +tion, we infer that the individual blades of the precipitates +correspond to distinct martensitic superdomains. In Fig. 3b, +these two blades are shaded by different colors, using the su- +perdomain color code defined in Fig. 4a. Similar but more +schematic graphical interpretation of Figs. 2a and 2b is given +in Fig. 2d and 2e. Our conjecture that the superdomains are +formed by the fine stripes of the primary tetragonal ferroelec- +tric domains is illustrated in Fig. 3c. +Let us stress that we have observed that this fine stripe pat- +tern within superdomains is gradually fading out. In other +words, the blades appear more and more homogeneous. Si- +multaneously, the contrast between adjacent blades observed +with the fixed orientation of crossed polarizers is enhanced +(see Fig. 2c). In the end, only one system of wide homoge- +neous parallel stripes persists over the whole sample, as it +is schematized in Fig. 2f. These resulting wide stripes obvi- +ously correspond to the primary tetragonal domains separated +charge domain walls, similar to those shown in Fig. 1a. +III. +THEORY +A. +Standard WLR theory of the phase front +Before addressing the properties of superdomain walls and +composed precipitates, let us summarize the already known +predictions of the martensitic WLR theory23. According to +our knowledge, the martensitic WLR theory has been first +applied to perovskite ferroelectrics in order to explain the +formation of the phase front between the paraelectric cu- +bic and ferroelectric tetragonal phase in the seminar work +of Ref. 24. Almost in parallel, it has been applied to high- +resisitivity BaTiO3 in Ref. 25. +Later the theory has been +further developed and independently tested or formulated by +many others26–31. +According to the martensitic theory, the ferroelectric phase +transition proceeds preferentially by propagation of a spe- + +200μm +200μm +200μm +[110] +[112] +[111]4 +FIG. 3: +Details of the transient domain structure shown in Fig. 2. +Top panels show awl-shaped ferroelectric precipitates surrounded by +the paraelectric phase. The region captured by the optical micro- +graph in panel (a) is graphically interpreted in terms of Zx and Yx +superdomains in panel (b). The fine-scale stripes due to the primary +ferroelectric domains within these superdomains form the resulting +pattern of panel (c), reminding of awl-shaped leaves with two blades. +Dashed lines highlight phase boundaries of one such precipitate. By +comparison with the Fig. 4 explained in Section III., the junction of +the two blades can be identified as a type I.A superdomain bound- +ary. The bottom panels (d-f) show the corresponding information +for an area fully covered by the ferroelectric phase. Note that the +assignment of the strain state in superdomains was done using the +predictions of the Section III., summarized in Fig. 4. The polariza- +tion content of the individual primary domains can be inferred from +a more subtle arguments given in section IV (see also Fig. 5 and color +code of Fig. 4). +cific coherent planar phase front, also termed as habit plane, +connecting the paraelectric part of the crystal with a suitably +twinned ferroelectric part of the crystal. In order to ensure the +mechanical compatibility between ferroelectric and paraelec- +tric phases, a martensitic twin is formed from two ferroelectric +domain states is a suitable volume ratio and acceptable phase +fronts are restricted to specific planes with a precisely defined +crystallographic orientation. +Let us consider the usual pseudocubic cartesian axes of the +parent perovskite structure, m¯3m > 4mm ferroelectric transi- +tion and a ferroelectric twin composed of the prevailing tetrag- +onal domain state with polarization along +z and a minority +tetragonal domain state with polarization along +x. It is con- +venient to denote such a twin as [Z,x]. According to the the- +ory, the optimal relative volume of the minority state +x in +the [Z,x] WLR twin, mechanically compatible with the cubic +phase, is given by +ξ = (a0 −a)/(c−a) , +(1) +where a0, a and c are lattice parameters of the cubic and +tetragonal phase at the point of the phase transition, respec- +tively. The average spontaneous strain of the representative +[Z,x] WLR twin, defined with respect to the cubic lattice pa- +rameter a0, can be expressed as +ε = +� +� +0 +0 +0 +0 ε⊥ +0 +0 +0 +ε∥ −ξ∆ , +� +� +(2) +where ε⊥ = (a − a0)/a0, ε∥ = (c − a0)/a0, are spontaneous +strain parameters of the primary tetragonal domain state and +∆ = ε∥ −ε⊥. The average polarization of the [Z,x] twin is then +P = (ξ,0,1−ξ)P0 , +(3) +where P0 is the equilibrium magnitude of the primary single +domain polarization at the phase transition. In this case, the +phase-front unit normal vector, defined as pointing towards +the side of the cubic phase, can have four possible orientations +n = (0,±α,± +� +1−α2) , +(4) +where +α = +� +−ε⊥ +∆(1−ξ) . +(5) +Interestingly, none of the vectors n in eq. (4) is perpen- +dicular to the average polarization of the WLR twin, listed +in in eq. (3). Therefore, the habit plane bears a net bound +charge density. In particular, the head-to-none phase front +of [Z,x] twin with n = (0,±α, +√ +1−α2) posses a positive +bound surface charge and the tail-to-none phase front n = +(0,±α,− +√ +1−α2) surface bears a negative bound surface +charge. In all cases, the absolute value of the bound surface +charge density on these phase fronts is σ = (1−ξ) +√ +1−α2P0. +In fact, it is comparable to the bound charge at the charged fer- +roelastic domain walls. +In case of BaTiO3, the volume ratio of primary domain +states in the WLR twin is close to 2:1 and the phase fronts are +approximately {056} crystallographic planes. These predic- +tions have been verified by direct observation of such macro- +scopic phase fronts26–28. +B. +Concept of superdomain boundaries +The observations reported in Figs. 2 and 3 suggest that in +the present experiments, conducted under the [111]-directed +poling field, there are many differently oriented phase fronts +and several distinct WLR twin states. Therefore, it is reason- +able to expect that several such secondary domain states, for +brevity denoted as superdomain states, might coexist in the +same crystal. Distinct superdomains can meet at superdomain +boundaries. +From the point of view of the average strain and polariza- +tion, the parent cubic point group symmetry allows 24 equiv- +alent orientational variants of the above described WLR twin +states. The exact macroscopic crystallographic symmetry19 +implies that these 24 WLR twin states can be considered as +24 domain states of the fully ferroelectric monoclinic species +m¯3m > m+ (see species 207 in Ref. 19 ). Nevertheless, the + +50μm +气 +[110] +50μm +[112] +[111]5 +Xy +Xz +Yx +Yz +Zx +Zy +Xy +(011) +(110) +- +- +(101) +(01¯1)∗ (1¯10)∗ +- +- +(10¯1)∗ +Xz (011) +- +(110) +(101) +- +(01¯1)∗ +- +(1¯10)∗ (10¯1)∗ +- +Yx (110) +- +(101) +(011) +- +(1¯10)∗ +- +(10¯1)∗ (01¯1)∗ +- +Yz +- +(110) +(101) +- +(011) +- +(1¯10)∗ (10¯1)∗ +- +(01¯1)∗ +Zx +- +(101) +(011) +- +(110) +- +(10¯1)∗ (01¯1)∗ +- +(1¯10)∗ +Zy (101) +- +- +(011) +(110) +(10¯1)∗ +- +- +(01¯1)∗ (1¯10)∗ +TABLE I: Orientations of the mechanically compatible superdomain +boundaries among the WLR superdomains. For each compatible do- +main pair, there are two permissible domain wall orientations. One +of them, marked by asterisk, is perpendicular to the (111) surface of +the sample. +average strain given by the eq. (2) distinguishes only 6 dis- +tinct average spontaneous superdomain strain tensors. There- +fore, for the considerations of the mechanical compatibility +of superdomain walls, it is sufficient to consider the effective +ferroelastic species m¯3m > mm+. In other words, for this pur- +pose it is possible to disregard the sign combinations of the +polarization components and consider only the six ferroelas- +tic superdomain states denoted in the following as Xy, Xz, Yx, +Yz, Zx, Zy. +The orientation of the mutually mechanically compatible +superdomain walls among these 6 WLR ferroelastic states was +determined from the average spontaneous strain tensors sim- +ilarly as in Ref. 32. There are three distinct types of mechan- +ically compatible ferroelastic superdomain walls, labeled as +type I., II. and III. (see the graph of Fig. 4a). Type I. super- +domain wall joins two superdomains with a common minori- +tary primary domain state (for example, superdomain states +Xy and Zy). Type II. superdomain wall connects two super- +domains with a common majority primary state (for example, +Xy and Xz) and type III. superdomain wall joins two superdo- +mains having their primary domain population interchanged +(for example, Xy and Yx pair). Each compatible pair allows +two orientations of the boundary, listed in Table I.). The other +pairs of the superdomain states are not mechanically compat- +ible, meaning that no mechanically permissible superdomain +walls can be constructed. +C. +Termination of superdomain walls at (111) surface +It turns out that in case of the investigated cubic to tetrag- +onal ferroelectric phase transition, not only the primary fer- +roelastic walls, but also the secondary ones are {011} type +planes. Consequently, these planes are are either perpendic- +ular to the (111) crystal surface, intersecting it along [¯211], +[1¯21], or [11¯2] directions, or they are oblique to the (111) sur- +FIG. 4: +Summary of theoretical predictions. The vertices of the +graph in panel (a) stand for the 6 types of ferroelastic superdomains +compatible with the paralectric cubic structure. Type of the connect- +ing line (single, double, tripple) indicate the type of the mechanically +permissible superdomain boundary of type I., II. and III, respectively. +Panel (b) displays the (111) surface sections of the hypothetical poly- +hedral ferroelectric precipitates delimited from the cubic phase by +mechanically compatible planar phase fronts. Same view direction +as in the Fig. 2 is assumed. Precipitates are labeled I., II. and III. ac- +cording to the type of the embedded central superdomain boundary. +Type I.A and I.B precipitates are composed of two infinite triangular +prisms (see Section III F). Type II. and III. precipitates have penta- +hedral pyramid shape that can terminate by an apex in the volume of +the sample. The outer hexagon is merely a guide for eye. +face, intersecting it along along [01¯1], [1¯10], or [¯101] direc- +tions. The superdomain walls of the former family are marked +by asterisk in Table I. It can be expected, that those with the +orientation perpendicular to the sample are more favorable for +reduction of the strain energy in the sample. + +Yz +[010] +a +X^ +[001] +[100] +Zy +x +Y +Z +Zx +Xy - +Xz - +X +Yx - +1. +Yz - +Xz +II. +Zy - +II. - +Zx - +b +(111) +1.A +Zx +Xz +1.B +I. +Yx +Xy +Z +Zy +X6 +D. +Termination of habit planes at (111) surface +By their incidence with respect to the (111) sample sur- +face, the habit planes of the phase fronts can be also di- +vided into two families. Members of the first family show +the incidence almost perpendicular to the surface, while the +others have rather an almost parallel position. +Intersec- +tion of the phase fronts with (111) sample surface obviously +falls along the cross product of the phase front and surface +normals. This yields a directional vector with components +(α − +√ +1−α2, +√ +1−α2,−α) or another vector symmetrically +equivalent to it. In this set of vectors, there are 6 directions +that are close to [¯211], [1¯21], or [11¯2] axes, and these corre- +spond to the phase fronts close to perpendicular to the (111) +surface. There are also 6 other orientations that are close to +[01¯1], [1¯10], or [¯101] directions, and these ones correspond +to the phase fronts almost parallel to the (111) surface. All +possible cases are also depicted in Fig. 4b. +E. +Single-superdomain precipitates +The canonical WLR scenario of the temperature driven +phase transition assumes a single phase front traversing the +crystal and leaving a single WLR twin behind. In contrast, the +[111]-directed poling field used here is expected to favor for- +mation of twinned precipitates of different types. Moreover, +the sides of the sample are typically subject of a lower field, so +that the transformation starts in the central part of the sample. +Let us first briefly consider what could be convenient shape of +one isolated precipitate formed by a single WLR twin. +We recall that there are just 2 compatible phase front planes +for a fixed twin, related to the 4 phase front normals listed in +eq. (4). It can be anticipated that the mechanically most conve- +nient shape of such a precipitate is a thin platelet, possibly of +a lentil-like shape, delimited by only two slightly bend phase +fronts, being close to one of the corresponding fully compat- +ible planar orientations. It can be expected, that the phase +fronts almost perpendicular to the platelet sample surface will +be energetically more convenient than the oblique ones. Since +the phase fronts on the opposite side of such lentils are neces- +sarily of opposite bound charge, the shape might be systemat- +ically somewhat asymmetric. +F. +Two-superdomain precipitates +It seems that the above described single superdomain pre- +cipitates of the same type would be mutually attracted and +merge together, while the parallel precipitates delimited by +phase fronts in a reversed order of charge densities would be +rather electrostatically repulsed from each other, at least be- +fore the bound charge is fully screened by mobile electronic +and ionic carriers. In either case, the existence of needle- +shape leaf-like motifs documented experimentally in Fig. 2 +and Fig. 3 clearly suggests that it might be more convenient +to form precipitates composed of two different WLR super- +domains. Guided by the experimental observations of Fig. 2, +our geometry considerations were further limited to such com- +posed precipitates that are symmetric with respect to the cen- +tral ferroelastic superdomain boundary, knowing also from the +observations that this boundary is a {110} oriented plane, and +that it is perpendicular to the (111) surface. In other words, we +assumed that the central superdomain boundary orientation is +one of those marked by asterisks in Table 1. +What could be a convenient shape and crystallographic ori- +entation for such a two-superdomain precipitate? The sim- +plest 3D objects delimited by planar walls are tetrahedra. +Therefore, we have been considering precipitates composed +by a pair of tetrahedra, each delimited by two phase fronts, a +common superdomain wall and the top surface. Let n1,n2,w +and s are the out-pointing unit vectors perpendicular to the +two phase fronts, to the superdomain wall, and to the (111) +surface normal, respectively. It can be shown that for a given +combination of such normals, the tetrahedron can be formed +if +[(n1 ×n2)·w][(n1 ×n2)·s] < 0 . +(6) +Clearly, if the surface normal is fixed and this tetrahedron for- +mation condition is satisfied for a given combination of n1,n2 +and w, it cannot be simultaneously satisfied with same n1,n2 +but with the opposite w. This implies that a given combina- +tion of n1 and n2 vectors allows the tetrahedra being formed +at only one side of the superdomain boundary. +It might happen that the vectors fulfill the limiting relation +((n1 × n2) · w) = 0. Then, the normals n1,n2 and w form an +infinite triangular prism instead, which could be limited by the +top and bottom (111) surface. In this case, another such prism +can be constructed with the same n1 and n2 pair on the other +side of the superdomain pair. +The inspection of available combinations of the phase front +normal vectors shows, that if the central superdomain bound- +ary is of type I., then two kinds prismatic precipitates can form +by a junction of two distinct triangular prisms. The prismatic +precipitate of the first kind has its sharp angle phase front +wedge pointing along the [¯211], [1¯21] or [11¯2] direction. We +denote this case as I.A. The second type of the prismatic pre- +cipitate has its sharp angle phase front wedge pointing in one +of the opposite directions. It is denoted as the I.B case. If +the central superdomain boundaries are of type II. or III., it +is possible to construct a pentahedral precipitate composed of +two real tetrahedra. For a given superdomain pair, the par- +ticular crystallograpic orientation of the individual facets of +such precipitate can be derived using the Table 1. For real- +istic values of lattice strain, corresponding to the {056} type +habit planes, the shapes and orientations of these precipitates +have been determined. The apparent shape of these precipi- +tates given by their termination on the top surface (view from +outside along the field direction) is given in Fig. 4b. +Nevertheless, it should be stressed that although the pre- +cipitates shown in the Fig. 4b are formed by the mechanically +compatible planar interfaces only, it can be expected that there +would be residual strain incompatibility at all their edges. In +reality, only the edges at the relatively sharp wedge-like junc- +tions are likely to be well accommodated by the elastic defor- +mation, while the junctions at obtuse angles are likely to be + +7 +energetically quite inconvenient. Therefore, when possible, +the precipitates are likely to join into more complex struc- +tures, such as the one shown in Fig. 2e, where the junctions +with obtuse angles are avoided only the sharp wedges are pre- +served. Note that this peculiar arrangement is formed by an +array of alternating head-to-head and tail-to-tail superdomain +walls, terminated by two inequivalent zig-zag shaped phase +fronts, one of the head-to-none type and one of the tail-to- +none type. +G. +Domain states promoted by the electric field +The electric field applied along the [111] crystallographic +direction is favoring the three primary domains states with +polarization along +x, +y and +z directions over the three +remaining domain states. Similarly, the P.E coupling of the +electric field to the polarization should favor superdomain +states composed of the +x, +y and +z primary domain states +only. In other words, one expect to the six superdomain states +[X,y], [X,z], [Y,x], [Y,z], [Z,x] and [Z,y] to be most likely +ones. Another 6 superdomain states [X,−y], [X,−z], [Y,−x], +[Y,−z], [Z,−x] and [Z,−y] could be considered as moder- +ately favored. The 6 superdomain states of [−X,y] type are +unlikely to occur and those of [−X,−y] form are probably +not formed under the [111] bias field at all. Examples of su- +perdomains with different degree of the alignment along the +[111]-directed electric field, in particular (a) the strongly fa- +vored superdomains [Z,x] and [Y,x], (b) moderately favored +superdomains [Z,−x] and [Y,−x], (c) moderately disfavored +superdomains [−Z,x] and [−Y,x], and (d) strongly disfavored +superdomains [−Z,−x] and [−Y,−x]. +FIG. 5: Possible primary domain polarization configurations within +a pair of adjacent martensitic twins (Zx and Yx), sharing a common +charged superdomain boundary of type I.B. With respect to the field +applied along the [111] direction, panels correspond to a strongly +favored (a), moderately favored (b), moderately disfavored (c) and +strongly disfavored configurations (d), respectively. The arguments +in section IV indicate that the transient structure of Fig. 2b and Fig. 3f +is that of the moderately favored case. +H. +Bound charge at superdomain boundaries +In general, the ferroelastic superdomain walls may be both +charged and uncharged ones. Considering that (i) the observed +superdomain walls are perpendicular to the (111) surface, and +that (ii) only the 6 strongly favored superdomains are present +in the sample, we are left with the charged superdomain walls +only. Let us stress that the same actually holds also if only the +weakly favored domain states are considered. The restriction +strogly favored superdomains weakly favored superdomains +type superdomain wall phase front superdomain wall phase front +I.A +-0.66 P0 +√ +2 ++0.74 P0 +-0.66 P0 +√ +2 ++0.74 P0 +I.B ++0.66 P0 +√ +2 +-0.74 P0 ++0.66 P0 +√ +2 +-0.74 P0 +II. ++0.25 P0 +√ +2 ++ 0.74 P0 +-0.25 P0 +√ +2 ++ 0.74 P0 +III. +-0.41 P0 +√ +2 ++ 0.74 P0 ++0.41 P0 +√ +2 ++ 0.74 P0 +TABLE II: Bound charge at principal interfaces of the idealized pre- +cipitate forms shown in the Fig. 4b. Indicated are values for the cen- +tral superdomain wall and for phase front facets having a longer in- +tersection with the (111) crystal surface. Type I.A corresponds to the +case with the sharp angle wedge vertex pointing to the [¯211] direction +(in Fig. 4b, the outer case), I.B stands for the sharp angle pointing to +the opposite direction. Results are listed separately for strongly and +weakly field-favored superdomains, respectively. +of the superdomain type allows to deduce also the charge den- +sities on the phase front facets of our composed precipitates. +The numerical estimates of the bound charge densities at the +permissible superdomain walls and at the principal phase front +facets is summarized in the Table II. +It is found that the prismatic precipitate of type I.A should +contain a charged central tail-to-tail superdomain wall sur- +rounded by two head-to-none phase fronts. Conversely, the +prismatic precipitate of type I.B should contain a charged cen- +tral superdomain wall of head-to-head type, surrounded by +two tail-to-none phase fronts. These conclusions are equally +valid for precipitates formed from the weakly favored super- +domain states, as well as for precipitates formed from the +strongly favored superdomain states. +Otherwise, in case of the strongly field-favored superdo- +mains, one can state that the permissible pentahedral precipi- +tates of type II. have their central superdomain wall of head- +to-head type and the phase fronts forming at the (111) sur- +faces the smaller angles are head-to-none type, while the other +pair of phase front facets are of tail-to-none type. In contrast, +the permissible pentahedral precipitates of type III. have their +central superdomain walls of tail-to-tail type and the phase +fronts are all of head-to-none type. In the case of precipi- +tates formed from the weakly favored superdomain states, the +superdomain walls and phase front facets of the pentahedral +precipitates of type II. and III. would have the opposite charge +densities in comparison to their strongly favored counterparts. +IV. +DISCUSSION +Process of the bulk precipitation under (111) poling field. +The above outlined theory allows to draw the most likely in- +terpretation of the principal observations shown in the Fig. 2. +Upon the reducing the temperature below the paraelectric +phase stability limit, the bulk free energy of the tetragonal +phase become lower than that of the cubic phase. The tran- +sition might be then initiated by the nucleation and growth +of single-superdomain lentil-shaped precipitates, which later +coalesce into pairs or larger agglomerates. +Inspired by the observations, we deduce that the initial pre- + +a8 +cipitates most likely have already the two-superdomain pre- +cipitate shape with the central wall of type I. (see Fig. 2d +and Fig. 4). Among others, the other plausible types (type +II. and III.) of the symmetric two-superdomain precipitates +would have much larger angles of the needle-like blade tips +than it is observed in Fig. 2b and Fig. 3a. Such individual two- +superdomain precipitates are growing and eventually merg- +ing into interconnected arrays of alternating wedges of type +I.A and I.B. as suggested in Fig. 2e. Even more likely, the +obtuse edges of the original two-superdomain precipitates +surrounded directly by the cubic phase are serving as natu- +ral nucleation spots stimulating growth of the next superdo- +main wall and the neigbouring superdomain blade. Such a +chain reaction would best explain the experimental fact that +these arrays are effectively growing in their lateral direction +as well. +Several additional pieces of evidence for the for- +mation of superdomain boundaries are collected in the Ap- +pendix/supplementary material (section VII.) +Conversion of superdomain boundaries into the charged +domain walls. When the whole volume of the sample turns +into the tetragonal phase, the primary domain state ratio in +the superdomains is no more controlled by the mechanical +compatibility with the cubic phase. Therefore, the minority +primary domains can gradually shrink and eventually vanish +completely. In the weakly favored superdomains, this process +can be promoted by the applied field. Since this process of +vanishing minority domains is rather systematic, we have con- +cluded that the transient superdomains are indeed only weakly +favored by the bias field. In fact, it is well in line with the rea- +sonable expectation that there is not enough mobile charge +in the crystal allowing to compensate the primary domain +walls in the superdomains if they would be charge ones. In +other words, the weakly favored superdomains with neutral +primary domain walls are preferred over the strongly favored +superdomains with charged primary domain walls. As a re- +sult, the superdomains with the partially field-favored content +are converted by the [111]-directed field to the primary do- +main states and the adjacent charged superdomain walls are +transformed to the primary charged domain walls with the full +bound charge. For example, once the paraelectric phase is ab- +sent, the partially field-favored transient superdomain pair of +Fig. 5b is naturally driven by the [111]-directed field to the +simple primary twin of Fig. 2f. A the same time, the super- +domain boundaries are converted to the regular, macroscopic +primary charged domain walls. +Transport of the charge carriers. The results of Fig. 2 and +its above described interpretation also offers a new insight in +the mechanism of the separation of the charge carriers, needed +for the compensation of the final distribution of bound charges +at charged domain walls. In fact, we can see that at the nucle- +ation stage, the phase fronts are originally very near to the +superdomain walls of the opposite charge and in fact, they +are directly connected. Thus, during the nucleation, the con- +ditions for electronic and ionic mobile charge carriers sepa- +ration along the boundaries or by the bulk diffusion are cer- +tainly much more favorable. Subsequently, the compensating +charge carriers are probably transported together with the dis- +placement of the phase fronts themselves. In other words, the +motion of the charge carriers is synchronized and governed +by the same stimuli that causes the phase fronts to move away +from their nucleation sites. The growing of the charged super- +domain walls is also interlinked with the motion of charged +phase fronts, because they are both connected by the common +wedges. Since all these boundaries are formally bearing size- +able bound charge, we believe that it is likely that they are +conductive and the charge transfer between the adjacent su- +perdomain walls of the opposite polarity might be facilitated +by a possible electric conduction along the superdomain walls +and the phase front zig-zags themselves. However, the de- +tailed theory of the transport of compensation charge carriers +is beyond the scope of this work. +Primary domain wall content of superdomains. In princi- +ple, all derivations made so far are independent on the type +of the primary ferroelastic domain walls in the superdomains. +Obviously, the primary ferroelastic domain walls can be ei- +ther charged or neutral ones. In general, the energy of the +neutral primary domain walls are much lower than that of the +charged ones. Therefore, the neutral primary walls seems to +be more likely to be formed. Interestingly, those that are ob- +served in the Fig. 2 clearly belong the family of {011} planes, +perpendicular the (111) direction. This orientation is certainly +favorable for decreasing of the overall elastic energy of sam- +ple with a (111) oriented thin platelet shape. Nevertheless, +this observation is leaving us with two competing options for +the interpretation of the type of the primary domain wall type. +The first possibility is that the superdomains of Fig. 2b are the +strongly favored ones. Then, the primary domain walls ob- +served there must be the charged domain walls. The second +possibility is that these observed superdomains are the moder- +ately favored ones. Then, the involved primary domain walls +are the neutral ones. Energetically, the superdomain with neu- +tral primary domain walls seems to be more plausible. +V. +CONCLUSION +In summary, by performing in-situ observations of the do- +main formation during the frustrative poling of BaTiO3 single +crystals near the ferroelectric phase transition, we have found +multiple evidences for the formation of charged superdomain +boundaries. These superdomain boundaries are formed al- +ready in the early stage of the phase transformation if not +simultaneously with the formation of the charged tetragonal- +cubic phase fronts themselves. While the charged phase fronts +necessarily perform translation motion trough the crystal in +order to suppress the residual cubic phase until the transforma- +tion is completed, the superdomain boundaries are relatively +immobile and they grow mainly in their length. Nevertheless, +translation motion of phase fronts and extension of superdo- +mains is interlinked as the phase fronts are directly attached +to the superdomain boundaries. +In the case of the volume nucleation scenario, which was +the most efficient one in the sense of the formation of mul- +tiple charged domain walls, the final charged ferroelastic do- +main walls were shown to be directly derived from these su- +perdomain boundaries. The triple junction of a superdomain + +9 +boundary and two adjacent phase fronts form a narrow wedge, +that probably plays an essential role in facilitating the trans- +port of the electronic and ionic charge carriers during the +propagation of these charged interfaces throughout the crys- +tal. The observed self-organized tertiary superdomain struc- +tures strongly suggest an efficient chain formation mechanism +consisting in nucleation of the new superdomain walls while +keeping a sharp angle termination at the backside of the last +narrow wedge triple junction in the band. +The ensemble of observations was successfully interpreted +in terms of the WLR theory, extended to describe the mul- +tiple superdomain structures and composed precipitates. In +particular, the theoretical analysis based on the mechanical +compatibility of WLR superdomains allowed to understand +the specific habitus and the fine structure of the ferroelectric +precipitates in the observed transient domain patterns. Simi- +lar theoretical framework might be applied also to other do- +main engineered states of polar perovskites, for example in +the broad family of relaxor ferroelectrics. We believe that the +insights in the mechanism of the charged domain formation +in BaTiO3 materials, which reveal the fundamental role of the +superdomain walls, will trigger a new interest in domain en- +gineering perspectives of tailoring the novel nanoscale based +properties of ferroic materials. +VI. +EXPERIMENTAL SECTION +Single crystal samples used in this work and their domain +structures have been carefully prepared using the methods de- +scribed in detail elsewhere9. The principal experiments of this +work presented in Figs. 1 and 2 were made using crystals with +a high effective mobile charge density about 104 C/m3. For +complementary observations shown in Fig. S1 we have used +samples of originally stoichiometric crystals substantially re- +duced in oxygen deficient atmosphere in order to achieve an +effective charge density of 102 − 105 C/m3. Finally, the re- +stricted charged domain formation reported in Figs. S2, S3a +and S3c have been investigated in nominally pure stoichio- +metric dark yellow crystals with effective charge density of +102 −103 C/m3. Charge density in all crystals was estimated +for the temperature 573K. All materials were supplied by GB +group http://www.crystalsland.com. For in-situ observations, +both main surfaces were polished to a 1 micron quality and +covered with 10 nm thick transparent platinum electrodes that +allowed both to engineer and to observe the domain structure +by optical means. The optical polarizing microscopy was op- +erated using both reflected and transmitted mode and polar- +ized and nonpolarized light. Observations were performed at +room temperature as well as during the cooling from the para- +electric to the tetragonal phase using the heating/cooling stage +equipped with electric leads allowing application of the poling +electric voltage. The poling field up to 1.2kV/mm was applied +along the viewing [111] direction using the high voltage DC +power supply SRS PS325. +VII. +ACKNOWLEDGEMENTS +We acknowledge the support from the Czech Grant Agency +(GACR project No.20-05167Y). Both authors are appreciat- +ing stimulating discussions about the subject with Prof. A. +Tagantsev from EPFL Laussane. Authors also appreciate the +support of Prof. N. Setter from EPFL Laussane who initi- +ated the investigation in this field under the EU 7th Frame- +work Programme (FP7/2007–2013)/ERC grant agreement n +[268058]. +VIII. +CONFLICT OF INTEREST +The authors declare no conflict of interest. +IX. +DATA AVAILABILITY STATEMENT +The data that support the findings of this study are available +from the corresponding author upon reasonable request. +∗ Electronic address: bednyakov@fzu.cz, hlinka@fzu.cz +1 Sluka, T., Tagantsev, A. K., Bednyakov, P. & Setter, N. Free- +electron gas at charged domain walls in insulating BaTiO3. Na- +ture Communications 4, 1808 (2013). +2 Werner, C. S. et al. Large and accessible conductivity of charged +domain walls in lithium niobate. +Scientific Reports 7, 9862 +(2017). +3 Bednyakov, P. S., Sturman, B. I., Sluka, T., Tagantsev, A. K. & +Yudin, P. V. Physics and applications of charged domain walls. +npj Computational Materials 4, 65 (2018). +4 Sturman, B., Podivilov, E., Stepanov, M., Tagantsev, A. & Setter, +N. +Quantum properties of charged ferroelectric domain walls. +Physical Review B 92, 214112 (2015). +5 Sturman, B. & Podivilov, E. Charged domain walls under super- +band-gap illumination. Physical Review B 95, 104102 (2017). +6 Sturman, B. & Podivilov, E. Ion and mixed electron-ion screening +of charged domain walls in ferroelectrics. Europhysics Letters +122, 67005 (2018). +7 Gureev, M. Y., Tagantsev, A. K. & Setter, N. Head-to-head and +tail-to-tail 180 degrees domain walls in an isolated ferroelectric. +Physical Review B 83, 184104 (2011). +8 Vul, B. M., Guro, G. M. & Ivanchik, I. I. Encountering domains +in ferroelectrics. Ferroelectrics 6, 29–31 (1973). +9 Bednyakov, P. S., Sluka, T., Tagantsev, A. K., Damjanovic, D. & +Setter, N. Formation of charged ferroelectric domain walls with +controlled periodicity. Scientific Reports 5, 15819 (2015). +10 Bednyakov, P., Sluka, T., Tagantsev, A., Damjanovic, D. & Set- +ter, N. Free-carrier-compensated charged domain walls produced +with super-bandgap illumination in insulating ferroelectrics. Ad- +vanced Materials 28, 9498–9503 (2016). +11 Crassous, A., Sluka, T., Tagantsev, A. K. & Setter, N. Polarization +charge as a reconfigurable quasi-dopant in ferroelectric thin films. +Nature Nanotechnology 10, 614 (2015). +12 Jiang, J. et al. Temporary formation of highly conducting domain + +10 +walls for non-destructive read-out of ferroelectric domain-wall re- +sistance switching memories. Nature Materials 17, 49 (2017). +13 Maksymovych, P. et al. Tunable metallic conductance in ferro- +electric nanodomains. Nano Letters 12, 209–213 (2012). +14 Schr¨oder, M. et al. Conducting domain walls in lithium niobate +single crystals. Advanced Functional Materials 22, 3936–3944 +(2012). +15 Li, L. et al. Giant resistive switching via control of ferroelectric +charged domain walls. Advnced Materials 28, 6574–6580 (2016). +16 Sharma, P. et al. Nonvolatile ferroelectric domain wall memory. +Science Advances 3, e1700512 (2017). +17 Mundy, J. A. et al. Functional electronic inversion layers at fer- +roelectric domain walls. Nature Materials 16, 622 (2017). +18 Sluka, T. & Tagantsev, A. Electronic elements based on quasi +two-dimensional electron/hole gas at charged domain walls in fer- +roelectrics. US Patent 9171602 B2 (27.10.2015). +19 Hlinka, J., Privratska, J., Ondrejkovic, P. & Janovec, V. Symme- +try guide to ferroaxial transitions. Physical Review Letters 116, +177602 (2016). +20 Beccard, H. et al. Nanoscale conductive sheets in ferroelectric +BaTiO3: Large hall electron mobilities at head-to-head domain +walls. ACS Applied Nano Materials 5, 8717–8722 (2022). +21 Sturman, B. & Podivilov, E. Ferroelectric domain reversal: The +role of domain wall conduction. JETP Letters 1–8 (2022). +22 Fousek, J. & Janovec, V. +The orientation of domain walls in +twinned ferroelectric crystals. +Journal of Applied Physics 40, +135–142 (1969). +23 Wechsler, M. S., Lieberman, D. S. & Read, T. A. On the theory of +the formation of martensite. Transations AIME 197, 1503–1515 +(1953). +24 DiDomenico Jr, M. & Wemple, S. H. Paraelectric-ferroelectric +phase boundaries in semiconducting perovskite-type crystals. +Physical Review 155, 539 (1967). +25 Parker, T. J. & Burfoot, J. C. Cubic-tetragonal phase transitions +in high resistivity BaTiO3 single crystals. Journal of Physics D: +Applied Physics 2, 1168–1170 (1969). +26 Fesenko, E. G., Gavrilyatchenko, V. G., Semenchev, A. F. & Yufa- +tova, S. M. Regularities in domain structure formation in multi- +axial ferroelectric crystals. Ferroelectrics 63, 289–298 (1985). +https://doi.org/10.1080/00150198508221411. +27 Fesenko, E. G., Gavrilyatchenko, V. G. & Semenchev, A. F. Do- +main structure of multiaxial ferroelectric crystals. Ferroelectrics +100, 195–207 (1989). +28 Dec, J. Paraelastic—ferroelastic interfaces. Phase Transitions: A +Multinational Journal 45, 35–58 (1993). +29 Jin, Y. M., Wang, Y. U., Khachaturyan, A. G., Li, J. & Viehland, +D. Adaptive ferroelectric states in systems with low domain wall +energy: Tetragonal microdomains. Journal of Applied Physics 94, +3629–3640 (2003). +30 Roytburd, A. L. Elastic domains and polydomain phases in solids. +Phase Transitions: A Multinational Journal 45, 1–34 (1993). +31 Tagantsev, A. K., Cross, L. E. & Fousek, J. Domains in ferroic +crystals and thin films (Springer, 2010). +32 Neuber, E. et al. Architecture of nanoscale ferroelectric domains +in GaMo4S8. Journal of Physics: Condensed Matter 30, 445402 +(2018). + +Supplementary information to the manuscript ”Charged domain walls in BaTiO3 crystals emerging +from superdomain boundaries” +Petr S. Bednyakov1 and Jiˇr´ı Hlinka1 +1FZU-Institute of Physics, The Czech Academy of Sciences, Na Slovance 2, 18221 Praha 8, Czech Republic∗ +(Dated: January 10, 2023) +PACS numbers: +This supplement contains description of additional observa- +tions supporting the presence of charged superdomain walls in +(111)-poled BaTiO3 crystal plates. +I. +SURFACE NUCLEATION +Obviously, the details of the nucleation mechanism may +vary depending on various specific conditions under which +the phase transition is induced. For example, some of our +originally less conductive samples were treated in the oxygen +reducing atmosphere in order to increase the conductivity of +the sample. Since the treatment is inducing more defects in +the near surface layer of the sample, it can be expected that +surface nucleation of the ferroelectric phase is enhanced in +such crystals. Moreover, structural surface defects and the +resulting surface nuclei in such sample also substantial local +strain gradients, as evidenced by the characteristic birefrin- +gence butterfly patterns seen in the optical image of one such +samples (Fig. S1a). By comparison of the apparent angles of +the surface image of the already grown nuclei with that of the +theoretical predictions summarized graphically in Fig.4, we +concluded that the observed nuclei involve type III. central +superdomain boundary. In some cases, several nuclei are con- +nected together (see Fig. S1b), again most likely in order to +avoid the mechanically incompatible obtuse wedges. For ex- +ample, the nuclei of Fig. S1b reminds a fragment of hypothet- +ical star-like pattern composed of all six superdomain types +(see Fig. S1d). In other cases, the nuclei was showing an ar- +row type geometry, possibly similar to the geometry sketched +in Fig. S1e. However, there was no tendency towards the con- +certed organization of the nuclei into a more ordered domain +structure type like in the case of bulk nucleation described +above. In fact, it turned out that this particular sample and +the related poling process scenario was not very efficient in +creation of multiple stable charged domain walls. Instead, it +either yielded the so-called zigzag domain walls that we shall +briefly address later on, or the resulting charged domain walls +were so unstable that they fully decomposed after the removal +of the poling electric field. +II. +EDGE NUCLEATION +Another interesting phase transformation scenario was ob- +served when the tetragonal phase nucleated at the edge of the +crystal. Fig. S2 shows such transient wedge domain imaged +at about 1 K below Tstart. This sample was a (111)-oriented, +FIG. S1: Formation of nuclei at the surface of the crystal. Surface +nuclei are typically formed at local defects located at the surface of +the samples with a sufficiently conductive surface layer. Positions of +surface nuclei show a marked butterfly birefringence pattern, show- +ing strong local strain gradients present even before the borders of the +nuclei are apparent (a). In this case the nuclei often expands arms in +more than a single direction (b). In all cases each arm appears to +be composed by two blades, suggesting the important role charged +superdomain walls. Panels (d) and (e) suggest a plausible idealized +structure of the surface nuclei considering the crystallographic orien- +tation of type III. pentagonal nuclei of Fig.4. +arXiv:2301.03409v1 [cond-mat.mtrl-sci] 9 Jan 2023 + +[110] +50μm +[112] +[111]2 +100 micron thick plate made of a crystal with a rather low +conductivity of the order of 0.01-0.1 S/m. +The nucleation +site was most likely directly related to the local electric con- +tact. Again, by comparison with the theoretical shapes of two- +superdomain precipitate images on the (111) sample surface, +we can assign it as the wedge domain with the central su- +perdomain of type III. In this case, the fine primary domains +could be seen to have the orientation indicated schematically +in Fig. S2c. They are not very clear on the micrograph because +these walls are oblique to the main surface. Assuming again +that the primary domain walls are electrically neutral ones, we +can conclude that here the involved superdomains formed are +the strongly field-favored ones. Obviously, the resolution of +the optical image was not sufficient to determine the micro- +scopic structure of the superdomain wall. Still, we can con- +clude that the overall bound charge at this superdomain wall +should be negative (in other words, it has a tail-to-tail ori- +entation of the average polarization). At the same time, the +phase fronts are of head-to-none type, implying that the nega- +tive mobile charge carriers are needed to screen these moving +phase fronts. In principle, this poling strategy can be suitable +when formation of just a single charged domain wall would +be desired. +FIG. S2: Charged superdomain wall formed at the edge crystal. (a) +Optical micrograph showing the transient wedge domain, separated +in two parts by a clear boundary, identified a superdomain wall as +schematically shown in the panel (b). The angle between the su- +perdomain wall and phase front boundary θ ≈ 21◦ corresponds best +to that of the type III. wedge geometry of Fig.4. Dark spot on the +top of the picture is a contact used for the electric field application, +the top and the bottom part of the image corresponds to the opposite +edges of the sample. Labels ”T” and ”C” stands for the tetragonal +and cubic phases, resp., crossed directors shows the orientation of +the polarizer and analyzer. The primary domain structure pattern (c) +at the sample surface can be deduced from the direction of the ob- +served stripes, mechanical compatibility and the assumption that the +majority primary domains are oriented favorably to the applied field. +III. +ZIG-ZAG DOMAIN WALLS +It has been repeatedly observed that frustrative poling of +low conductivity BaTiO3 samples results in formation of the +so called zigzag charged domain walls, in particular in case +of tail-to-tail charged walls that require charge compensation +FIG. S3: The three basic examples of the final charged domain walls +of the same macroscopic crystallographic orientation as observed in +various (111)-oriented BaTiO3 crystal plates. (a) Tail-to-tail zigzag +charged domain wall composed of nearly neutral domain walls and +its schematic drawing (b); an array of ideal planar charged domain +walls (c) and their schematic drawing (d); (e) charged superdomain +walls separating a superdomain region from the two primary domains +and its schematic drawing (f). Crossed black/white arrows in right- +upper corner of panels (a,c,e) correspond to the orientation of polar- +izers. +by the less available positive mobile carriers. On a macro- +scopic scale, such a zig-zag domain wall (example shown in +Fig. S3a) can be rationalized as a charged domain wall, differ- +ing from the normal one by its considerable thickness. How- +ever, on the microscopic scale, it is clearly formed by a zigzag +array of the almost neutral primary domain walls that are just +slightly tilted from their ideal neutral wall orientation. Due to +this small tilt, such primary walls carry a small bound charge, +which, however, in projection to the macroscale domain wall +orientation gives the usual bound charge density value. In the +present case, the nominal orientation of the neutral primary +walls involved in the zigzag are already inclined with respect +to the sample surface. Therefore, the zigzag image is partly +distorted. Nevertheless, by counting the number of the zigzag +turns per given length it can be estimated that the angle be- +tween adjacent zigzag lamellae is less than 10 degrees. This +corresponds to about at least 10 times smaller bound charge +density per surface area of the primary microscopic wall, or, +equivalently, it means that the surface of the zigzag micro- +scopic wall is at least 10 larger than that of the corresponding +macroscopic wall (or of its ideal planar counterpart, such as +the one in Fig. S3c,d). Interestingly, the macroscopic width +of the zigzag wall (the length of its tooth) is of the order of +the typical final distance between charged domain walls them- + +b +X +C +C +[112] +500μm +[110] +[111]b +a +WW +200μm +200μm +X + [112] +500μm +[110] +[111]3 +selves or even larger (see Fig. S3c). In this sense, the zig-zag +itself can be considered as a sort of secondary domain, al- +though here the volume ratio of the two primary domain states +is probably 1:1 and we could not see any tight relation to the +WLR superdomains discussed previously. We can speculate +that the effective width of the zig-zag wall can be related to the +volume needed to collect the required amount of charge, while +the width of the zig-zag tooth could be perhaps related to the +characteristic distance from which the compensation charge +can be collected by the usual diffusion processes. +IV. +SUPERDOMAIN WALL BETWEEN PRIMARY AND +SECONDARY DOMAINS. +Alternatively to zig-zag walls, in some cases we have ob- +served secondary domain walls connecting primary domains +and the twinned secondary domains, such as in Fig. S3e. Here +the primary fine stripe domains in the twinned central area are +most likely the neutral ones, in other words, those that are in- +clined with respect to the viewing direction. The sharp borders +between the dark and light area in the photograph Fig. S3e +thus correspond to the charged macroscopic tail-to-tail and +head-to-head charged domain walls, respectively. Because of +the twinning in the central area, the overall bound charge den- +sity is reduced in comparison with the primary charged walls +of Fig. S3c. Schematic interpretation of this structure is pro- +posed in Fig. S3f. Note that all large-area planar interfaces in +this domain structure are mechanically compatible ones. The +experiment does not reveal the microstructure the superdo- +main wall, simplest assumption is that it formed by an aligned +system of wedge domains ended on a common plane. In ei- +ther case, on the macroscopic scale, the superdomain wall of +Fig. S3f requires smaller overall amount of screening charge +then the primary domain of Fig. S3d. +In principle, we may speculate that the secondary (twinned) +domain in the center of Fig. S3e,f is probably residuum of +a WLR superdomain state, fully favored by the bias electric +field, which was surrounded by two superdomains only par- +tially favored by the bias electric field. As we have argued +already, it can be expected that in the course of the domain +structure coarsening, the bias field will promote the transfor- +mation of the partially favored superdomains into the primary +domains. On the other hand, there is no such obvious driv- +ing force present in case of the fully favored superdomains, +and therefore the process of elimination of the fine stripe pri- +mary domains is expected to be much slower, hindered or +completely frozen. +∗ Electronic address: bednyakov@fzu.cz, hlinka@fzu.cz + diff --git a/Y9E1T4oBgHgl3EQfwQWa/content/tmp_files/load_file.txt b/Y9E1T4oBgHgl3EQfwQWa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cc76ce83029590e032bf7f201a2551d52ad86943 --- /dev/null +++ b/Y9E1T4oBgHgl3EQfwQWa/content/tmp_files/load_file.txt @@ -0,0 +1,763 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf,len=762 +page_content='Charged domain walls in BaTiO3 crystals emerging from superdomain boundaries Petr S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Bednyakov1 and Jiˇr´ı Hlinka1 1FZU-Institute of Physics, The Czech Academy of Sciences, Na Slovance 2, 18221 Praha 8, Czech Republic∗ (Dated: January 10, 2023) Previous experiments with BaTiO3 single crystals have shown that application of the electric field in the vicinity of the ferroelectric phase transition can be used to introduce peculiar persisting ferroelectric domain walls, accompanied by the compensating charge in the form of two-dimensional electron gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The present in-situ optical observations of such electric poling process reveal formation of a transient coexistence of the cubic and ferroelectric phases, the latter one being broken into multiple martensitic superdomains, separated by superdomain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' It is revealed that as the transient superdomains convert into the regular ferroelectric domains, the superdomain boundaries transform into the desired charged domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In order to assign the observed transient domain patterns, to understand the shapes of the observed ferrolectric precipitates and their agglomerates as well as to provide the overall interpretation of the observed domain formation process, the implications of the mechanical compatibility of the coexisting superdomain states is derived in the framework of the Wechsler-Lieberman-Read theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The results also suggest that the transport of the compensating charge carriers towards the final charged domain wall location is directly associated with the electric conductivity and interlinked motion and growth of the superdomain walls and phase fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' PACS numbers: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' INTRODUCTION Ferroelectric charged domain walls have recently attracted lots interest due to their ultimate nanoscale thickness com- bined with promising charge transport properties1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Condi- tions of the stability of the charged domain walls in strong ferroelectrics like BaTiO3 has been discussed and clarified in terms of phenomenological and microscopic theory4–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Ex- perimentally, it has been shown that charged domain walls can be engineered with a controlled density9,10, written locally with scanning probe tools11 or using patterned electrodes12, and also repeatedly reconfigured13–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The possibility to ad- dress the conductivity of the domain wall has been even ex- ploited in various device proposals12,15–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' At ambient temperature, ferroelectric BaTiO3 crystal has tetragonal crystal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Its 6 primary ferroelectric do- main states are related to the m¯3m > 4mm macroscopic sym- metry breaking (species No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 198 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' When disre- garding the small angles associated with the elastic match- ing of domain states and adopting to the pseudocubic crys- tallographic notation, we can state that the polarization falls along one of the six {001} directions and that 180 degree and 90-degree ferroelectric domain wall can be formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The 90- degree walls are particularly attractive for domain engineering because they are also ferroelastic and can be induced by the frustrative poling that favors more than one ferroelectric do- main state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The existence of charged 90-degree domain walls always require almost perfect screening of the huge bound polar- ization charge at such a wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The usual BaTiO3 crystals are slightly n-doped semiconductors, so that the head-to-head charged domain walls can be compensated by the naturally present excess electrons6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The concentration of the com- pensation charge is so strikingly high that even the sponta- neous formation of the two-dimensional electron gas at this 90-degree head-to-head domain wall have been observed in BaTiO3 crystal1,20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 1: Charged domain walls in (111)-oriented BaTiO3 crystal plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Panels (a), (b) and (c) shows (01¯1), (10¯1) and (1¯10) sys- tems of charged domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' These ex-situ optical micrographs are taken in transmission mode, the viewing direction is that of the formerly applied [111]-directed poling field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Red arrows stand for the experimentally inferred polarization in the primary domain states favored by the poling electric field, while the the polarizer-analyzer orientation is indicated by the crossed directors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Panel (d) shows how the mechanically compatible neutral (NDW) and charged (CDW) pri- mary ferroelastic domain walls can be terminated on the (111) sur- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' It is worth noting that head-to-head and tail-to-tail ferroe- lastic domains walls form parallel planes alternating in the crystal (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Consequently, the positively charged car- riers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' oxygen vacancies or other ionized mobile defects or electron holes, also need to be present in the crystal in order to form a stable domain pattern with multiple domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The sufficient amount of mobile positive charge carriers have been secured using two alternative strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In the first ap- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='03409v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='mtrl-sci] 9 Jan 2023 a b (011) [010] (101) [001] [100] [001] 7 200μm C d (011) (011) [010] (110) [oT] (110) (101) [100] (101) (110) CDW NDW [112] [111]2 proach, crystals containing a sufficient amount of oxygen va- cancies were used9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In the second approach, the poling was performed under the UV light illumination1, suggesting that both photoexcited electrons and holes participated in screen- ing of the polarization bound charge10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Strictly speaking, the process of charged domain wall cre- ation in BaTiO3 requires not only the presence of charge carri- ers but also their efficient spatial redistribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' For example, it has been recently emphasized that formation of the charged domain walls in the polarization switching process is much more understandable when such are conductive and connected to the outer electrode21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' However, the exact mechanism of the 90-degree charge domain pattern formation process was not fully understood yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In particular, there is no obvious reason for an organized charge carrier separation before some sort of bound charge modulation is formed in the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Like- wise, the idea that the macroscopic domain patterns could be initially nucleated without the full compensation and only then stabilized by the carriers of both signs appears unrealistic when considering plausible charge carrier drift velocities and the micron-sized distances between the charged domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In this work, we draw the attention to the fact that the 90- degree charged domain patterns were so far created in bulk crystals of BaTiO3 by the process of frustrative electric poling in the vicinity of phase transitions9,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' With this in mind, we have been revisiting the problem of the charged domain walls creation by careful in-situ optical investigations of the early stage of their formation near the cubic to tetragonal phase transition9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' We have realized that the charged domain walls are actually originating from the charged superdomain walls, separating nanotwinned martensitic plates, here also called su- perdomains, that are formed in the process of the nucleation of the ferroelectric phase itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The optical observations are complemented by an in-detph theoretical analysis of these su- perdomain structures and walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', we present the main experimental observation that are documented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 1, 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Then, in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' we expose the result of the theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The part III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='A summarizes the known results of the Wechsler-Lieberman-Read (WLR) martensite theory of phase fronts in BaTiO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Among others, we clar- ify there what exactly we mean by superdomains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The rest of the Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' describes various properties of the super- domain walls and superdomain precipitates and the expected consequences for the adopted geometry of the frustrative pol- ing with [111] oriented field as we have derived them from the mechanical compatibility conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The main results are given in Tables I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' and in the summarizing Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', the theoretical background is used to analyze and interpret the principal results of the Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Next, we show that the so far derived conclusions are in agreement with some additional observations and several proposals for further investigation are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Finally, the paper is concluded by considerations about the role of possibly conductive phase fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' PRINCIPAL OBSERVATIONS The main results of the present experiments are summa- rized in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The samples used in all investigations were (111)-oriented platelets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Formation of straight charged do- main walls was most conveniently achieved in slightly off- stoichiometric BaTiO3 single crystals with conductivity of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='1-1 S/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Similarly as is the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 9, these charged walls were induced by frustrative poling method across the cu- bic to tetragonal phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' More precisely, the [111]- oriented dc electric field of about 1 kV/mm field was applied several K above the phase transition and then the sample was cooled down through the transition while the field was kept on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' After cooling the sample down to the ambient temper- ature and switching off the field, the contacts were removed and the resulting domain patterns were thoroughly investi- gated ex-situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' We have confirmed that most of the charged domain walls grown in this way were stable at least over sev- eral months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Typically, such samples have shown an array of parallel head-to-head and tail-to-tail domain walls separated by dis- tances of about 100 microns (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Interpretation of the results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 1 is rather straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Polarization vectors were identified by the orientation of the optical indi- catrix and the light extinction at the domain wall in the non- polarized light following the procedure described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Distinction between the head-to-head and tail-to-tail could be easily made based on the observation of the dark and light line at the domain wall location9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Comparison with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 1d confirms that crystallographic orientations of these charged domain walls agree with the three theoretically expected ori- entations of primary charged domain walls, as discussed for example in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 9,22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Repeating the same poling procedure using thin transpar- ent electrodes enabled us to observe also the transient domain structures formed at the early stage of the charge domain wall formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' This experiment, essential for the present paper, is reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' It was carried out with a similar sam- ple as that of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The precipitation started at temperature Tstart which was about 402 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Interestingly, we have observed that the transformation started by a simultaneous nucleation of many thin, lentil-shaped precipitates, appearing within the bulk of the optically homogeneous and isotropic paraelectric cubic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The boundaries of the precipitates perform back and forth jerky motion, typical for the dynamics of ferroelastic domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The image of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2a shows these precipitates at a tempera- ture of about about Tstart −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The cross section of the pre- cipitate with the surface appears as a needle-shaped leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The individual needles appear to be organized in clusters, suggest- ing a systematic preference or a direct interaction among the parallel precipitates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2b shows the same area at a later stage, corresponding to the temperature of about Tstart − 1 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' One can see that precipitates formed compact fan-like domain patterns delimited by two zig-zag boundaries (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2b and its schematic interpretation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Originally, three com- peting systems of parallel needle-shaped leaves were coexist- ing in the sample, each corresponding to of one of the charged 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2: Optical micrographs showing the charged domain wall formation process in (111)-oriented crystal plate under the electric field applied along the viewing direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Panel (a) suggests nucleation of the ferroelectric precipitates in the form of thin platelets embedded in the optically isotropic paralectric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In the viewing direction the platelet precipitates appear as needle-shaped leafs oriented similarly as the traces of the charged domain walls in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Panel (b) reveals that the needle-shaped leafs have two blades, each of them decorated with a fine stripe pattern, and that the precipitates become organized in a pattern with common zigzag phase fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Panel (c) shows that the fine stripe pattern is fading out and the that stable charged domain walls are formed at the central lines of the needle-shape leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The panels (d,e,f) suggest schematic representation of the corresponding observations of panels (a,b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Black spot in the center of images (a,b,c) is due to an electric contact on the top electrode;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' polarizer-analyzer orientation is indicated in upper right corner of panel (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' domain wall orientations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Progressively, only one of the three systems of precipitates extended over the whole crystal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' A closer look to one of the thicker precipitates shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2b allows to discern a central line, dividing each needle- shaped leaf into two adjacent blades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Let us note that for the given orientation of the crossed polarizers, one type of the blades is systematically darker, while the other is barely seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The relevant detail is thus enlarged in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 3a and schematized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' There one can also see that each of the adjacent blade is decorated inside with a differently oriented fine-stripe pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' From the analysis of this fine-stripe pattern and the overall geometry of the precipitates detailed in the next sec- tion, we infer that the individual blades of the precipitates correspond to distinct martensitic superdomains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 3b, these two blades are shaded by different colors, using the su- perdomain color code defined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Similar but more schematic graphical interpretation of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2a and 2b is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2d and 2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Our conjecture that the superdomains are formed by the fine stripes of the primary tetragonal ferroelec- tric domains is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Let us stress that we have observed that this fine stripe pat- tern within superdomains is gradually fading out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In other words, the blades appear more and more homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Si- multaneously, the contrast between adjacent blades observed with the fixed orientation of crossed polarizers is enhanced (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In the end, only one system of wide homoge- neous parallel stripes persists over the whole sample, as it is schematized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' These resulting wide stripes obvi- ously correspond to the primary tetragonal domains separated charge domain walls, similar to those shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' THEORY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Standard WLR theory of the phase front Before addressing the properties of superdomain walls and composed precipitates, let us summarize the already known predictions of the martensitic WLR theory23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' According to our knowledge, the martensitic WLR theory has been first applied to perovskite ferroelectrics in order to explain the formation of the phase front between the paraelectric cu- bic and ferroelectric tetragonal phase in the seminar work of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Almost in parallel, it has been applied to high- resisitivity BaTiO3 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Later the theory has been further developed and independently tested or formulated by many others26–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' According to the martensitic theory, the ferroelectric phase transition proceeds preferentially by propagation of a spe- 200μm 200μm 200μm [110] [112] [111]4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 3: Details of the transient domain structure shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Top panels show awl-shaped ferroelectric precipitates surrounded by the paraelectric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The region captured by the optical micro- graph in panel (a) is graphically interpreted in terms of Zx and Yx superdomains in panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The fine-scale stripes due to the primary ferroelectric domains within these superdomains form the resulting pattern of panel (c), reminding of awl-shaped leaves with two blades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Dashed lines highlight phase boundaries of one such precipitate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' By comparison with the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 4 explained in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', the junction of the two blades can be identified as a type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='A superdomain bound- ary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The bottom panels (d-f) show the corresponding information for an area fully covered by the ferroelectric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Note that the assignment of the strain state in superdomains was done using the predictions of the Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The polariza- tion content of the individual primary domains can be inferred from a more subtle arguments given in section IV (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 5 and color code of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' cific coherent planar phase front, also termed as habit plane, connecting the paraelectric part of the crystal with a suitably twinned ferroelectric part of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In order to ensure the mechanical compatibility between ferroelectric and paraelec- tric phases, a martensitic twin is formed from two ferroelectric domain states is a suitable volume ratio and acceptable phase fronts are restricted to specific planes with a precisely defined crystallographic orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Let us consider the usual pseudocubic cartesian axes of the parent perovskite structure, m¯3m > 4mm ferroelectric transi- tion and a ferroelectric twin composed of the prevailing tetrag- onal domain state with polarization along +z and a minority tetragonal domain state with polarization along +x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' It is con- venient to denote such a twin as [Z,x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' According to the the- ory, the optimal relative volume of the minority state +x in the [Z,x] WLR twin, mechanically compatible with the cubic phase, is given by ξ = (a0 −a)/(c−a) , (1) where a0, a and c are lattice parameters of the cubic and tetragonal phase at the point of the phase transition, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The average spontaneous strain of the representative [Z,x] WLR twin, defined with respect to the cubic lattice pa- rameter a0, can be expressed as ε = � � 0 0 0 0 ε⊥ 0 0 0 ε∥ −ξ∆ , � � (2) where ε⊥ = (a − a0)/a0, ε∥ = (c − a0)/a0, are spontaneous strain parameters of the primary tetragonal domain state and ∆ = ε∥ −ε⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The average polarization of the [Z,x] twin is then P = (ξ,0,1−ξ)P0 , (3) where P0 is the equilibrium magnitude of the primary single domain polarization at the phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In this case, the phase-front unit normal vector, defined as pointing towards the side of the cubic phase, can have four possible orientations n = (0,±α,± � 1−α2) , (4) where α = � −ε⊥ ∆(1−ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' (5) Interestingly, none of the vectors n in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' (4) is perpen- dicular to the average polarization of the WLR twin, listed in in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Therefore, the habit plane bears a net bound charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In particular, the head-to-none phase front of [Z,x] twin with n = (0,±α, √ 1−α2) posses a positive bound surface charge and the tail-to-none phase front n = (0,±α,− √ 1−α2) surface bears a negative bound surface charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In all cases, the absolute value of the bound surface charge density on these phase fronts is σ = (1−ξ) √ 1−α2P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In fact, it is comparable to the bound charge at the charged fer- roelastic domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In case of BaTiO3, the volume ratio of primary domain states in the WLR twin is close to 2:1 and the phase fronts are approximately {056} crystallographic planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' These predic- tions have been verified by direct observation of such macro- scopic phase fronts26–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Concept of superdomain boundaries The observations reported in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2 and 3 suggest that in the present experiments, conducted under the [111]-directed poling field, there are many differently oriented phase fronts and several distinct WLR twin states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Therefore, it is reason- able to expect that several such secondary domain states, for brevity denoted as superdomain states, might coexist in the same crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Distinct superdomains can meet at superdomain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' From the point of view of the average strain and polariza- tion, the parent cubic point group symmetry allows 24 equiv- alent orientational variants of the above described WLR twin states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The exact macroscopic crystallographic symmetry19 implies that these 24 WLR twin states can be considered as 24 domain states of the fully ferroelectric monoclinic species m¯3m > m+ (see species 207 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 19 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Nevertheless, the 50μm 气 [110] 50μm [112] [111]5 Xy Xz Yx Yz Zx Zy Xy (011) (110) (101) (01¯1)∗ (1¯10)∗ (10¯1)∗ Xz (011) (110) (101) (01¯1)∗ (1¯10)∗ (10¯1)∗ Yx (110) (101) (011) (1¯10)∗ (10¯1)∗ (01¯1)∗ Yz (110) (101) (011) (1¯10)∗ (10¯1)∗ (01¯1)∗ Zx (101) (011) (110) (10¯1)∗ (01¯1)∗ (1¯10)∗ Zy (101) (011) (110) (10¯1)∗ (01¯1)∗ (1¯10)∗ TABLE I: Orientations of the mechanically compatible superdomain boundaries among the WLR superdomains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' For each compatible do- main pair, there are two permissible domain wall orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' One of them, marked by asterisk, is perpendicular to the (111) surface of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' average strain given by the eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' (2) distinguishes only 6 dis- tinct average spontaneous superdomain strain tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' There- fore, for the considerations of the mechanical compatibility of superdomain walls, it is sufficient to consider the effective ferroelastic species m¯3m > mm+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In other words, for this pur- pose it is possible to disregard the sign combinations of the polarization components and consider only the six ferroelas- tic superdomain states denoted in the following as Xy, Xz, Yx, Yz, Zx, Zy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The orientation of the mutually mechanically compatible superdomain walls among these 6 WLR ferroelastic states was determined from the average spontaneous strain tensors sim- ilarly as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' There are three distinct types of mechan- ically compatible ferroelastic superdomain walls, labeled as type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' (see the graph of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' super- domain wall joins two superdomains with a common minori- tary primary domain state (for example, superdomain states Xy and Zy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' superdomain wall connects two super- domains with a common majority primary state (for example, Xy and Xz) and type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' superdomain wall joins two superdo- mains having their primary domain population interchanged (for example, Xy and Yx pair).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Each compatible pair allows two orientations of the boundary, listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The other pairs of the superdomain states are not mechanically compat- ible, meaning that no mechanically permissible superdomain walls can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Termination of superdomain walls at (111) surface It turns out that in case of the investigated cubic to tetrag- onal ferroelectric phase transition, not only the primary fer- roelastic walls, but also the secondary ones are {011} type planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Consequently, these planes are are either perpendic- ular to the (111) crystal surface, intersecting it along [¯211], [1¯21], or [11¯2] directions, or they are oblique to the (111) sur- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 4: Summary of theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The vertices of the graph in panel (a) stand for the 6 types of ferroelastic superdomains compatible with the paralectric cubic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Type of the connect- ing line (single, double, tripple) indicate the type of the mechanically permissible superdomain boundary of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' and III, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Panel (b) displays the (111) surface sections of the hypothetical poly- hedral ferroelectric precipitates delimited from the cubic phase by mechanically compatible planar phase fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Same view direction as in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2 is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Precipitates are labeled I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' ac- cording to the type of the embedded central superdomain boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='A and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='B precipitates are composed of two infinite triangular prisms (see Section III F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' precipitates have penta- hedral pyramid shape that can terminate by an apex in the volume of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The outer hexagon is merely a guide for eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' face, intersecting it along along [01¯1], [1¯10], or [¯101] direc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The superdomain walls of the former family are marked by asterisk in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' It can be expected, that those with the orientation perpendicular to the sample are more favorable for reduction of the strain energy in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Yz [010] a X^ [001] [100] Zy x Y Z Zx Xy - Xz - X Yx - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Yz - Xz II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Zy - II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' - Zx - b (111) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='A Zx Xz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='B I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Yx Xy Z Zy X6 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Termination of habit planes at (111) surface By their incidence with respect to the (111) sample sur- face, the habit planes of the phase fronts can be also di- vided into two families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Members of the first family show the incidence almost perpendicular to the surface, while the others have rather an almost parallel position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Intersec- tion of the phase fronts with (111) sample surface obviously falls along the cross product of the phase front and surface normals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' This yields a directional vector with components (α − √ 1−α2, √ 1−α2,−α) or another vector symmetrically equivalent to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In this set of vectors, there are 6 directions that are close to [¯211], [1¯21], or [11¯2] axes, and these corre- spond to the phase fronts close to perpendicular to the (111) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' There are also 6 other orientations that are close to [01¯1], [1¯10], or [¯101] directions, and these ones correspond to the phase fronts almost parallel to the (111) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' All possible cases are also depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Single-superdomain precipitates The canonical WLR scenario of the temperature driven phase transition assumes a single phase front traversing the crystal and leaving a single WLR twin behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In contrast, the [111]-directed poling field used here is expected to favor for- mation of twinned precipitates of different types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Moreover, the sides of the sample are typically subject of a lower field, so that the transformation starts in the central part of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Let us first briefly consider what could be convenient shape of one isolated precipitate formed by a single WLR twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' We recall that there are just 2 compatible phase front planes for a fixed twin, related to the 4 phase front normals listed in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' It can be anticipated that the mechanically most conve- nient shape of such a precipitate is a thin platelet, possibly of a lentil-like shape, delimited by only two slightly bend phase fronts, being close to one of the corresponding fully compat- ible planar orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' It can be expected, that the phase fronts almost perpendicular to the platelet sample surface will be energetically more convenient than the oblique ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Since the phase fronts on the opposite side of such lentils are neces- sarily of opposite bound charge, the shape might be systemat- ically somewhat asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Two-superdomain precipitates It seems that the above described single superdomain pre- cipitates of the same type would be mutually attracted and merge together, while the parallel precipitates delimited by phase fronts in a reversed order of charge densities would be rather electrostatically repulsed from each other, at least be- fore the bound charge is fully screened by mobile electronic and ionic carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In either case, the existence of needle- shape leaf-like motifs documented experimentally in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 3 clearly suggests that it might be more convenient to form precipitates composed of two different WLR super- domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Guided by the experimental observations of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2, our geometry considerations were further limited to such com- posed precipitates that are symmetric with respect to the cen- tral ferroelastic superdomain boundary, knowing also from the observations that this boundary is a {110} oriented plane, and that it is perpendicular to the (111) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In other words, we assumed that the central superdomain boundary orientation is one of those marked by asterisks in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' What could be a convenient shape and crystallographic ori- entation for such a two-superdomain precipitate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The sim- plest 3D objects delimited by planar walls are tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Therefore, we have been considering precipitates composed by a pair of tetrahedra, each delimited by two phase fronts, a common superdomain wall and the top surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Let n1,n2,w and s are the out-pointing unit vectors perpendicular to the two phase fronts, to the superdomain wall, and to the (111) surface normal, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' It can be shown that for a given combination of such normals, the tetrahedron can be formed if [(n1 ×n2)·w][(n1 ×n2)·s] < 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' (6) Clearly, if the surface normal is fixed and this tetrahedron for- mation condition is satisfied for a given combination of n1,n2 and w, it cannot be simultaneously satisfied with same n1,n2 but with the opposite w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' This implies that a given combina- tion of n1 and n2 vectors allows the tetrahedra being formed at only one side of the superdomain boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' It might happen that the vectors fulfill the limiting relation ((n1 × n2) · w) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Then, the normals n1,n2 and w form an infinite triangular prism instead, which could be limited by the top and bottom (111) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In this case, another such prism can be constructed with the same n1 and n2 pair on the other side of the superdomain pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The inspection of available combinations of the phase front normal vectors shows, that if the central superdomain bound- ary is of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', then two kinds prismatic precipitates can form by a junction of two distinct triangular prisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The prismatic precipitate of the first kind has its sharp angle phase front wedge pointing along the [¯211], [1¯21] or [11¯2] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' We denote this case as I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The second type of the prismatic pre- cipitate has its sharp angle phase front wedge pointing in one of the opposite directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' It is denoted as the I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='B case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' If the central superdomain boundaries are of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' or III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', it is possible to construct a pentahedral precipitate composed of two real tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' For a given superdomain pair, the par- ticular crystallograpic orientation of the individual facets of such precipitate can be derived using the Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' For real- istic values of lattice strain, corresponding to the {056} type habit planes, the shapes and orientations of these precipitates have been determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The apparent shape of these precipi- tates given by their termination on the top surface (view from outside along the field direction) is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Nevertheless, it should be stressed that although the pre- cipitates shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 4b are formed by the mechanically compatible planar interfaces only, it can be expected that there would be residual strain incompatibility at all their edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In reality, only the edges at the relatively sharp wedge-like junc- tions are likely to be well accommodated by the elastic defor- mation, while the junctions at obtuse angles are likely to be 7 energetically quite inconvenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Therefore, when possible, the precipitates are likely to join into more complex struc- tures, such as the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2e, where the junctions with obtuse angles are avoided only the sharp wedges are pre- served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Note that this peculiar arrangement is formed by an array of alternating head-to-head and tail-to-tail superdomain walls, terminated by two inequivalent zig-zag shaped phase fronts, one of the head-to-none type and one of the tail-to- none type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Domain states promoted by the electric field The electric field applied along the [111] crystallographic direction is favoring the three primary domains states with polarization along +x, +y and +z directions over the three remaining domain states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Similarly, the P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='E coupling of the electric field to the polarization should favor superdomain states composed of the +x, +y and +z primary domain states only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In other words, one expect to the six superdomain states [X,y], [X,z], [Y,x], [Y,z], [Z,x] and [Z,y] to be most likely ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Another 6 superdomain states [X,−y], [X,−z], [Y,−x], [Y,−z], [Z,−x] and [Z,−y] could be considered as moder- ately favored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The 6 superdomain states of [−X,y] type are unlikely to occur and those of [−X,−y] form are probably not formed under the [111] bias field at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Examples of su- perdomains with different degree of the alignment along the [111]-directed electric field, in particular (a) the strongly fa- vored superdomains [Z,x] and [Y,x], (b) moderately favored superdomains [Z,−x] and [Y,−x], (c) moderately disfavored superdomains [−Z,x] and [−Y,x], and (d) strongly disfavored superdomains [−Z,−x] and [−Y,−x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 5: Possible primary domain polarization configurations within a pair of adjacent martensitic twins (Zx and Yx), sharing a common charged superdomain boundary of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' With respect to the field applied along the [111] direction, panels correspond to a strongly favored (a), moderately favored (b), moderately disfavored (c) and strongly disfavored configurations (d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The arguments in section IV indicate that the transient structure of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 3f is that of the moderately favored case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Bound charge at superdomain boundaries In general, the ferroelastic superdomain walls may be both charged and uncharged ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Considering that (i) the observed superdomain walls are perpendicular to the (111) surface, and that (ii) only the 6 strongly favored superdomains are present in the sample, we are left with the charged superdomain walls only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Let us stress that the same actually holds also if only the weakly favored domain states are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The restriction strogly favored superdomains weakly favored superdomains type superdomain wall phase front superdomain wall phase front I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='66 P0 √ 2 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='74 P0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='66 P0 √ 2 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='74 P0 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='B +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='66 P0 √ 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='74 P0 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='66 P0 √ 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='74 P0 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='25 P0 √ 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='74 P0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='25 P0 √ 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='74 P0 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='41 P0 √ 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='74 P0 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='41 P0 √ 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='74 P0 TABLE II: Bound charge at principal interfaces of the idealized pre- cipitate forms shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Indicated are values for the cen- tral superdomain wall and for phase front facets having a longer in- tersection with the (111) crystal surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='A corresponds to the case with the sharp angle wedge vertex pointing to the [¯211] direction (in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 4b, the outer case), I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='B stands for the sharp angle pointing to the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Results are listed separately for strongly and weakly field-favored superdomains, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' of the superdomain type allows to deduce also the charge den- sities on the phase front facets of our composed precipitates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The numerical estimates of the bound charge densities at the permissible superdomain walls and at the principal phase front facets is summarized in the Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' It is found that the prismatic precipitate of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='A should contain a charged central tail-to-tail superdomain wall sur- rounded by two head-to-none phase fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Conversely, the prismatic precipitate of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='B should contain a charged cen- tral superdomain wall of head-to-head type, surrounded by two tail-to-none phase fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' These conclusions are equally valid for precipitates formed from the weakly favored super- domain states, as well as for precipitates formed from the strongly favored superdomain states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Otherwise, in case of the strongly field-favored superdo- mains, one can state that the permissible pentahedral precipi- tates of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' have their central superdomain wall of head- to-head type and the phase fronts forming at the (111) sur- faces the smaller angles are head-to-none type, while the other pair of phase front facets are of tail-to-none type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In contrast, the permissible pentahedral precipitates of type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' have their central superdomain walls of tail-to-tail type and the phase fronts are all of head-to-none type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In the case of precipi- tates formed from the weakly favored superdomain states, the superdomain walls and phase front facets of the pentahedral precipitates of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' would have the opposite charge densities in comparison to their strongly favored counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' DISCUSSION Process of the bulk precipitation under (111) poling field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The above outlined theory allows to draw the most likely in- terpretation of the principal observations shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Upon the reducing the temperature below the paraelectric phase stability limit, the bulk free energy of the tetragonal phase become lower than that of the cubic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The tran- sition might be then initiated by the nucleation and growth of single-superdomain lentil-shaped precipitates, which later coalesce into pairs or larger agglomerates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Inspired by the observations, we deduce that the initial pre- a8 cipitates most likely have already the two-superdomain pre- cipitate shape with the central wall of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2d and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Among others, the other plausible types (type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=') of the symmetric two-superdomain precipitates would have much larger angles of the needle-like blade tips than it is observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Such individual two- superdomain precipitates are growing and eventually merg- ing into interconnected arrays of alternating wedges of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='A and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' as suggested in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Even more likely, the obtuse edges of the original two-superdomain precipitates surrounded directly by the cubic phase are serving as natu- ral nucleation spots stimulating growth of the next superdo- main wall and the neigbouring superdomain blade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Such a chain reaction would best explain the experimental fact that these arrays are effectively growing in their lateral direction as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Several additional pieces of evidence for the for- mation of superdomain boundaries are collected in the Ap- pendix/supplementary material (section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=') Conversion of superdomain boundaries into the charged domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' When the whole volume of the sample turns into the tetragonal phase, the primary domain state ratio in the superdomains is no more controlled by the mechanical compatibility with the cubic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Therefore, the minority primary domains can gradually shrink and eventually vanish completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In the weakly favored superdomains, this process can be promoted by the applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Since this process of vanishing minority domains is rather systematic, we have con- cluded that the transient superdomains are indeed only weakly favored by the bias field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In fact, it is well in line with the rea- sonable expectation that there is not enough mobile charge in the crystal allowing to compensate the primary domain walls in the superdomains if they would be charge ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In other words, the weakly favored superdomains with neutral primary domain walls are preferred over the strongly favored superdomains with charged primary domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' As a re- sult, the superdomains with the partially field-favored content are converted by the [111]-directed field to the primary do- main states and the adjacent charged superdomain walls are transformed to the primary charged domain walls with the full bound charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' For example, once the paraelectric phase is ab- sent, the partially field-favored transient superdomain pair of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 5b is naturally driven by the [111]-directed field to the simple primary twin of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' A the same time, the super- domain boundaries are converted to the regular, macroscopic primary charged domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Transport of the charge carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The results of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2 and its above described interpretation also offers a new insight in the mechanism of the separation of the charge carriers, needed for the compensation of the final distribution of bound charges at charged domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In fact, we can see that at the nucle- ation stage, the phase fronts are originally very near to the superdomain walls of the opposite charge and in fact, they are directly connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Thus, during the nucleation, the con- ditions for electronic and ionic mobile charge carriers sepa- ration along the boundaries or by the bulk diffusion are cer- tainly much more favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Subsequently, the compensating charge carriers are probably transported together with the dis- placement of the phase fronts themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In other words, the motion of the charge carriers is synchronized and governed by the same stimuli that causes the phase fronts to move away from their nucleation sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The growing of the charged super- domain walls is also interlinked with the motion of charged phase fronts, because they are both connected by the common wedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Since all these boundaries are formally bearing size- able bound charge, we believe that it is likely that they are conductive and the charge transfer between the adjacent su- perdomain walls of the opposite polarity might be facilitated by a possible electric conduction along the superdomain walls and the phase front zig-zags themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' However, the de- tailed theory of the transport of compensation charge carriers is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Primary domain wall content of superdomains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In princi- ple, all derivations made so far are independent on the type of the primary ferroelastic domain walls in the superdomains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Obviously, the primary ferroelastic domain walls can be ei- ther charged or neutral ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In general, the energy of the neutral primary domain walls are much lower than that of the charged ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Therefore, the neutral primary walls seems to be more likely to be formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Interestingly, those that are ob- served in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2 clearly belong the family of {011} planes, perpendicular the (111) direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' This orientation is certainly favorable for decreasing of the overall elastic energy of sam- ple with a (111) oriented thin platelet shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Nevertheless, this observation is leaving us with two competing options for the interpretation of the type of the primary domain wall type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The first possibility is that the superdomains of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2b are the strongly favored ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Then, the primary domain walls ob- served there must be the charged domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The second possibility is that these observed superdomains are the moder- ately favored ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Then, the involved primary domain walls are the neutral ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Energetically, the superdomain with neu- tral primary domain walls seems to be more plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' CONCLUSION In summary, by performing in-situ observations of the do- main formation during the frustrative poling of BaTiO3 single crystals near the ferroelectric phase transition, we have found multiple evidences for the formation of charged superdomain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' These superdomain boundaries are formed al- ready in the early stage of the phase transformation if not simultaneously with the formation of the charged tetragonal- cubic phase fronts themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' While the charged phase fronts necessarily perform translation motion trough the crystal in order to suppress the residual cubic phase until the transforma- tion is completed, the superdomain boundaries are relatively immobile and they grow mainly in their length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Nevertheless, translation motion of phase fronts and extension of superdo- mains is interlinked as the phase fronts are directly attached to the superdomain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In the case of the volume nucleation scenario, which was the most efficient one in the sense of the formation of mul- tiple charged domain walls, the final charged ferroelastic do- main walls were shown to be directly derived from these su- perdomain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The triple junction of a superdomain 9 boundary and two adjacent phase fronts form a narrow wedge, that probably plays an essential role in facilitating the trans- port of the electronic and ionic charge carriers during the propagation of these charged interfaces throughout the crys- tal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The observed self-organized tertiary superdomain struc- tures strongly suggest an efficient chain formation mechanism consisting in nucleation of the new superdomain walls while keeping a sharp angle termination at the backside of the last narrow wedge triple junction in the band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The ensemble of observations was successfully interpreted in terms of the WLR theory, extended to describe the mul- tiple superdomain structures and composed precipitates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In particular, the theoretical analysis based on the mechanical compatibility of WLR superdomains allowed to understand the specific habitus and the fine structure of the ferroelectric precipitates in the observed transient domain patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Simi- lar theoretical framework might be applied also to other do- main engineered states of polar perovskites, for example in the broad family of relaxor ferroelectrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' We believe that the insights in the mechanism of the charged domain formation in BaTiO3 materials, which reveal the fundamental role of the superdomain walls, will trigger a new interest in domain en- gineering perspectives of tailoring the novel nanoscale based properties of ferroic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' EXPERIMENTAL SECTION Single crystal samples used in this work and their domain structures have been carefully prepared using the methods de- scribed in detail elsewhere9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The principal experiments of this work presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 1 and 2 were made using crystals with a high effective mobile charge density about 104 C/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' For complementary observations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S1 we have used samples of originally stoichiometric crystals substantially re- duced in oxygen deficient atmosphere in order to achieve an effective charge density of 102 − 105 C/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Finally, the re- stricted charged domain formation reported in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S2, S3a and S3c have been investigated in nominally pure stoichio- metric dark yellow crystals with effective charge density of 102 −103 C/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Charge density in all crystals was estimated for the temperature 573K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' All materials were supplied by GB group http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='crystalsland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' For in-situ observations, both main surfaces were polished to a 1 micron quality and covered with 10 nm thick transparent platinum electrodes that allowed both to engineer and to observe the domain structure by optical means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The optical polarizing microscopy was op- erated using both reflected and transmitted mode and polar- ized and nonpolarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Observations were performed at room temperature as well as during the cooling from the para- electric to the tetragonal phase using the heating/cooling stage equipped with electric leads allowing application of the poling electric voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The poling field up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='2kV/mm was applied along the viewing [111] direction using the high voltage DC power supply SRS PS325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We acknowledge the support from the Czech Grant Agency (GACR project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='20-05167Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Both authors are appreciat- ing stimulating discussions about the subject with Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Tagantsev from EPFL Laussane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Authors also appreciate the support of Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Setter from EPFL Laussane who initi- ated the investigation in this field under the EU 7th Frame- work Programme (FP7/2007–2013)/ERC grant agreement n [268058].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' CONFLICT OF INTEREST The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' DATA AVAILABILITY STATEMENT The data that support the findings of this study are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' ∗ Electronic address: bednyakov@fzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='cz, hlinka@fzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='cz 1 Sluka, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Tagantsev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Bednyakov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Setter, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Free- electron gas at charged domain walls in insulating BaTiO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Na- ture Communications 4, 1808 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 2 Werner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Large and accessible conductivity of charged domain walls in lithium niobate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Scientific Reports 7, 9862 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 3 Bednyakov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Sturman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Sluka, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Tagantsev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Yudin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Physics and applications of charged domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' npj Computational Materials 4, 65 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 4 Sturman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Podivilov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Stepanov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Tagantsev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Setter, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Quantum properties of charged ferroelectric domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Physical Review B 92, 214112 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 5 Sturman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Podivilov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Charged domain walls under super- band-gap illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Physical Review B 95, 104102 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 6 Sturman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Podivilov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Ion and mixed electron-ion screening of charged domain walls in ferroelectrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Europhysics Letters 122, 67005 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 7 Gureev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Tagantsev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Setter, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Head-to-head and tail-to-tail 180 degrees domain walls in an isolated ferroelectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Physical Review B 83, 184104 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 8 Vul, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Guro, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Ivanchik, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Encountering domains in ferroelectrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Ferroelectrics 6, 29–31 (1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 9 Bednyakov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Sluka, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Tagantsev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Damjanovic, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Setter, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Formation of charged ferroelectric domain walls with controlled periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Scientific Reports 5, 15819 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 10 Bednyakov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Sluka, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Tagantsev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Damjanovic, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Set- ter, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Free-carrier-compensated charged domain walls produced with super-bandgap illumination in insulating ferroelectrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Ad- vanced Materials 28, 9498–9503 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 11 Crassous, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Sluka, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Tagantsev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Setter, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Polarization charge as a reconfigurable quasi-dopant in ferroelectric thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Nature Nanotechnology 10, 614 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 12 Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Temporary formation of highly conducting domain 10 walls for non-destructive read-out of ferroelectric domain-wall re- sistance switching memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Nature Materials 17, 49 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 13 Maksymovych, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Tunable metallic conductance in ferro- electric nanodomains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Nano Letters 12, 209–213 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 14 Schr¨oder, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Conducting domain walls in lithium niobate single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Advanced Functional Materials 22, 3936–3944 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 15 Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Giant resistive switching via control of ferroelectric charged domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Advnced Materials 28, 6574–6580 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 16 Sharma, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Nonvolatile ferroelectric domain wall memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Science Advances 3, e1700512 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 17 Mundy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Functional electronic inversion layers at fer- roelectric domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Nature Materials 16, 622 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 18 Sluka, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Tagantsev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Electronic elements based on quasi two-dimensional electron/hole gas at charged domain walls in fer- roelectrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' US Patent 9171602 B2 (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 19 Hlinka, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Privratska, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Ondrejkovic, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Janovec, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Symme- try guide to ferroaxial transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Physical Review Letters 116, 177602 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 20 Beccard, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Nanoscale conductive sheets in ferroelectric BaTiO3: Large hall electron mobilities at head-to-head domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' ACS Applied Nano Materials 5, 8717–8722 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 21 Sturman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Podivilov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Ferroelectric domain reversal: The role of domain wall conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' JETP Letters 1–8 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 22 Fousek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Janovec, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The orientation of domain walls in twinned ferroelectric crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Journal of Applied Physics 40, 135–142 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 23 Wechsler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Lieberman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Read, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' On the theory of the formation of martensite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Transations AIME 197, 1503–1515 (1953).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 24 DiDomenico Jr, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Wemple, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Paraelectric-ferroelectric phase boundaries in semiconducting perovskite-type crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Physical Review 155, 539 (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 25 Parker, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Burfoot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Cubic-tetragonal phase transitions in high resistivity BaTiO3 single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Journal of Physics D: Applied Physics 2, 1168–1170 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 26 Fesenko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Gavrilyatchenko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Semenchev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Yufa- tova, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Regularities in domain structure formation in multi- axial ferroelectric crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Ferroelectrics 63, 289–298 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='1080/00150198508221411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 27 Fesenko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Gavrilyatchenko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Semenchev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Do- main structure of multiaxial ferroelectric crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Ferroelectrics 100, 195–207 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 28 Dec, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Paraelastic—ferroelastic interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Phase Transitions: A Multinational Journal 45, 35–58 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 29 Jin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Khachaturyan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Viehland, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Adaptive ferroelectric states in systems with low domain wall energy: Tetragonal microdomains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Journal of Applied Physics 94, 3629–3640 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 30 Roytburd, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Elastic domains and polydomain phases in solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Phase Transitions: A Multinational Journal 45, 1–34 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 31 Tagantsev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', Cross, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' & Fousek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Domains in ferroic crystals and thin films (Springer, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' 32 Neuber, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Architecture of nanoscale ferroelectric domains in GaMo4S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Journal of Physics: Condensed Matter 30, 445402 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Supplementary information to the manuscript ”Charged domain walls in BaTiO3 crystals emerging from superdomain boundaries” Petr S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Bednyakov1 and Jiˇr´ı Hlinka1 1FZU-Institute of Physics, The Czech Academy of Sciences, Na Slovance 2, 18221 Praha 8, Czech Republic∗ (Dated: January 10, 2023) PACS numbers: This supplement contains description of additional observa- tions supporting the presence of charged superdomain walls in (111)-poled BaTiO3 crystal plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' SURFACE NUCLEATION Obviously, the details of the nucleation mechanism may vary depending on various specific conditions under which the phase transition is induced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' For example, some of our originally less conductive samples were treated in the oxygen reducing atmosphere in order to increase the conductivity of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Since the treatment is inducing more defects in the near surface layer of the sample, it can be expected that surface nucleation of the ferroelectric phase is enhanced in such crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Moreover, structural surface defects and the resulting surface nuclei in such sample also substantial local strain gradients, as evidenced by the characteristic birefrin- gence butterfly patterns seen in the optical image of one such samples (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' By comparison of the apparent angles of the surface image of the already grown nuclei with that of the theoretical predictions summarized graphically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='4, we concluded that the observed nuclei involve type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' central superdomain boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In some cases, several nuclei are con- nected together (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S1b), again most likely in order to avoid the mechanically incompatible obtuse wedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' For ex- ample, the nuclei of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S1b reminds a fragment of hypothet- ical star-like pattern composed of all six superdomain types (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In other cases, the nuclei was showing an ar- row type geometry, possibly similar to the geometry sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S1e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' However, there was no tendency towards the con- certed organization of the nuclei into a more ordered domain structure type like in the case of bulk nucleation described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In fact, it turned out that this particular sample and the related poling process scenario was not very efficient in creation of multiple stable charged domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Instead, it either yielded the so-called zigzag domain walls that we shall briefly address later on, or the resulting charged domain walls were so unstable that they fully decomposed after the removal of the poling electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' EDGE NUCLEATION Another interesting phase transformation scenario was ob- served when the tetragonal phase nucleated at the edge of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S2 shows such transient wedge domain imaged at about 1 K below Tstart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' This sample was a (111)-oriented, FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S1: Formation of nuclei at the surface of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Surface nuclei are typically formed at local defects located at the surface of the samples with a sufficiently conductive surface layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Positions of surface nuclei show a marked butterfly birefringence pattern, show- ing strong local strain gradients present even before the borders of the nuclei are apparent (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In this case the nuclei often expands arms in more than a single direction (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In all cases each arm appears to be composed by two blades, suggesting the important role charged superdomain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Panels (d) and (e) suggest a plausible idealized structure of the surface nuclei considering the crystallographic orien- tation of type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' pentagonal nuclei of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='03409v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='mtrl-sci] 9 Jan 2023 [110] 50μm [112] [111]2 100 micron thick plate made of a crystal with a rather low conductivity of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='01-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='1 S/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The nucleation site was most likely directly related to the local electric con- tact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Again, by comparison with the theoretical shapes of two- superdomain precipitate images on the (111) sample surface, we can assign it as the wedge domain with the central su- perdomain of type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In this case, the fine primary domains could be seen to have the orientation indicated schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' They are not very clear on the micrograph because these walls are oblique to the main surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Assuming again that the primary domain walls are electrically neutral ones, we can conclude that here the involved superdomains formed are the strongly field-favored ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Obviously, the resolution of the optical image was not sufficient to determine the micro- scopic structure of the superdomain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Still, we can con- clude that the overall bound charge at this superdomain wall should be negative (in other words, it has a tail-to-tail ori- entation of the average polarization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' At the same time, the phase fronts are of head-to-none type, implying that the nega- tive mobile charge carriers are needed to screen these moving phase fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In principle, this poling strategy can be suitable when formation of just a single charged domain wall would be desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S2: Charged superdomain wall formed at the edge crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' (a) Optical micrograph showing the transient wedge domain, separated in two parts by a clear boundary, identified a superdomain wall as schematically shown in the panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The angle between the su- perdomain wall and phase front boundary θ ≈ 21◦ corresponds best to that of the type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' wedge geometry of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Dark spot on the top of the picture is a contact used for the electric field application, the top and the bottom part of the image corresponds to the opposite edges of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Labels ”T” and ”C” stands for the tetragonal and cubic phases, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=', crossed directors shows the orientation of the polarizer and analyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The primary domain structure pattern (c) at the sample surface can be deduced from the direction of the ob- served stripes, mechanical compatibility and the assumption that the majority primary domains are oriented favorably to the applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' ZIG-ZAG DOMAIN WALLS It has been repeatedly observed that frustrative poling of low conductivity BaTiO3 samples results in formation of the so called zigzag charged domain walls, in particular in case of tail-to-tail charged walls that require charge compensation FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S3: The three basic examples of the final charged domain walls of the same macroscopic crystallographic orientation as observed in various (111)-oriented BaTiO3 crystal plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' (a) Tail-to-tail zigzag charged domain wall composed of nearly neutral domain walls and its schematic drawing (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' an array of ideal planar charged domain walls (c) and their schematic drawing (d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' (e) charged superdomain walls separating a superdomain region from the two primary domains and its schematic drawing (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Crossed black/white arrows in right- upper corner of panels (a,c,e) correspond to the orientation of polar- izers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' by the less available positive mobile carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' On a macro- scopic scale, such a zig-zag domain wall (example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S3a) can be rationalized as a charged domain wall, differ- ing from the normal one by its considerable thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' How- ever, on the microscopic scale, it is clearly formed by a zigzag array of the almost neutral primary domain walls that are just slightly tilted from their ideal neutral wall orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Due to this small tilt, such primary walls carry a small bound charge, which, however, in projection to the macroscale domain wall orientation gives the usual bound charge density value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In the present case, the nominal orientation of the neutral primary walls involved in the zigzag are already inclined with respect to the sample surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Therefore, the zigzag image is partly distorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Nevertheless, by counting the number of the zigzag turns per given length it can be estimated that the angle be- tween adjacent zigzag lamellae is less than 10 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' This corresponds to about at least 10 times smaller bound charge density per surface area of the primary microscopic wall, or, equivalently, it means that the surface of the zigzag micro- scopic wall is at least 10 larger than that of the corresponding macroscopic wall (or of its ideal planar counterpart, such as the one in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S3c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Interestingly, the macroscopic width of the zigzag wall (the length of its tooth) is of the order of the typical final distance between charged domain walls them- b X C C [112] 500μm [110] [111]b a WW 200μm 200μm X [112] 500μm [110] [111]3 selves or even larger (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In this sense, the zig-zag itself can be considered as a sort of secondary domain, al- though here the volume ratio of the two primary domain states is probably 1:1 and we could not see any tight relation to the WLR superdomains discussed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' We can speculate that the effective width of the zig-zag wall can be related to the volume needed to collect the required amount of charge, while the width of the zig-zag tooth could be perhaps related to the characteristic distance from which the compensation charge can be collected by the usual diffusion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' SUPERDOMAIN WALL BETWEEN PRIMARY AND SECONDARY DOMAINS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Alternatively to zig-zag walls, in some cases we have ob- served secondary domain walls connecting primary domains and the twinned secondary domains, such as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S3e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Here the primary fine stripe domains in the twinned central area are most likely the neutral ones, in other words, those that are in- clined with respect to the viewing direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The sharp borders between the dark and light area in the photograph Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S3e thus correspond to the charged macroscopic tail-to-tail and head-to-head charged domain walls, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Because of the twinning in the central area, the overall bound charge den- sity is reduced in comparison with the primary charged walls of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Schematic interpretation of this structure is pro- posed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S3f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' Note that all large-area planar interfaces in this domain structure are mechanically compatible ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' The experiment does not reveal the microstructure the superdo- main wall, simplest assumption is that it formed by an aligned system of wedge domains ended on a common plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In ei- ther case, on the macroscopic scale, the superdomain wall of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S3f requires smaller overall amount of screening charge then the primary domain of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' In principle, we may speculate that the secondary (twinned) domain in the center of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' S3e,f is probably residuum of a WLR superdomain state, fully favored by the bias electric field, which was surrounded by two superdomains only par- tially favored by the bias electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' As we have argued already, it can be expected that in the course of the domain structure coarsening, the bias field will promote the transfor- mation of the partially favored superdomains into the primary domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' On the other hand, there is no such obvious driv- ing force present in case of the fully favored superdomains, and therefore the process of elimination of the fine stripe pri- mary domains is expected to be much slower, hindered or completely frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content=' ∗ Electronic address: bednyakov@fzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='cz, hlinka@fzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} +page_content='cz' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfwQWa/content/2301.03409v1.pdf'} diff --git a/ZNE2T4oBgHgl3EQfZQdS/content/2301.03862v1.pdf b/ZNE2T4oBgHgl3EQfZQdS/content/2301.03862v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..11cdf2bc76119ee8c99cf969cd8419a030e04fdc --- /dev/null +++ b/ZNE2T4oBgHgl3EQfZQdS/content/2301.03862v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1143b5e48e575fee247011aca60a8b8361eb591a37696ec0ab50379fa1041049 +size 440234 diff --git a/ZNE2T4oBgHgl3EQfZQdS/vector_store/index.faiss b/ZNE2T4oBgHgl3EQfZQdS/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..c9e6458b63cbde9dd640f47d6445bf796b3b1729 --- /dev/null +++ b/ZNE2T4oBgHgl3EQfZQdS/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ef42613f651bc4434cb0497eb6caefb6390c964ed95c0e762e4c125eaf92a069 +size 3211309 diff --git a/_9AyT4oBgHgl3EQf3vnw/vector_store/index.pkl b/_9AyT4oBgHgl3EQf3vnw/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..45d489a98b1f354c4582a8a2eafd391e3ae02e4e --- /dev/null +++ b/_9AyT4oBgHgl3EQf3vnw/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4cc2202282e4e8c402821dac62196b5515707259c8ee3483b467b914b086a27f +size 185102 diff --git a/_dE2T4oBgHgl3EQfmgfK/content/2301.04000v1.pdf b/_dE2T4oBgHgl3EQfmgfK/content/2301.04000v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5d9621013ca04b4fa16ab0a44e19b49bd64725bd --- /dev/null +++ b/_dE2T4oBgHgl3EQfmgfK/content/2301.04000v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8c3f03f9b24a792ffa9c03c7434846522e9a5604d3498b79b42b6796f4cf2b97 +size 2211591 diff --git a/_dE2T4oBgHgl3EQfmgfK/vector_store/index.pkl b/_dE2T4oBgHgl3EQfmgfK/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..995461d9c7293ebb4bbf9e3e02b0ebd1a5d5c91f --- /dev/null +++ b/_dE2T4oBgHgl3EQfmgfK/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:57f7884c701ad91da2061c7097626781739ad6d774650d193cfa58e46b62b7e9 +size 183883 diff --git a/_dFKT4oBgHgl3EQfUy2I/vector_store/index.faiss b/_dFKT4oBgHgl3EQfUy2I/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..759c9b460f580ca946df37362f93100c4a341519 --- /dev/null +++ b/_dFKT4oBgHgl3EQfUy2I/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9a0e95ad85563d6e95d8f6d1bedeaef4d5e703a1e24dc8ac718d65b49677bb1a +size 3538989 diff --git a/_dFKT4oBgHgl3EQfUy2I/vector_store/index.pkl b/_dFKT4oBgHgl3EQfUy2I/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..74f94c03f37ffdad100ab8b721f9a0330fbc5d19 --- /dev/null +++ b/_dFKT4oBgHgl3EQfUy2I/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9d107778a39564c41f3927d90472ef326b953cb12992dd4850d14d3252040429 +size 125540 diff --git a/atFAT4oBgHgl3EQf4x5l/content/tmp_files/2301.08728v1.pdf.txt b/atFAT4oBgHgl3EQf4x5l/content/tmp_files/2301.08728v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..31b4a567f75740d52d3282ca2e68b17c626e2ca5 --- /dev/null +++ b/atFAT4oBgHgl3EQf4x5l/content/tmp_files/2301.08728v1.pdf.txt @@ -0,0 +1,2136 @@ +arXiv:2301.08728v1 [math-ph] 20 Jan 2023 +New Mexico Tech (January 20, 2023) +Spectral Asymptotics +of Elliptic Operators +on Manifolds +Ivan G. Avramidi +Department of Mathematics +New Mexico Institute of Mining and Technology +Socorro, NM 87801, USA +E-mail: ivan.avramidi@nmt.edu +Abstract +The study of spectral properties of natural geometric elliptic partial dif- +ferential operators acting on smooth sections of vector bundles over Rie- +mannian manifolds is a central theme in global analysis, differential ge- +ometry and mathematical physics. Instead of studying the spectrum of a +differential operator L directly one usually studies its spectral functions, +that is, spectral traces of some functions of the operator, such as the spec- +tral zeta function ζ(s) = TrL−s and the heat trace Θ(t) = Tr exp(−tL). The +kernel U(t; x, x′) of the heat semigroup exp(−tL), called the heat kernel, +plays a major role in quantum field theory and quantum gravity, index the- +orems, non-commutative geometry, integrable systems and financial math- +ematics. We review some recent progress in the study of spectral asymp- +totics. +We study more general spectral functions, such as Tr f(tL), that +we call quantum heat traces. Also, we define new invariants of differen- +tial operators that depend not only on the their eigenvalues but also on the +eigenfunctions, and, therefore, contain much more information about the ge- +ometry of the manifold. Furthermore, we study some new invariants, such +as Tr exp(−tL+)exp(−sL−), that contain relative spectral information of two +differential operators. Finally we show how the convolution of the semi- +groups of two different operators can be computed by using purely algebraic +methods. + +1 +1 +Introduction +The study of spectral properties of natural geometric partial differential opera- +tors is a central theme in global analysis, differential geometry and mathematical +physics. In particular, the basic question of spectral geometry is: “To what extent +does the spectrum of an elliptic partial differential operator determine the geom- +etry of the underlying manifold?”, or as M. Kac put it in his famous paper [24]: +“Can one hear the shape of a drum?” In general, the answer to Kac’s question is +“no” [28, 26]. Instead of studying the spectrum of a differential operator directly +one usually studies its spectral functions, that is, spectral traces of some functions +of the operator, such as the spectral zeta function and the heat trace. The heat +trace is the trace of the heat kernel, which is the fundamental solution of the heat +equation for an elliptic partial differential operator with a positive leading sym- +bol. The existence of non-isometric isospectral manifolds demonstrates that the +spectrum alone does not determine the geometry. That is why, it makes sense to +study more general invariants of partial differential operators, maybe even such +invariants that are not spectral invariants, that is, invariants that depend not only +on the eigenvalues but also on the eigenfunctions, and, therefore, contain much +more information about the geometry of the manifold. +Another motivation to study the heat kernel comes from quantum field theory +and statistical physics. The main objects of interest are the effective action (or +the partition function) and the Green functions (or the correlation functions). All +these objects are expressed in terms of the functional determinants of some self- +adjoint elliptic partial differential operators, their heat traces and their resolvents. +It turns out that all of them can be expressed in terms of the heat kernels of those +operators. +In financial mathematics one uses stochastic differential equations to model +the random behavior of some financial assets. Then the behavior of the corre- +sponding derivative securities is determined by deterministic parabolic partial dif- +ferential equation (such as the diffusion or heat equation) with an elliptic partial +differential operator of second order. The conditional probability density is then +nothing else but the heat kernel. +Many problems in mathematics and physics naturally lead to the presence of +boundaries and to the corresponding boundary value problems for partial differen- +tial operators. The type of the boundary conditions is not limited to the classical +Dirichlet and Neumann ones. In some applications, such as quantum gravity and +applied mathematics, there appear mixed boundary conditions on vector bundles +(that mix the Dirichlet and the Neumann one), oblique boundary conditions (that +hkrev.tex; January 23, 2023; 1:22; p. 1 + +2 +involve the tangential derivatives to the boundary), and even discontinuous bound- +ary conditions, so called Zaremba boundary conditions, (that jump from Dirichlet +to Neumann across a co-dimension 2 submanifold in the boundary). +Most natural elliptic partial differential operators are second order operators +with scalar leading symbols, so called Laplace type operators, or first order op- +erators whose square is a Laplace type operator, so called Dirac type operators. +However, in some applications, such as in gauge field theories and quantum grav- +ity, there appear second-order elliptic partial differential operators with non-scalar +leading symbols, so called non-Laplace type operators. Another motivation for +studying such operators is non-commutative geometry where the metric tensor, +that is, the inner product in the tangent bundle, is endomorphism valued in some +vector bundle. +In the generic situation when it is impossible to compute the heat kernel ex- +actly, it becomes very important to study various asymptotic regimes. Of special +interest is the study of the short-time asymptotic expansion of the heat kernel. +This expansion is closely related to the semi-classical expansion in quantum the- +ory and the high-temperature expansion in statistical physics. The coefficients of +this expansion, called the heat invariants, are spectral invariants associated with +the asymptotic properties of the spectrum. There are also non-trivial links be- +tween spectral invariants and non-linear completely integrable systems, such as +the Korteweg-de Vries hierarchy. In many interesting cases such systems are, in +fact, infinite-dimensional Hamiltonian systems, and the infinite set of integrals +of motion of these systems is related to the spectral invariants of a linear elliptic +partial differential operator. +2 +Heat Kernel +2.1 +Elliptic Operators +Let (M,g) be a smooth compact Riemannian manifold of dimension n equipped +with a positive definite Riemannian metric g. We denote the local coordinates by +xµ, with Greek indices running over 1,...,n. The Riemannian volume element is +defined as usual by dvol = dx g1/2, where g = detgµν and dx = dx1 ∧ ··· ∧ dxn is +the standard Lebesgue measure. +Let V be a smooth vector bundle over M with the typical fiber V of dimension +N. Let C∞(V) be the space of smooth sections of the bundle V. The completion +of the space C∞(V) defines the Hilbert space L2(V) of square integrable sections. +hkrev.tex; January 23, 2023; 1:22; p. 2 + +3 +Let +∇V : C∞(V) → C∞(T ∗M ⊗V) +(2.1) +be a compatible connection on the vector bundle V. By using the Levi-Civita +connection of the metric g the connection is given its unique natural extension to +bundles in the tensor algebra over V, its dual V∗ and the tangent and cotangent +bundles, T M and T ∗M; the resulting connection will usually be denoted just by ∇. +The fiber inner product ⟨ , ⟩ defines a natural L2 inner product, ( , ), on the bundle +V and the L2-trace, Tr, using the invariant Riemannian measure on the manifold +M. +Let +∇∗ : C∞(T M ⊗V) → C∞(V) +(2.2) +be the formal adjoint of the connection. Let a, B and Q be smooth maps +a : T ∗M ⊗V → T M ⊗V, +(2.3) +B : T ∗M ⊗V → V, +(2.4) +Q : V → V. +(2.5) +defined by endomorphism-valued tensors satisfying +aµν = aνµ, +(2.6) +(aµν)∗ = aµν, +(Bµ)∗ = −Bµ, +Q∗ = Q +(2.7) +Every formally self-adjoint second-order partial differential operator L :C∞(V) → +C∞(V) has the form +L += +−∇µaµν∇ν + Bµ∇µ +∇µBµ + Q +(2.8) +Natural non-Laplace type operators can be constructed as follows. Let +Γ : T ∗M ⊗V → V +(2.9) +be a smooth map; this defines the Dirac type operator +D = Γ∇ : C∞(V) → C∞(V) +(2.10) +and the operator +L = D∗D : C∞(V) → C∞(V); +(2.11) +hkrev.tex; January 23, 2023; 1:22; p. 3 + +4 +in this case +aµν = 1 +2 +�Γµ ∗Γν +Γν ∗Γµ�. +(2.12) +More generally, let Vj, j = 1,..., s, be some vector bundles and +Pj : T ∗M ⊗V → Vj +(2.13) +be some smooth maps. This defines the gradients +G j = Pj∇ : C∞(V) → C∞(Vj), +(2.14) +and the operator +L = +s +� +j=1 +αjG∗ +jG j : C∞(V) → C∞(V), +(2.15) +where αj are some real constants. In this case +aµν = +s +� +j=1 +αj +1 +2 +� +Pµ ∗ +j Pν +j + Pν ∗ +j Pµ +j +� +. +(2.16) +The leading symbol of the operator L is given by the endomorphism +H(x,ξ) = aµν(x)ξµξν, +(2.17) +with x ∈ M and ξ ∈ T ∗ +x M. Since this matrix is self-adjoint all its eigenvalues must +be real. The operator L is elliptic if all eigenvalues are positive, in other words, the +matrix H(x,ξ) is positive. The operator L is also self-adjoint; strictly speaking, it +is essentially self-adjoint, that is, it has a unique self-adjoint extension (from now +on, we will just say that the operator L is self-adjoint). +If all eigenvalues of the leading symbol of the operator L are equal, that is, the +operator L has a scalar positive definite leading symbol defined by the Riemannian +metric +H(x,ξ) = |ξ|2I, +(2.18) +with |ξ|2 = gµν(x)ξµξν, then the operator is called of Laplace type. In this case the +connection can be redefined to absorb the vector Bµ, so that every Laplace type +operator has the form +L = −∆+ Q, +(2.19) +hkrev.tex; January 23, 2023; 1:22; p. 4 + +5 +where +∆ += +gµν∇µ∇ν += +g−1/2(∂µ +Aµ)g1/2gµν(∂ν +Aν) +(2.20) +is the Laplacian. Here and below ∂µ = ∂/∂xµ denotes the partial derivative. A +Laplace type operator is defined in terms of three pieces of local information: the +Riemannian metric g, the connection one-form A and the potential Q. +2.2 +Heat Kernel +Let L be a self-adjoint elliptic positive partial differential operator of second order +on the Hilbert space L2(V). For manifolds with boundary the domain of the op- +erator L has to be supplemented with some suitable elliptic boundary conditions. +For compact manifolds the spectrum of the operator L is an increasing real se- +quence of eigenvalues {λk}k∈Z+ with the corresponding orthonormal eigensections +{ϕk}k∈Z+ (counted with multiplicities) defined by +(L−λk)ϕk = 0, +(ϕi,ϕ j)L2 = δi j. +(2.21) +For noncompact manifolds the spectrum of the operator L is continuous, it goes +from a positive real constant c to ∞. In general, it is impossible to compute the +spectrum exactly. That is why, it becomes of special importance the study of the +asymptotics of the eigenvalues (and the eigensections) as k → ∞. Rather than +doing this directly it is more convenient to study the asymptotics of some spectral +functions and special traces such as the heat trace and the zeta function. +Let V ⊠V∗ be the external tensor product of the bundles V and V∗ over the +product manifold M × M. The heat kernel U(t; x, x′) of the operator L is a one- +parameter family of smooth sections of V ⊠V∗ defined by requiring it to satisfy +the heat equation +(∂t + L)U(t; x, x′) = 0 +(2.22) +for t > 0 with the initial condition +U(0+; x, x′) = δ(x, x′), +(2.23) +that is, +U(t; x, x′) = exp(−tL)δ(x, x′), +(2.24) +where δ(x, x′) is the covariant Dirac delta-distribution. +hkrev.tex; January 23, 2023; 1:22; p. 5 + +6 +The heat kernel is regular at the diagonal with a well defined diagonal value +Udiag(t; x) = U(t; x, x), +(2.25) +which is a section of the endomorphism bundle End(V). The heat kernel diag- +onal is well defined on both compact and noncompact manifolds. For compact +manifolds the heat kernel can be computed in terms of the spectral data +U(t; x, x′) += +∞ +� +k=1 +e−tλkϕk(x)ϕ∗ +k(x′). +(2.26) +and has a well defined heat trace Θ(t) = Tr exp(−tL), +Θ(t) += +∞ +� +k=1 +e−tλk += +� +M +dvol trUdiag(t); +(2.27) +here and below tr denotes the fiber trace. This also enables one to define the +spectral zeta function by the Mellin-Laplace transform of the heat trace +ζ(s,λ) = +∞ +� +k=1 +1 +(λk −λ)s = +1 +Γ(s) +∞ +� +0 +ts−1etλΘ(t), +(2.28) +where λ is a complex parameter with a sufficiently large negative real part, and +the regularized spectral determinant +Det(L−λ) = exp +� +−∂sζ(s,λ) +���s=0 +� +. +(2.29) +The resolvent G(λ; x, x′) of the operator L is a section of V ⊠ V∗ depending +on a complex parameter λ defined by +(L−λ)G(λ; x, x′) = δ(x, x′). +(2.30) +The resolvent is related to the heat kernel by the Laplace transform +G(λ; x, x′) += +∞ +� +0 +dt etλU(t; x, x′). +(2.31) +hkrev.tex; January 23, 2023; 1:22; p. 6 + +7 +For compact manifolds there is the spectral representation of the resolvent +G(λ; x, x′) += +∞ +� +k=1 +1 +λk −λϕk(x)ϕ∗ +k(x′). +(2.32) +Off diagonal, that is, for x � x′, the resolvent is an analytic function of λ for Reλ < +c with sufficiently large negative real constant c. It has a diagonal singularity as +x → x′. +On a general manifold the heat kernel cannot be computed exactly, and that is +why the short-time asymptotic expansion as t → 0 of the heat kernel and its heat +trace is studied. For Laplace type operators on a it has a rather simple form. Let +x′ be a fixed point in the interior of the manifold; we consider a sufficiently small +geodesic ball centered at x′, so that each point x of the ball can be connected by a +unique geodesic with the point x′. This can be always done if the size of the ball +is smaller than the injectivity radius of the manifold rinj(M). +Let σ = σ(x, x′) be the Ruse-Synge function defined by +σ(x, x′) = 1 +2r2(x, x′), +(2.33) +where r(x, x′) is the geodesic distance between the points x and x′, and ∆(x, x′) be +the Van Vleck-Morette determinant +∆(x, x′) = g−1/2(x)det +� +−∂2σ(x, x′) +∂xµ∂xν′ +� +g−1/2(x′). +(2.34) +Near the diagonal of M × M these two-point functions are smooth single-valued +functions of the coordinates of the points x and x′. +In the interior of the manifold (on a finite distance from the boundary, if +present) there is a local short time asymptotic expansion of the heat kernel of +a Laplace type operator as t → 0+ +U(t; x, x′) ∼ (4πt)−n/2∆1/2(x, x′)exp +� +−σ(x, x′) +2t +� ∞ +� +k=0 +tka2k(x, x′), +(2.35) +where ak(x, x′) are the so-called local off-diagonal heat kernel coefficients. Notice +that there are no odd-order coefficients here, that is, a2k+1(x, x′) = 0. The heat +kernel coefficients can be computed in form of a covariant Taylor series from the +recurence relations obtained by substituting this ansatz in the heat equation [1] +with the initial condition +a0(x, x′) = P(x, x′), +(2.36) +hkrev.tex; January 23, 2023; 1:22; p. 7 + +8 +where P(x, x′) is the operator of parallel transport of sections along the geodesic +from the point x′ to the point x. +This expansion immediately gives the asymptotic expansion as t → 0+ of the +heat kernel diagonal +Udiag(t; x) ∼ (4πt)−n/2 +∞ +� +k=0 +tkadiag +2k (x, x), +(2.37) +where adiag +k +(x) = ak(x, x) are the diagonal local heat kernel coefficients. +For manifolds without boundary this also gives the asymptotic expansion of +the heat trace +Θ(t) ∼ (4πt)−n/2 +∞ +� +k=0 +tkA2k, +(2.38) +where +A2k = +� +M +dvoltradiag +2k +(2.39) +are the global heat kernel coefficients. The heat trace as well as the global heat +kernel coefficients are obviously spectral invariants of the operator L. These co- +efficients were computed up to adiag +8 +(see [1]). The low-order asymptotics of the +heat trace of a Laplace type operator has the form +Θ(t) += +(4πt)−n/2 +� +Nvol(M)+t +� +M +dvol +�N +6 R−trQ +� ++O +� +t2�� +, +(2.40) +where R is the scalar curvature. +3 +Boundary Value Problems +3.1 +Mixed Boundary Value Problem +Now, let M be a compact Riemannian manifold with smooth boundary ∂M. Let N +be the inward-pointing unit normal vector field to the boundary ∂M. To make the +operator L elliptic we need to impose some boundary conditions of the boundary +data. The classical boundary conditions are +ϕ +���∂M += +0, +(Dirichlet), +(3.1) +∇Nϕ +���∂M += +0, +(Neumann). +(3.2) +hkrev.tex; January 23, 2023; 1:22; p. 8 + +9 +The Robin boundary condition is a slight generalization of the Neumann one +(∇N +S )ϕ +����∂M = 0, +(Robin). +(3.3) +where S is a smooth self-adjoint endomorphism. One can go further and mix these +boundary conditions as follows +(I −Π)ϕ +���∂M += +0, +(3.4) +Π(∇N +S )Πϕ +���∂M += +0, +(3.5) +where Π is a self-adjoint projection. A Laplace type operator L equipped with +mixed boundary conditions is essentially self-adjoint and elliptic. +The heat kernel asymptotics as t → 0 does not depend on the boundary con- +ditions in the interior of the manifold and has the same form as the heat kernel +for manifolds without boundary. Near the boundary there is a narrow strip of the +width of order t1/2 where one should add an additional term, whose role is to +satisfy the boundary conditions. In a narrow strip near the boundary this compen- +sating term in the heat kernel diagonal behaves as a distribution near the boundary +as t → 0. It is precisely this feature that leads to the presence of boundary terms in +the global heat kernel coefficients. The heat trace asymptotic expansion as t → 0+ +has the half-integer powers of t, +Θ(t) ∼ (4πt)−n/2 +∞ +� +k=0 +tk/2Ak; +(3.6) +where +Ak = +� +M +dvol tr adiag +k ++ +� +∂M +dvolbk , +(3.7) +where the boundary heat kernel coefficients bk are local invariants constructed +polynomially from the jets of the symbols of both the operator L and the boundary +operator. +The low-order asymptotics have the form +Θ(t) += +(4πt)−n/2 +� +Nvol(M)+t1/2 +� +∂M +dvol +√π +2 (2trΠ− N) +(3.8) ++t + +� +M +dvol +�N +6 R−trQ +� ++ +� +∂M +dvol +�N +3 K +2trΠS +�+O +� +t3/2�� +, +where K is the trace of the extrinsic curvature of the boundary. +hkrev.tex; January 23, 2023; 1:22; p. 9 + +10 +3.2 +Grubb-Gilkey-Smith Boundary Value Problem +Let ˆ∇i be the tangential covariant derivative, Γi be a vector-valued anti-self-adjoint +endomorphism and Λ be a first-order formally self-adjoint tangential differential +operator defined by +Λ = 1 +2 +� +Γi ˆ∇i + ˆ∇iΓi� ++S . +(3.9) +The Grubb-Gilkey-Smith boundary conditions then read +(I −Π)ϕ +����∂M += +0, +(3.10) +Π(∇N +Λ)Πϕ +����∂M += +0. +(3.11) +A Laplace type operator L equipped with such boundary conditions is essen- +tially self-adjoint but not necessarily elliptic. To be elliptic the operator Λ has to +satisfy the strong ellipticity condition. Let T be a matrix defined by the leading +symbol of the operator Λ, +T(ξ) = Γj ˆξ j, +(3.12) +where ˆξ ∈ T ∗∂M is a covector on the boundary. Since the matrices Γi are anti- +self-adjoint, the matrix T 2(ξ) is self-adjoint and negative, T 2(ξ) < 0, for any ˆξ � 0. +Then the oblique boundary value problem is elliptic if the matrix +I|ˆξ|2 +T 2(ξ) > 0 +(3.13) +is positive for any ˆξ � 0. Here |ˆξ|2 = ˆgi jˆξiˆξ j is defined with the metric ˆgi j on the +boundary. +When the boundary value problem is elliptic the heat trace asymptotic ex- +pansion has the canonical form (3.6). Contrary to the classical boundary value +problems (Dirichlet, Neumann, mixed), because of the non-commutativity of the +matrices Γi, the explicit form of the coefficients bk is unknown, in general; the +low-order asymptotic expansion has the form [19, 18] +Θ(t) = (4πt)−n/2 + +Nvol(M)+t1/2 +� +∂M +dvol +√π +2 +� +2trΠ−3N +2γ +� ++O(t) + +, +(3.14) +where +γ = +� +Rn−1 +dˆξ +π(n−1)/2 tr exp +� +−|ˆξ|2 −T 2(ξ) +� +(3.15) +hkrev.tex; January 23, 2023; 1:22; p. 10 + +11 +This can be computed explicitly in special cases. If the matrices Γi commute then +γ = tr +� +I +Γ2�−1/2, +(3.16) +where Γ2 = ˆgi jΓiΓj. One can also compute the coefficient γ explicitly in the non- +commutative case when the matrices Γi form a Clifford algebra +ΓiΓj +ΓjΓi = −2κΠˆgi j +(3.17) +with a real parameter κ. This problem is elliptic if κ < 1. we obtain +γ = (1−κ)−(n−1)/2trΠ. +(3.18) +3.3 +Zaremba Boundary Value Problem +Let the boundary ∂M be decomposed as a disjoint union ∂M = Σ1∪Σ2∪Σ0, where +Σ1 and Σ2 are smooth compact submanifolds of ∂M of dimension (n−1), with the +boundary Σ0 which is a smooth compact manifold without boundary of dimension +(n − 2). Zaremba boundary value problem is defined by the following boundary +conditions +ϕ +���Σ1 += +0, +(3.19) +(∇N +S )ϕ +���Σ2 += +0, +(3.20) +where S is a smooth self-adjoint endomorphism. +Since the boundary operator is discontinuous, this problem is a singular bound- +ary value problem. It is well known that in this case the heat trace asymptotic +expansion as t → 0+ contains, in general, logarithmic terms +Θ(t) ∼ (4πt)−n/2 +∞ +� +k=0 +tk/2Ak +logt +∞ +� +k=0 +tk/2Hk . +(3.21) +However, Seeley [27] has shown that for Zaremba problem the logarithmic terms +do not appear, that is, all +Hk = 0. +(3.22) +Therefore, the heat trace asymptotics has the canonical form (3.6). However, the +global heat kernel coefficients get a contribution from the codimension 2 subman- +ifold Σ0 +Ak = +� +M +dvoltr adiag +k ++ +� +Σ1 +dvolb(1) +k + +� +Σ2 +dvolb(2) +k + +� +Σ0 +dvolck . +(3.23) +hkrev.tex; January 23, 2023; 1:22; p. 11 + +12 +It turns out [7] that the boundary conditions on the open sets Σ1 and Σ2 are +not enough to fix the problem, and an additional boundary condition along the +singular set Σ0 is needed. This additional boundary condition can be considered +formally as an extension of Dirichlet conditions from Σ1 to Σ0, (regular boundary +condition) +� √ρϕ�����Σ0 = 0, +(3.24) +where ρ is the normal geodesic distance to the singular set Σ0, or an extension of +Neumann (or Robin) conditions from Σ2 to Σ0, +(∂ρ −h)� √ρϕ�����Σ0 = 0, +(3.25) +where h is a real parameter. However, strictly speaking the boundary condition on +Σ0 does not follow from the boundary conditions on Σ1 and Σ2 and can be chosen +rather arbitrarily. +The coefficients of the asymptotic expansion can be computed by constructing +asymptotic solutions: +1. in the interior, +2. in the thin shell near the Σ1 and Σ2, and +3. in a thin strip close to the singular submanifold Σ0. +The trace of the heat kernel of the Zaremba boundary value problem has the fol- +lowing asymptotic expansion as t → 0+ +Θ(t) += +(4πt)−n/2 +� +Nvol(M)+t1/2 +√π +2 N [vol(Σ2)−vol(Σ1)] ++t +�� +M +dvol +�N +6 R−trQ +� ++ N +3 +� +Σ1 +dvolK + +� +Σ2 +dvol +�N +3 K +2trS +�� ++α π +4Nvol(Σ0)+O +� +t3/2�� +, +(3.26) +where α = −1 for the boundary condition (3.24) and α = 7 for the boundary con- +dition (3.25). +hkrev.tex; January 23, 2023; 1:22; p. 12 + +13 +4 +Non-Laplace Type Operators +We consider a non-Laplace type operator of the form +L = −∇µaµν∇ν + Q, +(4.1) +where aµν is an endomorphism-valued symmetric tensor. If we assume, in addi- +tion, that the leading symbol of the operator L is positive then the operator L is +self-adjoint and elliptic and, therefore, has the same canonical heat trace expan- +sion of the form (3.6) +Θ(t) ∼ (4πt)−n/2 +∞ +� +k=0 +tk/2Ak. +(4.2) +For manifolds without boundary one can get the asymptotic expansion of the +heat kernel as follows. Let ξ ∈ T ∗M be a covector and ⟨ξ, x⟩ = ξµxµ. It is easy to +see that +exp(−i⟨ξ, x⟩)Lexp(i⟨ξ, x⟩) = H + K + L, +(4.3) +where +H(x,ξ) = aµν(x)ξµξν +(4.4) +is the leading symbol of the operator L and K is the first order operator defined by +K = −iξµ +�aµν∇ν +∇νaµν�. +(4.5) +Then the asymptotics of the heat trace as t → 0 are given by +Θ(t) ∼ (4πt)−n/2 +� +M +dx +� +Rn +dξ +πn/2tr exp +� +−H − +√ +tK −tL +� +· I , +(4.6) +By using the Volterra series for the heat semigroup we get +exp +� +−H − +√ +tK −tL +� += e−H −t1/2 +1 +� +0 +dτ1e−(1−τ1)HKe−τ1H ++t +� 1 +� +0 +dτ2 +τ2 +� +0 +dτ1e−(1−τ2)HKe−(τ2−τ1)HKe−τ1H − +− +1 +� +0 +dτ1e−(1−τ1)HLe−τ1H +� ++O(t2). +(4.7) +hkrev.tex; January 23, 2023; 1:22; p. 13 + +14 +Since K is linear in ξ the term proportional to t1/2 vanishes after integration over +ξ and we obtain the first two coefficients of the asymptotic expansion of the heat +kernel diagonal in the form +A0 += +� +M +dx +� +Rn +dξ +πn/2tr exp�−H(x,ξ)�, +(4.8) +A1 += +0, +(4.9) +A2 += +� +M +dx +� +Rn +dξ +πn/2 tr +� 1 +� +0 +dτ2 +τ2 +� +0 +dτ1e−(1−τ2)HKe−(τ2−τ1)HKe−τ1H − +− +1 +� +0 +dτ1e−(1−τ1)HLe−τ1H +� +. +(4.10) +Since there is no Riemannian metric, the spectral invariants of a non-Laplace type +operator are not expressed in terms of the invariants of the curvature. It is very +important to develop the corresponding differential-geometric language based on +the non-scalar leading symbol of a non-Laplace type operator. +Boundary value problems for non-Laplace type operators are more compli- +cated. For the Dirichlet boundary value problem the asymptotics are as follows. +Let x = (r, ˆx) be local coordinates near the boundary where r is the normal geodesic +distance to the boundary and ˆxi are local coordinates on the boundary. We decom- +pose a covector ξ ∈ T ∗M as ξ = (ω, ˆξ) where ω is a real number and ˆξ ∈ T ∗∂M is a +covector on the boundary. Let Φ(λ; ˆx, ˆξ) be a function defined by +Φ(λ; ˆx, ˆξ) = +� +R +dω +2π +� +H(0, ˆx,ω, ˆξ)−λI +�−1 +(4.11) +and Ψ(x,ξ) be a function defined by +Ψ(x,ξ) = +c+i∞ +� +c−i∞ +dλ +2πie−λ ∂ +∂λ logdetΦ(λ; ˆx, ˆξ), +(4.12) +where c is a sufficiently large positive real constant. Then the heat trace boundary +coefficient A1 for the Dirichlet boundary conditions has the form +A1 = − √π +� +∂M +d ˆx +� +Rn−1 +dˆξ +π(n−1)/2Ψ(ˆx, ˆξ). +(4.13) +hkrev.tex; January 23, 2023; 1:22; p. 14 + +15 +5 +Non-perturbative Spectral Asymptotics +Let M be a compact Riemannian manifold without boundary and V be a complex +vector bundle over M realizing a representation of the group G ×U(1). Let ϕ be +a section of the bundle V and ∇ be the total connection on the bundle S (includ- +ing the G-connection as well as the U(1)-connection). Then the commutator of +covariant derivatives defines the curvatures +[∇µ,∇ν]ϕ = (Rµν +iFµν)ϕ, +(5.1) +where Rµν is the curvature of the G-connection and Fµν is the curvature of the +U(1)-connection. +We assume that the U(1)-connection is parallel, that is, +∇µFαβ = 0. +(5.2) +This equation puts severe algebraic restriction on the curvature tensor +RλαµνFλβ −RλβµνFλα = 0 +(5.3) +and, therefore, gives powerful restriction on the holonomy group of the manifold. +For example, F could be a simplectic form of a K¨ahler manifold. +Let U(t; x, x′) be the heat kernel of the Laplacian ∆ = gµν∇µ∇ν. We rescale the +curvature by +F �→ 1 +ε2 F, +(5.4) +where ε > 0 is a small positive parameter; let ∆ε be the rescaled Laplacian and +Uε(εt; x, x′) be the rescaled heat kernel. Then as ε → 0 there is the new asymptotic +expansion of the off-diagonal heat kernel [20] +Uε(ε2t; x, x′) ∼ U0(t; x, x′) +∞ +� +k=0 +εk−ntk/2bk(t; x, x′), +(5.5) +where +U0(t; x, x′) = (4πt)−n/2∆1/2(x, x′)det +� +tiF +sinh(tiF) +�1/2 +exp +� +− 1 +4t ⟨u,tiF coth(tiF)u⟩ +� +, +(5.6) +F is the matrix F = (Fµν) and uµ are normal coordinates with origin at x′. Here +the coefficients bk(t; x, x′) are analytic functions of t that depend on F only in the +hkrev.tex; January 23, 2023; 1:22; p. 15 + +16 +dimensionless combination tF. Of course, for t = 0 they are equal to the standard +heat kernel coefficients, that is, +bk(0; x, x) = ak(x, x′), +(5.7) +and, therefore, the odd-order coefficients at t = 0 vanish, b2k+1(0; x, x′) = 0. More- +over, the odd-order coefficients vanish also for any t on the diagonal x = x′, that +is, +bdiag +2k+1(t) = 0 +(5.8) +Then the asymptotic expansion of the heat kernel diagonal and the heat trace are +Tr exp(ε2t∆ε) ∼ (4πt)−n/2 +∞ +� +k=0 +εk−ntk/2Bk(t), +(5.9) +where +Bk(t) = +� +M +dvol det +� +tiF +sinh(tiF) +�1/2 +tr bdiag +k +(t). +(5.10) +The coefficients Bk are new spectral invariants of the Laplacian. They are dif- +ferential polynomials in the Riemann curvature tensor (and the curvature of the +G-connection) and its derivatives with universal coefficients depending in a non- +polynomial but analytic way on the curvature F, more precisely, on tF. +We explicitly computed the coefficients bk (both off diagonal and the diagonal +values) for k = 0,1,2,3,4. These functions generate all terms quadratic and linear +in the Riemann curvature and of arbitrary order in F in the standard heat kernel +coefficients adiag +k +. In that sense, we effectively sum up the usual short time heat +kernel asymptotic expansion to all orders of the curvature F. The first two non- +zero coefficients have the form +bdiag +0 +(t) += +I, +(5.11) +bdiag +2 +(t) += +Jαβµν(t)RαβµνI + 1 +2Hµν(t)Rµν, +(5.12) +where +H(t) = coth(tiF)− 1 +itF +(5.13) +and Jαβµν(t) is a more complicated tensor constructed from the matrix F that is +analytic in t +Jαβµν(t) = 1 +6δα[µδβν] +O(t2) +(5.14) +hkrev.tex; January 23, 2023; 1:22; p. 16 + +17 +6 +Heat Determinant +The existence of non-isometric isospectral manifolds demonstrates that the spec- +trum alone does not determine the geometry. That is why it is worth studying +new invariants that depend not only on the eigenvalues but also on the eigenfunc- +tions, and, therefore, contain much more information about the geometry of the +manifold. +Let {λk,ϕk}∞ +k=1 be the eigenvalues and the eigensections of a Laplace type +operator L acting on sections of a N-dimensional vector bundle V over a n- +dimensional compact Riemannian manifold M without boundary. Let Φk +l be the +one-forms defined by +Φk +l = ⟨ϕk,Dϕl⟩ = +� +ϕk,∇µϕl +� +dxµ , +(6.1) +where ⟨·,·⟩ is the fiber inner product, and D = d+A is the covariant exterior deriva- +tive. Then the coefficients +Ψk1...kn +l1...ln += +� +M +Φk1 +l1 ∧···∧Φkn +ln +(6.2) +measure the correlations between the eigensections. We define a new invariant +(called the heat determinant) by +K(t) = 1 +n! +∞ +� +k1,l1,...,kn,ln=1 +exp�−t(λk1 +λl1 +···+λkn +λln)�����Ψk1...kn +l1...ln +���� +2 +. +(6.3) +One can show that this invariant can be expressed directly in terms of the heat +kernel U(t; x, x′), +K(t) = +� +M×M +dxdx′ det +� +tr +� +U∗(t; x, x′)∇µ∇ν′U(t; x, x′) +�� +. +(6.4) +We prove that there is an asymptotic expansion as t → 0 +K(t) ∼ 1 +2Nn(4π)−n2 � π +2n +�n/2 +t−n +� +n+ 1 +2 +� ∞ +� +k=0 +tk/2Ck, +(6.5) +where +Ck = +� +M +dvolck, +(6.6) +hkrev.tex; January 23, 2023; 1:22; p. 17 + +18 +and ck are differential polynomials in the Riemann curvature, the curvature of the +bundle connection and the potential Q with some universal numerical coefficients +that depend only on the dimensions n and N. On manifolds without boundary all +odd-order coefficients vanish +C2k+1 = 0. +(6.7) +In particular, +C0 = vol(M) +(6.8) +and the coefficients c2 and c4 are computed explicitly in our paper [17]. +7 +Quantum Heat Traces +We initiate the study of new invariants of second-order elliptic partial differential +operators acting on sections of vector bundles over compact Riemannian mani- +folds without boundary. We draw a deep analogy between the spectral invariants +of elliptic operators and the statistical physics. We consider an elliptic self-adjoint +positive partial differential operator H and its square root, +ω = H1/2, +(7.1) +which is an elliptic self-adjoint positive pseudo-differential operator of first order. +We interpret the classical heat trace +Θ(β) = Tr exp(−βH) +(7.2) +as the partition function for the Boltzman distribution with β = 1/T being the +inverse temperature. By analogy, we define the relativistic heat trace +Θr(β) += +Tr exp(−βω) +(7.3) +and the quantum heat traces +Θb(β,µ) += +Tr +1 +exp[β(ω−µ)]−1, +(7.4) +Θf (β,µ) += +Tr +1 +exp[β(ω−µ)]+1, +(7.5) +where µ is a parameter that plays the role of the chemical potential. We show that +these new invariants can be reduced to some integrals of the classical heat trace +and compute the high-temperature asymptotics of these invariants as β → 0. +hkrev.tex; January 23, 2023; 1:22; p. 18 + +19 +We introduce a function Aq of a complex variable q defined by the Mellin +transform of the heat trace +Aq += +(4π)n/2 +1 +Γ(−q) +� ∞ +0 +dt t−q−1+n/2Θ(t) += +(4π)n/2Γ +�n +2 −q +� +Γ(−q) ζ +�n +2 −q +� +. +(7.6) +Then we show that for a positive operator H: +1. the function Aq is an entire function of q, +2. its values at non-negative integer points Ak, with k ∈ Z+, are equal to the +standard heat trace coefficients, which are locally computable, +3. while the values of the function Ak+1/2 at the half-integer points k + 1/2, +with k a positive integer, as well as the values of its derivative A′ +k = ∂qAq|q=k +at the positive integer points k ∈ Z+ are new global invariants that are not +locally computable. +We use the integral +exp(−x) = (4π)−1/2 +� ∞ +0 +dt t−3/2exp +� +− 1 +4t −tx2 +� +, +(7.7) +valid for x ≥ 0, to reduce the relativistic heat trace to the classical heat trace +Θr(β) = (4π)−1/2 +� ∞ +0 +dt t−3/2exp +� +− 1 +4t +� +Θ(tβ2). +(7.8) +Next, we express the relativistic heat trace in terms of the function Aq via a Mellin- +Barnes integral +Θr(β) = 2(4π)−(n+1)/2 1 +2πi +c+i∞ +� +c−i∞ +dqΓ(−q)Γ�−q+(n+1)/2��β +2 +�2q−n +Aq, +(7.9) +We compute the asymptotics of the relativistic heat trace Θr(β) as β → 0. We +obtained in even dimension n = 2m, +Θr(β) +∼ +∞ +� +k=0 +β2k−2mb(1) +k Ak + +∞ +� +k=0 +β2k+1b(2) +k Ak+m+1/2, +(7.10) +hkrev.tex; January 23, 2023; 1:22; p. 19 + +20 +and in odd dimension n = 2m+1, +Θr(β) +∼ +∞ +� +k=0 +β2k−2m−1b(3) +k Ak +logβ +∞ +� +k=0 +β2k+1b(4) +k Ak+m+1 + +∞ +� +k=0 +β2k+1b(5) +k A′ +k+m+1, +(7.11) +and computed all numerical coefficients b(i) +k . Notice that the coefficients of the +singular part containing the inverse powers of β and the logarithm are locally +computable. +We express the quantum heat traces in terms of the classical one +Θb, f (β,µ) += +∞ +� +0 +dt hb, f (t,βµ)Θ +� +tβ2� +, +(7.12) +where +hf (t,βµ) += +(4π)−1/2t−3/2 +∞ +� +k=1 +(−1)k+1kexp +� +−k2 +4t +kβµ +� +, +(7.13) +hb(t,βµ) += +(4π)−1/2t−3/2 +∞ +� +k=1 +kexp +� +−k2 +4t +kβµ +� +, +(7.14) +This gives the Mellin-Barnes representation of the quantum heat traces +Θb, f (β,µ) += +2(4π)−(n+1)/2 1 +2πi +c+i∞ +� +c−i∞ +dq Γ(−q)Γ�−q+(n+1)/2��β +2 +�2q−n +×Fb, f (n−2q,βµ)Aq, +(7.15) +where c < 0 and +Fb(s,βµ) += +∞ +� +k=1 +ekβµ +ks = +1 +Γ(s) +� ∞ +0 +dt +ts−1 +et−βµ −1, +(7.16) +F f (s,βµ) += +∞ +� +k=1 +(−1)k+1ekβµ +ks = +1 +Γ(s) +� ∞ +0 +dt +ts−1 +et−βµ +1. +(7.17) +We compute their asymptotics as β → 0. +hkrev.tex; January 23, 2023; 1:22; p. 20 + +21 +For µ = 0 we obtain an asymptotic expansion as β → 0: in even dimension +n = 2m, +Θf (β,0) +∼ +m +� +k=0 +β2k−2mc(1) +k Ak + +∞ +� +k=0 +β2k+1c(2) +k Ak+m+1/2, +(7.18) +Θb(β,0) += +m +� +k=0 +β2k−2mc(3) +k Ak + +∞ +� +k=−1 +β2k+1c(4) +k Ak+m+1/2, +(7.19) +and in odd dimension n = 2m+1, +Θf (β,0) +∼ +∞ +� +k=0 +β2k−2m−1c(5) +k Ak +logβ +∞ +� +k=0 +β2k+1c(6) +k Ak+m+1 + +∞ +� +k=0 +β2k+1c(7) +k A′ +k+m+1, +(7.20) +Θb(β,0) += +m−1 +� +k=0 +β2k−2m−1c(8) +k Ak +logβ +∞ +� +k=−1 +β2k+1c(9) +k Ak+m+1 + +∞ +� +k=−1 +β2k+1c(10) +k +A′ +k+m+1. +(7.21) +and computed all numerical coefficients c(i) +k . +8 +Relative Spectral Invariants +Let L± be two self-adjoint elliptic second-order partial differential operators acting +on smooth sections of the vector bundle V with a positive definite scalar leading +symbols of Laplace type, +L± = −g−1/4 +± +(∂i +A± +i )g1/2 +± gi j +±(∂j +A± +j )g−1/4 +± ++ Q±. +(8.1) +Here g± +i j are two metrics, A± +i are two connection one-forms and Q± are two endo- +morphisms; also gi j +± are the inverse metrics and g± = detg± +i j. +We assume that V is a Clifford bundle. Let D± be two self-adjoint elliptic +first-order partial differential operators acting on smooth sections of the vector +bundle V of Dirac type, +D± = g1/4 +± iγaej +±a(∂j +A± +j )g−1/4 +± ++S ±, +(8.2) +where γa are the Dirac matrices satisfying +γaγb +γbγa = 2δabI, +(8.3) +hkrev.tex; January 23, 2023; 1:22; p. 21 + +22 +ei +±a are two orthonormal frames for the metrics gi j +± satisfying +gi j +± = δabei +±aei +±b, +(8.4) +A± +j are two connections and S ± are two endomorphisms. We require that the +endomorphisms S ± commute with the Dirac matrices so that the square of the +Dirac type operators, D2 +±, are operators of Laplace type. +The spectral information about the operators L± and D± is contained in the +classical heat traces +Θ±(t) += +Tr exp(−tL±), +(8.5) +H±(t) += +TrD± exp(−tD2 +±). +(8.6) +The relative spectral information is contained in the relative spectral invariants +Ψ(t, s) += +Tr �exp(−tL+)−exp(−tL−)��exp(−sL+)−exp(−sL−)� +(8.7) +Φ(t, s) += +Tr +� +D+ exp(−tD2 ++)− D− exp(−tD2 +−) +�� +D+ exp(−sD2 ++)− D− exp(−sD2 +−) +� +(8.8) +which can be expressed in terms of the combined heat traces +X(t, s) += +Tr �exp(−tL+)exp(−sL−)� += +� +M×M +dx dx′ tr �U+(t; x, x′)U−(s; x′, x)�, +(8.9) +Y(t, s) += +Tr +� +D+ exp(−tD2 ++)D− exp(−sD2 +−) +� += +� +M×M +dx dx′ tr �D+U+(t; x, x′)D−U−(s; x′, x)�. +(8.10) +We study the asymptotics of the combined heat traces (8.9) and (8.10). We +define the time-dependent metric gi j = gi j(t, s) as the inverse of the matrix +gi j = tgi j ++ + sgi j +−, +(8.11) +with t, s > 0; throughout the paper we use the notation g = detgi j for the determi- +nant of the metric. Also, we define the time-dependent connection Ai = Ai(t, s) +by +Ai = gi j +� +tgjk ++ A+ +k + sgjk +− A− +k +� +. +(8.12) +and the vectors +C± +i = A± +i −Ai. +(8.13) +hkrev.tex; January 23, 2023; 1:22; p. 22 + +23 +Theorem 1 There are asymptotic expansions as ε → 0 +X(εt,εs) +∼ +(4πε)−n/2 +∞ +� +k=0 +εkBk(t, s), +(8.14) +Y(εt,εs) +∼ +(4πε)−n/2 +∞ +� +k=0 +εk−1Ck(t, s), +(8.15) +where +Bk(t, s) += +� +M +dx g1/2(t, s)bk(t, s), +(8.16) +Ck(t, s) += +� +M +dx g1/2(t, s)ck(t, s). +(8.17) +1. The coefficients bk(t, s) and ck(t, s) are scalar invariants built polynomially +from the covariant derivatives (defined with respect to the metric gi j and the +connection Ai) of the metrics g± +i j, the vectors C± +i and the potentials Q± and +S ±. +2. The coefficients bk(t, s) are homogeneous functions of t and s of degree k +and the coefficients ck(t, s) are homogeneous functions of t and s of degree +(k −1). +The first coefficients of the asymptotic expansion of the combined heat traces +are +b0(t, s) += +N, +(8.18) +c0(t, s) += +N 1 +2δabei ++agi j(t, s)ej +−b, +(8.19) +where N = dimV. +9 +Bogolyubov Invariant +Further, let V be a twisted spin-tensor bundle. Let ξ and η be two self-adjoint +anti-commuting involutive endomorphisms of the bundle V, +ξ2 = η2 = I, +ξ∗ = ξ, +η∗ = η, +ξη = −ηξ. +(9.1) +hkrev.tex; January 23, 2023; 1:22; p. 23 + +24 +Let D± be a self-adjoint elliptic first-order partial differential operators of Dirac +type acting on smooth sections of the bundle V that anti-commute with η and +commute with ξ, +D±η = −ηD±, +D±ξ = ξD±. +(9.2) +Suppose that the square of the operators D± are Laplace type operators +D2 +± = H± +(9.3) +Then the operators D± + mη are Dirac type operators whose square are Laplace +type operators +(D± +mη)2 = H± +m2I +(9.4) +The bosonic and fermionic Bogolyubov invariants are determined by the fol- +lowing traces +Bb(β) += +Tr +� +1 +exp(βω+)+1 − +1 +exp(βω−)+1 +�� +1 +exp(βω+)−1 − +1 +exp(βω−)−1 +� +, +(9.5) +Bf(β) += +β2Tr +�(D+ +mη) +sinh(βω+) − (D− +mη) +sinh(βω−) +�2 +. +(9.6) +where +ω± = (H± +m2)1/2, +(9.7) +The Bogolyubov invariants can be expressed in terms of the the relative spec- +tral invariants Ψ(t, s) and Φ(t, s) defined in (8.7) and (??). Let hb, f,0 be the func- +tions defined by +hf (t) += +(4π)−1/2t−3/2 +∞ +� +k=1 +(−1)k+1kexp +� +−k2 +4t +� +, +(9.8) +hb(t) += +(4π)−1/2t−3/2 +∞ +� +k=1 +kexp +� +−k2 +4t +� +, +(9.9) +h0(t) += +(4π)−1/2t−3/2 +∞ +� +k=0 +(2k +1)exp +� +−(2k +1)2 +4t +� +. +(9.10) +Then we obtain the heat trace representation for the Bogolyubov invariant. For +the bosonic case we get from (9.5) +Bb(β) = +∞ +� +0 +dt +∞ +� +0 +ds hf (t)hb(s)exp +� +−m2β2(s+t) +� +Ψ +� +β2t,β2s +� +, +(9.11) +hkrev.tex; January 23, 2023; 1:22; p. 24 + +25 +where +Ψ(t, s) = Tr �exp(−tH+)−exp(−tH−)��exp(−sH+)−exp(−sH−)�. +(9.12) +For the fermionic case we get from (9.6) +Bf(β) += +∞ +� +0 +dt +∞ +� +0 +ds h0(s)h0 (t)exp +� +−m2β2(s+t) +� +×β2� +Φ +� +β2t,β2s +� ++m2Ψ +� +β2t,β2s +�� +, +(9.13) +where +Φ(t, s) = Tr �D+ exp(−tH+)− D− exp(−tH−)��D+ exp(−sH+)− D− exp(−sH−)�. +(9.14) +It is easy to see that the integrals for the Bogolyubov invariant converge both as +t, s → 0 and (for sufficiently large m2) also as t, s → ∞. +Now we can compute the asymptotics of Bogolyubov invariant as β → 0. We +obtain om odd dimension n = 2m+1 +Bb(β) ∼ +∞ +� +k=0 +β2k−nc(1) +k + +∞ +� +k=0 +β2kc(2) +k , +(9.15) +and in the even dimension n = 2m +Bb(β) ∼ +m−1 +� +k=0 +β2k−nc(3) +k + +∞ +� +k=0 +β2kc(4) +k +logβ2 +∞ +� +k=0 +β2kc(5) +k , +(9.16) +Here c(i) +k +are some coefficients. Notice that the coefficients c(1) +k +of the all odd +powers of β as well as the coefficients c(3) +k +and c(5) +k +of the singular part and the +logarithmic part are locally computable invariants whereas the coefficients c(2) +k +of +the even non-negative powers of β as well as the coefficients c(4) +k +of the regular +part are non-locally computable global invariants. The leading asymptotics have +the form +Bb(β) = β−nc(1) +0 +β−n+2c(1) +1 +O(β−n+4)+O(logβ). +(9.17) +Similarly, for the Dirac operator we obtain: in the odd dimension n = 2m+1: +Bf(β) ∼ +∞ +� +k=0 +β2k−nd(1) +k + +∞ +� +k=0 +β2kd(2) +k , +(9.18) +hkrev.tex; January 23, 2023; 1:22; p. 25 + +26 +and in the even dimension n = 2m +Bf(β) ∼ +m−1 +� +k=0 +β2k−nd(3) +k + +∞ +� +k=0 +β2kd(4) +k +logβ2 +∞ +� +k=0 +β2kd(5) +k , +(9.19) +Here the coefficients d(1) +k +of the all odd powers of β as well as the coefficients +d(3) +k +and d(5) +k +of the singular part and the logarithmic part are locally computable +invariants whereas the coefficients d(2) +k +of the even non-negative powers of β as +well as the coefficients d(4) +k +of the regular part are non-locally computable global +invariants. The leading asymptotics have the form +Bf (β) = β−nd(1) +0 +β−n+2d(1) +1 +O(β−n+4)+O(logβ). +(9.20) +10 +Heat Semigroups on Weyl Algebra +Let xi be the coordinates of the Euclidean space Rn and ∂i be the corresponding +partial derivatives. More precisely, we consider the partial derivative operators +acting on the space C∞ +0 (Rn) of smooth functions of compact support. Recall that +this space is dense in the Hilbert space L2(Rn), which defines the extension of +these operators to the whole Hilbert space L2(Rn). Moreover, the partial deriva- +tives are unbounded (essentially) anti-self-adjoint operators in this Hilbert space. +Then the Weyl algebra (the universal enveloping algebra of the Heisenberg alge- +bra) is simply the ring of all differential operators with polynomial coefficients. +The operators ∇k are anti-self-adjoint operators of the form +∇k = ∂k − 1 +2iRk jxj. +(10.1) +Given a positive matrix g we introduce an operator called the Laplacian by +∆g = gi j∇i∇j, +(10.2) +where gi j is the inverse matrix. +We consider two Laplacians ∆± defined with two different matrices R± +i j and +two different matrices g± +i j and study the integral kernel of the product of the corre- +sponding heat semigroups +U(t, s; x, x′) += +exp(t∆+)exp(s∆−)δ(x− x′). +(10.3) +hkrev.tex; January 23, 2023; 1:22; p. 26 + +27 +The (2n + 1) operators (∇1,...,∇n, x1,..., xn,i) form the Lie algebra hn of the +Heisenberg group H2n+1 with the commutation relations +[∇k, xj] += +δj +k, +(10.4) +[∇k,∇j] += +iRk j , +(10.5) +[∇k,i] += +[xk,i] = 0. +(10.6) +The universal enveloping algebra U(hn) of the Heisenberg algebra (called the Weyl +algebra) is the set of all polynomials in these operators subject to these commuta- +tion relations. +The operator exp⟨ξ,∇⟩ : C∞ +0 (Rn) → C∞ +0 (Rn) acts by +�exp⟨ξ,∇⟩ f �(x) += +exp +� +−1 +2 ⟨ξ,iRx⟩ +� +f (x+ξ). +(10.7) +and is an isometry. +First, we show that the heat semigroup can be represented as a non-commutative +Gaussian integral +exp(t∆g) = (4π)−n/2Ω(t) +� +Rn +dξexp +� +−1 +4 ⟨ξ,D(t)ξ⟩ +� +exp⟨ξ,∇⟩ , +(10.8) +where D(t) = (Di j) is a symmetric matrix defined by +D(t) = iRcoth +� +tg−1iR +� +, +(10.9) +and +Ω(t) += +det +� +g−1sinh(tg−1iR) +g−1iR +�−1/2 +. +(10.10) +Next, we consider two sets of such operators ∇+ +i and ∇− +j forming the Lie alge- +bra +[∇+ +i ,∇+ +j ] += +iR+ +i j, +(10.11) +[∇− +i ,∇− +j ] += +iR− +i j, +(10.12) +[∇+ +i ,∇− +j ] += +iRi j. +(10.13) +where +Ri j = 1 +2 +� +R+ +i j +R− +i j +� +. +(10.14) +hkrev.tex; January 23, 2023; 1:22; p. 27 + +28 +Then we compute the product of the semigroups exp(t∆+)exp(s∆−). Let ∇i +and Xi be the operators defined by +∇i += +1 +2(∇+ +i +∇− +i ), +(10.15) +Xi += +∇+ +i −∇− +i . +(10.16) +Then +exp(t∆+)exp(s∆−) = (4π)−n/2Ω(t, s)exp +� +X,D−1(t, s)X +� +(10.17) +× +� +Rn +dξ exp +� +−1 +4 ⟨ξ,H(t, s)ξ⟩− 1 +2 +� +ξ,ZT(t, s)D−1(t, s)X +�� +exp⟨ξ,∇⟩, +where T±(t), D(t, s), Z(t, s) and H(t, s) are the matrices defined by +T±(t) += +D±(t)+iR, +(10.18) +D(t, s) += +D+(t)+ D−(s), +(10.19) +Z(t, s) += +D+(t)− D−(s)−2iR−, +(10.20) +H(t, s) += +1 +4 +� +D(t, s)−ZT(t, s)D−1(t, s)Z(t, s) +� +, +(10.21) +and +Ω(t, s) = +�detT+(t)detT−(s) +detD(t, s) +�1/2 +. +(10.22) +Finally, we compute the convolution of the heat kernels +U(t, s; x, x′) += +(4π)−n/2(detB(t, s))1/2 +(10.23) +×exp +� +−1 +4 ⟨x,A+(t, s)x⟩− 1 +4 +�x′,A−(t, s)x′�+ 1 +2 +�x,B(t, s)x′�. +� +, +where A±(t, s) and B(t, s) be matrices defined by +A+(t, s) += +D+(t)−T T ++ (t)D−1(t, s)T+(t), +(10.24) +A−(t, s) += +D−(s)−T T +− (s)D−1(t, s)T−(s), +(10.25) +B(t, s) += +T+(t)D−1(t, s)T−(s), +(10.26) +hkrev.tex; January 23, 2023; 1:22; p. 28 + +29 +References +[1] I. G. Avramidi, A covariant technique for the calculation of the one-loop +effective action, Nucl. Phys. B 355 (1991) 712–754 +[2] I. G. Avramidi, A method for calculating the heat kernel for manifolds with +boundary, Yadernaya Fizika 56 (1993) 245–252, [Russian]; Phys. Atom. +Nucl. 56 (1993) 138–142 [English] +[3] I. G. Avramidi, A new algebraic approach for calculating the heat kernel +in gauge theories, Phys. Lett. B 305 (1993) 27–34 +[4] I. G. Avramidi, Heat Kernel and Quantum Gravity, (Berlin-New York: +Springer 2000) +[5] I. G. Avramidi, Matrix general relativity: a new look at old problems, Class. +Quant. Grav. 21 (2004) 103–120 +[6] I. G. Avramidi, Gauged gravity via spectral asymptotics of non-Laplace +type operators, J. High Energy Phys. 07 (2004) 030, 36 pp. +[7] I. G. Avramidi, Heat kernel asymptotics of Zaremba boundary value prob- +lem, Math. Phys., Analysis and Geom. 7 (2004) 9–46 +[8] I. G. Avramidi, Dirac operator in matrix geometry, Int. J. Geom. Methods +Mod. Phys. 2 (2005), 227–264 +[9] I. G. Avramidi, Heat kernel on homogeneous bundles over symmetric +spaces, Commun. Math. Phys. 288 (2009) 963-1006 +[10] I. G. Avramidi, Mathemathical tools for calculation of the effective ac- +tion in quantum gravity, in: New Paths Towards Quantum Gravity, Eds. B. +Booss-Bavnbek, G. Esposito and M. Lesch, (Berlin, Springer, 2010), pp. +193-259 +[11] I. G. Avramidi, Heat Kernel Method and its Applications, (Heidelberg-New +York: Birkh¨auser, 2015) +[12] I. G. Avramidi, Quantum heat traces, J. Geom. Phys. 112 (2017) 271-288 +[13] I. G. Avramidi, Bogolyubov invariant via relative spectral invariants on +manifolds, J. Math. Phys. 61 (2020) 032303 +hkrev.tex; January 23, 2023; 1:22; p. 29 + +30 +[14] I. G. Avramidi, Relative spectral invariants of elliptic operators on mani- +folds, J. Geom. Phys. 150 (2020) 103599 +[15] I. G. Avramidi, Heat semigroups on Weyl algebra, J. Geom. Phys. 161 +(2021) 104044 +[16] I. G. Avramidi and T. Branson, Heat kernel asymptotics of operators with +non-Laplace principal part, Rev. Math. Phys. 13 (2001) 847–890 +[17] I. G. Avramidi and B. J. Buckman, Heat determinant on manifolds, J. +Geom. Phys. 104 (2016) 64-88 +[18] I. G. Avramidi and G. Esposito, New invariants in the one-loop divergences +on manifolds with boundary, Class. Quant. Grav. 15 (1998) 281–297 +[19] I. G. Avramidi and G. Esposito, Gauge theories on manifolds with bound- +ary, Commun. Math. Phys. 200 (1999) 495–543 +[20] I. G. Avramidi and G. Fucci, Non-perturbative heat kernel asymptotics on +homogeneous Abelian bundles, Commun. Math. Phys. 291 (2009) 543-577 +[21] R. Camporesi, Harmonic analysis and propagators on homogeneous +spaces, Phys. Rep. 196, (1990), 1–134. +[22] P. B. Gilkey, The spectral geometry of Riemannian manifold, J. Diff. Geom. +10 (1975) 601–618. +[23] P. B. Gilkey, Invariance Theory, the Heat Equation and the Atiyah-Singer +Index Theorem, CRC Press, Boca Raton, 1995. +[24] M. Kac, Can one hear the shape of a drum? Am. Math. Monthly, 73 (1966) +1-23 +[25] K. Kirsten, Spectral Functions in Mathematics and Physics, CRC Press, +Boca Raton, 2001. +[26] D. Schueth, Continuous families of isospectral metrics on simply connected +manifolds, Annals of Mathematics, 149 (1999) 287–308, +[27] R. T. Seeley, Trace expansions for the Zaremba problem, Comm. Part. Diff. +Eqs. 27 (2002), 2403–2421. +hkrev.tex; January 23, 2023; 1:22; p. 30 + +31 +[28] T. Sunada, Riemannian coverings and isospectral manifolds, Ann. Math. +121 (1985) 169–186 +[29] D. V. Vassilevich, Heat kernel expansion: user’s manual, Phys. Rep., 388 +(2003), 279–360. +hkrev.tex; January 23, 2023; 1:22; p. 31 + diff --git a/bNFLT4oBgHgl3EQfXy8v/vector_store/index.faiss b/bNFLT4oBgHgl3EQfXy8v/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..997064d160add070d7a414a8565c41d6a860c506 --- /dev/null +++ b/bNFLT4oBgHgl3EQfXy8v/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:dd5f6312b2d617174f163707be24a232a6172982ba34981c3836219bdfc61d98 +size 3670061 diff --git a/cdE1T4oBgHgl3EQfdgTc/content/tmp_files/2301.03197v1.pdf.txt b/cdE1T4oBgHgl3EQfdgTc/content/tmp_files/2301.03197v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8f3966a6d8db4ee6e93902eb34e25218a4b69288 --- /dev/null +++ b/cdE1T4oBgHgl3EQfdgTc/content/tmp_files/2301.03197v1.pdf.txt @@ -0,0 +1,1146 @@ +arXiv:2301.03197v1 [math.AP] 9 Jan 2023 +ON THE PRINCIPAL FREQUENCY OF NON-HOMOGENEOUS +MEMBRANES +V. GOL’DSHTEIN, V. PCHELINTSEV +Abstract. We obtained estimates for first eigenvalues of the Dirichlet bound- +ary value problem for elliptic operators in divergence form (i.e. for the principal +frequency of non-homogeneous membranes) in bounded domains Ω ⊂ C satis- +fying quasihyperbolic boundary conditions. The suggested method is based on +the quasiconformal composition operators on Sobolev spaces and their applica- +tions to constant estimates in the corresponding Sobolev-Poincaré inequalities. +We also prove a variant of the Rayleigh-Faber-Khran inequality for a special +case of these elliptic operators. +1. Introduction +In this paper we give applications of the theory of quasiconformal mappings to +the Dirichlet eigenvalue problem for two-dimensional elliptic operators in divergence +form +(1.1) +LAf(z) = −div[A(z)∇f(z)], +z = (x, y) ∈ Ω, +f(x, y) = 0 on ∂Ω, +in bounded domains Ω ⊂ C satisfying quasihyperbolic boundary conditions [19, 20]. +We assume that A ∈ M 2×2(Ω), where M 2×2(Ω) is the class of all 2 × 2 symmetric +matrix functions A(z) = {akl(z)}, detA = 1 a. e. in Ω, with measurable entries +satisfying to the uniform ellipticity condition +(1.2) +1 +K |ξ|2 ≤ ⟨A(w)ξ, ξ⟩ ≤ K|ξ|2 a. e. in Ω, +for every ξ ∈ C, where 1 ≤ K < ∞. Such elliptic operators in divergence form arise +in various problems of mathematical physics (see, for example, [3]). +Under these conditions the matrix A induces a quasiconformal homeomorphism +ϕA : Ω → �Ω which we call the A-quasiconformal mapping [11]. This allow us to +reduce (by quasiconformal change of variable) this elliptic operator in divergence +form to the Laplace operator. +A domain Ω satisfies the γ-quasihyperbolic boundary condition with some γ > 0 +if the growth condition on the quasihyperbolic metric +kΩ(x0, x) ≤ 1 +γ log dist(x0, ∂Ω) +dist(x, ∂Ω) + C0 +satisfied for all x ∈ Ω, where x0 ∈ Ω is a fixed base point and C0 = C0(x0) < ∞, +[7, 17]. Here +kΩ(x1, x2) := inf +ˆ +γ +ds +dist(x, ∂Ω), +0Key words and phrases: Elliptic equations, Sobolev spaces, quasiconformal mappings. +02010 Mathematics Subject Classification: 35P15, 46E35, 30C65. +1 + +DIRICHLET EIGENVALUES PROBLEM +2 +where the infimum is taken over all rectifiable curves γ joining x1 and x2. +This class of domains includes, in particular, domains with Lipschitz boundary, +some domains with Hölder singularities, and domains of snowflakes type [23]. +A function f ∈ W 1,2 +0 +(Ω, A) is a solution to the generalized spectral problem +for the elliptic operator in divergence form LAf(z) with the Dirichlet boundary +condition if +¨ +Ω +⟨A(z)∇f(z), ∇g(z)⟩ dxdy = λ +¨ +Ω +f(z)g(z) dxdy, ∀g ∈ W 1,2 +0 +(Ω, A). +It is known [16, 22] that in a bounded domain Ω ⊂ C the operator LAf(z) +with the Dirichlet boundary condition has discrete spectrum represented as the +non-decreasing sequence +0 < λ1(A, Ω) ≤ λ2(A, Ω) ≤ . . . ≤ λn(A, Ω) ≤ . . . , +where each eigenvalue is repeated as many time as its multiplicity. By the min-max +principle, the first eigenvalue λ1(A, Ω) is defined by +λ1(A, Ω) = +inf +f∈W 1,2 +0 +(Ω,A)\{0} +∥f | L1,2 +A (Ω)∥2 +∥f | L2(Ω)∥2 . +The lower estimates for the first eigenvalues of the Laplace operator with the +Dirichlet boundary condition in a bounded domain are connected by the Rayleigh- +Faber-Khran inequality [6, 21] which means that the first Dirichlet eigenvalue in a +bounded domain Ω is not less than the corresponding Dirichlet eigenvalue in the +disc of the same area Ω∗ with R∗ as its radius, i.e., +λ1(I, Ω) := λ1(Ω) ≥ λ1(Ω∗) = j2 +0,1 +R2∗ +, +where j0,1 ≈ 2.4048 is the first positive zero of the Bessel function J0. This inequal- +ity was improved by the method based on the capacity theory [22]. +Unfortunately, for the first eigenvalue λ1(A, Ω) Rayleigh-Faber-Khran type in- +equality has not been proven. However, lower estimates for the first eigenvalues the +operator LAf(z) with the Dirichlet boundary condition in bounded domains can +be obtained easily by using the Rayleigh-Faber-Krahn inequality and the uniform +ellipticity condition (1.2): Let Ω ⊂ C be a bounded domain such that |Ω| = |Ω∗| and +K is the ellipticity constant of the matrix A. Then +(1.3) +λ1(A, Ω) ≥ λ1(Ω) +K +≥ λ1(Ω∗) +K += j2 +0,1 +KR2∗ +. +In this paper we obtain lower estimates for the first Dirichlet eigenvalues of the +divergence form elliptic operators LAf(z) in bounded domains with some quasi- +hyperbolic boundary condition. We call such domains as β-regular domains for +some β ∈ (1, ∞]. The class of all β-regular domains coincides with the class of all +domains with quasihyperbolic boundary conditions. +Our machinery is based on connections between A-quasiconformal mappings +[3, 4, 15] and composition operators on Sobolev spaces [11]. + +DIRICHLET EIGENVALUES PROBLEM +3 +One of the main results of the article states the following estimate for ∞-regular +domains: +If a simply connected bounded domain Ω ⊂ C satisfies to the quasihy- +perbolic boundary condition, then +(1.4) +λ1(A, Ω) ≥ +λ1(�Ω) +∥Jϕ−1 +A | L∞(�Ω)∥ +, +where λ1(�Ω) is the first Dirichlet eigenvalue of the Laplace operator and Jϕ−1 +A +is a +Jacobian of the inverse mapping to the A-quasiconformal mapping ϕA : Ω → �Ω. +A detailed discussion about β-regular domains can be found in Section 3. +Note that if �Ω = Ω∗ and ∥Jϕ−1 +A | L∞(Ω∗)∥ < K then estimate (1.4) is better than +estimates (1.3). For example, this condition is satisfied for measure preserving A- +quasiconformal mappings ϕA : Ω → Ω∗ (|J(z, ϕA)| = 1 a.e. in Ω). Some examples +can be found at the end of this paper. +Taking into account the domain monotonicity property for the Dirichlet eigen- +values of the operator LAf(z) (see, for example, [16]) and estimate (1.4) we obtain +estimates for variations of the first Dirichlet eigenvalues of the operator LAf(z) un- +der quasiconformal deformations of the domain. Namely: Let Ω ⊂ C be a bounded +∞-regular domain. We assume that ϕA(Ω) := �Ω ⊃ Ω, then +λ1(A, Ω) − λ1(�Ω) ≥ +1 − ∥Jϕ−1 +A | L∞(�Ω)∥ +∥Jϕ−1 +A | L∞(�Ω)∥ +λ1(�Ω), +where λ1(�Ω) is the first Dirichlet eigenvalue of the Laplace operator and Jϕ−1 +A +is a +Jacobian of the inverse mapping to the A-quasiconformal mapping ϕA : Ω → �Ω. +In the case of the measure preserving A-quasiconformal mappings ϕA : Ω → D +(where D ⊂ C is the unit disc) we prove Rayleigh-Faber-Khran type inequality for +the operator LAf(z): Let Ω ⊂ C be a simply connected bounded domain such that +there exists a measure preserving A-quasiconformal mapping ϕA : Ω → D. Then +λ1(A, Ω) ≥ λ1(A, D). +2. Sobolev spaces and A-quasiconformal mappings +Let E ⊂ C be a measurable set on the complex plane and h : E → R be a +positive almost everywhere (a.e.) locally integrable function, i.e. a weight. The +weighted Lebesgue space Lp(E, h), 1 ≤ p < ∞, is the space of all locally integrable +functions endowed with the following norm +∥f | Lp(E, h)∥ = + + +¨ +E +|f(z)|ph(z) dxdy + + +1 +p +< ∞. +The two-weighted Sobolev space W 1,p(Ω, h, 1), 1 ≤ p < ∞, is defined as the +normed space of all locally integrable weakly differentiable functions f : Ω → R +endowed with the following norm: +∥f | W 1,p(Ω, h, 1)∥ = ∥f | Lp(Ω, h)∥ + ∥∇f | Lp(Ω)∥. + +DIRICHLET EIGENVALUES PROBLEM +4 +In the case h = 1 this weighted Sobolev space coincides with the classical Sobolev +space W 1,p(Ω). The seminormed Sobolev space L1,p(Ω), 1 ≤ p < ∞, is the space +of all locally integrable weakly differentiable functions f : Ω → R endowed with the +following seminorm: +∥f | L1,p(Ω)∥ = ∥∇f | Lp(Ω)∥, 1 ≤ p < ∞. +We also need a weighted seminormed Sobolev space L1,2 +A (Ω) (associated with +the matrix A), defined as the space of all locally integrable weakly differentiable +functions f : Ω → R with the finite seminorm given by: +∥f | L1,2 +A (Ω)∥ = + + +¨ +Ω +⟨A(z)∇f(z), ∇f(z)⟩ dxdy + + +1 +2 +. +The corresponding Sobolev space W 1,2(Ω, A) is defined as the normed space of +all locally integrable weakly differentiable functions f : Ω → R endowed with the +following norm: +∥f | W 1,2(Ω, A)∥ = ∥f | L2(Ω)∥ + ∥f | L1,2 +A (Ω)∥. +The Sobolev space W 1,2 +0 +(Ω, A) is the closure in the W 1,2(Ω, A)-norm of the space +C∞ +0 (Ω). +We consider the Sobolev spaces as Banach spaces of equivalence classes of func- +tions up to a set of p-capacity zero [22]. +Recall that a homeomorphism ϕ : Ω → �Ω, Ω, �Ω ⊂ C, is called a Q-quasiconformal +mapping if ϕ ∈ W 1,2 +loc (Ω) and there exists a constant 1 ≤ Q < ∞ such that +|Dϕ(z)|2 ≤ Q|J(z, ϕ)| for almost all z ∈ Ω. +Note that quasiconformal mappings have a finite distortion and possesses the +Luzin N-property (i.e. a image of any set of measure zero has measure zero) [26]. +If ϕ : Ω → �Ω is a Q-quasiconformal mapping then ϕ is differentiable almost +everywhere in Ω and +|J(z, ϕ)| = Jϕ(z) := lim +r→0 +|ϕ(B(z, r))| +|B(z, r)| +for almost all z ∈ Ω. +Now we give a construction of A-quasiconformal mappings connected with the +matrix A. +Recall that matrix functions A(z) = {akl(z)} with measurable entries akl(z) +belongs to a class M 2×2(Ω) of all 2 × 2 symmetric matrix functions that satisfy to +an additional condition detA = 1 a.e. and to the uniform ellipticity condition: +(2.1) +1 +K |ξ|2 ≤ ⟨A(z)ξ, ξ⟩ ≤ K|ξ|2 a.e. in Ω, +for every ξ ∈ C and for some 1 ≤ K < ∞. The basic idea is that every positive +quadratic form +ds2 = a11(x, y)dx2 + 2a12(x, y)dxdy + a22(x, y)dy2 +defined in a planar domain Ω can be reduced, by means of a quasiconformal change +of variables, to the canonical form +ds2 = Λ(du2 + dv2), Λ ̸= 0, a.e. in �Ω, +given that a11a22 − a2 +12 ≥ κ0 > 0, a11 > 0, almost everywhere in Ω [1, 4]. + +DIRICHLET EIGENVALUES PROBLEM +5 +By [4] any matrix A of the type under discussion induces a quasiconformal home- +omorphism as a solution to the corresponding Beltrami equation. +The detailed +procedure is described below +Let ξ(z) = Re ϕ(z) be a real part of a quasiconformal mapping ϕ(z) = ξ(z) + +iη(z), which satisfies to the Beltrami equation: +(2.2) +ϕz(z) = µ(z)ϕz(z), a.e. in Ω, +where +ϕz = 1 +2 +�∂ϕ +∂x − i∂ϕ +∂y +� +and +ϕz = 1 +2 +�∂ϕ +∂x + i∂ϕ +∂y +� +, +with the complex dilatation µ(z) given by +(2.3) +µ(z) = a22(z) − a11(z) − 2ia12(z) +det(I + A(z)) +, +I = +� +1 +0 +0 +1 +� +. +We call this quasiconformal mapping (with the complex dilatation µ defined by +(2.3)) as an A-quasiconformal mapping and we will use the notation ϕA for this +quasiconformal mapping. +Note that the uniform ellipticity condition (2.1) can be written as +(2.4) +|µ(z)| ≤ K − 1 +K + 1, a.e. in Ω. +Conversely from (2.3) (see, for example, [3], p. 412) one can recover the matrix +A : +(2.5) +A(z) = +� |1−µ|2 +1−|µ|2 +−2 Im µ +1−|µ|2 +−2 Im µ +1−|µ|2 +|1+µ|2 +1−|µ|2 +� +, a.e. in Ω. +Thus, for any A ∈ M 2×2(Ω) by (2.4) can be produced the complex dilatation +µ(z), for which, in turn, the Beltrami equation (2.2) induces an A-quasiconformal +homeomorphism ϕ : Ω → �Ω as its solution (by the Riemann measurable mapping +theorem (see, for example, [1])). Let us briefly say that A and ϕA are agreed. +Therefore with the given A-divergent form elliptic operator defined in a domain +Ω ⊂ C can be assochiated the A-quasiconformal mapping ϕA : Ω → �Ω with the +quasiconformality coefficient +QA = 1 + ∥µ | L∞(Ω)∥ +1 − ∥µ | L∞(Ω)∥, +where µ defined by (2.3). +From the estimate |µ(z)| ≤ K−1 +K+1 immediately follows that QA ≤ K. +Note that the inverse mapping to the A-quasiconformal mapping ϕA : Ω → �Ω is +the A−1-quasiconformal mapping [11]. +In [11] the relationship between composition operators on Sobolev spaces and +A-quasiconformal mappings was studied and the following theorem was proved. +Theorem 2.1. Let Ω, �Ω be domains in C. Then a homeomorphism ϕA : Ω → �Ω +is an A-quasiconformal mapping if and only if ϕ induces, by the composition rule +ϕ∗(f) = f ◦ ϕ, an isometry of Sobolev spaces L1,2 +A (Ω) and L1,2(�Ω) i.e. +∥ϕ∗ +A(f) | L1,2 +A (Ω)∥ = ∥f | L1,2(�Ω)∥ +for any f ∈ L1,2(�Ω). + +DIRICHLET EIGENVALUES PROBLEM +6 +This theorem generalizes the well known property of conformal mappings gener- +ate the isometry of uniform Sobolev spaces L1 +2(Ω) and L1 +2(�Ω) (see, for example, [5]). +It is also refines (in the case n = 2) the functional characterization of quasiconformal +mappings in the terms of isomorphisms of uniform Sobolev spaces [25]. +3. Estimate of the constant in Sobolev-Poincaré inequality +In [12] was proved the following weighted Sobolev-Poincaré inequality for a +bounded domain Ω ⊂ C. +We denote by h(z) = |J(z, ϕA)| the quasihyperbolic +weight defined by an A-quasiconformal mapping ϕA : Ω → �Ω. +Theorem 3.1. Let A belongs to a class M 2×2(Ω) and Ω be a bounded simply +connected planar domain. +Then for any function f ∈ W 1,2 +0 +(Ω, A) the following +weighted Sobolev-Poincaré inequality + + +¨ +Ω +|f(z)|rh(z)dxdy + + +1 +r +≤ Cr,2(h, A, Ω) + + +¨ +Ω +⟨A(z)∇f(z), ∇f(z)⟩ dxdy + + +1 +2 +holds for any r ≥ 2 with the constant Cr,2(h, A, Ω) = Cr,2(�Ω). +Here Cr,2(�Ω) is the best constant in the (non-weight) Sobolev-Poincaré inequality +in a bounded domain �Ω ⊂ C with the upper estimate (see [10]): +(3.1) +Cr,2(�Ω) ≤ +inf +p∈( 2r +r+2 ,2) +�p − 1 +2 − p +� p−1 +p +�√π · +p√ +2 +�−1 |�Ω| +1 +r +� +Γ(2/p)Γ(3 − 2/p) +. +Using Theorem 3.1 we give an upper estimate for the constant in the Sobolev- +Poincaré inequality in domains with the quasihyperbolic boundary condition. As +shown in [2], the Jacobians of quasiconformal mappings ψ : �Ω → Ω belong to Lβ(�Ω) +for some β > 1 if and only if Ω has the γ-quasihyperbolic boundary condition for +some γ. We note that β depends only on �Ω and the quasiconformal coefficient +K(ψ). +Since we need the exact value of the integrability exponent β for the Jacobians +of quasiconformal mappings, we consider an equivalent description of domains with +quasihyperbolic boundary in terms of the integrability of Jacobians [9]. +A simply connected domain Ω ⊂ C is called an A-quasiconformal β-regular +domain about a simply connected domain �Ω ⊂ C if +¨ +�Ω +|J(w, ϕ−1 +A )|β dudv < ∞ +for some β > 1, where ϕA : Ω → �Ω is a corresponding A-quasiconformal mapping. +The property of the quasiconformal β-regularity implies the integrability of a Ja- +cobian of quasiconformal mappings and therefore for any quasiconformal β-regular +domain we have the embedding of weighted Lebesgue spaces Lr(Ω, h) into non- +weighted Lebesgue spaces Ls(Ω) for s = β−1 +β r [9]. + +DIRICHLET EIGENVALUES PROBLEM +7 +Lemma 3.2. [9] Let Ω be an A-quasiconformal β-regular domain. Then for any +function f ∈ Lr(Ω, h), β/(β − 1) ≤ r < ∞, the inequality +∥f | Ls(Ω)∥ ≤ + + + +¨ +�Ω +|J(w, ϕ−1 +A )|βdudv + + + +1 +β · 1 +s +∥f | Lr(Ω, h)∥ +holds for s = β−1 +β r. +We ready to prove the upper estimate for the constant in the Sobolev-Poincaré +inequality in quasiconformal regular domains. +Theorem 3.3. Let A belong to a class M 2×2(Ω) and a domain Ω be A-quasi- +conformal β-regular about �Ω. Then: +(1) for any function f ∈ W 1,2 +0 +(Ω, A) and for any s ≥ 1, the Sobolev-Poincaré +inequality +∥f | Ls(Ω)∥ ≤ Cs,2(A, Ω)∥f | L1,2 +A (Ω)∥ +holds with the constant +Cs,2(A, Ω) ≤ C βs +β−1 ,2(�Ω)∥Jϕ−1 +A | Lβ(�Ω)∥ +1 +s , +1 < β < ∞; +(2) if β = ∞ then for any function f ∈ W 1,2 +0 +(Ω, A), the Sobolev-Poincaré +inequality +∥f | L2(Ω)∥ ≤ C2,2(A, Ω)∥f | L1,2 +A (Ω)∥ +holds with the constant +C2,2(A, Ω) ≤ C2,2(�Ω) +��Jϕ−1 +A | L∞(�Ω) +�� +1 +2 . +Here Jϕ−1 +A +is a Jacobian of the inverse mapping to the A-quasiconformal +mapping ϕA : Ω → �Ω. +Remark 3.4. The constant C2 +2,2(�Ω) = 1/λ1(�Ω), where λ1(�Ω) is the first Dirichlet +eigenvalue of Laplacian in a domain �Ω ⊂ C. +Proof. 1. Let +β +β−1 ≤ r and s = β−1 +β r. It means that such r exists for any s ≥ 1. +By Lemma 3.2 and Theorem 3.1 we get +∥f | Ls(Ω)∥ = + + +¨ +Ω +|f(z)|sdxdy + + +1 +s +≤ + + + +¨ +�Ω +|J(w, ϕ−1 +A )|βdudv + + + +1 +β · 1 +s  + +¨ +Ω +|f(z)|r|J(ϕA, x, y)|dxdy + + +1 +r +≤ Cr,2(�Ω) + + + +¨ +�Ω +|J(w, ϕ−1 +A )|βdudv + + + +1 +β · 1 +s  + +¨ +Ω +⟨A(z)∇f(z), ∇f(z)⟩ dxdy + + +1 +2 +for any s ≥ 1. + +DIRICHLET EIGENVALUES PROBLEM +8 +2. Let a function f ∈ L2(Ω). Since quasiconformal mappings possess the Luzin +N-property, then |J(z, ϕA)|−1 = |J(w, ϕ−1 +A )| for almost all z ∈ Ω and for almost all +w = ϕA(z) ∈ �Ω. Hence + + +¨ +Ω +|f(z)|2 dxdy + + +1 +2 += + + +¨ +Ω +|f(z)|2|J(z, ϕA)|−1|J(z, ϕA)| dxdy + + +1 +2 +≤ ∥JϕA | L∞(Ω)∥− 1 +2 + + +ˆ +Ω +|f(z)|2|J(z, ϕA)| dxdy + + +1 +2 +. +Applying the change of variable formula for quasiconformal mappings [26], (non- +weighed) Sobolev-Poincaré inequality [10], and Theorem 2.1 we obtain + + +¨ +Ω +|f(z)|2 dxdy + + +1 +2 +≤ ∥Jϕ−1 +A | L∞(�Ω)∥ +1 +2 + + + +¨ +�Ω +|f ◦ ϕ−1 +A (w)|2 dudv + + + +1 +2 +≤ C2,2(�Ω)∥Jϕ−1 +A | L∞(�Ω)∥ +1 +2 + + + +¨ +�Ω +|∇(f ◦ ϕ−1 +A (w))|2 dudv + + + +1 +2 += C2,2(�Ω)∥Jϕ−1 +A | L∞(�Ω)∥ +1 +2 + + +¨ +Ω +⟨A(z)∇f(z), ∇f(z)⟩ dxdy + + +1 +2 +for any f ∈ W 1,2 +0 +(Ω, A). +□ +4. Lower estimates for λ1(A, Ω) +We consider the generalized formulation of the Dirichlet eigenvalue problem (1.1): +¨ +Ω +⟨A(z)∇f(z), ∇g(z)⟩ dxdy = λ +¨ +Ω +f(z)g(z) dxdy, ∀g ∈ W 1,2 +0 +(Ω, A). +By the min-max principle (see, for example, [16, 22]) the first Dirichlet eigenvalue +λ1(A, Ω) of the elliptic operator in divergence form LAf(z) is defined by +λ1(A, Ω) = +inf +f∈W 1,2 +0 +(Ω,A)\{0} +∥f | L1,2 +A (Ω)∥2 +∥f | L2(Ω)∥2 . +In other words, λ +− 1 +2 +1 +(A, Ω) is the exact constant C2,2(A, Ω) in the Sobolev-Poincaré +inequality +∥f | L2(Ω)∥ ≤ C2,2(A, Ω)∥f | L1,2 +A (Ω)∥, f ∈ W 1,2 +0 +(Ω, A). + +DIRICHLET EIGENVALUES PROBLEM +9 +Theorem 4.1. Let A belong to the class M 2×2(Ω) and Ω be an A-quasiconformal +β-regular domain about a domain �Ω. Then +1 +λ1(A, Ω) ≤ C2 +2β +β−1 ,2(�Ω)∥Jϕ−1 +A | Lβ(�Ω)∥, +where Jϕ−1 +A is a Jacobian of the inverse mapping to the A-quasiconformal mapping +ϕA : Ω → �Ω and +C 2β +β−1 ,2(�Ω) ≤ +inf +p∈( +2β +2β−1 ,2) +�p − 1 +2 − p +� �√π · +p√ +2 +�−1 |�Ω| +β−1 +2β +� +Γ(2/p)Γ(3 − 2/p) +. +Proof. By the min-max principle and Theorem 3.3 in the case s = 2, we have +¨ +Ω +|f(z)|2 dxdy ≤ C2 +2,2(A, Ω) +¨ +Ω +⟨A(z)∇f(z), ∇f(z)⟩ dxdy, +where +C2,2(A, Ω) ≤ C 2β +β−1 ,2(�Ω) + + + +¨ +�Ω +|J(w, ϕ−1 +A )|β dudv + + + +1 +2β +. +Thus +1 +λ1(A, Ω) ≤ C2 +2β +β−1 ,2(�Ω) + + + +¨ +�Ω +|J(w, ϕ−1 +A )|β dudv + + + +1 +β +. +□ +In the limit case β = ∞ we have the following assertion: +Theorem 4.2. Let A belong to a class M 2×2(Ω) and Ω be an A-quasiconformal +∞-regular domain about a domain �Ω. Then +(4.1) +1 +λ1(A, Ω) ≤ C2 +2,2(�Ω)∥Jϕ−1 +A | L∞(�Ω)∥ = +∥Jϕ−1 +A | L∞(�Ω)∥ +λ1(�Ω) +, +where Jϕ−1 +A is a Jacobian of the inverse mapping to the A-quasiconformal mapping +ϕA : Ω → �Ω. +As an application of Theorem 4.2 we consider several examples. +Example 4.3. The homeomorphism +ϕ(z) = +a +a2 − b2 z − +b +a2 − b2 z, +z = x + iy, +a > b ≥ 0, +is an A-quasiconformal and maps the right triangle with arbitrary angels +Ω = +� +(x, y) ∈ R2 : 0 ≤ x ≤ a + b, 0 ≤ y ≤ −a − b +a + bx + (a − b) +� +onto the 45◦ right triangle +�Ω = +� +(u, v) ∈ R2 : 0 ≤ u ≤ 1, 0 ≤ v ≤ 1 − u +� +. + +DIRICHLET EIGENVALUES PROBLEM +10 +The mapping ϕ satisfies the Beltrami equation with +µ(z) = ϕz +ϕz += − b +a +and the Jacobian J(z, ϕ) = |ϕz|2 − |ϕz|2 = 1/(a2 − b2). It is easy to verify that µ +induces, by formula (2.5), the matrix function A(z) form +A(z) = +� a+b +a−b +0 +0 +a−b +a+b +� +. +Given that |J(w, ϕ−1)| = |J(z, ϕ)|−1 = a2 − b2 and λ1(�Ω) = 5π2 (see, for example, +[14]). Then by Theorem 4.2 we have +λ1(A, Ω) ≥ +λ1(�Ω) +∥Jϕ−1 | L∞(�Ω)∥ += +5π2 +a2 − b2 . +Example 4.4. The homeomorphism +ϕ(z) = 2 · z +3 +8 +z +1 +8 +− 1, ϕ(0) = −1, +z = x + iy, +is an A-quasiconformal and maps the interior of the non-convex domain +Ω := +� +(ρ, θ) ∈ R2 : ρ = cos4 +�θ +2 +� +, +−π ≤ θ ≤ π +� +onto the unit disc D. The mapping ϕ satisfies the Beltrami equation with +µ(z) = ϕz +ϕz += −1 +3 +z +z +and the Jacobian +J(z, ϕ) = |ϕz|2 − |ϕz|2 = +1 +2 · |z| +3 +2 . +We see that µ induces, by formula (2.5), the matrix function A(z) form +A(z) = +� |3z+z|2 +8|z|2 +3 +4 Im z +z +3 +4 Im z +z +|3z−z|2 +8|z|2 +� +. +Given that |J(w, ϕ−1)| = |J(z, ϕ)|−1 = 2 · |z| +3 +2 and λ1(D) = (j0,1)2. +Then by +Theorem 4.2 we have +λ1(A, Ω) ≥ +λ1(D) +∥Jϕ−1 | L∞(D)∥ ≥ (j0,1)2 +2 +. +Taking into account Theorem 4.2 and the domain monotonicity property of the +Dirichlet eigenvalues for the elliptic operator in divergence form LAf(z), we obtain +the following result for the special case ∥Jϕ−1 +A | L∞(�Ω)∥ < 1. +Proposition 4.5. Let Ω be an A-quasiconformal ∞-regular domain about �Ω. We +assume that �Ω ⊃ Ω, then +λ1(A, Ω) − λ1(�Ω) ≥ +1 − ∥Jϕ−1 +A | L∞(�Ω)∥ +∥Jϕ−1 +A | L∞(�Ω)∥ +λ1(�Ω). + +DIRICHLET EIGENVALUES PROBLEM +11 +Proof. Since �Ω ⊃ Ω, we have λ1(A, Ω) ≥ λ1(�Ω). Taking into account the inequal- +ity (4.1) in Theorem 4.2 and making elementary calculation, we get +λ1(A, Ω) − λ1(�Ω) ≥ +1 − ∥Jϕ−1 +A | L∞(�Ω)∥ +∥Jϕ−1 +A | L∞(�Ω)∥ +λ1(�Ω). +□ +4.1. The Rayleigh-Faber-Krahn type inequality. The theory of composition +operators [11] allows us to reduce the spectral problem for the divergence form +elliptic operator (1.1) defined in simply connected bounded domain Ω ⊂ C to a +weighted spectral problem for the Laplace operator in a simply connected bounded +domain �Ω ⊂ C. By the chain rule applied to a function f(z) = g ◦ ϕA(z) [15], we +have +−div[A(z)∇f(z)] = −div[A(z)∇g ◦ ϕA(z)] = −h(w)∆g(w), +where the weight h(w) = |J(w, ϕ−1 +A )|−1 is a Jacobian of the inverse mapping to the +A-quasiconformal mapping ϕA : Ω → �Ω. +From here we can point out that +λ1(A, Ω) = +inf +f∈W 1,2 +0 +(Ω,A)\{0} +˜ +Ω +⟨A(z)∇f, ∇f⟩ dxdy +˜ +Ω +|f|2dxdy += +inf +g∈W 1,2 +0 +(�Ω,h,1)\{0} +˜ +�Ω +|∇g|2dudv +˜ +�Ω +|g|2h(w)dudv = λ1(h, �Ω). +Let ϕA : Ω → �Ω be A-quasiconformal mappings for which |J(z, ϕA)| = 1 for +almost all z ∈ Ω. In this case quasiconformal weights h(w) = |J(w, ϕ−1 +A )| = 1 for +almost all w ∈ �Ω. Hence we get that λ1(A, Ω) = λ1(�Ω). +Using this equality and the classical Rayleigh-Faber-Krahn inequality we imme- +diately obtain a version of this inequality for elliptic operators in divergence form. +Namely: +Theorem 4.6. Let Ω ⊂ C be a simply connected bounded domain such that there +exists a measure preserving A-quasiconformal mapping ϕA : Ω → D. Then +λ1(A, Ω) ≥ λ1(A, D). +A description of the class of measure preserving A-quasiconformal mappings +and/or corresponding divergence form elliptic equations is an open problem. Let +us give simple examples of such mappings. +Example 4.7. Let ϕ(z) = (ax, 1 +ay), z = x + iy and a > 1. Then J(z, ϕ) = 1, the +quasiconformality coefficient Qϕ = a2 and µϕ = a2−1 +a2+1. The matrix can be easily +recontracted +A(z) = +� 1 +a2 +0 +0 +a2 +� +. +A little bit more complicated example. + +DIRICHLET EIGENVALUES PROBLEM +12 +Example 4.8. Let ϕ(z) = (ax + by, 1 +ay), z = x + iy and a > 1. Then J(z, ϕ) = 1, +the quasiconformality coefficient Qϕ = a2. Calculation of µϕ is more complicated. +We use the Beltrami equation, i.e +ϕz = 1 +2(a + 1 +a) − i b +2, +ϕz = 1 +2(a − 1 +a) + i b +2. +Hence +µϕ = (a2 − 1)(a2 + 1) − a2b2 +(a2 + 1)2 + a2b2 ++ i +2a3b +(a2 + 1)2 + a2b2 . +The matrix A can be easily reconstracted by elementary calculations. +Example 4.9. Let f ∈ L1 +∞(R). +Then ϕ(z) = (x + f(y), y), z = x + iy, is a +quasiconformal mapping with |J(z, ϕ)| = 1 (see, [11]). A basic calculation implies +ϕz = 1 − if ′(y) +2 +, +ϕz = if ′(y) +2 +. +Hence +µϕ = − +(f ′(y))2 +4 + (f ′(y))2 + i +2f ′(y) +4 + (f ′(y))2 . +The matrix can be easily recontracted +A(z) = +� +1 +−f ′(y) +−f ′(y) +1 + (f ′(y))2 +� +. +This algorithm can be used for more complicated examples. +5. Appendix +Few remarks about measure preserving and quasi-preserving (bi-Lipschitz) qua- +siconformal mappings andA-quasiconformal mappings. +A quasiconformal mapping ϕ is a solution of the corresponding Beltrami equation +(5.1) +ϕz(z) = µ(z)ϕz(z). +Because J(z, ϕ) = |ϕ2 +z|(z) − |ϕ2 +z|(z), the condition J(z, ϕ) = 1 can be written as +(5.2) +|ϕz|2(z)(1 − |µ(z)|2) = 1. +Recall that for A-quasiconformal mappings we have +µ(z) = a22(z) − a11(z) − 2ia12(z) +det(I + A(z)) +. +By elementary calculations it is easy to verify that any measure preserving qua- +siconformal mapping ϕ : Ω → �Ω is a bi-Lipschitz mapping in the following sense: +D(ϕ) ∈ L1 +∞(Ω) and D(ϕ−1) ∈ L1 +∞(�Ω) (or Dϕ ∈ L∞(Ω) and Dϕ−1 ∈ L∞(�Ω)). An +equivalent geometric description is the following. The mapping ϕ and its inverse +are locally Lipschitz homeomorphisms with uniformly bounded Lipschitz constants. +The condition (5.2) can be soften in the spirit of quasiconformality up to the +inequality +1 +C < J(z, ϕ) < C +for some positive constant C. We call such homeomorphism as quasi-preserving +measure homeomorphisms. + +DIRICHLET EIGENVALUES PROBLEM +13 +For quasiconformal quasi-preserving measure homeomorphisms this condition +can be written as +(5.3) +1 +C < |ϕ2 +z|(z)(1 − |µ(z)|2) < C. +The class of quasiconformal quasi-preserving measure homeomorphisms coincide +with the class of bi-Lipschitz homeomorphisms. +The constant C can be easily +recalculated in terms of uniform Lipschitz constants for the corresponding homeo- +morphism and its inverse. +We do not have any geometric description of such homeomorphisms i.e to solution +of systems ((5.1),(5.2)) or ((5.1),(5.3)). Let us look for some simplified cases. +Suppose the matrix A is a diagonal matrix. Then the quasiconformality coeffi- +cient µ(z) is a real number. If a11(z) > 0 then 0 < µ(z) < 1 and the condition (5.2) +can be simplified: +(5.4) +|ϕz|2(z)(1 − µ(z)2) = 1. +By simple calculations +µ(z) = +a11(z) + a−1 +11 (z) +2 + a11(z) + a−1 +11 (z). +Let us give an example of such homeomorphisms. +Example 5.1. Suppose Ω := (0, 1) × (0, 1) and ϕ(x, y) = a(x) + ib(y) where a(x) +and b(y) belong to C1(Ω). We also suppose that +inf +x∈(0,1)(da/dx) > +sup +y∈(0,1) +(db/dy) +and I1 := +inf +y∈(0,1)(db/dy) > 0. +Any such mapping is a quasi-preserving measure one, because +I1 ≤ J(z, ϕ) ≤ +� +|da/dx|C1(Ω) × |db/dy|C1(Ω) +� +and quasiconformal one because +Q ≤ +sup +x∈(0,1) +(da/dx) +inf +y∈(0,1)(db/dy) . +The corresponding matrix A can be easily reconstructed +(5.5) +A(z) = +� 1−µ(z) +1+µ(z) +0 +0 +1+µ(z) +1−µ(z) +� +, a.e. in Ω, +where for z = x + iy +µ(z) = da/dx(x) − db/dy(y) +da/dx(x) + db/dy(y) . +Acknowledgements. +The second author was supported by the Ministry of Science and Higher Edu- +cation of Russia (agreement No. 075-02-2022-884). + +DIRICHLET EIGENVALUES PROBLEM +14 +References +[1] Ahlfors, L., Lectures on quasiconformal mappings, D. Van Nostrand Co., Inc., Toronto, Ont.- +New York-London, 1966. +[2] Astala, K., Koskela, P., Quasiconformal mappings and global integrability of the derivative, +J. Anal. Math. 57 (1991), 203–220. +[3] Astala, K., Iwaniec, T., Martin, G., Elliptic partial differential equations and quasiconformal +mappings in the plane, Princeton University Press, Princeton and Oxford, 2008. +[4] Bojarski, B., Gutlyanski˘ı, V., Martio, O, Ryazanov, V., Infinitesimal geometry of quasicon- +formal and bi-Lipschitz mappings in the plane, EMS, Zurich (2013). +[5] Courant, R., Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces. Springer- +Verlag, Berlin-Heidelberg-New York (1977). +[6] Faber, G., “ Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher +Spannung die kreisförmige den tiefsten Grundton gibt”, Sitz. ber. bayer. Akad. Wiss. 169–172 +(1923). +[7] Gehring, F.W., Martio, O., Lipschitz classes and quasiconformal mappings. Ann. Acad. Sci. +Fenn. Ser. A I Math., 10, 203–219 (1985). +[8] Gol’dshtein, V., Pchelintsev, V., Ukhlov, A., Integral estimates of conformal derivatives and +spectral properties of the Neumann-Laplacian, J. Math. Anal. Appl. 463 (2018), 19–39. +[9] Gol’dshtein, V., Pchelintsev, V., Ukhlov, A., Spectral properties of the Neumann-Laplace +operator in quasiconformal regular domains, Differential Equations, Mathematical Physics, +and Applications. Selim Grigorievich Krein Centennial, Contemporary Mathematics, AMS, +734 (2019), 129–144. +[10] Gol’dshtein, V., Pchelintsev, V., Ukhlov, A., Spectral stability estimates of Dirichlet diver- +gence form elliptic operators, Anal. Math. Phys. 10 (2020), 74. +[11] Gol’dshtein, V., Pchelintsev, V., Ukhlov, A., Quasiconformal mappings and Neumann eigen- +values of divergent elliptic operators, Complex Var. Elliptic Equ., 67, No. 9 (2022), 2281– +2302. +[12] Gol’dshtein, V., Pchelintsev, V., Ukhlov, A., Estimates of Dirichlet eigenvalues of divergent +elliptic operators in non-Lipschitz domains, Journal of Mathematical Sciences, 268, No. 3 +(2022), 343–354. +[13] Gol’dshtein, V., Ukhlov, A., Weighted Sobolev spaces and embedding theorems, Trans. Amer. +Math. Soc. 361 (2009), 3829–3850. +[14] Grebenkov, D. S., Nguyen, B.-T., Geometrical Structure of Laplacian Eigenfunctions, SIAM +Review 55(4) (2013), 601–667. +[15] Gutlyanski˘ı, V., Nesmelova, O., Ryazanov, V., On quasiconformal maps and semi-linear +equations in the plane, J. Math. Sci. 229(1) (2018), 7–29. +[16] Henrot A. Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathe- +matics, Birkhäuser, 2006. +[17] Hurri, R., Poincaré domains in Rn, Ann. Acad. Sci. Fenn., Ser. A, I. Math., Dissertationes, +71 (1988), 1–42. +[18] Hurri-Syrjänen, R., Marola, N., Vähäkangas, A. V., Poincaré inequalities in quasihyperbolic +boundary condition domains, Manuscripta Math. 148 (2015), 99–118. +[19] Koskela, P., Onninen, J., Tyson, J. T., Quasihyperbolic boundary conditions and capacity: +Hölder continuity of quasiconformal mappings, Comment. Math. Helv. 76 (2001), 416–435. +[20] Koskela, P., Onninen, J., Tyson, J. T., Quasihyperbolic boundary conditions and capacity: +Poincaré domains, Math. Ann. 323 (2002), 811–830. +[21] Krahn, E., “Uber eine von Rayleigh formulierte Minimaleigenschaft des Kreises”, Math. Ann. +94, 97–100 (1925). +[22] Maz’ya, V., Sobolev Spaces. With Applications to Elliptic Partial Differential Equations, +Springer, Berlin (2011). +[23] Rohde, S., Quasicircles modulo bilipschitz maps, Rev. Mat. Iberoam., 17 (2001), 643–659. +[24] Ukhlov, A., On mappings, which induce embeddings of Sobolev spaces, Siberian Math. J. 34 +(1993), 185–192. +[25] Vodop’yanov, S.K., Gol’dstein, V.M., Lattice isomorphisms of the spaces W 1 +n and quasicon- +formal mappings. Siberian Math. J. 16, 224–246 (1975). +[26] Vodop’yanov, S. K., Gol’dstein, V. M., Reshetnyak, Yu. G., On geometric properties of +functions with generalized first derivatives, Uspekhi Mat. Nauk 34 (1979), 17–65. + +DIRICHLET EIGENVALUES PROBLEM +15 +[27] Vodop’yanov, S. K., Ukhlov, A. D., Superposition operators in Sobolev spaces, Izvestiya VUZ +46 (2002), 11–33. +Department of Mathematics, Ben-Gurion University of the Negev, P.O.Box 653, +Beer Sheva, 8410501, Israel +E-mail address: vladimir@math.bgu.ac.il +Regional Scientific and Educational Mathematical Center, Tomsk State Univer- +sity, 634050 Tomsk, Lenin Ave. 36, Russia +E-mail address: vpchelintsev@vtomske.ru + diff --git a/cdE1T4oBgHgl3EQfdgTc/content/tmp_files/load_file.txt b/cdE1T4oBgHgl3EQfdgTc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..944e65395e16e07b6f121d4d05cc55fc1aa4d39f --- /dev/null +++ b/cdE1T4oBgHgl3EQfdgTc/content/tmp_files/load_file.txt @@ -0,0 +1,476 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf,len=475 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='03197v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='AP] 9 Jan 2023 ON THE PRINCIPAL FREQUENCY OF NON-HOMOGENEOUS MEMBRANES V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' GOL’DSHTEIN, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' PCHELINTSEV Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' We obtained estimates for first eigenvalues of the Dirichlet bound- ary value problem for elliptic operators in divergence form (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' for the principal frequency of non-homogeneous membranes) in bounded domains Ω ⊂ C satis- fying quasihyperbolic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The suggested method is based on the quasiconformal composition operators on Sobolev spaces and their applica- tions to constant estimates in the corresponding Sobolev-Poincaré inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' We also prove a variant of the Rayleigh-Faber-Khran inequality for a special case of these elliptic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Introduction In this paper we give applications of the theory of quasiconformal mappings to the Dirichlet eigenvalue problem for two-dimensional elliptic operators in divergence form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='1) LAf(z) = −div[A(z)∇f(z)], z = (x, y) ∈ Ω, f(x, y) = 0 on ∂Ω, in bounded domains Ω ⊂ C satisfying quasihyperbolic boundary conditions [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' We assume that A ∈ M 2×2(Ω), where M 2×2(Ω) is the class of all 2 × 2 symmetric matrix functions A(z) = {akl(z)}, detA = 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' in Ω, with measurable entries satisfying to the uniform ellipticity condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='2) 1 K |ξ|2 ≤ ⟨A(w)ξ, ξ⟩ ≤ K|ξ|2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' in Ω, for every ξ ∈ C, where 1 ≤ K < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Such elliptic operators in divergence form arise in various problems of mathematical physics (see, for example, [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Under these conditions the matrix A induces a quasiconformal homeomorphism ϕA : Ω → �Ω which we call the A-quasiconformal mapping [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' This allow us to reduce (by quasiconformal change of variable) this elliptic operator in divergence form to the Laplace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' A domain Ω satisfies the γ-quasihyperbolic boundary condition with some γ > 0 if the growth condition on the quasihyperbolic metric kΩ(x0, x) ≤ 1 γ log dist(x0, ∂Ω) dist(x, ∂Ω) + C0 satisfied for all x ∈ Ω, where x0 ∈ Ω is a fixed base point and C0 = C0(x0) < ∞, [7, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Here kΩ(x1, x2) := inf ˆ γ ds dist(x, ∂Ω), 0Key words and phrases: Elliptic equations, Sobolev spaces, quasiconformal mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 02010 Mathematics Subject Classification: 35P15, 46E35, 30C65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 1 DIRICHLET EIGENVALUES PROBLEM 2 where the infimum is taken over all rectifiable curves γ joining x1 and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' This class of domains includes, in particular, domains with Lipschitz boundary, some domains with Hölder singularities, and domains of snowflakes type [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' A function f ∈ W 1,2 0 (Ω, A) is a solution to the generalized spectral problem for the elliptic operator in divergence form LAf(z) with the Dirichlet boundary condition if ¨ Ω ⟨A(z)∇f(z), ∇g(z)⟩ dxdy = λ ¨ Ω f(z)g(z) dxdy, ∀g ∈ W 1,2 0 (Ω, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' It is known [16, 22] that in a bounded domain Ω ⊂ C the operator LAf(z) with the Dirichlet boundary condition has discrete spectrum represented as the non-decreasing sequence 0 < λ1(A, Ω) ≤ λ2(A, Ω) ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' ≤ λn(A, Ω) ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' , where each eigenvalue is repeated as many time as its multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' By the min-max principle, the first eigenvalue λ1(A, Ω) is defined by λ1(A, Ω) = inf f∈W 1,2 0 (Ω,A)\\{0} ∥f | L1,2 A (Ω)∥2 ∥f | L2(Ω)∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The lower estimates for the first eigenvalues of the Laplace operator with the Dirichlet boundary condition in a bounded domain are connected by the Rayleigh- Faber-Khran inequality [6, 21] which means that the first Dirichlet eigenvalue in a bounded domain Ω is not less than the corresponding Dirichlet eigenvalue in the disc of the same area Ω∗ with R∗ as its radius, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', λ1(I, Ω) := λ1(Ω) ≥ λ1(Ω∗) = j2 0,1 R2∗ , where j0,1 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='4048 is the first positive zero of the Bessel function J0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' This inequal- ity was improved by the method based on the capacity theory [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Unfortunately, for the first eigenvalue λ1(A, Ω) Rayleigh-Faber-Khran type in- equality has not been proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' However, lower estimates for the first eigenvalues the operator LAf(z) with the Dirichlet boundary condition in bounded domains can be obtained easily by using the Rayleigh-Faber-Krahn inequality and the uniform ellipticity condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='2): Let Ω ⊂ C be a bounded domain such that |Ω| = |Ω∗| and K is the ellipticity constant of the matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='3) λ1(A, Ω) ≥ λ1(Ω) K ≥ λ1(Ω∗) K = j2 0,1 KR2∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' In this paper we obtain lower estimates for the first Dirichlet eigenvalues of the divergence form elliptic operators LAf(z) in bounded domains with some quasi- hyperbolic boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' We call such domains as β-regular domains for some β ∈ (1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The class of all β-regular domains coincides with the class of all domains with quasihyperbolic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Our machinery is based on connections between A-quasiconformal mappings [3, 4, 15] and composition operators on Sobolev spaces [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' DIRICHLET EIGENVALUES PROBLEM 3 One of the main results of the article states the following estimate for ∞-regular domains: If a simply connected bounded domain Ω ⊂ C satisfies to the quasihy- perbolic boundary condition, then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='4) λ1(A, Ω) ≥ λ1(�Ω) ∥Jϕ−1 A | L∞(�Ω)∥ , where λ1(�Ω) is the first Dirichlet eigenvalue of the Laplace operator and Jϕ−1 A is a Jacobian of the inverse mapping to the A-quasiconformal mapping ϕA : Ω → �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' A detailed discussion about β-regular domains can be found in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Note that if �Ω = Ω∗ and ∥Jϕ−1 A | L∞(Ω∗)∥ < K then estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='4) is better than estimates (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' For example, this condition is satisfied for measure preserving A- quasiconformal mappings ϕA : Ω → Ω∗ (|J(z, ϕA)| = 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' in Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Some examples can be found at the end of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Taking into account the domain monotonicity property for the Dirichlet eigen- values of the operator LAf(z) (see, for example, [16]) and estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='4) we obtain estimates for variations of the first Dirichlet eigenvalues of the operator LAf(z) un- der quasiconformal deformations of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Namely: Let Ω ⊂ C be a bounded ∞-regular domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' We assume that ϕA(Ω) := �Ω ⊃ Ω, then λ1(A, Ω) − λ1(�Ω) ≥ 1 − ∥Jϕ−1 A | L∞(�Ω)∥ ∥Jϕ−1 A | L∞(�Ω)∥ λ1(�Ω), where λ1(�Ω) is the first Dirichlet eigenvalue of the Laplace operator and Jϕ−1 A is a Jacobian of the inverse mapping to the A-quasiconformal mapping ϕA : Ω → �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' In the case of the measure preserving A-quasiconformal mappings ϕA : Ω → D (where D ⊂ C is the unit disc) we prove Rayleigh-Faber-Khran type inequality for the operator LAf(z): Let Ω ⊂ C be a simply connected bounded domain such that there exists a measure preserving A-quasiconformal mapping ϕA : Ω → D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Then λ1(A, Ω) ≥ λ1(A, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Sobolev spaces and A-quasiconformal mappings Let E ⊂ C be a measurable set on the complex plane and h : E → R be a positive almost everywhere (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=') locally integrable function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' a weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The weighted Lebesgue space Lp(E, h), 1 ≤ p < ∞, is the space of all locally integrable functions endowed with the following norm ∥f | Lp(E, h)∥ = \uf8eb \uf8ed ¨ E |f(z)|ph(z) dxdy \uf8f6 \uf8f8 1 p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The two-weighted Sobolev space W 1,p(Ω, h, 1), 1 ≤ p < ∞, is defined as the normed space of all locally integrable weakly differentiable functions f : Ω → R endowed with the following norm: ∥f | W 1,p(Ω, h, 1)∥ = ∥f | Lp(Ω, h)∥ + ∥∇f | Lp(Ω)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' DIRICHLET EIGENVALUES PROBLEM 4 In the case h = 1 this weighted Sobolev space coincides with the classical Sobolev space W 1,p(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The seminormed Sobolev space L1,p(Ω), 1 ≤ p < ∞, is the space of all locally integrable weakly differentiable functions f : Ω → R endowed with the following seminorm: ∥f | L1,p(Ω)∥ = ∥∇f | Lp(Ω)∥, 1 ≤ p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' We also need a weighted seminormed Sobolev space L1,2 A (Ω) (associated with the matrix A), defined as the space of all locally integrable weakly differentiable functions f : Ω → R with the finite seminorm given by: ∥f | L1,2 A (Ω)∥ = \uf8eb \uf8ed ¨ Ω ⟨A(z)∇f(z), ∇f(z)⟩ dxdy \uf8f6 \uf8f8 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The corresponding Sobolev space W 1,2(Ω, A) is defined as the normed space of all locally integrable weakly differentiable functions f : Ω → R endowed with the following norm: ∥f | W 1,2(Ω, A)∥ = ∥f | L2(Ω)∥ + ∥f | L1,2 A (Ω)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The Sobolev space W 1,2 0 (Ω, A) is the closure in the W 1,2(Ω, A)-norm of the space C∞ 0 (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' We consider the Sobolev spaces as Banach spaces of equivalence classes of func- tions up to a set of p-capacity zero [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Recall that a homeomorphism ϕ : Ω → �Ω, Ω, �Ω ⊂ C, is called a Q-quasiconformal mapping if ϕ ∈ W 1,2 loc (Ω) and there exists a constant 1 ≤ Q < ∞ such that |Dϕ(z)|2 ≤ Q|J(z, ϕ)| for almost all z ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Note that quasiconformal mappings have a finite distortion and possesses the Luzin N-property (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' a image of any set of measure zero has measure zero) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' If ϕ : Ω → �Ω is a Q-quasiconformal mapping then ϕ is differentiable almost everywhere in Ω and |J(z, ϕ)| = Jϕ(z) := lim r→0 |ϕ(B(z, r))| |B(z, r)| for almost all z ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Now we give a construction of A-quasiconformal mappings connected with the matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Recall that matrix functions A(z) = {akl(z)} with measurable entries akl(z) belongs to a class M 2×2(Ω) of all 2 × 2 symmetric matrix functions that satisfy to an additional condition detA = 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' and to the uniform ellipticity condition: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='1) 1 K |ξ|2 ≤ ⟨A(z)ξ, ξ⟩ ≤ K|ξ|2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' in Ω, for every ξ ∈ C and for some 1 ≤ K < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The basic idea is that every positive quadratic form ds2 = a11(x, y)dx2 + 2a12(x, y)dxdy + a22(x, y)dy2 defined in a planar domain Ω can be reduced, by means of a quasiconformal change of variables, to the canonical form ds2 = Λ(du2 + dv2), Λ ̸= 0, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' in �Ω, given that a11a22 − a2 12 ≥ κ0 > 0, a11 > 0, almost everywhere in Ω [1, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' DIRICHLET EIGENVALUES PROBLEM 5 By [4] any matrix A of the type under discussion induces a quasiconformal home- omorphism as a solution to the corresponding Beltrami equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The detailed procedure is described below Let ξ(z) = Re ϕ(z) be a real part of a quasiconformal mapping ϕ(z) = ξ(z) + iη(z), which satisfies to the Beltrami equation: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='2) ϕz(z) = µ(z)ϕz(z), a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' in Ω, where ϕz = 1 2 �∂ϕ ∂x − i∂ϕ ∂y � and ϕz = 1 2 �∂ϕ ∂x + i∂ϕ ∂y � , with the complex dilatation µ(z) given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='3) µ(z) = a22(z) − a11(z) − 2ia12(z) det(I + A(z)) , I = � 1 0 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' We call this quasiconformal mapping (with the complex dilatation µ defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='3)) as an A-quasiconformal mapping and we will use the notation ϕA for this quasiconformal mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Note that the uniform ellipticity condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='1) can be written as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='4) |µ(z)| ≤ K − 1 K + 1, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Conversely from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='3) (see, for example, [3], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 412) one can recover the matrix A : (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='5) A(z) = � |1−µ|2 1−|µ|2 −2 Im µ 1−|µ|2 −2 Im µ 1−|µ|2 |1+µ|2 1−|µ|2 � , a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Thus, for any A ∈ M 2×2(Ω) by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='4) can be produced the complex dilatation µ(z), for which, in turn, the Beltrami equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='2) induces an A-quasiconformal homeomorphism ϕ : Ω → �Ω as its solution (by the Riemann measurable mapping theorem (see, for example, [1])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Let us briefly say that A and ϕA are agreed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Therefore with the given A-divergent form elliptic operator defined in a domain Ω ⊂ C can be assochiated the A-quasiconformal mapping ϕA : Ω → �Ω with the quasiconformality coefficient QA = 1 + ∥µ | L∞(Ω)∥ 1 − ∥µ | L∞(Ω)∥, where µ defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' From the estimate |µ(z)| ≤ K−1 K+1 immediately follows that QA ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Note that the inverse mapping to the A-quasiconformal mapping ϕA : Ω → �Ω is the A−1-quasiconformal mapping [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' In [11] the relationship between composition operators on Sobolev spaces and A-quasiconformal mappings was studied and the following theorem was proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Let Ω, �Ω be domains in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Then a homeomorphism ϕA : Ω → �Ω is an A-quasiconformal mapping if and only if ϕ induces, by the composition rule ϕ∗(f) = f ◦ ϕ, an isometry of Sobolev spaces L1,2 A (Ω) and L1,2(�Ω) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' ∥ϕ∗ A(f) | L1,2 A (Ω)∥ = ∥f | L1,2(�Ω)∥ for any f ∈ L1,2(�Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' DIRICHLET EIGENVALUES PROBLEM 6 This theorem generalizes the well known property of conformal mappings gener- ate the isometry of uniform Sobolev spaces L1 2(Ω) and L1 2(�Ω) (see, for example, [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' It is also refines (in the case n = 2) the functional characterization of quasiconformal mappings in the terms of isomorphisms of uniform Sobolev spaces [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Estimate of the constant in Sobolev-Poincaré inequality In [12] was proved the following weighted Sobolev-Poincaré inequality for a bounded domain Ω ⊂ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' We denote by h(z) = |J(z, ϕA)| the quasihyperbolic weight defined by an A-quasiconformal mapping ϕA : Ω → �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Let A belongs to a class M 2×2(Ω) and Ω be a bounded simply connected planar domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Then for any function f ∈ W 1,2 0 (Ω, A) the following weighted Sobolev-Poincaré inequality \uf8eb \uf8ed ¨ Ω |f(z)|rh(z)dxdy \uf8f6 \uf8f8 1 r ≤ Cr,2(h, A, Ω) \uf8eb \uf8ed ¨ Ω ⟨A(z)∇f(z), ∇f(z)⟩ dxdy \uf8f6 \uf8f8 1 2 holds for any r ≥ 2 with the constant Cr,2(h, A, Ω) = Cr,2(�Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Here Cr,2(�Ω) is the best constant in the (non-weight) Sobolev-Poincaré inequality in a bounded domain �Ω ⊂ C with the upper estimate (see [10]): (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='1) Cr,2(�Ω) ≤ inf p∈( 2r r+2 ,2) �p − 1 2 − p � p−1 p �√π · p√ 2 �−1 |�Ω| 1 r � Γ(2/p)Γ(3 − 2/p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='1 we give an upper estimate for the constant in the Sobolev- Poincaré inequality in domains with the quasihyperbolic boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' As shown in [2], the Jacobians of quasiconformal mappings ψ : �Ω → Ω belong to Lβ(�Ω) for some β > 1 if and only if Ω has the γ-quasihyperbolic boundary condition for some γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' We note that β depends only on �Ω and the quasiconformal coefficient K(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Since we need the exact value of the integrability exponent β for the Jacobians of quasiconformal mappings, we consider an equivalent description of domains with quasihyperbolic boundary in terms of the integrability of Jacobians [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' A simply connected domain Ω ⊂ C is called an A-quasiconformal β-regular domain about a simply connected domain �Ω ⊂ C if ¨ �Ω |J(w, ϕ−1 A )|β dudv < ∞ for some β > 1, where ϕA : Ω → �Ω is a corresponding A-quasiconformal mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The property of the quasiconformal β-regularity implies the integrability of a Ja- cobian of quasiconformal mappings and therefore for any quasiconformal β-regular domain we have the embedding of weighted Lebesgue spaces Lr(Ω, h) into non- weighted Lebesgue spaces Ls(Ω) for s = β−1 β r [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' DIRICHLET EIGENVALUES PROBLEM 7 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [9] Let Ω be an A-quasiconformal β-regular domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Then for any function f ∈ Lr(Ω, h), β/(β − 1) ≤ r < ∞, the inequality ∥f | Ls(Ω)∥ ≤ \uf8eb \uf8ec \uf8ed ¨ �Ω |J(w, ϕ−1 A )|βdudv \uf8f6 \uf8f7 \uf8f8 1 β · 1 s ∥f | Lr(Ω, h)∥ holds for s = β−1 β r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' We ready to prove the upper estimate for the constant in the Sobolev-Poincaré inequality in quasiconformal regular domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Let A belong to a class M 2×2(Ω) and a domain Ω be A-quasi- conformal β-regular about �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Then: (1) for any function f ∈ W 1,2 0 (Ω, A) and for any s ≥ 1, the Sobolev-Poincaré inequality ∥f | Ls(Ω)∥ ≤ Cs,2(A, Ω)∥f | L1,2 A (Ω)∥ holds with the constant Cs,2(A, Ω) ≤ C βs β−1 ,2(�Ω)∥Jϕ−1 A | Lβ(�Ω)∥ 1 s , 1 < β < ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' (2) if β = ∞ then for any function f ∈ W 1,2 0 (Ω, A), the Sobolev-Poincaré inequality ∥f | L2(Ω)∥ ≤ C2,2(A, Ω)∥f | L1,2 A (Ω)∥ holds with the constant C2,2(A, Ω) ≤ C2,2(�Ω) ��Jϕ−1 A | L∞(�Ω) �� 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Here Jϕ−1 A is a Jacobian of the inverse mapping to the A-quasiconformal mapping ϕA : Ω → �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The constant C2 2,2(�Ω) = 1/λ1(�Ω), where λ1(�Ω) is the first Dirichlet eigenvalue of Laplacian in a domain �Ω ⊂ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Let β β−1 ≤ r and s = β−1 β r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' It means that such r exists for any s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='1 we get ∥f | Ls(Ω)∥ = \uf8eb \uf8ed ¨ Ω |f(z)|sdxdy \uf8f6 \uf8f8 1 s ≤ \uf8eb \uf8ec \uf8ed ¨ �Ω |J(w, ϕ−1 A )|βdudv \uf8f6 \uf8f7 \uf8f8 1 β · 1 s \uf8eb \uf8ed ¨ Ω |f(z)|r|J(ϕA, x, y)|dxdy \uf8f6 \uf8f8 1 r ≤ Cr,2(�Ω) \uf8eb \uf8ec \uf8ed ¨ �Ω |J(w, ϕ−1 A )|βdudv \uf8f6 \uf8f7 \uf8f8 1 β · 1 s \uf8eb \uf8ed ¨ Ω ⟨A(z)∇f(z), ∇f(z)⟩ dxdy \uf8f6 \uf8f8 1 2 for any s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' DIRICHLET EIGENVALUES PROBLEM 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Let a function f ∈ L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Since quasiconformal mappings possess the Luzin N-property, then |J(z, ϕA)|−1 = |J(w, ϕ−1 A )| for almost all z ∈ Ω and for almost all w = ϕA(z) ∈ �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Hence \uf8eb \uf8ed ¨ Ω |f(z)|2 dxdy \uf8f6 \uf8f8 1 2 = \uf8eb \uf8ed ¨ Ω |f(z)|2|J(z, ϕA)|−1|J(z, ϕA)| dxdy \uf8f6 \uf8f8 1 2 ≤ ∥JϕA | L∞(Ω)∥− 1 2 \uf8eb \uf8ed ˆ Ω |f(z)|2|J(z, ϕA)| dxdy \uf8f6 \uf8f8 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Applying the change of variable formula for quasiconformal mappings [26], (non- weighed) Sobolev-Poincaré inequality [10], and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='1 we obtain \uf8eb \uf8ed ¨ Ω |f(z)|2 dxdy \uf8f6 \uf8f8 1 2 ≤ ∥Jϕ−1 A | L∞(�Ω)∥ 1 2 \uf8eb \uf8ec \uf8ed ¨ �Ω |f ◦ ϕ−1 A (w)|2 dudv \uf8f6 \uf8f7 \uf8f8 1 2 ≤ C2,2(�Ω)∥Jϕ−1 A | L∞(�Ω)∥ 1 2 \uf8eb \uf8ec \uf8ed ¨ �Ω |∇(f ◦ ϕ−1 A (w))|2 dudv \uf8f6 \uf8f7 \uf8f8 1 2 = C2,2(�Ω)∥Jϕ−1 A | L∞(�Ω)∥ 1 2 \uf8eb \uf8ed ¨ Ω ⟨A(z)∇f(z), ∇f(z)⟩ dxdy \uf8f6 \uf8f8 1 2 for any f ∈ W 1,2 0 (Ω, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Lower estimates for λ1(A, Ω) We consider the generalized formulation of the Dirichlet eigenvalue problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='1): ¨ Ω ⟨A(z)∇f(z), ∇g(z)⟩ dxdy = λ ¨ Ω f(z)g(z) dxdy, ∀g ∈ W 1,2 0 (Ω, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' By the min-max principle (see, for example, [16, 22]) the first Dirichlet eigenvalue λ1(A, Ω) of the elliptic operator in divergence form LAf(z) is defined by λ1(A, Ω) = inf f∈W 1,2 0 (Ω,A)\\{0} ∥f | L1,2 A (Ω)∥2 ∥f | L2(Ω)∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' In other words, λ − 1 2 1 (A, Ω) is the exact constant C2,2(A, Ω) in the Sobolev-Poincaré inequality ∥f | L2(Ω)∥ ≤ C2,2(A, Ω)∥f | L1,2 A (Ω)∥, f ∈ W 1,2 0 (Ω, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' DIRICHLET EIGENVALUES PROBLEM 9 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Let A belong to the class M 2×2(Ω) and Ω be an A-quasiconformal β-regular domain about a domain �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Then 1 λ1(A, Ω) ≤ C2 2β β−1 ,2(�Ω)∥Jϕ−1 A | Lβ(�Ω)∥, where Jϕ−1 A is a Jacobian of the inverse mapping to the A-quasiconformal mapping ϕA : Ω → �Ω and C 2β β−1 ,2(�Ω) ≤ inf p∈( 2β 2β−1 ,2) �p − 1 2 − p � �√π · p√ 2 �−1 |�Ω| β−1 2β � Γ(2/p)Γ(3 − 2/p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' By the min-max principle and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='3 in the case s = 2, we have ¨ Ω |f(z)|2 dxdy ≤ C2 2,2(A, Ω) ¨ Ω ⟨A(z)∇f(z), ∇f(z)⟩ dxdy, where C2,2(A, Ω) ≤ C 2β β−1 ,2(�Ω) \uf8eb \uf8ec \uf8ed ¨ �Ω |J(w, ϕ−1 A )|β dudv \uf8f6 \uf8f7 \uf8f8 1 2β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Thus 1 λ1(A, Ω) ≤ C2 2β β−1 ,2(�Ω) \uf8eb \uf8ec \uf8ed ¨ �Ω |J(w, ϕ−1 A )|β dudv \uf8f6 \uf8f7 \uf8f8 1 β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' □ In the limit case β = ∞ we have the following assertion: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Let A belong to a class M 2×2(Ω) and Ω be an A-quasiconformal ∞-regular domain about a domain �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='1) 1 λ1(A, Ω) ≤ C2 2,2(�Ω)∥Jϕ−1 A | L∞(�Ω)∥ = ∥Jϕ−1 A | L∞(�Ω)∥ λ1(�Ω) , where Jϕ−1 A is a Jacobian of the inverse mapping to the A-quasiconformal mapping ϕA : Ω → �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' As an application of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='2 we consider several examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The homeomorphism ϕ(z) = a a2 − b2 z − b a2 − b2 z, z = x + iy, a > b ≥ 0, is an A-quasiconformal and maps the right triangle with arbitrary angels Ω = � (x, y) ∈ R2 : 0 ≤ x ≤ a + b, 0 ≤ y ≤ −a − b a + bx + (a − b) � onto the 45◦ right triangle �Ω = � (u, v) ∈ R2 : 0 ≤ u ≤ 1, 0 ≤ v ≤ 1 − u � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' DIRICHLET EIGENVALUES PROBLEM 10 The mapping ϕ satisfies the Beltrami equation with µ(z) = ϕz ϕz = − b a and the Jacobian J(z, ϕ) = |ϕz|2 − |ϕz|2 = 1/(a2 − b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' It is easy to verify that µ induces, by formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='5), the matrix function A(z) form A(z) = � a+b a−b 0 0 a−b a+b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Given that |J(w, ϕ−1)| = |J(z, ϕ)|−1 = a2 − b2 and λ1(�Ω) = 5π2 (see, for example, [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Then by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='2 we have λ1(A, Ω) ≥ λ1(�Ω) ∥Jϕ−1 | L∞(�Ω)∥ = 5π2 a2 − b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The homeomorphism ϕ(z) = 2 · z 3 8 z 1 8 − 1, ϕ(0) = −1, z = x + iy, is an A-quasiconformal and maps the interior of the non-convex domain Ω := � (ρ, θ) ∈ R2 : ρ = cos4 �θ 2 � , −π ≤ θ ≤ π � onto the unit disc D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The mapping ϕ satisfies the Beltrami equation with µ(z) = ϕz ϕz = −1 3 z z and the Jacobian J(z, ϕ) = |ϕz|2 − |ϕz|2 = 1 2 · |z| 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' We see that µ induces, by formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='5), the matrix function A(z) form A(z) = � |3z+z|2 8|z|2 3 4 Im z z 3 4 Im z z |3z−z|2 8|z|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Given that |J(w, ϕ−1)| = |J(z, ϕ)|−1 = 2 · |z| 3 2 and λ1(D) = (j0,1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Then by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='2 we have λ1(A, Ω) ≥ λ1(D) ∥Jϕ−1 | L∞(D)∥ ≥ (j0,1)2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Taking into account Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='2 and the domain monotonicity property of the Dirichlet eigenvalues for the elliptic operator in divergence form LAf(z), we obtain the following result for the special case ∥Jϕ−1 A | L∞(�Ω)∥ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Let Ω be an A-quasiconformal ∞-regular domain about �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' We assume that �Ω ⊃ Ω, then λ1(A, Ω) − λ1(�Ω) ≥ 1 − ∥Jϕ−1 A | L∞(�Ω)∥ ∥Jϕ−1 A | L∞(�Ω)∥ λ1(�Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' DIRICHLET EIGENVALUES PROBLEM 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Since �Ω ⊃ Ω, we have λ1(A, Ω) ≥ λ1(�Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Taking into account the inequal- ity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='1) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='2 and making elementary calculation, we get λ1(A, Ω) − λ1(�Ω) ≥ 1 − ∥Jϕ−1 A | L∞(�Ω)∥ ∥Jϕ−1 A | L∞(�Ω)∥ λ1(�Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The Rayleigh-Faber-Krahn type inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The theory of composition operators [11] allows us to reduce the spectral problem for the divergence form elliptic operator (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='1) defined in simply connected bounded domain Ω ⊂ C to a weighted spectral problem for the Laplace operator in a simply connected bounded domain �Ω ⊂ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' By the chain rule applied to a function f(z) = g ◦ ϕA(z) [15], we have −div[A(z)∇f(z)] = −div[A(z)∇g ◦ ϕA(z)] = −h(w)∆g(w), where the weight h(w) = |J(w, ϕ−1 A )|−1 is a Jacobian of the inverse mapping to the A-quasiconformal mapping ϕA : Ω → �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' From here we can point out that λ1(A, Ω) = inf f∈W 1,2 0 (Ω,A)\\{0} ˜ Ω ⟨A(z)∇f, ∇f⟩ dxdy ˜ Ω |f|2dxdy = inf g∈W 1,2 0 (�Ω,h,1)\\{0} ˜ �Ω |∇g|2dudv ˜ �Ω |g|2h(w)dudv = λ1(h, �Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Let ϕA : Ω → �Ω be A-quasiconformal mappings for which |J(z, ϕA)| = 1 for almost all z ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' In this case quasiconformal weights h(w) = |J(w, ϕ−1 A )| = 1 for almost all w ∈ �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Hence we get that λ1(A, Ω) = λ1(�Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Using this equality and the classical Rayleigh-Faber-Krahn inequality we imme- diately obtain a version of this inequality for elliptic operators in divergence form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Namely: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Let Ω ⊂ C be a simply connected bounded domain such that there exists a measure preserving A-quasiconformal mapping ϕA : Ω → D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Then λ1(A, Ω) ≥ λ1(A, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' A description of the class of measure preserving A-quasiconformal mappings and/or corresponding divergence form elliptic equations is an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Let us give simple examples of such mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Let ϕ(z) = (ax, 1 ay), z = x + iy and a > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Then J(z, ϕ) = 1, the quasiconformality coefficient Qϕ = a2 and µϕ = a2−1 a2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The matrix can be easily recontracted A(z) = � 1 a2 0 0 a2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' A little bit more complicated example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' DIRICHLET EIGENVALUES PROBLEM 12 Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Let ϕ(z) = (ax + by, 1 ay), z = x + iy and a > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Then J(z, ϕ) = 1, the quasiconformality coefficient Qϕ = a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Calculation of µϕ is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' We use the Beltrami equation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='e ϕz = 1 2(a + 1 a) − i b 2, ϕz = 1 2(a − 1 a) + i b 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Hence µϕ = (a2 − 1)(a2 + 1) − a2b2 (a2 + 1)2 + a2b2 + i 2a3b (a2 + 1)2 + a2b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The matrix A can be easily reconstracted by elementary calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Let f ∈ L1 ∞(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Then ϕ(z) = (x + f(y), y), z = x + iy, is a quasiconformal mapping with |J(z, ϕ)| = 1 (see, [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' A basic calculation implies ϕz = 1 − if ′(y) 2 , ϕz = if ′(y) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Hence µϕ = − (f ′(y))2 4 + (f ′(y))2 + i 2f ′(y) 4 + (f ′(y))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The matrix can be easily recontracted A(z) = � 1 −f ′(y) −f ′(y) 1 + (f ′(y))2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' This algorithm can be used for more complicated examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Appendix Few remarks about measure preserving and quasi-preserving (bi-Lipschitz) qua- siconformal mappings andA-quasiconformal mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' A quasiconformal mapping ϕ is a solution of the corresponding Beltrami equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='1) ϕz(z) = µ(z)ϕz(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Because J(z, ϕ) = |ϕ2 z|(z) − |ϕ2 z|(z), the condition J(z, ϕ) = 1 can be written as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='2) |ϕz|2(z)(1 − |µ(z)|2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Recall that for A-quasiconformal mappings we have µ(z) = a22(z) − a11(z) − 2ia12(z) det(I + A(z)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' By elementary calculations it is easy to verify that any measure preserving qua- siconformal mapping ϕ : Ω → �Ω is a bi-Lipschitz mapping in the following sense: D(ϕ) ∈ L1 ∞(Ω) and D(ϕ−1) ∈ L1 ∞(�Ω) (or Dϕ ∈ L∞(Ω) and Dϕ−1 ∈ L∞(�Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' An equivalent geometric description is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The mapping ϕ and its inverse are locally Lipschitz homeomorphisms with uniformly bounded Lipschitz constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='2) can be soften in the spirit of quasiconformality up to the inequality 1 C < J(z, ϕ) < C for some positive constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' We call such homeomorphism as quasi-preserving measure homeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' DIRICHLET EIGENVALUES PROBLEM 13 For quasiconformal quasi-preserving measure homeomorphisms this condition can be written as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='3) 1 C < |ϕ2 z|(z)(1 − |µ(z)|2) < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The class of quasiconformal quasi-preserving measure homeomorphisms coincide with the class of bi-Lipschitz homeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The constant C can be easily recalculated in terms of uniform Lipschitz constants for the corresponding homeo- morphism and its inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' We do not have any geometric description of such homeomorphisms i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='e to solution of systems ((5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='1),(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='2)) or ((5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='1),(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Let us look for some simplified cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Suppose the matrix A is a diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Then the quasiconformality coeffi- cient µ(z) is a real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' If a11(z) > 0 then 0 < µ(z) < 1 and the condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='2) can be simplified: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='4) |ϕz|2(z)(1 − µ(z)2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' By simple calculations µ(z) = a11(z) + a−1 11 (z) 2 + a11(z) + a−1 11 (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Let us give an example of such homeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Suppose Ω := (0, 1) × (0, 1) and ϕ(x, y) = a(x) + ib(y) where a(x) and b(y) belong to C1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' We also suppose that inf x∈(0,1)(da/dx) > sup y∈(0,1) (db/dy) and I1 := inf y∈(0,1)(db/dy) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Any such mapping is a quasi-preserving measure one, because I1 ≤ J(z, ϕ) ≤ � |da/dx|C1(Ω) × |db/dy|C1(Ω) � and quasiconformal one because Q ≤ sup x∈(0,1) (da/dx) inf y∈(0,1)(db/dy) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The corresponding matrix A can be easily reconstructed (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='5) A(z) = � 1−µ(z) 1+µ(z) 0 0 1+µ(z) 1−µ(z) � , a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' in Ω, where for z = x + iy µ(z) = da/dx(x) − db/dy(y) da/dx(x) + db/dy(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' The second author was supported by the Ministry of Science and Higher Edu- cation of Russia (agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 075-02-2022-884).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' DIRICHLET EIGENVALUES PROBLEM 14 References [1] Ahlfors, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Lectures on quasiconformal mappings, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Van Nostrand Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Toronto, Ont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='- New York-London, 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [2] Astala, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Koskela, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Quasiconformal mappings and global integrability of the derivative, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 57 (1991), 203–220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [3] Astala, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Iwaniec, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Martin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton University Press, Princeton and Oxford, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [4] Bojarski, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Gutlyanski˘ı, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Martio, O, Ryazanov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Infinitesimal geometry of quasicon- formal and bi-Lipschitz mappings in the plane, EMS, Zurich (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [5] Courant, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Springer- Verlag, Berlin-Heidelberg-New York (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [6] Faber, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', “ Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt”, Sitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' ber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' bayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Wiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 169–172 (1923).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [7] Gehring, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Martio, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Lipschitz classes and quasiconformal mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Fenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' A I Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', 10, 203–219 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [8] Gol’dshtein, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Pchelintsev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Ukhlov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Integral estimates of conformal derivatives and spectral properties of the Neumann-Laplacian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 463 (2018), 19–39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [9] Gol’dshtein, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Pchelintsev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Ukhlov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Spectral properties of the Neumann-Laplace operator in quasiconformal regular domains, Differential Equations, Mathematical Physics, and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Selim Grigorievich Krein Centennial, Contemporary Mathematics, AMS, 734 (2019), 129–144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [10] Gol’dshtein, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Pchelintsev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Ukhlov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Spectral stability estimates of Dirichlet diver- gence form elliptic operators, Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 10 (2020), 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [11] Gol’dshtein, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Pchelintsev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Ukhlov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Quasiconformal mappings and Neumann eigen- values of divergent elliptic operators, Complex Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Elliptic Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', 67, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 9 (2022), 2281– 2302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [12] Gol’dshtein, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Pchelintsev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Ukhlov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Estimates of Dirichlet eigenvalues of divergent elliptic operators in non-Lipschitz domains, Journal of Mathematical Sciences, 268, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 3 (2022), 343–354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [13] Gol’dshtein, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Ukhlov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Weighted Sobolev spaces and embedding theorems, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 361 (2009), 3829–3850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [14] Grebenkov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Nguyen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Geometrical Structure of Laplacian Eigenfunctions, SIAM Review 55(4) (2013), 601–667.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [15] Gutlyanski˘ı, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Nesmelova, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Ryazanov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', On quasiconformal maps and semi-linear equations in the plane, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 229(1) (2018), 7–29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [16] Henrot A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathe- matics, Birkhäuser, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [17] Hurri, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Poincaré domains in Rn, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Fenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' A, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Dissertationes, 71 (1988), 1–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [18] Hurri-Syrjänen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Marola, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Vähäkangas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Poincaré inequalities in quasihyperbolic boundary condition domains, Manuscripta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 148 (2015), 99–118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [19] Koskela, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Onninen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Tyson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Quasihyperbolic boundary conditions and capacity: Hölder continuity of quasiconformal mappings, Comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Helv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 76 (2001), 416–435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [20] Koskela, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Onninen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Tyson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Quasihyperbolic boundary conditions and capacity: Poincaré domains, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 323 (2002), 811–830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [21] Krahn, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', “Uber eine von Rayleigh formulierte Minimaleigenschaft des Kreises”, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 94, 97–100 (1925).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [22] Maz’ya, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Sobolev Spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' With Applications to Elliptic Partial Differential Equations, Springer, Berlin (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [23] Rohde, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Quasicircles modulo bilipschitz maps, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Iberoam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', 17 (2001), 643–659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [24] Ukhlov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', On mappings, which induce embeddings of Sobolev spaces, Siberian Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 34 (1993), 185–192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [25] Vodop’yanov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Gol’dstein, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Lattice isomorphisms of the spaces W 1 n and quasicon- formal mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Siberian Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 16, 224–246 (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' [26] Vodop’yanov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Gol’dstein, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Reshetnyak, Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', On geometric properties of functions with generalized first derivatives, Uspekhi Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Nauk 34 (1979), 17–65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' DIRICHLET EIGENVALUES PROBLEM 15 [27] Vodop’yanov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Ukhlov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=', Superposition operators in Sobolev spaces, Izvestiya VUZ 46 (2002), 11–33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' Department of Mathematics, Ben-Gurion University of the Negev, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='Box 653, Beer Sheva, 8410501, Israel E-mail address: vladimir@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='bgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='il Regional Scientific and Educational Mathematical Center, Tomsk State Univer- sity, 634050 Tomsk, Lenin Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content=' 36, Russia E-mail address: vpchelintsev@vtomske.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} +page_content='ru' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfdgTc/content/2301.03197v1.pdf'} diff --git a/e9AzT4oBgHgl3EQfMPso/content/tmp_files/2301.01127v1.pdf.txt b/e9AzT4oBgHgl3EQfMPso/content/tmp_files/2301.01127v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f897aab6cf8c23bdcd1069ade9f75b48a96b99bd --- /dev/null +++ b/e9AzT4oBgHgl3EQfMPso/content/tmp_files/2301.01127v1.pdf.txt @@ -0,0 +1,2341 @@ + + +Gender Diversity in Ownership and Firm Innovativeness in Emerging Markets. The +Mediating Roles of R&D Investments and External Capital +Vartuhi Tonoyan +California State University, Fresno +Craig School of Business +4245 N. Backer Avenue, M/S PB7 +Fresno, CA 93740 US +vtonoyan@csufresno.edu + +Christopher J. Boudreaux +Florida Atlantic University +College of Business +777 Glades Road, Kaye Hall 145 +Boca Raton, FL 33431 US +cboudreaux@fau.edu + +Abstract. Despite recent evidence linking gender diversity in the firm with firm innovativeness, +we know little about the underlying mechanisms. Building on and extending the Upper Echelon +and entrepreneurship literature, we address two lingering questions: why and how does gender +diversity in firm ownership affect firm innovativeness? We use survey data collected from 7,848 +owner-managers of SMEs across 29 emerging markets to test our hypotheses. Our findings +demonstrate that firms with higher gender diversity in ownership are more likely to invest in +R&D and rely upon a breadth of external capital, with such differentials explaining sizeable +proportions of the higher likelihood of overall firm innovativeness, product and process, as well +as organizational and marketing innovations exhibited by their firms. Our findings are robust to +corrections for alternative measurement of focal variables, sensitivity to outliers and subsamples, +and endogenous self-selection concerns. + +JEL codes: J16, L26, M13, M14, O31, O32 + +Keywords: Gender diversity, firm innovation, mediating mechanisms, R&D, external capital + + +1 + +1. Introduction +A growing body of literature has been accumulating in two management disciplines that share a +common focus on gender and firm innovativeness: Upper Echelon and entrepreneurship. Despite +their common interest, the two research streams have mainly developed in parallel. As a result, +the findings from one area have rarely informed the other. +Studies examining the impact of gender diversity in the firm on firm innovativeness is an +example of the accelerating work within the first stream, the Upper Echelon literature. Our +review of the burgeoning research (summarized in the online appendix A1-A2) suggests, +however, that extant research faces two limitations. First and most important, the majority of +scholarship has not studied the question of why gender diversity influences firm innovativeness. +Understanding "the nuts and bolts" of a managerial phenomenon, i.e., interdependencies among +focal constructs and their underlying mechanisms, is a fundamental feature of theory +development (Pelled, 1996). This knowledge is also essential for managers and entrepreneurs +interested in designing processes to trigger specific organizational outcomes. Yet, virtually all of +the investigations conducted to date (Almor et al., 2019; Biga-Diambeidou, 2019; Faems and +Subramanian, 2013; Garcia-Martinez et al., 2017; Horbach and Jacob, 2018; Ruiz-Jiménez et al., +2016; Ritter-Hayashi, 2019; Talke et al., 2010; Teruel and Segarra, 2017; Xie et al., 2020), have +not examined potential explanatory factors. Therefore, Pelled's (1996, p. 616) conclusion from +almost 25 years ago that demographic diversity research takes "intervening processes for +granted" appears to remain true today. Second, despite a growing recognition within both +scholarly (Audretsch et al., 2011; Rosenbusch et al., 2011) and policy-oriented research (OECD, +2005, 2011) that innovation should not be conceived solely in terms of new products or +technological process innovations, of the existing studies (reviewed in the online appendix), only + +2 + +Teruel and Segara (2017), Garcia-Martinez et al. (2017), and Díaz-García et al. (2013) examined +additional innovative outputs such as the introduction of new marketing or organizational +methods. +Although emerging scholarship from the second stream, on the "gendering" of +entrepreneurship and innovation, has been more informative in unpacking some of the +mechanisms explaining male-female differentials in the innovativeness of small and/or +entrepreneurial firms by focusing on either the characteristics of the entrepreneur or their +institutional environments (Strohmeyer and Tonoyan, 2005; Strohmeyer, Tonoyan, and Jennings, +2017; Thébaud, 2015a and 2015b), this literature has largely neglected to study the implications +of gender diversity. That is, prior entrepreneurship research, with a few exceptions (Horbach and +Jacob, 2018; Dai et al., 2019; Na and Shin, 2019; Ritter-Hayashi et al., 2019), has generally +ignored businesses owned by the mixed-gender-teams as a focus group (Kim, 2006). This +omission is unfortunate not only because it neglects a significant share of firms owned-led by +both males and females (Parker, 2018), but also because it fails to answer the question of +whether and why mixed-gender-owner teams differ from gender-homogeneous teams (i.e., those +owned by all-males and all-females) with regard to their innovativeness. +Our research overcomes these limitations by both synthesizing insights from and +extending the Upper Echelon and entrepreneurship literature. We address the above-noted gaps +by theorizing why and how gender diversity in the ownership of small-and-medium-sized +enterprises (SMEs) affects firm innovativeness. We delineate some of the mechanisms behind +the “gender diversity in firm ownership and firm innovativeness” nexus considering the role of a +firm's investments in research and development (R&D) and its financial resource endowments. +The crux of our argument is that mixed-gender-owner teams will exhibit higher overall levels of + +3 + +firm innovativeness, and this tendency will be partially attributable to their higher likelihood of +investing in R&D and relying on a breadth of external capital. Second, we focus on distinct +dimensions of a firm's innovation output not yet examined in prior work — overall +innovativeness, product and process innovations, and organizational and marketing innovations. +Such inclusive conceptualization of firm innovativeness is essential, as it attends to various calls +in both scholarly (Rosenbush et al., 2011) and policy literature (OECD, 2005, 2011) to study +outputs of innovation beyond technological product and process innovation (Ayyagari et al., +2011; Strohmeyer et al., 2017; Boudreaux et al., 2019). We use the Business Environment and +Enterprise Performance Survey (BEEPS) collected during 2012-2016 comprising a large sample +of 7,848 randomly-selected public (i.e., stock-listed) and private (i.e., non-listed) firms from 29 +emerging markets. Such a sample increases generalizability of our findings to the "understudied +private firms" (Tyrowicz et al., 2019) in the literature and institutional contexts different from +those of a developed economy (Jennings et al., 2013) mostly examined in prior work. +We test our arguments using multilevel (i.e., hierarchical) linear regression to account for +the nesting of firms within countries and adjusting for numerous controls at the firm and country- +levels. We conduct multiple robustness checks to assess the stability and reliability of our +findings. Our analyses provide reliable and robust support for our hypotheses contributing to a +more comprehensive understanding of the relationship between gender diversity and firm +innovativeness. Specifically, they reveal differences in the innovativeness of firms led by the +mixed-gender versus gender-homogeneous owning teams, documenting that the former exhibit +higher likelihoods of overall firm innovativeness, product and process innovations, as well as +organizational and marketing innovations, even after correcting for the firm’s selection into +higher- versus lower-innovation industries. Most fundamentally, our findings unearth some of + +4 + +the underlying mechanisms that have been overlooked in prior work, demonstrating the +intervening roles played by a firm’s investment in R&D and reliance upon a breadth of external +capital. Combined, these findings offer the merit of considering alternatives to the predominant +argument linking mixed-gender-owning teams with firm financial performance (for meta- +analyses and reviews see, e.g., Jeong and Harrison, 2017, and Eagly, 2016). They reveal +important implications for the Upper Echelon and entrepreneurship literature, as well as +managers and public policy. +2. Literature review +It has been traditional for scholars in the Upper Echelon literature to study the +implications of the gender diversity on corporate boards for firm financial performance, +including stock market prices and volatility, return on assets, sales, and Tobin’s Q (for reviews +see Terjesen et al. 2009; Post and Byron, 2014; Jeong and Harrison, 2017). Grounded in +management, learning theories, and social psychology, the crux of the argument is that gender +diversity enhances firm financial performance due to complementarity in skills, networks, and +behaviors of male and female executives. However, rigorously conducted meta-analyses +(Terjesen et al. 2009; Post and Byron, 2014; Jeong and Harrison, 2017) and reviews (Eagly, +2016) concluded that beyond finding "small" and "weak" associations (Jeong and Harrison, +2017), empirical studies had not provided strong and robust support for the theory's central +tenets. Some suggested female executives have small "decision latitude" on the financial +performance of publicly-traded firms (Jeong and Harrison, 2017) since their appointment to +corporate boards is often symbolic to "window-dress" the organization as a workplace promoting +gender equality (Eagly, 2016). Also, in-group favoritism marginalizes female executives' +influence on decision-making due to their "tokenism" or outgroup minority status (Kanter, 1977). + +5 + +In light of these findings, scholars have suggested: (a) studying organizational contexts +different from corporate boards of public firms that are more gender-inclusive providing +considerable "decision latitude" to women executives and thus accentuating their impact on firm +strategy (Jeong and Harrison, 2017), and (b) examining performance outcomes different from +financial measures since gains of profit and productivity "are not the most appropriate place to +look for diversity's benefits" (Eagly, 2013:224). Many have called to study firm innovation +suggesting that gender diversity likely leads to more creativity and novel thinking within a firm, +increasing its overall innovativeness (Dezso¨ and Ross, 2012; Eagly, 2016, 2013). +Indeed, emerging research has begun to analyze the question of whether gender diversity +in the firm increases firm innovativeness, and if so, how and why. While most of the burgeoning +literature (summarized in online appendix tables A1-A2) suggests it does, others do not (Biga- +Diambeidou et al., 2019). It is important to note, however, that scholars employed different +measures complicating the comparability of their findings. While some examined gender +diversity on corporate boards (e.g., Horbach and Jacob, 2018), others explored gender diversity +among employees (e.g., Østergaard et al., 2011). Many studies also adopted relatively narrow +measures of innovation mostly focusing either on new product and technological process +innovations (e.g., Dai et al., 2019; Ruiz-Jiménez et al., 2016), patent applications (e.g., Faems +and Subramanian, 2013), or inputs to innovation process such as R&D intensity (e.g., Almor et +al., 2019; Xie et al., 2020). +Unfortunately, our current understanding of why gender diversity influences firm +innovation is also limited. As indicated in our review of the emerging literature (summarized in +online appendix tables A1-A2), the overwhelming majority of the studies have not studied the + +6 + +mediating constructs linking gender diversity with firm innovativeness to explain the nature of +that relationship. +3. Theory and hypotheses +Heeding the calls in the Upper Echelon literature to study gender-inclusive organizational +contexts (Jeong and Harrison, 2017; Eagly, 2016 and 2013; Dezso and Ross, 2012), we focus on +gender diversity in firm ownership. Compared to corporate boards and firm workforce, firm +ownership seems a more suitable context for studying the implications of gender diversity for +firm innovativeness. A firm’s ownership structure aligns both incentives and actions of owners +(Fama and Jensen, 1983). Hence, women’s involvement in the ownership of a firm implies that +they will have "decision latitude" being able to shape a firm’s strategic choices (Jeong and +Harrison, 2017). Also, while some studies noted potential conflicts between CEOs and in work +teams that are likely to arise due to diversity of backgrounds, expertise, or values, such conflicts +are less likely to occur between owner-managers because such members self-select into founding +and running the firm encouraging a reciprocal exchange and collaboration (Dai et al., 2019). +We suggest that firm innovativeness should be conceptualized according to the product, +process, organizational, and marketing innovation domains in which the firm has introduced +something novel (Ayyagari et al., 2011; Strohmeyer et al. 2017). Such broader focus accords +with Schumpeter's conceptualization of firm innovation encompassing new products, markets, +production processes, and organizing methods (Schumpeter, 1934). It is also consistent with +OECD guidelines (OECD, 2005, 2011) and recent calls to study novel outputs beyond new +product developments or those that are primarily technological (Rosenbush et al., 2011; +Strohmeyer et al., 2017; Boudreaux et al., 2019). + +7 + + We posit two factors as key explanatory mechanisms underlying the relationship +between gender diversity in firm ownership and firm innovativeness. The first concerns the +firm's internal dynamics focusing on the decision to allocate resources to R&D. The second +discusses the role of external capital. +R&D investments are a critical precursor to firm innovation (Schilling and Green, 2011). +Innovations from R&D need not be technological inventions: they also include commercial +discoveries, upgrades to existing products and processes, and new business models (Parker 2018: +587). Although R&D often fails to yield commercial outcomes locking firms into strategies that +are difficult to change, firms across a variety of industries invest in R&D to develop new +products and services and replace those threatened by substitutes (Shane 2009). +We contend that mixed-gender owning teams are more likely to invest in R&D for +several reasons (Parker, 2018). First, by increasing the range of perspectives and cognitive +resources available internally within the firm (Schilling and Green, 2011), gender diversity +broadens the firm's knowledge base, thereby facilitating R&D undertakings (Almor et al., 2019; +Miller and Triana, 2009). Second, gender diversity influences how firms recombine internal +knowledge in novel ways through interaction, discussion, and mutual learning (e.g., Xie et al., +2020). Third, it expands an organization's search activities and external reach, thereby boosting +the firm's absorptive capacity (Cohen and Levinthal, 1990) — a key determinant of R&D +investments (Parker, 2018). Fourth, gender diversity also increases dispositional preferences for +novelty and change, committing managers and employees to continuous exploration and +implementation of creative ideas (Baron and Tang, 2011; Strohmeyer et al., 2017). +Although a handful of studies provide empirical support for the above-noted arguments +demonstrating positive associations between various measures of gender diversity (e.g., in + +8 + +executive positions or R&D workforce) and R&D intensity at the firm level (Miller and Triana, +2009; Biga-Diambeidou et al., 2019; Almor et al., 2019; Xie et al., 2020), none have investigated +the link between gender diversity in firm ownership, R&D investments, and firm innovation. +We posit external capital as a second intervening mechanism. External capital includes +funding from banks, equity funds, micro-finance organizations, as well as suppliers, customers, +family, and friends (Manolova et al., 2006). It is the "lifeblood" of firm innovation as it provides +adequate capitalization vital for the development and diffusion of innovation (Gorodnichenko +and Schnitzer, 2013). It also signals legitimacy, i.e., credibility and acceptance of the business by +the stakeholders (Godwin, Stevens and Brenner, 2006; Beckman, Burton, and O'Reilly, 2007). +Under-capitalization creates bottlenecks at various stages of innovation development, including +the inability to hire and retain qualified personnel, conduct R&D, and market products and +services (Shane, 2009). Innovating firms relying too heavily on internal capital often come to +discover that "capital is not enough" (Bradley et al., 2012; Boudreaux and Nikolaev, 2019). +Unfortunately, liquidity constraints are often more severe in emerging markets where financial +systems cannot meet firms' financial needs: A survey of owner-managers of more than 15,500 +firms across 29 emerging markets indicated firms’ inability to access credit at reasonable terms +as one of the top three business constraints (BEEPS, 2016). +We suggest that firms led by the mixed-gender-owning teams will be more likely to rely +upon external funding. First, men network predominantly with other men, "important others" in +business and finance, and form weak ties with former colleagues and employers (Tonoyan et al., +2020). In contrast, women often network with other women, rely on strong ties with family (for +review see Jennings and Brush, 2013), and engage in support groups to overcome traditional +male dominance in their industry (Vogel et al., 2014). Such network diversity increases the + +9 + +likelihood of obtaining both institutional and bootstrapping financing from friends and family +(Manolova et al., 2006). Second, diversity due to gender differences in professional experiences +and functional backgrounds is likely to send positive "signals" to financial-resource providers +about the "team's completeness" (Beckman et al., 2007:123), increasing the firm’s likelihood of +success and hence the perception of its investment-worthiness. Mixed-gender-owner teams have +the best of gender-specific behaviors and characteristics: they exhibit both a greater "task-related +diversity" (Lee and Beckman, 2019), i.e., range of diverse skills and abilities, and superior +relational skills combining men's agency and goal orientation with women's communality and +nurturing of relationships with various stakeholders (Eagly, 2016). +A handful of studies offer empirical support for our conjectures. Vogel et al. (2014) +showed that both task-oriented and relations-oriented diversity of mixed-gender entrepreneurial +teams were positively related to the hypothetical resource providers' willingness to provide +external capital. Beckman et al. (2007) demonstrated that task-related diversity resulting from the +top management team's demographic diversity increased the start-up's ability to attract equity +capital in Silicon Valley. To the best of our knowledge, the literature has not yet considered the +linkages between mixed-gender-owner teams, external capital, and firm innovativeness. +In light of the preceding discussion, we develop our hypotheses as follows: +Hypothesis 1: Firms with higher gender diversity in ownership will exhibit higher firm +innovativeness. +Hypothesis 2: Higher innovativeness anticipated for firms with higher gender diversity in +ownership will be partly attributable to their higher likelihood of investing in R&D. +Hypothesis 3: Higher innovativeness anticipated for firms with higher gender diversity in +ownership will be partly attributable to their higher likelihood of relying on external capital. + +10 + +3. Methodology + +3.1. Data and sample +Our data come from the Business Environment and Enterprise Performance Survey +(BEEPS). Conducted by the World Bank and European Bank for Research and Development +during 2012-2016, it is the largest firm-level survey studying managerial perceptions of the +business environment in emerging markets of the post-Soviet Union, Central-Eastern Europe, +Baltic, and Asia. The data are collected through face-to-face interviews with the firm owner- +managers using a simple random or randomly-stratified sampling and standardized survey +instruments to ensure comparability across countries (BEEPS, 2016). +BEEPS has been featured in prominent management and economics journals (Ayyagari et +al., 2011; McCann and Bahl, 2017). It is well suited for testing our hypotheses for several +reasons. First and most important, it is the only firm-level data containing questions pertaining to +our dependent variable, focal independent variable, and proposed mediators across 29 countries. +Second, it provides rich information on firm characteristics (e.g., size, age, industry sector, and +establishment type) likely to influence firm innovation, which is required to test the net effects of +gender diversity on firm innovativeness. A final advantage is its multi-country design. Scholars +have called for cross-country studies of this nature to enhance our understanding of what +determines firm innovation across emerging markets (Bradley et al., 2012; Jennings et al., 2013; +Boudreaux et al., 2019). Our final sample —after removing missing observations— consists of +7,848 cross-sectional firm observations from 29 emerging markets. +3.2. Measures +3.2.1. Dependent variables +We measure firm innovativeness, our dependent variable, several different ways. Our +first measure, overall firm innovativeness, captures firm innovation according to the dimension + +11 + +of width —i.e., product, process, organizational, and marketing areas— in which the firm has +introduced something new (Ayygari et al., 2011; Strohmeyer et al., 2017). To measure Ioverall, we +created a formative index as follows: +𝐼𝑜𝑣𝑒𝑟𝑎𝑙𝑙 = 𝑖𝑝𝑟𝑜𝑑𝑢𝑐𝑡 + 𝑖𝑝𝑟𝑜𝑐𝑒𝑠𝑠 + 𝑖𝑚𝑎𝑟𝑘𝑒𝑡𝑖𝑛𝑔 + 𝑖𝑜𝑟𝑔𝑎𝑛𝑖𝑧𝑎𝑡𝑖𝑜𝑛𝑎𝑙 +The index’s four constituent items were derived from the survey questions about whether +the firm had introduced any of the following initiatives during the last three years: (1) "new or +significantly improved products or services" (excluding a "simple resale of new goods purchased +from others and changes of a solely aesthetic nature") (product innovation); (2) "new or +significantly improved methods for the production or supply of products or services" (process +innovation); (3) "new or significantly improved organizational or management practices or +structures" (organizational innovation), and (4) "new or significantly improved marketing +methods" (marketing innovation). Each was coded dichotomously as 1=yes, 0=no. The index +thus ranges from 0 to 4. +Our second measure is product and process innovations. Unlike many intermediary input +indicators of firm innovation (such as patents or R&D intensity), new products and processes +capture the commercial value of a firm's novel offerings across various industries (Audretsch et +al., 2011). Its two constituent items include new product development and process innovations +(each coded 1 if the firm had introduced them, 0 if not). The index thus ranges from 0 to 2. +Our third measure of a firm’s innovativeness is organizational and marketing +innovations. An organizational restructuring consists of changes in business practices, quality +and human resource management, and external relations; marketing innovation includes changes +to design or packaging, product promotion, placement, and pricing (OECD, 2005, 2011). Both +innovation types are critical for SME competitiveness, often substituting rather than + +12 + +complementing product and process innovations (Audretsch et al., 2011). Its two constituent +items include organizational and marketing innovations (each coded 1 if the firm had introduced +these, 0 if not). The index ranges to 0 to 2. +3.2.2. Independent and mediating variables +Following prior literature (Talke et al., 2010; Triana et al., 2019; Xie et al., 2020; Zhang, +2020), we measure our independent variable, gender diversity in firm ownership, using Blau's +index of heterogeneity, 2 x (1 – ΣPi2), where Pi denotes the percentage of the population in each +gender group (Blau, 1977). 1 To create this variable, we combined two questions from the +BEEPS survey questionnaire: whether there are any females “amongst the owner of the firm”, +and if yes, “what percentage of the firm is owned by females?” The index ranges from 0 +indicating complete gender homogeneity (describing all-male and all-female-owned firms) to the +highest value of 1 indicating complete gender diversity (equal ownership by males and females). +Our first mediator, a firm’s R&D investments, is coded 1 if the firm has invested in +research and development activities —either in-house or contracted with other companies +(outsourced) (1=yes; 0=no). +To measure our second hypothesized mediator, a firm's breadth of external capital, +Capitalbreadth, we created a formative index as follows: +𝐸𝑥𝑡𝑒𝑟𝑛𝑎𝑙 𝐶𝑎𝑝𝑖𝑡𝑎𝑙𝑏𝑟𝑒𝑎𝑑𝑡ℎ = 𝑖𝑏𝑎𝑛𝑘𝑠 + 𝑖𝑛𝑜𝑛−𝑏𝑎𝑛𝑘𝑠 + 𝑖𝑡𝑟𝑎𝑑𝑒 𝑐𝑟𝑒𝑑𝑖𝑡 + 𝑖𝑜𝑡ℎ𝑒𝑟 +The four constituent items were derived from the survey questions on sources of funding +the firm uses to finance its working capital: (1) "banks (private and state-owned)" (banks); (2) + +1 For example, if women and men owned 90% and 10% of firm ownership, respectively, the Blau index would be +0.36 (2 x (1 - (0.92 + 0.12)). If each of the genders possessed 50% of the firm ownership, the Blau index would be +equal to one indicating gender parity (2 x (1-(0.52 + 0.52)). + + +13 + +"non-banks financial institutions” including “microfinance institutions, credit cooperatives, +credit unions or finance companies" (non-banks); (3) "purchases on credit from suppliers and +advances from customers" (trade credit); and (4) "other" (moneylenders, friends, relatives, etc.)" +(other). Each was coded dichotomously as 1=yes, 0=no. The index ranges from 0 to 4. +3.2.3. Control variables +To better isolate the effects of our hypothesized predictors in HLM, we included various +controls at the firm- and country-levels. Our firm-level controls capture both contemporaneous +and imprinting determinants of firm innovation. Prior work has shown that larger firms, younger +firms, foreign-owned firms, private firms, exporting firms, firms with specific characteristics of +the Top Management, public firms, and those exposed to international markets and knowledge +spillovers are more innovative (Audretsch et al., 2011). +SME is coded 1 if the firm employed less than 250 full-time employees at the time of the +survey (0 otherwise). Start-up is coded 1 if the firm was less than six years of age at the time of +the survey (0 otherwise). Foreign ownership is coded 1 if private foreign individuals or +organizations owned more than 50 percent of the firm (0 otherwise). State ownership is coded 1 +if the government held more than 50 percent of firm ownership (0 otherwise). Firm exporting is +coded 1 if direct exports comprised part of the firm revenues. Top Manager describes if the +firm’s Top Manager was a female (coded 1 if yes, 0 if not). Industry experience is a continuous +variable measuring years of industry experience of the firm’s Top Manager. +International quality certification is coded 1 if the firm had an internationally-recognized +quality certificate at the time of the survey (0 if not). Technology license is coded 1 if the firm +had technology licenses from a foreign-owned company (“excluding office software”) at the time +of the survey (0 if not). Private firm/JV is coded 1 if the firm was initially established either as +"private, from the time of start-up" or "joint venture with a foreign partner(s)" (0 if it was + +14 + +established as a "state-owned firm", "privatization of a state-owned firm", or "private subsidiary +of a formerly state-owned firm"). Part of larger business is coded 1 if the firm is part of a larger +business (0 if no). Public company is coded 1 if the firm is a shareholding company with (non-) +publicly traded shares (0 if a sole proprietorship, partnership, or limited partnership). +We included various country-level controls to adjust for the institutional environments +that are likely to influence cross-country differences in firm innovation (Boudreaux, 2017). +Although we explored many such possible controls, we retained only three in our main models to +have sufficient statistical power at level two of our multilevel modeling of 29 countries +(Raudenbush and Bryk, 1992). 2 We included the country's per capita GDP as the country's +overall economy influences both the likelihood of firm innovation and gender diversity in the +firm's ownership-management (Xie et al., 2020). The country's total expenditures on research +and development (R&D) (% of GDP) is a proxy for a nation's technological prowess and +absorptive capacity for innovation. We also adjusted for the country's business density, which +influences firm innovation either directly by encouraging the introduction of novel products or +technologies as a firm differentiation strategy or indirectly through knowledge spillovers from +other firms (Audretsch et al., 2011). 3All country indicators are lagged for two years to establish +causality and standardized to minimize multicollinearity. +3.3. Analytic techniques +We used hierarchical linear modeling (HLM) as our primary analytic technique +(Raudenbush and Bryk, 2002) to control for the intra-class correlation (ICC) that was evident + +2 To obtain our country controls, we used the World Bank’s Development Indicators data, EBRD’s Science, +Technology, and Innovation Indicators, and the US Patent and Trademark Office’s patent data. + +3 Additional country indicators that we controlled for included the country’s overall gender egalitarianism, women’s +percentage in the labor force, political instability, corruption, and voice and accountability. Models that controlled +for these measures produced similar results to those presented in the main analysis (available upon request). + +15 + +and attributable to the nesting of firms within countries (ICC = 0.126, p ≤ .001 in model 1c; ICC += 0.086, p ≤ .01 in model 2c; and ICC=0.110, p ≤ .001 in model 3c, see Table 2). Because of the +continuous nature of our innovation measures, we used multilevel linear regression models. We +examined the variance inflation factor within each model to check the potential for +multicollinearity. All were well below the threshold value of 10, with a mean of only 4.94. +We supplemented the HLM results with the KHB mediation model, a rigorous approach +for mediation analysis (Kohler, Karlson, and Holm, 2011) featured in prominent management +journals (e.g., Tonoyan, Strohmeyer, and Jennings, 2020).4 +4. Results + +4.1. Key descriptive findings +Table 1 reports descriptive statistics and correlations. Of the particular interest are the +overall means for our innovation measures. The means for the overall innovativeness (0.87), +product and process innovations (0.43), and organizational and marketing innovations (0.44) are +well below these variables’ theoretical maximums of 4.00, 2.00, and 2.00, respectively, +suggesting that a large share of firms in our sample have not exhibited overall firm +innovativeness or introduced product and process and/or organizational and marketing +innovations. This finding is consistent with evidence that most SMEs are not very innovative in +practice (Ruef, 2010; Thébaud, 2015a, b; Strohmeyer et al., 2017). + [Insert Table 1] +The means for our focal variables are also revealing. Approximately 11 percent of the +firms invested in R&D. The mean for the breadth of external capital, at 0.70, is well below its + +4 The KHB mediation model has several advantages over structural equation modeling (SEM). Unlike SEM, KHB +mediation permits multiple mediators and does not require latent constructs (Kohler et al., 2011). + + +16 + +theoretical maximum of 4.0, suggesting that only a small proportion of emerging-market firms +used various sources of external capital. This finding is consistent with evidence on financing +patterns of innovation in emerging markets (Manolova et al., 2006; Ayyagari et al., 2011). +The bivariate comparisons between firms with and without gender-diverse ownership +structures are also revealing. Consistent with our theorizing, the former are significantly more +likely than that latter to exhibit overall firm innovativeness, develop product and process, as well +as organizational and marketing innovations. The former are also more likely than the latter to +allocate resources to R&D and rely on a breadth of external capital. +4.2. HLM and mediation results +Table 2 presents our HLM results. The baseline multilevel models 1a, 2a, and 3a examine +the gross effects of gender diversity in firm ownership omitting control variables. We observe a +positive and highly significant coefficient on gender diversity, our focal independent variable. +Models 1b, 2b, and 3b test whether the net effects of our independent variable on the dependent +variables remain positive and significant in the presence of the controls. The positive and highly +significant coefficients for the gender diversity in firm ownership variable in these models +demonstrate that they did. These findings strongly support Hypothesis 1. Models 1c, 2c, and 3c +test the effects of our independent variable on firm innovation after introducing our focal +mediators. The coefficients for gender diversity in firm ownership are positive and highly +significant in models 1c and 3c and insignificant in model 2c. The coefficients for R&D +investments and breadth of external capital are positive and highly significant. This pattern of +findings suggests that higher innovativeness of firms with higher gender diversity in firm +ownership is either partially or fully attributable to differences in likelihoods of R&D + +17 + +investments and reliance upon external capital. These results provide strong initial support for +Hypotheses 2 and 3. +[Insert Table 2] +Table 3 presents the results from a more sophisticated KHB mediation analysis (Kohler et +al., 2011). We observe an indirect effect of gender diversity on firm innovativeness through both +of the hypothesized mediators. More precisely, R&D investments and breadth of external capital +jointly account for a total of 37.06%, 45.43%, and 33.02% of the observed differentials in overall +firm innovativeness, product and process innovations, and organizational and marketing +innovations, respectively. These findings lend strong additional support for Hypotheses 2 and 3. +[Insert Table 3] +4.3. Robustness checks + +4.3.1. Alternative measures of our focal constructs and mediators +We conducted numerous robustness checks to assess the stability of our results pertaining +to alternative measures of our dependent, independent variables as well as mediators, which we +summarize in Tables 4 and 5. +Table 4 summarizes the findings for each type of innovation as a separate dependent +variable. Because of the dichotomous nature of the separate innovation measures, we used +multilevel logistic regression models. The findings lend additional support for our Hypotheses 1- +3, which predicted a higher likelihood of firm innovativeness for firms with higher gender +diversity in firm ownership, with such differences being partially attributable to a firm’s +likelihood of R&D investments and reliance upon external capital. The total amount of mediation +explained by our focal mediators corresponds to 45.83%, 35.85%, 28.21%, and 29.38% for new +products, processes, organizational innovation, and marketing innovation, respectively. +[Insert Table 4] + +18 + +In Table 5, we estimated multilevel models using alternative measures of our focal +independent variable and the two mediators: a dichotomous measure of the gender diversity +index coded 1 if the firm had a minimum of 59% of gender diversity in its ownership structure +(model 1), an alternate dichotomous measure for a firm’s R&D investments describing whether +or a not the firm used human-resource-management practices likely to enhance firm +innovativeness (model 2), a continuous measure including only funding from banks and non- +bank financial institutions as the first alternate measure for external capital (model 3a), and a +dichotomous measure capturing whether or not the firm received “any subsidies from the +national, regional, local governments, or European Union” as the second alternate measure for +external capital (model 3b). Our results are robust to these alternative specifications. +[Insert Table 5] +4.3.2. Sensitivity to outliers and subsamples + +We assessed the sensitivity of our findings by re-estimating our multilevel models: +excluding a small percentage of the solo owner-manager-led firms (model 4), checking for +individual firm outlier cases (model 5), on subsamples of firms from higher-innovation industries +(model 6) and lower-innovation industries (model 7). The findings provide strong additional +evidence attesting to the robustness of our results. + +4.3.3. Selection correction +We estimated the Heckman selection model (Heckman, 1979) to correct for the firm's +potential self-selection into the decision to invest in R&D, on which we elaborate in online +Appendix table A3 and its notes. The results strongly supported findings pertaining to our focal +independent variable and external capital. +4.4. Analytic extensions + +19 + +Instead of using a continuous variable to measure gender diversity in firm ownership, we +created a dichotomous variable describing the “mixed-gender-owner teams” (i.e., firms owned +equally by males and females and those with either male-dominated or female-dominated +ownership structures) versus all-male-owned and all-female-owned firms. This analysis (models +1a-1c of online appendix Table A4) produced the same pattern of results demonstrating that the +mixed-gender-owned teams (N=1,881) exhibited a higher likelihood of overall firm +innovativeness than all-male-owned and all-female-owned teams combined (N=6,008), with the +total amount of mediation explained by the mediators corresponding to 32.17%. +We then compared only all-male-owned teams versus only all-female-owned teams with +the mixed-gender-owned teams. Our findings showed that all-male-owned teams (N=5,271) +exhibited a significantly lower overall innovativeness than the mixed-gender-owned teams, with +a lower likelihood of R&D investments and reliance on external capital mediating 36.16% of the +relationship (models 2a-c, Table 6). Although all-female-owned firms (N=737) exhibited lower +overall innovativeness than the mixed-gender-owned teams, and R&D investments and external +capital were significant predictors of their firm innovation, these mediators did not explain the +hypothesized differences with the mixed-gender-owner teams (see models 3a-c in online +appendix Table A4). +We assessed the possible “treatment effects” (Rosenbaum and Rubin, 1983) to correct for +the potential selection bias stemming from the fact the “treated” firms might differ from “non- +treated” for reasons other than the treatment. We defined firms with a score in the upper 80-100th +percentile of the Blau index as the treated, those with no gender diversity as the untreated. We +matched the two groups on various observable characteristics pertaining to firm, industry, and +country likely to impact firm innovativeness using one-to-one matching, caliper (0.01), and + +20 + +Mahalanobis algorithms. The treated firms had a higher likelihood of overall firm innovativeness +than the non-treated. See Online Appendix Tables A5 and A6 for more information. +5. Discussion +The study’s objective was to provide theoretical and empirical insights into the questions +of how and why gender diversity in firm ownership influences firm innovativeness. With respect +to the how question, our analysis of secondary data of 7,848 owner-managers of SMEs in 29 +emerging markets indicates that firms with higher gender diversity in firm ownership are more +likely to exhibit higher firm innovativeness. With respect to the why question, our findings +provide strong and robust support for the theorized mediating roles played by the firm’s R&D +investments and reliance upon a breadth of external capital. We could demonstrate that sizeable +proportions of the observed differentials in firm innovation—more precisely, 37.06% in overall +firm innovativeness, 45.43% in product and process innovations, and 33.2% in organizational +and marketing innovations—could be explained by our proposed mediators. +5.1. Contributions to and implications for the Upper Echelon literature +Our results extend the Upper Echelon literature in several ways. First, building on a small +body of nascent work (e.g., Dai et al., 2019; Na and Shin, 2019; Ritter-Hayashi et al., 2019), we +studied a highly relevant and important diversity construct —gender diversity in firm +ownership—whereas most existing work (summarized in our online appendix) has examined the +construct of gender diversity in other organizational contexts (such as the board of directors, +senior management, task teams, and employees). We further examined firm innovativeness as a +performance metric while most prior literature has studied the implications of gender diversity +for firm financial performance (for a meta-analysis see Jeong and Harrison, 2017). +Our study is a first step in understanding some of the mechanisms underlying the “gender +diversity and firm innovativeness” nexus that pertain to the firm’s strategic allocation of + +21 + +resources into research and development as well as its financial resource endowments. To extend +our single-mediation model, we invite scholars to test a unified "double-mediation" model +(Kohler et al., 2011) that would combine the mediating effects of both the owners' individual +characteristics, i.e., various types of skills, experiences, networks, and dispositional traits male +and female owners bring to the table (e.g., Lee and Beckman, 2019), and their strategic practices +on firm innovativeness. Future work could also examine other mediating channels (such as +recruitment of specific personnel, establishing strategic alliances with business and government, +and employing strategies preventing imitators from capturing the firm’s profits from innovation) +through which gender diversity in firm ownership is likely to influence firm innovativeness. +The socio-economic significance of our key finding about gender diversity sets the stage +for a variety of related research questions. Scholars could investigate if ethnically-diverse and/or +age-diverse owner teams are more innovative due to skill complementarity and strategic choices. +Although emerging evidence suggests this might be the case (Hoogendoorn and van Praag, 2012; +Brixy et al., 2020), we need more research. +Future studies could also study the linkage between gender diversity in ownership and +firm growth through the mediating effects of firm innovativeness. Are firms headed by the +mixed-gender-owner teams not only more innovative but also more likely to survive and grow? +Studies of the potential downsides associated with too much diversity in firm ownership also +hold promise. We invite researchers to test a curvilinear relationship to examine whether too +little or too much diversity in firm ownership might hinder firm performance. Emerging evidence +from a field experiment with student teams (that were required to start a business venture, elect +CEOs, conduct meetings, produce, sell, make money, and liquidate within a year as part of an +entrepreneurship class) demonstrated clear benefits of gender diversity. But, that relationship + +22 + +was curvilinear with the peak reached when women’s share in the team became 0.55 +(Hoogendoorn, Oosterbeek, and van Praag, 2013). Too much gender diversity is likely to +become detrimental by increasing costs of coordination and communication, delaying team +decisions, and slowing market moves. Yet, too little gender diversity might also hinder team +performance by increasing team complacency with the information exchanged among the team +members becoming too homogeneous and communication too easy. +5.2 Contributions to and implications for the entrepreneurship literature +Our study also extends and possesses implications for entrepreneurship research. Much +current work views innovativeness as a function of the firm's organizational characteristics, +alliances with venture capitalists, large corporations, and universities, as well as embeddedness +in technology clusters, industries, and national innovation systems (Audretsch et al., 2011). This +stream ignores, however, the question of who participates in innovation activity. As a result, +some scholars have noted our limited understanding of how individual owners and entrepreneurs +influence firm innovation (Baron and Tang, 2011; Strohmeyer et al., 2017). We contribute to the +small but increasing body of work on this topic demonstrating the role played by individual-level +characteristics —mixed-gender-owner teams— not much examined within existing +entrepreneurship research. +We also extend nascent work by offering a more inclusive conceptualization of firm +innovation that recognizes not only new products and technological process innovation but also +overall firm innovativeness along with the separate individual domains (i.e., product, process, +organizational, and marketing) in which the firm has introduced something novel (Ayyagari et +al., 2011; Strohmeyer et al., 2017). That said, future researchers could examine the consequences +of gender diversity in firm ownership for "innovation depth" or the radicalness of the firm's new + +23 + +offerings regardless of the individual domains where they were introduced. Radical innovations +typically emerge after recombining existing ideas from different domains that previously seemed +unrelated (Hargadon, 2003; Schilling and Green, 2011). As mixed-gender-owned teams are more +likely to possess knowledge and experience from multiple unrelated domains, they might also be +more likely to introduce radical innovation. +We also contribute to the literature on women's entrepreneurship. Although recent +research started to study entrepreneurship as a "gendered process" (Brush, 1992; Jennings and +Brush, 2013), only a few studies (Horbach and Jacob, 2018; Dai et al., 2019; Na and Shin, 2019; +Ritter-Hayashi et al., 2019) examine mixed-gender ownership as a focus group. Yet, ignoring +businesses owned by mixed-gender teams is problematic, since such omission overlooks how +complements in skills, networks, and characteristics of male and female entrepreneurs shape +firm-level outcomes and evaluations of resource providers (Kim, 2006; Parker, 2018). We +demonstrated that a male-female partnership was more beneficial for firm innovativeness than +all-male partnership because such diversity enhanced the likelihood of R&D investments and +reliance upon external capital. We offered a theory suggesting that the addition of a partner to the +firm ownership outside the “in” network who can bring with them skills or attributes that the +other partners lack (Godwin et al., 2006) will enhance the firm innovativeness. Unexpectedly, +however, our proposed mediators did not explain the gaps in the innovativeness of firms owned +by all females and the mixed genders. Such a finding merits more detailed investigation in future +research. +We also encourage future research on the moderating effects of a country’s institutional +environment: given that the degree of sex-based segregation in business ownership and +innovation varies across countries (Thébaud, 2015a; Tonoyan et al., 2020), partnering with a + +24 + +male (or female) business owner might confer more benefits in securing critical resources and +organizational legitimacy (Godwin et al., 2006) in some institutional contexts than others. +As some suggested that female owner-managers might prefer creating organizations that +are more egalitarian and less hierarchical (Jennings and Brush, 2013), future research is also +warranted for studying the importance of the organizational culture. Gender diversity in +ownership might create climates for inclusion, fostering employees’ participation in problem- +solving, diversity in opinions, and encouraging exchange between the firm and its constituents +(e.g., customers and regulators) (Cropley and Cropley, 2017). Such channels are likely to +enhance employee creativity, firm absorptive capacity, and hence innovativeness. +6. Limitations and suggested directions +The results of our research are subject to a few limitations, mainly stemming from the +secondary nature of our focal dataset, that should be addressed in future research. Although our +results were robust to various sensitivity checks and self-selection concerns, due to the cross- +sectional nature of our data, we encourage the replication of our findings with longitudinal firm +data. Since our sample mostly comprised emerging markets from the post-Socialist countries, we +also encourage replication of our results on samples of different countries. Finally, we focused +solely upon the firm's lead owner-managers assuming that such individuals as heads of firms are +those who "call the shots", i.e., make strategic decisions likely to impact firm innovativeness. +But, innovation is an inclusive process often involving the input of highly-capable employees +from various functional areas (such as engineering, marketing, and manufacturing) (Østergaard +et al., 2011). Scholars should thus study the interplay between skill diversity of owner-managers +and specialization amongst the firm employees to extend our understanding of the antecedents of +firm innovation. + +25 + +7. Conclusion +Returning to our guiding questions of how and why gender diversity in firm ownership +influences firm innovativeness, we conclude with the following responses. With respect to the +how questions, firms with higher gender diversity in ownership in our sample differed +significantly in their degree of innovativeness, exhibiting higher likelihoods of overall +innovativeness, product and process, as well as organizational and marketing innovations. As for +the why question, sizeable proportions of these differentials were attributable to a higher +likelihood of investing in R&D and relying upon external capital. +We would like to provide a final note about our study’s implications for managers and +policymakers. A managerial implication is that both male and female business owners should +consider partnering with the opposite sex, as gender diversity in ownership significantly +increases the odds that the firm will become more innovative along the dimensions of new +products, processes, organizational practices, and marketing methods. A policy implication is +that policymakers might want to focus on desegregating business ownership. Providing +institutional, social, and resource-based support for increasing the share of the mixed-gender- +owner teams will go a long way in enhancing firm innovativeness. + +References + +Almor, T., Bazel-Shoham, O., Lee, S.M., 2019. The dual effect of board gender diversity on +R&D investments. Long Range Planning 10188.https://doi.org/10.1016/j.lrp.2019.05.004 +Audretsch, D.B., Falck, O., Heblich, S., 2011. Handbook of research on innovation and +entrepreneurship. Edward Elgar Publishing. +Ayyagari, M., Demirgüç-Kunt, A., Maksimovic, V., 2011. Firm innovation in emerging markets: +The role of finance, governance, and competition. Journal of Financial and Quantitative +Analysis 46, 1545–1580. https://doi.org/10.1017/S0022109011000378 +Baron, Tang, J., 2011. The role of entrepreneurs in firm-level innovation: Joint effects of positive +affect, creativity, and environmental dynamism. Journal of Business Venturing 26, 49– +60. https://doi.org/10.1016/j.jbusvent.2009.06.002 + +26 + +Beckman, C.M., Burton, M.D., O'Reilly, C., 2007. Early teams: The impact of team demography +on VC financing and going public. Journal of Business Venturing 22, 147–173. +BEEPS, 2016. The business environment in the transition region. https://www.ebrd.com +Biga-Diambeidou, M., Bruna, M.G., Dang, R., Houanti, L., 2019. Does gender diversity among +new venture team matter for R&D intensity in technology-based new ventures? Evidence +Blau, P.M., 1977. Inequality and heterogeneity: A primitive theory of social structure. Free Press +New York. +Boudreaux, C.J., 2017. Institutional quality and innovation: some cross-country evidence. +Journal of Entrepreneurship and Public Policy 6, 26–40. https://doi.org/10.1108/JEPP-04- +2016-0015 +Boudreaux, C.J., Nikolaev, B., 2019. Capital is not enough: opportunity entrepreneurship and +formal institutions. Small Business Economics 53, 709–738. +https://doi.org/10.1007/s11187-018-0068-7 +Boudreaux, C.J., Nikolaev, B.N., Klein, P., 2019. Socio-cognitive traits and entrepreneurship: +The moderating role of economic institutions. Journal of Business Venturing 34, 178– +196. +Bradley, S.W., McMullen, J.S., Artz, K., Simiyu, E.M., 2012. Capital is not enough: Innovation +in developing economies. Journal of Management Studies 49, 684–717. +https://doi.org/10.1111/j.1467-6486.2012.01043.x +Brixy, U., Brunow, S., D'Ambrosio, A., 2020. The unlikely encounter: Is ethnic diversity in start- +ups associated with innovation? Research Policy 49, 103950. +https://doi.org/10.1016/j.respol.2020.103950 +Brush, C.G., 1992. Research on women business owners: past trends, a new perspective and +future directions. Entrepreneurship: Theory and Practice 16, 5–31. +Cohen, W.M., Levinthal, D.A., 1990. Absorptive capacity: A new perspective on learning and +innovation. Administrative Science Quarterly 35, 128–152. +Cropley, D. and Cropley, A. 2017. Innovation capacity, organisational culture and gender. +European Journal of Innovation Management, 20, 493-510. https://doi.org/10.1108/EJIM- +12-2016-0120. https://doi.org/10.2307/2393553. +Dai, Y., Byun, G., Ding, F., 2019. The direct and indirect impact of gender diversity in new +venture teams on innovation performance. Entrepreneurship Theory and Practice 43, +505–528. https://doi.org/10.1177/1042258718807696 +Dezsö, C.L., Ross, D.G., 2012. Does female representation in top management improve firm +performance? A panel data investigation. Strategic Management Journal 33, 1072–1089. +https://doi.org/10.1002/smj.1955 +Díaz-García, C., González-Moreno, A., Sáez-Martínez, F.J., 2013. Gender diversity within R&D +teams: Its impact on radicalness of innovation. Innovation 15, 149–160. +https://doi.org/10.5172/impp.2013.15.2.149 +Eagly, A.H., 2016. When passionate advocates meet research on diversity, does the honest +broker stand a chance? Journal of Social Issues 72, 199–222. doi.org/10.1111/josi.12163 +Eagly, A.H., 2013. Sex differences in social behavior: A social-role interpretation. Psychology +Press. +Faems, D., Subramanian, A.M., 2013. R&D manpower and technological performance: The +impact of demographic and task-related diversity. Research Policy 42, 1624–1633. +https://doi.org/10.1016/j.respol.2013.06.001 + +27 + +Fama, E.F., Jensen, M.C., 1983. Agency problems and residual claims. The Journal of Law and +Economics 26, 327–349. https://doi.org/10.1086/467038 +Garcia-Martinez, M., Zouaghi, F., Garcia-Marco, T., 2017. Diversity is strategy: the effect of +R&D team diversity on innovative performance. R&D Management 47, 311–329. +https://doi.org/10.1111/radm.12244 +Godwin, L. N., Stevens, C. E., & Brenner, N. L. (2006). Forced to play by the rules? Theorizing +how mixed–sex founding teams benefit women entrepreneurs in male–dominated +contexts. Entrepreneurship Theory and Practice, 30, 623-642. +https://doi.org/10.1111/j.1540-6520.2006.00139. +Gorodnichenko, Y., Schnitzer, M., 2013. Financial constraints and innovation: Why poor +countries don't catch up. Journal of the European Economic Association 11, 1115–1152. +https://doi.org/10.1111/jeea.12033 +Hargadon, A., 2003. How breakthroughs happen: The surprising truth about how companies +innovate. Harvard Business Press. +Heckman, J.J., 1979. Sample selection bias as a specification error. Econometrica 47, 153–161. +https://doi.org/10.2307/1912352 +Hoogendoorn, S., van Praag, M., 2012. Ethnic diversity and team performance. A field experiment. +Tinbergen Institute Discussion Paper 068-2. +Hoogendoorn, S., Oosterbeek, H., van Praag, M., 2013. The impact of gender diversity on the +performance of business teams: Evidence from a field experiment. Management Science, +59, 1514-1528. +Horbach, J., Jacob, J., 2018. The relevance of personal characteristics and gender diversity for +(eco-)innovation activities at the firm-level: Results from a linked employer–employee +database in Germany. Business Strategy and the Environment 27, 924–934. +https://doi.org/10.1002/bse.2042 +Jennings, D., R. Greenwood, M.D., Lounsbury, and R. Suddaby, 2013. Institutions, +entrepreneurs, and communities: A special issue on entrepreneurship. Journal of Business +Venturing, 28, 1-9. +Jennings, J.E., Brush, C.G., 2013. Research on women entrepreneurs: challenges to (and from) +the broader entrepreneurship literature? Academy of Management Annals 7, 663–715. +Jeong, S.-H., Harrison, D.A., 2017. Glass breaking, strategy making, and value creating: Meta- +analytic outcomes of women as CEOs and TMT members. AMJ 60, 1219–1252. +https://doi.org/10.5465/amj.2014.0716 +Kanter, R. M., 1977. Some effects of proportion on group life. American Journal of Sociology, +82, 965–90. +Kim, G.O., 2006. Do equally-owned small businesses have equal access to credit? Small +Business Economics 27, 369–386. https://doi.org/10.1007/s11187-005-2558-7 +Kohler, U., Karlson, K.B., Holm, A., 2011. Comparing coefficients of nested nonlinear +probability models. The Stata Journal 11, 420–438. +Lee, H., Beckman, C. 2019. Unpacking board diversity: Women director experience and +corporate social responsibility. Paper presented at the Academy of Management Meeting. +Manolova, T.., Manev, I.M., Carter, N.M., Gyoshev, B.S. 2006. Breaking the family and friends' +circle: Predictors of external financing usage among men and women entrepreneurs in a +transitional economy. Venture Capital, 8, 109-132. + +28 + +McCann, B.T., Bahl, M., 2017. The influence of competition from informal firms on new +product development. Strategic Management Journal 38, 1518–1535. +https://doi.org/10.1002/smj.2585 +Miller, T., Triana, M.D.C., 2009. Demographic diversity in the boardroom: Mediators of the +board diversity–firm performance relationship. Journal of Management Studies 46, 755– +786. https://doi.org/10.1111/j.1467-6486.2009.00839.x +Na, K., Shin, K., 2019. The gender effect on a firm's innovative activities in the emerging +economies. Sustainability 11, 1992. https://doi.org/10.3390/su11071992 +OECD, 2011. Education at a Glance, OECD Indicators. +OECD, 2005. Oslo Manual: Guidelines for Collecting and Interpreting Innovation Data, 3rd +edition. +Østergaard, C.R., Timmermans, B., Kristinsson, K., 2011. Does a different view create +something new? The effect of employee diversity on innovation. Research Policy 40, +500–509. https://doi.org/10.1016/j.respol.2010.11.004 +Parker, S.C., 2018. The Economics of Entrepreneurship. Cambridge: Cambridge University +Press. +Pelled, L. H. 1996. Demographic diversity, conflict, and work group outcomes: An intervening +process theory. Organization Science, 7:615-631. https://doi.org/10.1287/orsc.7.6.615 +Post, C., Byron, K., 2014. Women on boards and firm financial performance: A meta-analysis. +Academy of Management Journal 58, 1546–1571. https://doi.org/10.5465/amj.2013.0319 +Raudenbush, S.W., Bryk, A.S., 2002. Hierarchical linear models: Applications and data analysis +methods. Sage. +Ritter-Hayashi, D., Vermeulen, P., Knoben, J., 2019. Is this a man's world? The effect of gender +diversity and gender equality on firm innovativeness. PLoS One 14. +https://doi.org/10.1371/journal.pone.0222443 +Rosenbaum, P.R., Rubin, D.B., 1983. The central role of the propensity score in observational +studies for causal effects. Biometrika 70, 41–55. https://doi.org/10.1093/biomet/70.1.41 +Rosenbusch, N., Brinckmann, J., Bausch, A., 2011. Is innovation always beneficial? A meta- +analysis of the relationship between innovation and performance in SMEs. Journal of +Business Venturing 26, 441–457. https://doi.org/10.1016/j.jbusvent.2009.12.002 +Ruef, M., 2010. The entrepreneurial group: Social identities, relations, and collective action. +Princeton University Press. +Ruiz-Jiménez, J.M., Fuentes-Fuentes, M. del M., Ruiz-Arroyo, M., 2016. Knowledge +combination capability and innovation: The effects of gender diversity on Top +Management teams in technology-based Firms. Journal of Business Ethics 135, 503–515. +https://doi.org/10.1007/s10551-014-2462-7 +Schilling, M.A., Green, E., 2011. Recombinant search and breakthrough idea generation: An +analysis of high impact papers in the social sciences. Research Policy 40, 1321–1331. +https://doi.org/10.1016/j.respol.2011.06.009 +Schumpeter, J.A., 1934. The Theory of Economic Development: An Inquiry Into Profits, Capital, +Credit, Interest, and the Business Cycle. Transaction Publishers. +Shane, S.A., 2009. Technology strategy for managers and entrepreneurs. Pearson/Prentice Hall. +Strohmeyer, R., Tonoyan, V., 2005. Bridging the gender gap in employment growth: On the role +of innovativeness and occupational segregation. The International Journal of +Entrepreneurship and Innovation 6, 259-273. httpsdoi.org/10.5367/000000005775179865 + +29 + +Strohmeyer, R., Tonoyan, V., Jennings, J.E., 2017. Jacks-(and Jills)-of-all-trades: On whether, +how and why gender influences firm innovativeness. Journal of Business Venturing 32, +498–518. https://doi.org/10.1016/j.jbusvent.2017.07.001 +Talke, K., Salomo, S., Rost, K., 2010. How top management team diversity affects +innovativeness and performance via the strategic choice to focus on innovation fields. +Research Policy 39, 907–918. https://doi.org/10.1016/j.respol.2010.04.001 +Terjesen, S., Sealy, R., Singh, V., 2009. Women directors on corporate boards: A review and +research agenda. Corporate Governance: An International Review 17, 320–337. +https://doi.org/10.1111/j.1467-8683.2009.00742.x +Teruel, M., Segarra, A., 2017. The link between gender diversity and innovation: What is the +role of firm size? International Review of Entrepreneurship 15, 319–340. +Thébaud, S., 2015a. Business as plan B: Institutional foundations of gender inequality in +entrepreneurship across 24 industrialized countries. Administrative Science Quarterly 60, +671–711. https://doi.org/10.1177/0001839215591627 +Thébaud, S., 2015b. Status beliefs and the spirit of capitalism: Accounting for gender biases in +entrepreneurship and innovation. Social Forces, 94, 61-86. +Tonoyan, V., Strohmeyer, R., Jennings, J.E., 2020. Gender gaps in perceived start-up ease: +Implications of sex-based labor market segregation for entrepreneurship across 22 +European countries. Administrative Science Quarterly 65, 181–225. +Triana, M. del C., Richard, O.C., Su, W., 2019. Gender diversity in senior management, strategic +change, and firm performance: Examining the mediating nature of strategic change in +high tech firms. Research Policy 48, 1681–1693. +https://doi.org/10.1016/j.respol.2019.03.013 +Tyrowicz, J., Terjesen, S., Mazurek, J., 2020. All on board? New evidence on board gender +diversity from a large panel of European firms. European Management Journal. +https://doi.org/10.1016/j.emj.2020.01.001 +Vogel, R., Puhan, T.X., Shehu, E., Kliger, D., Beese, H., 2014. Funding decisions and +entrepreneurial team diversity: A field study. Journal of Economic Behavior & +Organization 107, 595–613. https://doi.org/10.1016/j.jebo.2014.02.021 +Xie, L., Zhou, J., Zong, Q., Lu, Q., 2020. Gender diversity in R&D teams and innovation +efficiency: Role of the innovation context. Research Policy 49, 103885. +https://doi.org/10.1016/j.respol.2019.103885 +Zhang, L., 2020. An institutional approach to gender diversity and firm performance. +Organization Science. https://doi.org/10.1287/orsc.2019.1297 + +30 + +Table 1: Descriptive statistics and correlations + + + + +Gender diverse +ownership + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +Mean SD Firms +with +Firms +without Diff. stat. +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 17 +18 +19 +20 +21 +22 +23 +24 +25 +26 +1 Over’ll firm innov. +0.87 0.01 1.04 +0.81 +t=6.66*** +1 + + + + + + + + + + + + + + + + + + + + + + + + +2 Prod. & proc. innov. +0.43 0.01 0.51 +0.41 +t=5.15*** 0.88 +1 + + + + + + + + + + + + + + + + + + + + + + + +3 Org. & markt. innov. +0.44 0.01 0.53 +0.41 +t=6.48*** 0.89 0.56 +1 + + + + + + + + + + + + + + + + + + + + + + +4 Product innov. + +0.24 0.00 0.28 +0.23 +t=4.18*** 0.75 0.88 0.45 +1 + + + + + + + + + + + + + + + + + + + + + +5 Process innov. +0.19 0.00 0.23 +0.18 +t=4.80*** 0.77 0.85 0.51 0.49 +1 + + + + + + + + + + + + + + + + + + + + +6 Organizational innov. +0.21 0.00 0.25 +0.19 +t=5.70*** 0.79 0.51 0.88 0.40 0.49 +1 + + + + + + + + + + + + + + + + + + + +7 Marketing innovation +0.23 0.00 0.28 +0.21 +t=5.76*** 0.78 0.48 0.89 0.40 0.42 0.57 +1 + + + + + + + + + + + + + + + + + + +8 Gender-diverse owner. +0.18 0.00 n/a +n/a +n/a +0.07 0.05 0.06 0.05 0.05 0.05 0.05 +1 + + + + + + + + + + + + + + + + + +9 R&D activity +0.11 0.00 0.14 +0.10 +t=4.91*** 0.41 0.37 0.35 0.32 0.32 0.32 0.30 0.05 +1 + + + + + + + + + + + + + + + + +10 Breadth of extern. +capital +0.70 0.01 0.78 +0.67 +t=5.10*** 0.17 0.15 0.15 0.13 0.12 0.13 0.14 0.06 0.10 +1 + + + + + + + + + + + + + + + +11 SME +0.96 0.00 0.95 +0.96 +t=-2.09** -0.08 -0.05 -0.09 -0.04 -0.04 -0.10 -0.06 0.00 -0.07 -0.05 +1 + + + + + + + + + + + + + + +12 Startup +0.16 0.00 0.12 +0.17 +t=-5.19*** -0.05 -0.06 -0.04 -0.05 -0.05 -0.04 -0.03 -0.05 -0.04 -0.07 0.05 +1 + + + + + + + + + + + + + +13 Top Manager: Female +0.19 0.00 0.30 +0.15 t=14.16*** 0.02 0.02 0.02 0.01 0.01 0.01 0.03 0.17 -0.03 -0.02 0.07 0.01 +1 + + + + + + + + + + + + +14 Years of industry exper +experience +17.99 0.12 19.44 17.54 t=6.82*** 0.05 0.05 0.03 0.05 0.03 0.04 0.02 0.08 0.08 0.03 -0.06 -0.27 -0.07 +1 + + + + + + + + + + + +15 Foreign ownership +0.05 0.00 0.04 +0.05 +t=-2.46** 0.07 0.05 0.07 0.05 0.04 0.06 0.06 -0.04 0.04 -0.02 -0.12 0.02 -0.02 -0.04 +1 + + + + + + + + + + +16 State ownership +0.01 0.00 0.01 +0.01 +t=0.93 +-0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.01 -0.02 -0.01 -0.02 -0.15 -0.02 -0.01 -0.05 -0.03 +1 + + + + + + + + + + +17 Exporting activity +0.20 0.00 0.23 +0.19 +t=4.01*** 0.16 0.16 0.11 0.15 0.13 0.12 0.08 0.04 0.18 0.13 -0.16 -0.07 -0.08 0.11 0.14 -0.03 +1 + + + + + + + + + +18 International quality +certification +0.27 0.01 0.30 +0.25 +t=4.36*** 0.15 0.15 0.12 0.13 0.12 0.13 0.09 0.04 0.17 0.13 -0.18 -0.10 -0.08 0.11 0.10 -0.01 0.28 +1 + + + + + + + + +19 Technology licenses +0.17 0.00 0.17 +0.17 +t=0.17 +0.15 0.14 0.12 0.13 0.12 0.11 0.10 0.00 0.14 0.07 -0.12 -0.01 -0.08 0.03 0.08 -0.01 0.12 0.21 +1 + + + + + + +20 Private firm/JV +0.87 0.00 0.81 +0.90 +t=-9.80*** -0.02 -0.02 -0.02 0.00 -0.03 -0.03 0.00 -0.03 -0.02 0.03 0.13 0.08 0.01 -0.05 -0.02 -0.27 -0.01 -0.04 0.02 +1 + + + + + +21 Part of larger business +0.09 0.00 0.09 +0.08 +t=1.46 +0.05 0.02 0.07 0.01 0.01 0.07 0.05 -0.01 0.05 0.03 -0.17 0.00 -0.02 0.02 0.13 0.04 0.12 0.15 0.11 -0.04 +1 + + + + +22 Public company +0.86 0.00 0.90 +0.84 +t=5.51*** 0.03 0.02 0.03 0.02 0.02 0.04 0.01 0.04 0.05 0.03 -0.06 0.01 -0.04 -0.03 0.07 0.01 0.08 0.11 0.06 -0.05 0.04 +1 + + + +23 Metropolitan area +0.35 0.01 0.34 +0.35 +t=-0.91 -0.02 -0.03 0.00 -0.03 -0.03 0.02 -0.01 -0.01 -0.01 -0.06 -0.03 0.06 0.00 0.00 0.04 -0.06 0.00 0.01 0.09 0.07 0.02 0.10 +1 + + +24 Cntry’s business dens. +4.29 0.04 4.68 +4.17 +t=5.11*** 0.12 0.10 0.11 0.08 0.09 0.10 0.09 0.07 0.05 0.09 0.05 -0.07 0.10 0.06 0.04 -0.09 0.02 0.02 -0.05 0.09 -0.09 0.16 -0.10 +1 + +25 Cntry’s p. cap. GDP +-0.13 0.01 -0.05 -0.15 +t=3.24*** 0.02 0.05 -0.02 0.07 0.01 -0.02 -0.01 0.06 0.09 0.12 0.00 -0.13 0.01 0.19 0.05 -0.09 0.24 0.12 0.02 0.13 0.02 -0.06 -0.15 0.27 +1 + +26 Cntry’s R&D exp. +-0.29 0.01 -0.15 -0.33 +t=6.30*** 0.02 0.04 0.00 0.06 0.01 0.00 0.00 0.08 0.10 0.11 -0.02 -0.12 0.01 0.16 0.05 -0.05 0.24 0.10 0.02 0.07 0.03 -0.06 -0.09 0.26 0.85 +1 +Notes: correlations above |.01| are statistically significant, p ≤ .05; a 16 industry dummies are exclusive. Hence, correlations are not reported. The industry distribution of innovation is statistically significant (overall +χ2= 155.80***) with details as follows: IT (45%), machinery and equipment (38%), electronics (37%), plastics and rubber (36%), other manufacturing (31%), food (31%), basic metals and furniture (30%), wholesale +(26%), non-metallic minerals (25%), chemicals (24%), textiles and garment (23%), construction (22%), retail (20%), other services (19%), transport (18%), and hotel and restaurants (17%). + +31 + +Table 2: Results of multilevel regression analyses for Hypothesis 1 through 3 +Multilevel model: +GLS + +GLS + +GLS +Dependent variable: +Overall Firm Innovativeness + +Product and Process +Innovation + Organizational and Marketing +Innovation +Model: +1a +1b +1c + +2a +2b +2c + +3a +3b +3c +Independent variable + + + + + + + + + + + +Gender diversity +0.171*** 0.144*** +0.089* + 0.076*** 0.057** +0.031 + 0.094*** 0.084*** +0.055* +in firm ownership +(0.04) +(0.04) +(0.04) + +(0.02) +(0.02) +(0.02) + +(0.02) +(0.02) +(0.02) +Mediators + + + + + + + + + + + +R&D investments + + +1.362*** + + + +0.677*** + + + +0.680*** + + + +(0.04) + + + +(0.02) + + + +(0.02) +Breadth of external capital + + +0.160*** + + + +0.067*** + + + +0.093*** + + + +(0.02) + + + +(0.01) + + + +(0.01) +Control variables + + + + + + + + + + + +SME + +-0.338*** -0.273*** + + +-0.068 +-0.037 + + +-0.265*** -0.230*** + + +(0.07) +(0.06) + + +(0.04) +(0.04) + + +(0.04) +(0.04) +Start-up + +-0.056 +-0.045 + + +-0.027 +-0.023 + + +-0.029 +-0.023 + + +(0.04) +(0.04) + + +(0.02) +(0.02) + + +(0.02) +(0.02) +Top Manager: Female + +-0.020 +0.002 + + +-0.000 +0.010 + + +-0.019 +-0.008 + + +(0.04) +(0.03) + + +(0.02) +(0.02) + + +(0.02) +(0.02) +Years of industry experience +experience + +0.002 +0.001 + + +0.001 +0.001 + + +0.001 +0.001 + + +(0.00) +(0.00) + + +(0.00) +(0.00) + + +(0.00) +(0.00) +Foreign ownership + +-0.004 +0.043 + + +-0.035 +-0.014 + + +0.031 +0.057 + + +(0.06) +(0.06) + + +(0.04) +(0.03) + + +(0.04) +(0.04) +State ownership + +-0.077 +-0.118 + + +-0.026 +-0.045 + + +-0.054 +-0.075 + + +(0.13) +(0.12) + + +(0.07) +(0.07) + + +(0.07) +(0.07) +Exporting activity + +0.322*** +0.195*** + + +0.185*** 0.125*** + + +0.136*** +0.071*** + + +(0.04) +(0.04) + + +(0.02) +(0.02) + + +(0.02) +(0.02) +Intern. quality certification + +0.252*** +0.146*** + + +0.129*** 0.078*** + + +0.123*** +0.069*** + + +(0.03) +(0.03) + + +(0.02) +(0.02) + + +(0.02) +(0.02) +Technology license + +0.430*** +0.302*** + + +0.230*** 0.167*** + + +0.198*** +0.133*** + + +(0.04) +(0.03) + + +(0.02) +(0.02) + + +(0.02) +(0.02) +Private firm/JV + +0.009 +0.030 + + +-0.004 +0.007 + + +0.011 +0.021 + + +(0.04) +(0.04) + + +(0.02) +(0.02) + + +(0.03) +(0.02) +Part of larger business + +0.100* +0.072 + + +-0.001 +-0.014 + + +0.107*** +0.093*** + + +(0.05) +(0.05) + + +(0.03) +(0.03) + + +(0.03) +(0.03) +Public company + +0.019 +-0.019 + + +-0.005 +-0.023 + + +0.025 +0.005 + + +(0.04) +(0.04) + + +(0.02) +(0.02) + + +(0.02) +(0.02) +16 Industry dummies +No +Yes +Yes + +No +Yes +Yes + +No +Yes +Yes + + + + + + + + + + + + +Overall model statistics + + + + + + + + + + + +Constant +0.874*** 1.129*** +0.871*** + 0.436** +* +0.546** +* +0.428** +* + 0.437** +* +0.577*** 0.438*** + +(0.09) +(0.18) +(0.16) + +(0.04) +(0.09) +(0.08) + +(0.05) +(0.10) +(0.09) +ICC1 + + +0.126*** + + + +0.086** +* + + + +0.110*** + + + + + (0.03) + + + + (0.02) + + + +(0.03) + +Snijders/Bosker R2 level (1) +0.0038 0.0719 +0.2127 + +0.0026 0.0729 +0.1811 + +0.0032 0.0558 +0.1653 +Snijders/Bosker R2 level (2) +0.0124 0.0317 +0.2624 + +0.0130 0.0361 +0.2743 + +0.0103 0.0502 +0.2491 +No. of cases level 1 +7614 +7614 +7614 + +7636 +7637 +7637 + +7639 +7639 +7639 +No. of cases level 2 +29 +29 +29 + +29 +29 +29 + +29 +29 +29 +Notes: standard errors in parentheses.*** p ≤ .001, ** p ≤ .01, * p ≤ .05 (two-tailed tests) +1ICC=intra-class correlation + + + +32 + +Table 3. KHB Mediation Analysis Results for Hypotheses 2 and 3 + Dependent Variable: +Overall Firm +Innovativeness +Product and +Process Innovation +Organizational and +Marketing Innovation + Model: +(1) +(2) +(3) +Summary of effects for specified predictor X on firm +innovation + + + + Total direct and indirect effect +0.16*** +0.063*** +0.095*** + +(0.04) +(0.02) +(0.02) + Direct effect +0.10** +0.034 +0.063*** + +(0.04) +(0.02) +(0.02) + Combined indirect effect +0.059** +0.029** +0.031*** + +(0.02) +(0.01) +(0.01) + Total amount mediated +37.06% +45.43% +33.02% + + + + +Indirect effects of specified predictor X on firm +innovation through proposed mediator + + + + R&D investments +0.043* +0.021* +0.022* + +(0.02) +(0.01) +(0.01) + +26.84% +34.01% +23.09% +Breadth of external capital +0.016*** +0.007*** +0.009*** + +(0.00) +(0.00) +(0.00) + +10.23% +11.42% +9.94% + + + + +Notes: unstandardized coefficients are displayed in the first row, standard errors in parentheses in the second row, and +percentage reduced due to mediation in the third row; all control variables from table 2 are included in the mediation +models. *** p ≤ .001, ** p ≤ .01, * p ≤ .05 (two-tailed tests) + + + + + + + + + + + +33 + +Table 4: HLM Robustness Checks and Mediation for Separate Types of Innovation + + New product +development +New processes + Organizational +innovation + Marketing innovation +Independent Variable: Gender +diversity in firm +ownership + 0.176* + 0.102 + 0.180* + 0.104 +0.280** +0.210* + 0.240*** +0.165 + (0.08) + (0.09) + (0.09) + (0.09) +(0.09) +(0.09) + (0.08) +(0.09) +Mediator 1: R&D investments + + 1.707*** + + 1.669*** + +1.712*** + +1.707*** + + + (0.09) + + (0.09) + +(0.09) + + (0.09) +Mediator 2: Breadth of external +capital + + 0.226*** + + 0.248*** + +0.302*** + +0.325*** + + + (0.04) + + (0.04) + +(0.04) + + (0.04) +Control variables +all +included +all +included +all +included +all +included +all +included +all +included +all +included +all +included +Total amount of mediation in +KHB +n/a + 45.83% +n/a + 35.85% +n/a +28.21% +n/a + 29.38% +McKelvey & Zavoina's R2 + 0.1016 + 0.1681 + 0.0916 + 0.1535 + 0.0894 +0.1563 + 0.0557 + 0.1269 +N of cases level1; level 2 + 7,651; 29 + 7,651; 29 + 7,651; 29 + 7,651; 29 +7,653; +29 +7,653; +29 + 7,651; 29 + 7,651; 29 +Notes: Full results available upon request. *** p ≤ .001, ** p ≤ .01, * p ≤ .05 (two-tailed tests) + + + + + + + + + + + + + +34 + +Table 5: Additional HLM Robustness Check and Mediation Results for Hypotheses 1-3 + +Model 1: +Alternate +IV2 +Model 2: +Alternate +for M13 +Model 3a: +Alternate +(1) for +M24a +Model 3b: +Alternate +(2) for +M24b +Model 4: +Without +solo- +owned +firms5 + +Model 5: +Without +case +outliers6 +Model 6: +Higher- +innovation +industries7 +Model 7: +Lower- +innovation +industries8 +Model 8: +Heckman +two-part +model +with +HLM9 +IV: Gender +diversity in firm +ownership1 +[0.142***] +0.071* +[0.176***] +0.103** +[0.171***] +0.087* +[0.164***] +0.093** +[0.161***] +0.089* +[0.171***] +0.089* + + +[0.224***] +0.116 + +[0.139**] +0.083 + +0.090* +n/a +M1: R&D +1.363*** +0.989*** +1.361*** +1.371*** +1.360*** +1.362*** +1.374*** +1.359*** +n/a +M2:External capital +0.160*** +0.127*** +0.235*** +0.289*** +0.158*** +0.160*** +0.175*** +0.145*** +0.111*** +Control variables +All +included +All +included +All +included +All +included +All +included +All +included +All +included +All +included +All +included +Total amount +mediated +43.80% +36.39% +36.35% +31.96% +36.95% +37.06% +33.39% +42.85% +n/a +Snijders/Bosker +R2 level (1) +0.2124 +0.2473 +0.2127 +0.2078 +0.2154 +0.2127 +0.2483 +0.1737 +0.2612 +Snijders/Bosker +R2 level (2) +0.2609 +0.4141 +0.2663 +0.2632 +0.2716 +0.2624 +0.3737 +0.2001 +0.5226 +N of cases +level1; level 2 +7614; 29 +7544; 29 +7617; 29 +7850; 29 +6873; 29 +7614; 29 +3025; 29 +4589; 29 +7279;28 +*** p ≤ .001, ** p ≤ .01, * p ≤ .05 (two-tailed tests) +1 DV=overall innovativeness; Regression coefficients for the “gender diversity in firm ownership” without standard errors are displayed in the table (full results are available upon request): values reported +with brackets are the regression coefficients based on the analysis that does not include any mediators or controls (analogous to models 1a, 2a, and 3a in Table 3); values without square brackets are the +regression coefficients based on the analysis with all mediators and controls included (analogous to model 1c, 2c, and 3c). +2 Alternate gender diversity index= coded 1 if the firm has a minimum of 59% gender diversity in the ownership structure (else 0). +3 Alternate measurement for “R&D investments”=HR-pertinent practices enhancing firm innovativeness. Coded 1 if the firm has given its “employees some time to develop or try out a new approach or +new idea about products or services, business process, firm management, or marketing” (else 0). +4a Alternate measure (1) for external capital=a two-item formative index derived from questions about whether the company had used any of the following external sources for financing its working +capital: (1) “borrowed from banks (private and state-owned)”, (2) “borrowed from non-bank financial institutions which include microfinance institutions, credit cooperatives, credit unions, or finance +companies”. Each was coded dichotomously as 1 = yes, 0 = no. Our alternate index for financial capital is thus a count variable ranging from 0 to 2. +4b Alternate measure (2) for external capital= Coded 1 if a firm received any “subsidies from the national, regional, or local governments or European Union sources” over the last three years (else 0). +5 Without solo-owned firms = analysis excludes all firms with sole proprietorship (N=839). +6 Without case outliers = The “dfbeta” analysis indicated that there were no individual case outliers. +7 Higher-innovation industries =Industries with above-population-means for innovation intensity of 0.8680 (machinery and equipment, plastics and rubber, basic metals, furniture, non-metallic mineral +products, other manufacturing, electronics and IT, wholesale, and food). +8 Lower-innovation industries = Industries with below-population-means for innovation intensity 0.8680 (retail, hotel and restaurants, other services, construction, transport, textiles and garments, +chemicals). +9 Heckman two-part model with hierarchical linear modeling (HLM). We used the Heckman (1979) model to correct for a potential self-selection bias into R&D activities. The Heckman correction +model is a two-step process. In our case, the first step involved estimating a model for explaining an owner-manager’s self-selection into R&D activities. We used firm size, firm age, firm industries, +location in a metropolitan area, country-fixed effects, firm subsidies, HR-pertinent innovation enhancement practices, foreign ownership, and state ownership, as the predictor variables. In the second +step, we estimated a multilevel model for the focal dependent variable, including the inverse Mill’s ratio or Rho (i.e., the predicted probability of self-selection into R&D activities) from the selection +equation. See the online appendix for more information. + + diff --git a/e9AzT4oBgHgl3EQfMPso/content/tmp_files/load_file.txt b/e9AzT4oBgHgl3EQfMPso/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..84e8e398f70a97365e789e1d7f84c4c86f2e145b --- /dev/null +++ b/e9AzT4oBgHgl3EQfMPso/content/tmp_files/load_file.txt @@ -0,0 +1,2047 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf,len=2046 +page_content='Gender Diversity in Ownership and Firm Innovativeness in Emerging Markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The Mediating Roles of R&D Investments and External Capital Vartuhi Tonoyan California State University, Fresno Craig School of Business 4245 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Backer Avenue, M/S PB7 Fresno, CA 93740 US vtonoyan@csufresno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='edu Christopher J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Boudreaux Florida Atlantic University College of Business 777 Glades Road, Kaye Hall 145 Boca Raton, FL 33431 US cboudreaux@fau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='edu Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Despite recent evidence linking gender diversity in the firm with firm innovativeness, we know little about the underlying mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Building on and extending the Upper Echelon and entrepreneurship literature, we address two lingering questions: why and how does gender diversity in firm ownership affect firm innovativeness?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We use survey data collected from 7,848 owner-managers of SMEs across 29 emerging markets to test our hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Our findings demonstrate that firms with higher gender diversity in ownership are more likely to invest in R&D and rely upon a breadth of external capital, with such differentials explaining sizeable proportions of the higher likelihood of overall firm innovativeness, product and process, as well as organizational and marketing innovations exhibited by their firms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Our findings are robust to corrections for alternative measurement of focal variables, sensitivity to outliers and subsamples, and endogenous self-selection concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' JEL codes: J16, L26, M13, M14, O31, O32 Keywords: Gender diversity, firm innovation, mediating mechanisms, R&D, external capital 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Introduction A growing body of literature has been accumulating in two management disciplines that share a common focus on gender and firm innovativeness: Upper Echelon and entrepreneurship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Despite their common interest, the two research streams have mainly developed in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' As a result, the findings from one area have rarely informed the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Studies examining the impact of gender diversity in the firm on firm innovativeness is an example of the accelerating work within the first stream, the Upper Echelon literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Our review of the burgeoning research (summarized in the online appendix A1-A2) suggests, however, that extant research faces two limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' First and most important, the majority of scholarship has not studied the question of why gender diversity influences firm innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Understanding "the nuts and bolts" of a managerial phenomenon, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', interdependencies among focal constructs and their underlying mechanisms, is a fundamental feature of theory development (Pelled, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' This knowledge is also essential for managers and entrepreneurs interested in designing processes to trigger specific organizational outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Yet, virtually all of the investigations conducted to date (Almor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Biga-Diambeidou, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Faems and Subramanian, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Garcia-Martinez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Horbach and Jacob, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Ruiz-Jiménez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Ritter-Hayashi, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Talke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Teruel and Segarra, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2020), have not examined potential explanatory factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" Therefore, Pelled's (1996, p." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 616) conclusion from almost 25 years ago that demographic diversity research takes "intervening processes for granted" appears to remain true today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Second, despite a growing recognition within both scholarly (Audretsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Rosenbusch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011) and policy-oriented research (OECD, 2005, 2011) that innovation should not be conceived solely in terms of new products or technological process innovations, of the existing studies (reviewed in the online appendix), only 2 Teruel and Segara (2017), Garcia-Martinez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' (2017), and Díaz-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' (2013) examined additional innovative outputs such as the introduction of new marketing or organizational methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Although emerging scholarship from the second stream, on the "gendering" of entrepreneurship and innovation, has been more informative in unpacking some of the mechanisms explaining male-female differentials in the innovativeness of small and/or entrepreneurial firms by focusing on either the characteristics of the entrepreneur or their institutional environments (Strohmeyer and Tonoyan, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Strohmeyer, Tonoyan, and Jennings, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Thébaud, 2015a and 2015b), this literature has largely neglected to study the implications of gender diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' That is, prior entrepreneurship research, with a few exceptions (Horbach and Jacob, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Na and Shin, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Ritter-Hayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019), has generally ignored businesses owned by the mixed-gender-teams as a focus group (Kim, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' This omission is unfortunate not only because it neglects a significant share of firms owned-led by both males and females (Parker, 2018), but also because it fails to answer the question of whether and why mixed-gender-owner teams differ from gender-homogeneous teams (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', those owned by all-males and all-females) with regard to their innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Our research overcomes these limitations by both synthesizing insights from and extending the Upper Echelon and entrepreneurship literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We address the above-noted gaps by theorizing why and how gender diversity in the ownership of small-and-medium-sized enterprises (SMEs) affects firm innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" We delineate some of the mechanisms behind the “gender diversity in firm ownership and firm innovativeness” nexus considering the role of a firm's investments in research and development (R&D) and its financial resource endowments." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The crux of our argument is that mixed-gender-owner teams will exhibit higher overall levels of 3 firm innovativeness, and this tendency will be partially attributable to their higher likelihood of investing in R&D and relying on a breadth of external capital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" Second, we focus on distinct dimensions of a firm's innovation output not yet examined in prior work — overall innovativeness, product and process innovations, and organizational and marketing innovations." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Such inclusive conceptualization of firm innovativeness is essential, as it attends to various calls in both scholarly (Rosenbush et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011) and policy literature (OECD, 2005, 2011) to study outputs of innovation beyond technological product and process innovation (Ayyagari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Strohmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Boudreaux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We use the Business Environment and Enterprise Performance Survey (BEEPS) collected during 2012-2016 comprising a large sample of 7,848 randomly-selected public (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', stock-listed) and private (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', non-listed) firms from 29 emerging markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Such a sample increases generalizability of our findings to the "understudied private firms" (Tyrowicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019) in the literature and institutional contexts different from those of a developed economy (Jennings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2013) mostly examined in prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We test our arguments using multilevel (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', hierarchical) linear regression to account for the nesting of firms within countries and adjusting for numerous controls at the firm and country- levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We conduct multiple robustness checks to assess the stability and reliability of our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Our analyses provide reliable and robust support for our hypotheses contributing to a more comprehensive understanding of the relationship between gender diversity and firm innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Specifically, they reveal differences in the innovativeness of firms led by the mixed-gender versus gender-homogeneous owning teams, documenting that the former exhibit higher likelihoods of overall firm innovativeness, product and process innovations, as well as organizational and marketing innovations, even after correcting for the firm’s selection into higher- versus lower-innovation industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Most fundamentally, our findings unearth some of 4 the underlying mechanisms that have been overlooked in prior work, demonstrating the intervening roles played by a firm’s investment in R&D and reliance upon a breadth of external capital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Combined, these findings offer the merit of considering alternatives to the predominant argument linking mixed-gender-owning teams with firm financial performance (for meta- analyses and reviews see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Jeong and Harrison, 2017, and Eagly, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' They reveal important implications for the Upper Echelon and entrepreneurship literature, as well as managers and public policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Literature review It has been traditional for scholars in the Upper Echelon literature to study the implications of the gender diversity on corporate boards for firm financial performance, including stock market prices and volatility, return on assets, sales, and Tobin’s Q (for reviews see Terjesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Post and Byron, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Jeong and Harrison, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Grounded in management, learning theories, and social psychology, the crux of the argument is that gender diversity enhances firm financial performance due to complementarity in skills, networks, and behaviors of male and female executives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' However, rigorously conducted meta-analyses (Terjesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Post and Byron, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Jeong and Harrison, 2017) and reviews (Eagly, 2016) concluded that beyond finding "small" and "weak" associations (Jeong and Harrison, 2017), empirical studies had not provided strong and robust support for the theory\'s central tenets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Some suggested female executives have small "decision latitude" on the financial performance of publicly-traded firms (Jeong and Harrison, 2017) since their appointment to corporate boards is often symbolic to "window-dress" the organization as a workplace promoting gender equality (Eagly, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Also, in-group favoritism marginalizes female executives\' influence on decision-making due to their "tokenism" or outgroup minority status (Kanter, 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 5 In light of these findings,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' scholars have suggested: (a) studying organizational contexts different from corporate boards of public firms that are more gender-inclusive providing considerable "decision latitude" to women executives and thus accentuating their impact on firm strategy (Jeong and Harrison,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 2017),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' and (b) examining performance outcomes different from financial measures since gains of profit and productivity "are not the most appropriate place to look for diversity\'s benefits" (Eagly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 2013:224).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Many have called to study firm innovation suggesting that gender diversity likely leads to more creativity and novel thinking within a firm, increasing its overall innovativeness (Dezso¨ and Ross, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Eagly, 2016, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Indeed, emerging research has begun to analyze the question of whether gender diversity in the firm increases firm innovativeness, and if so, how and why.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' While most of the burgeoning literature (summarized in online appendix tables A1-A2) suggests it does, others do not (Biga- Diambeidou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' It is important to note, however, that scholars employed different measures complicating the comparability of their findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' While some examined gender diversity on corporate boards (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Horbach and Jacob, 2018), others explored gender diversity among employees (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Østergaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Many studies also adopted relatively narrow measures of innovation mostly focusing either on new product and technological process innovations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Ruiz-Jiménez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2016), patent applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Faems and Subramanian, 2013), or inputs to innovation process such as R&D intensity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Almor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Unfortunately, our current understanding of why gender diversity influences firm innovation is also limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' As indicated in our review of the emerging literature (summarized in online appendix tables A1-A2), the overwhelming majority of the studies have not studied the 6 mediating constructs linking gender diversity with firm innovativeness to explain the nature of that relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Theory and hypotheses Heeding the calls in the Upper Echelon literature to study gender-inclusive organizational contexts (Jeong and Harrison, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Eagly, 2016 and 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Dezso and Ross, 2012), we focus on gender diversity in firm ownership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Compared to corporate boards and firm workforce, firm ownership seems a more suitable context for studying the implications of gender diversity for firm innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' A firm’s ownership structure aligns both incentives and actions of owners (Fama and Jensen, 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Hence, women’s involvement in the ownership of a firm implies that they will have "decision latitude" being able to shape a firm’s strategic choices (Jeong and Harrison, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Also, while some studies noted potential conflicts between CEOs and in work teams that are likely to arise due to diversity of backgrounds, expertise, or values, such conflicts are less likely to occur between owner-managers because such members self-select into founding and running the firm encouraging a reciprocal exchange and collaboration (Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We suggest that firm innovativeness should be conceptualized according to the product, process, organizational, and marketing innovation domains in which the firm has introduced something novel (Ayyagari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Strohmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" Such broader focus accords with Schumpeter's conceptualization of firm innovation encompassing new products, markets, production processes, and organizing methods (Schumpeter, 1934)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' It is also consistent with OECD guidelines (OECD, 2005, 2011) and recent calls to study novel outputs beyond new product developments or those that are primarily technological (Rosenbush et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Strohmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Boudreaux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 7 We posit two factors as key explanatory mechanisms underlying the relationship between gender diversity in firm ownership and firm innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" The first concerns the firm's internal dynamics focusing on the decision to allocate resources to R&D." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The second discusses the role of external capital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' R&D investments are a critical precursor to firm innovation (Schilling and Green, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Innovations from R&D need not be technological inventions: they also include commercial discoveries, upgrades to existing products and processes, and new business models (Parker 2018: 587).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Although R&D often fails to yield commercial outcomes locking firms into strategies that are difficult to change, firms across a variety of industries invest in R&D to develop new products and services and replace those threatened by substitutes (Shane 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We contend that mixed-gender owning teams are more likely to invest in R&D for several reasons (Parker, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" First, by increasing the range of perspectives and cognitive resources available internally within the firm (Schilling and Green, 2011), gender diversity broadens the firm's knowledge base, thereby facilitating R&D undertakings (Almor et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Miller and Triana, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Second, gender diversity influences how firms recombine internal knowledge in novel ways through interaction, discussion, and mutual learning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" Third, it expands an organization's search activities and external reach, thereby boosting the firm's absorptive capacity (Cohen and Levinthal, 1990) — a key determinant of R&D investments (Parker, 2018)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Fourth, gender diversity also increases dispositional preferences for novelty and change, committing managers and employees to continuous exploration and implementation of creative ideas (Baron and Tang, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Strohmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Although a handful of studies provide empirical support for the above-noted arguments demonstrating positive associations between various measures of gender diversity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', in 8 executive positions or R&D workforce) and R&D intensity at the firm level (Miller and Triana, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Biga-Diambeidou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Almor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2020), none have investigated the link between gender diversity in firm ownership, R&D investments, and firm innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We posit external capital as a second intervening mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' External capital includes funding from banks, equity funds, micro-finance organizations, as well as suppliers, customers, family, and friends (Manolova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' It is the "lifeblood" of firm innovation as it provides adequate capitalization vital for the development and diffusion of innovation (Gorodnichenko and Schnitzer, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' It also signals legitimacy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', credibility and acceptance of the business by the stakeholders (Godwin, Stevens and Brenner, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" Beckman, Burton, and O'Reilly, 2007)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Under-capitalization creates bottlenecks at various stages of innovation development, including the inability to hire and retain qualified personnel, conduct R&D, and market products and services (Shane, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Innovating firms relying too heavily on internal capital often come to discover that "capital is not enough" (Bradley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Boudreaux and Nikolaev, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" Unfortunately, liquidity constraints are often more severe in emerging markets where financial systems cannot meet firms' financial needs: A survey of owner-managers of more than 15,500 firms across 29 emerging markets indicated firms’ inability to access credit at reasonable terms as one of the top three business constraints (BEEPS, 2016)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We suggest that firms led by the mixed-gender-owning teams will be more likely to rely upon external funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' First, men network predominantly with other men, "important others" in business and finance, and form weak ties with former colleagues and employers (Tonoyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' In contrast, women often network with other women, rely on strong ties with family (for review see Jennings and Brush, 2013), and engage in support groups to overcome traditional male dominance in their industry (Vogel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Such network diversity increases the 9 likelihood of obtaining both institutional and bootstrapping financing from friends and family (Manolova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Second, diversity due to gender differences in professional experiences and functional backgrounds is likely to send positive "signals" to financial-resource providers about the "team\'s completeness" (Beckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2007:123), increasing the firm’s likelihood of success and hence the perception of its investment-worthiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Mixed-gender-owner teams have the best of gender-specific behaviors and characteristics: they exhibit both a greater "task-related diversity" (Lee and Beckman, 2019), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=", range of diverse skills and abilities, and superior relational skills combining men's agency and goal orientation with women's communality and nurturing of relationships with various stakeholders (Eagly, 2016)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' A handful of studies offer empirical support for our conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Vogel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" (2014) showed that both task-oriented and relations-oriented diversity of mixed-gender entrepreneurial teams were positively related to the hypothetical resource providers' willingness to provide external capital." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Beckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" (2007) demonstrated that task-related diversity resulting from the top management team's demographic diversity increased the start-up's ability to attract equity capital in Silicon Valley." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' To the best of our knowledge, the literature has not yet considered the linkages between mixed-gender-owner teams, external capital, and firm innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' In light of the preceding discussion, we develop our hypotheses as follows: Hypothesis 1: Firms with higher gender diversity in ownership will exhibit higher firm innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Hypothesis 2: Higher innovativeness anticipated for firms with higher gender diversity in ownership will be partly attributable to their higher likelihood of investing in R&D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Hypothesis 3: Higher innovativeness anticipated for firms with higher gender diversity in ownership will be partly attributable to their higher likelihood of relying on external capital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Data and sample Our data come from the Business Environment and Enterprise Performance Survey (BEEPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Conducted by the World Bank and European Bank for Research and Development during 2012-2016, it is the largest firm-level survey studying managerial perceptions of the business environment in emerging markets of the post-Soviet Union, Central-Eastern Europe, Baltic, and Asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The data are collected through face-to-face interviews with the firm owner- managers using a simple random or randomly-stratified sampling and standardized survey instruments to ensure comparability across countries (BEEPS, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' BEEPS has been featured in prominent management and economics journals (Ayyagari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' McCann and Bahl, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' It is well suited for testing our hypotheses for several reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' First and most important, it is the only firm-level data containing questions pertaining to our dependent variable, focal independent variable, and proposed mediators across 29 countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Second, it provides rich information on firm characteristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', size, age, industry sector, and establishment type) likely to influence firm innovation, which is required to test the net effects of gender diversity on firm innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' A final advantage is its multi-country design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Scholars have called for cross-country studies of this nature to enhance our understanding of what determines firm innovation across emerging markets (Bradley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Jennings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Boudreaux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Our final sample —after removing missing observations— consists of 7,848 cross-sectional firm observations from 29 emerging markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Measures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Dependent variables We measure firm innovativeness, our dependent variable, several different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Our first measure, overall firm innovativeness, captures firm innovation according to the dimension 11 of width —i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', product, process, organizational, and marketing areas— in which the firm has introduced something new (Ayygari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Strohmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' To measure Ioverall, we created a formative index as follows: 𝐼𝑜𝑣𝑒𝑟𝑎𝑙𝑙 = 𝑖𝑝𝑟𝑜𝑑𝑢𝑐𝑡 + 𝑖𝑝𝑟𝑜𝑐𝑒𝑠𝑠 + 𝑖𝑚𝑎𝑟𝑘𝑒𝑡𝑖𝑛𝑔 + 𝑖𝑜𝑟𝑔𝑎𝑛𝑖𝑧𝑎𝑡𝑖𝑜𝑛𝑎𝑙 The index’s four constituent items were derived from the survey questions about whether the firm had introduced any of the following initiatives during the last three years: (1) "new or significantly improved products or services" (excluding a "simple resale of new goods purchased from others and changes of a solely aesthetic nature") (product innovation);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' (2) "new or significantly improved methods for the production or supply of products or services" (process innovation);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' (3) "new or significantly improved organizational or management practices or structures" (organizational innovation), and (4) "new or significantly improved marketing methods" (marketing innovation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Each was coded dichotomously as 1=yes, 0=no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The index thus ranges from 0 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Our second measure is product and process innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" Unlike many intermediary input indicators of firm innovation (such as patents or R&D intensity), new products and processes capture the commercial value of a firm's novel offerings across various industries (Audretsch et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Its two constituent items include new product development and process innovations (each coded 1 if the firm had introduced them, 0 if not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The index thus ranges from 0 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Our third measure of a firm’s innovativeness is organizational and marketing innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' An organizational restructuring consists of changes in business practices, quality and human resource management, and external relations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' marketing innovation includes changes to design or packaging, product promotion, placement, and pricing (OECD, 2005, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Both innovation types are critical for SME competitiveness, often substituting rather than 12 complementing product and process innovations (Audretsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Its two constituent items include organizational and marketing innovations (each coded 1 if the firm had introduced these, 0 if not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The index ranges to 0 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Independent and mediating variables Following prior literature (Talke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Triana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" Zhang, 2020), we measure our independent variable, gender diversity in firm ownership, using Blau's index of heterogeneity, 2 x (1 – ΣPi2), where Pi denotes the percentage of the population in each gender group (Blau, 1977)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 1 To create this variable, we combined two questions from the BEEPS survey questionnaire: whether there are any females “amongst the owner of the firm”, and if yes, “what percentage of the firm is owned by females?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The index ranges from 0 indicating complete gender homogeneity (describing all-male and all-female-owned firms) to the highest value of 1 indicating complete gender diversity (equal ownership by males and females).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Our first mediator, a firm’s R&D investments, is coded 1 if the firm has invested in research and development activities —either in-house or contracted with other companies (outsourced) (1=yes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 0=no).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' To measure our second hypothesized mediator, a firm\'s breadth of external capital, Capitalbreadth, we created a formative index as follows: 𝐸𝑥𝑡𝑒𝑟𝑛𝑎𝑙 𝐶𝑎𝑝𝑖𝑡𝑎𝑙𝑏𝑟𝑒𝑎𝑑𝑡ℎ = 𝑖𝑏𝑎𝑛𝑘𝑠 + 𝑖𝑛𝑜𝑛−𝑏𝑎𝑛𝑘𝑠 + 𝑖𝑡𝑟𝑎𝑑𝑒 𝑐𝑟𝑒𝑑𝑖𝑡 + 𝑖𝑜𝑡ℎ𝑒𝑟 The four constituent items were derived from the survey questions on sources of funding the firm uses to finance its working capital: (1) "banks (private and state-owned)" (banks);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' (2) 1 For example, if women and men owned 90% and 10% of firm ownership, respectively, the Blau index would be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='36 (2 x (1 - (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='92 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' If each of the genders possessed 50% of the firm ownership, the Blau index would be equal to one indicating gender parity (2 x (1-(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='52 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='52)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 13 "non-banks financial institutions” including “microfinance institutions, credit cooperatives, credit unions or finance companies" (non-banks);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' (3) "purchases on credit from suppliers and advances from customers" (trade credit);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' and (4) "other" (moneylenders, friends, relatives, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=')" (other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Each was coded dichotomously as 1=yes, 0=no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The index ranges from 0 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Control variables To better isolate the effects of our hypothesized predictors in HLM, we included various controls at the firm- and country-levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Our firm-level controls capture both contemporaneous and imprinting determinants of firm innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Prior work has shown that larger firms, younger firms, foreign-owned firms, private firms, exporting firms, firms with specific characteristics of the Top Management, public firms, and those exposed to international markets and knowledge spillovers are more innovative (Audretsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' SME is coded 1 if the firm employed less than 250 full-time employees at the time of the survey (0 otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Start-up is coded 1 if the firm was less than six years of age at the time of the survey (0 otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Foreign ownership is coded 1 if private foreign individuals or organizations owned more than 50 percent of the firm (0 otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' State ownership is coded 1 if the government held more than 50 percent of firm ownership (0 otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Firm exporting is coded 1 if direct exports comprised part of the firm revenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Top Manager describes if the firm’s Top Manager was a female (coded 1 if yes, 0 if not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Industry experience is a continuous variable measuring years of industry experience of the firm’s Top Manager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' International quality certification is coded 1 if the firm had an internationally-recognized quality certificate at the time of the survey (0 if not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Technology license is coded 1 if the firm had technology licenses from a foreign-owned company (“excluding office software”) at the time of the survey (0 if not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Private firm/JV is coded 1 if the firm was initially established either as "private, from the time of start-up" or "joint venture with a foreign partner(s)" (0 if it was 14 established as a "state-owned firm", "privatization of a state-owned firm", or "private subsidiary of a formerly state-owned firm").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Part of larger business is coded 1 if the firm is part of a larger business (0 if no).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Public company is coded 1 if the firm is a shareholding company with (non-) publicly traded shares (0 if a sole proprietorship, partnership, or limited partnership).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We included various country-level controls to adjust for the institutional environments that are likely to influence cross-country differences in firm innovation (Boudreaux, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Although we explored many such possible controls, we retained only three in our main models to have sufficient statistical power at level two of our multilevel modeling of 29 countries (Raudenbush and Bryk, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" 2 We included the country's per capita GDP as the country's overall economy influences both the likelihood of firm innovation and gender diversity in the firm's ownership-management (Xie et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" The country's total expenditures on research and development (R&D) (% of GDP) is a proxy for a nation's technological prowess and absorptive capacity for innovation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" We also adjusted for the country's business density, which influences firm innovation either directly by encouraging the introduction of novel products or technologies as a firm differentiation strategy or indirectly through knowledge spillovers from other firms (Audretsch et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 3All country indicators are lagged for two years to establish causality and standardized to minimize multicollinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Analytic techniques We used hierarchical linear modeling (HLM) as our primary analytic technique (Raudenbush and Bryk, 2002) to control for the intra-class correlation (ICC) that was evident 2 To obtain our country controls, we used the World Bank’s Development Indicators data, EBRD’s Science, Technology, and Innovation Indicators, and the US Patent and Trademark Office’s patent data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 3 Additional country indicators that we controlled for included the country’s overall gender egalitarianism, women’s percentage in the labor force, political instability, corruption, and voice and accountability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Models that controlled for these measures produced similar results to those presented in the main analysis (available upon request).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 15 and attributable to the nesting of firms within countries (ICC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='126, p ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='001 in model 1c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' ICC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='086, p ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 in model 2c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' and ICC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='110, p ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='001 in model 3c, see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Because of the continuous nature of our innovation measures, we used multilevel linear regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We examined the variance inflation factor within each model to check the potential for multicollinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' All were well below the threshold value of 10, with a mean of only 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We supplemented the HLM results with the KHB mediation model, a rigorous approach for mediation analysis (Kohler, Karlson, and Holm, 2011) featured in prominent management journals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Tonoyan, Strohmeyer, and Jennings, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Key descriptive findings Table 1 reports descriptive statistics and correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Of the particular interest are the overall means for our innovation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The means for the overall innovativeness (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='87), product and process innovations (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='43), and organizational and marketing innovations (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='44) are well below these variables’ theoretical maximums of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00, respectively, suggesting that a large share of firms in our sample have not exhibited overall firm innovativeness or introduced product and process and/or organizational and marketing innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' This finding is consistent with evidence that most SMEs are not very innovative in practice (Ruef, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Thébaud, 2015a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Strohmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' [Insert Table 1] The means for our focal variables are also revealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Approximately 11 percent of the firms invested in R&D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The mean for the breadth of external capital, at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='70, is well below its 4 The KHB mediation model has several advantages over structural equation modeling (SEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Unlike SEM, KHB mediation permits multiple mediators and does not require latent constructs (Kohler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 16 theoretical maximum of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='0, suggesting that only a small proportion of emerging-market firms used various sources of external capital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' This finding is consistent with evidence on financing patterns of innovation in emerging markets (Manolova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Ayyagari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The bivariate comparisons between firms with and without gender-diverse ownership structures are also revealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Consistent with our theorizing, the former are significantly more likely than that latter to exhibit overall firm innovativeness, develop product and process, as well as organizational and marketing innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The former are also more likely than the latter to allocate resources to R&D and rely on a breadth of external capital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' HLM and mediation results Table 2 presents our HLM results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The baseline multilevel models 1a, 2a, and 3a examine the gross effects of gender diversity in firm ownership omitting control variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We observe a positive and highly significant coefficient on gender diversity, our focal independent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Models 1b, 2b, and 3b test whether the net effects of our independent variable on the dependent variables remain positive and significant in the presence of the controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The positive and highly significant coefficients for the gender diversity in firm ownership variable in these models demonstrate that they did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' These findings strongly support Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Models 1c, 2c, and 3c test the effects of our independent variable on firm innovation after introducing our focal mediators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The coefficients for gender diversity in firm ownership are positive and highly significant in models 1c and 3c and insignificant in model 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The coefficients for R&D investments and breadth of external capital are positive and highly significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' This pattern of findings suggests that higher innovativeness of firms with higher gender diversity in firm ownership is either partially or fully attributable to differences in likelihoods of R&D 17 investments and reliance upon external capital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' These results provide strong initial support for Hypotheses 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' [Insert Table 2] Table 3 presents the results from a more sophisticated KHB mediation analysis (Kohler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We observe an indirect effect of gender diversity on firm innovativeness through both of the hypothesized mediators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' More precisely, R&D investments and breadth of external capital jointly account for a total of 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06%, 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='43%, and 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02% of the observed differentials in overall firm innovativeness, product and process innovations, and organizational and marketing innovations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' These findings lend strong additional support for Hypotheses 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' [Insert Table 3] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Robustness checks 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Alternative measures of our focal constructs and mediators We conducted numerous robustness checks to assess the stability of our results pertaining to alternative measures of our dependent, independent variables as well as mediators, which we summarize in Tables 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Table 4 summarizes the findings for each type of innovation as a separate dependent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Because of the dichotomous nature of the separate innovation measures, we used multilevel logistic regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The findings lend additional support for our Hypotheses 1- 3, which predicted a higher likelihood of firm innovativeness for firms with higher gender diversity in firm ownership, with such differences being partially attributable to a firm’s likelihood of R&D investments and reliance upon external capital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The total amount of mediation explained by our focal mediators corresponds to 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='83%, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='85%, 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='21%, and 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='38% for new products, processes, organizational innovation, and marketing innovation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' [Insert Table 4] 18 In Table 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' we estimated multilevel models using alternative measures of our focal independent variable and the two mediators: a dichotomous measure of the gender diversity index coded 1 if the firm had a minimum of 59% of gender diversity in its ownership structure (model 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' an alternate dichotomous measure for a firm’s R&D investments describing whether or a not the firm used human-resource-management practices likely to enhance firm innovativeness (model 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' a continuous measure including only funding from banks and non- bank financial institutions as the first alternate measure for external capital (model 3a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' and a dichotomous measure capturing whether or not the firm received “any subsidies from the national,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' regional,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' local governments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' or European Union” as the second alternate measure for external capital (model 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Our results are robust to these alternative specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' [Insert Table 5] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Sensitivity to outliers and subsamples We assessed the sensitivity of our findings by re-estimating our multilevel models: excluding a small percentage of the solo owner-manager-led firms (model 4), checking for individual firm outlier cases (model 5), on subsamples of firms from higher-innovation industries (model 6) and lower-innovation industries (model 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The findings provide strong additional evidence attesting to the robustness of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" Selection correction We estimated the Heckman selection model (Heckman, 1979) to correct for the firm's potential self-selection into the decision to invest in R&D, on which we elaborate in online Appendix table A3 and its notes." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The results strongly supported findings pertaining to our focal independent variable and external capital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Analytic extensions 19 Instead of using a continuous variable to measure gender diversity in firm ownership, we created a dichotomous variable describing the “mixed-gender-owner teams” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', firms owned equally by males and females and those with either male-dominated or female-dominated ownership structures) versus all-male-owned and all-female-owned firms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' This analysis (models 1a-1c of online appendix Table A4) produced the same pattern of results demonstrating that the mixed-gender-owned teams (N=1,881) exhibited a higher likelihood of overall firm innovativeness than all-male-owned and all-female-owned teams combined (N=6,008), with the total amount of mediation explained by the mediators corresponding to 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='17%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We then compared only all-male-owned teams versus only all-female-owned teams with the mixed-gender-owned teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Our findings showed that all-male-owned teams (N=5,271) exhibited a significantly lower overall innovativeness than the mixed-gender-owned teams, with a lower likelihood of R&D investments and reliance on external capital mediating 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='16% of the relationship (models 2a-c, Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Although all-female-owned firms (N=737) exhibited lower overall innovativeness than the mixed-gender-owned teams, and R&D investments and external capital were significant predictors of their firm innovation, these mediators did not explain the hypothesized differences with the mixed-gender-owner teams (see models 3a-c in online appendix Table A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We assessed the possible “treatment effects” (Rosenbaum and Rubin, 1983) to correct for the potential selection bias stemming from the fact the “treated” firms might differ from “non- treated” for reasons other than the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We defined firms with a score in the upper 80-100th percentile of the Blau index as the treated, those with no gender diversity as the untreated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We matched the two groups on various observable characteristics pertaining to firm, industry, and country likely to impact firm innovativeness using one-to-one matching, caliper (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01), and 20 Mahalanobis algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The treated firms had a higher likelihood of overall firm innovativeness than the non-treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' See Online Appendix Tables A5 and A6 for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Discussion The study’s objective was to provide theoretical and empirical insights into the questions of how and why gender diversity in firm ownership influences firm innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' With respect to the how question, our analysis of secondary data of 7,848 owner-managers of SMEs in 29 emerging markets indicates that firms with higher gender diversity in firm ownership are more likely to exhibit higher firm innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' With respect to the why question, our findings provide strong and robust support for the theorized mediating roles played by the firm’s R&D investments and reliance upon a breadth of external capital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We could demonstrate that sizeable proportions of the observed differentials in firm innovation—more precisely, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06% in overall firm innovativeness, 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='43% in product and process innovations, and 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2% in organizational and marketing innovations—could be explained by our proposed mediators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Contributions to and implications for the Upper Echelon literature Our results extend the Upper Echelon literature in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' First, building on a small body of nascent work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Na and Shin, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Ritter-Hayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019), we studied a highly relevant and important diversity construct —gender diversity in firm ownership—whereas most existing work (summarized in our online appendix) has examined the construct of gender diversity in other organizational contexts (such as the board of directors, senior management, task teams, and employees).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We further examined firm innovativeness as a performance metric while most prior literature has studied the implications of gender diversity for firm financial performance (for a meta-analysis see Jeong and Harrison, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Our study is a first step in understanding some of the mechanisms underlying the “gender diversity and firm innovativeness” nexus that pertain to the firm’s strategic allocation of 21 resources into research and development as well as its financial resource endowments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' To extend our single-mediation model, we invite scholars to test a unified "double-mediation" model (Kohler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=", 2011) that would combine the mediating effects of both the owners' individual characteristics, i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', various types of skills, experiences, networks, and dispositional traits male and female owners bring to the table (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Lee and Beckman, 2019), and their strategic practices on firm innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Future work could also examine other mediating channels (such as recruitment of specific personnel, establishing strategic alliances with business and government, and employing strategies preventing imitators from capturing the firm’s profits from innovation) through which gender diversity in firm ownership is likely to influence firm innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The socio-economic significance of our key finding about gender diversity sets the stage for a variety of related research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Scholars could investigate if ethnically-diverse and/or age-diverse owner teams are more innovative due to skill complementarity and strategic choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Although emerging evidence suggests this might be the case (Hoogendoorn and van Praag, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Brixy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2020), we need more research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Future studies could also study the linkage between gender diversity in ownership and firm growth through the mediating effects of firm innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Are firms headed by the mixed-gender-owner teams not only more innovative but also more likely to survive and grow?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Studies of the potential downsides associated with too much diversity in firm ownership also hold promise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We invite researchers to test a curvilinear relationship to examine whether too little or too much diversity in firm ownership might hinder firm performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Emerging evidence from a field experiment with student teams (that were required to start a business venture, elect CEOs, conduct meetings, produce, sell, make money, and liquidate within a year as part of an entrepreneurship class) demonstrated clear benefits of gender diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' But, that relationship 22 was curvilinear with the peak reached when women’s share in the team became 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='55 (Hoogendoorn, Oosterbeek, and van Praag, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Too much gender diversity is likely to become detrimental by increasing costs of coordination and communication, delaying team decisions, and slowing market moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Yet, too little gender diversity might also hinder team performance by increasing team complacency with the information exchanged among the team members becoming too homogeneous and communication too easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2 Contributions to and implications for the entrepreneurship literature Our study also extends and possesses implications for entrepreneurship research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" Much current work views innovativeness as a function of the firm's organizational characteristics, alliances with venture capitalists, large corporations, and universities, as well as embeddedness in technology clusters, industries, and national innovation systems (Audretsch et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' This stream ignores, however, the question of who participates in innovation activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' As a result, some scholars have noted our limited understanding of how individual owners and entrepreneurs influence firm innovation (Baron and Tang, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Strohmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We contribute to the small but increasing body of work on this topic demonstrating the role played by individual-level characteristics —mixed-gender-owner teams— not much examined within existing entrepreneurship research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We also extend nascent work by offering a more inclusive conceptualization of firm innovation that recognizes not only new products and technological process innovation but also overall firm innovativeness along with the separate individual domains (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', product, process, organizational, and marketing) in which the firm has introduced something novel (Ayyagari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Strohmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' That said, future researchers could examine the consequences of gender diversity in firm ownership for "innovation depth" or the radicalness of the firm\'s new 23 offerings regardless of the individual domains where they were introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Radical innovations typically emerge after recombining existing ideas from different domains that previously seemed unrelated (Hargadon, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Schilling and Green, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' As mixed-gender-owned teams are more likely to possess knowledge and experience from multiple unrelated domains, they might also be more likely to introduce radical innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" We also contribute to the literature on women's entrepreneurship." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Although recent research started to study entrepreneurship as a "gendered process" (Brush, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Jennings and Brush, 2013), only a few studies (Horbach and Jacob, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Na and Shin, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Ritter-Hayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019) examine mixed-gender ownership as a focus group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Yet, ignoring businesses owned by mixed-gender teams is problematic, since such omission overlooks how complements in skills, networks, and characteristics of male and female entrepreneurs shape firm-level outcomes and evaluations of resource providers (Kim, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Parker, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We demonstrated that a male-female partnership was more beneficial for firm innovativeness than all-male partnership because such diversity enhanced the likelihood of R&D investments and reliance upon external capital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We offered a theory suggesting that the addition of a partner to the firm ownership outside the “in” network who can bring with them skills or attributes that the other partners lack (Godwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2006) will enhance the firm innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Unexpectedly, however, our proposed mediators did not explain the gaps in the innovativeness of firms owned by all females and the mixed genders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Such a finding merits more detailed investigation in future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We also encourage future research on the moderating effects of a country’s institutional environment: given that the degree of sex-based segregation in business ownership and innovation varies across countries (Thébaud, 2015a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Tonoyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2020), partnering with a 24 male (or female) business owner might confer more benefits in securing critical resources and organizational legitimacy (Godwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2006) in some institutional contexts than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' As some suggested that female owner-managers might prefer creating organizations that are more egalitarian and less hierarchical (Jennings and Brush, 2013), future research is also warranted for studying the importance of the organizational culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Gender diversity in ownership might create climates for inclusion, fostering employees’ participation in problem- solving, diversity in opinions, and encouraging exchange between the firm and its constituents (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', customers and regulators) (Cropley and Cropley, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Such channels are likely to enhance employee creativity, firm absorptive capacity, and hence innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Limitations and suggested directions The results of our research are subject to a few limitations, mainly stemming from the secondary nature of our focal dataset, that should be addressed in future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Although our results were robust to various sensitivity checks and self-selection concerns, due to the cross- sectional nature of our data, we encourage the replication of our findings with longitudinal firm data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Since our sample mostly comprised emerging markets from the post-Socialist countries, we also encourage replication of our results on samples of different countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Finally, we focused solely upon the firm\'s lead owner-managers assuming that such individuals as heads of firms are those who "call the shots", i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', make strategic decisions likely to impact firm innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' But, innovation is an inclusive process often involving the input of highly-capable employees from various functional areas (such as engineering, marketing, and manufacturing) (Østergaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Scholars should thus study the interplay between skill diversity of owner-managers and specialization amongst the firm employees to extend our understanding of the antecedents of firm innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Conclusion Returning to our guiding questions of how and why gender diversity in firm ownership influences firm innovativeness, we conclude with the following responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' With respect to the how questions, firms with higher gender diversity in ownership in our sample differed significantly in their degree of innovativeness, exhibiting higher likelihoods of overall innovativeness, product and process, as well as organizational and marketing innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' As for the why question, sizeable proportions of these differentials were attributable to a higher likelihood of investing in R&D and relying upon external capital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We would like to provide a final note about our study’s implications for managers and policymakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' A managerial implication is that both male and female business owners should consider partnering with the opposite sex, as gender diversity in ownership significantly increases the odds that the firm will become more innovative along the dimensions of new products, processes, organizational practices, and marketing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' A policy implication is that policymakers might want to focus on desegregating business ownership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Providing institutional, social, and resource-based support for increasing the share of the mixed-gender- owner teams will go a long way in enhancing firm innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' References Almor, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Bazel-Shoham, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The dual effect of board gender diversity on R&D investments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Long Range Planning 10188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='lrp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='004 Audretsch, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Falck, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Heblich, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Handbook of research on innovation and entrepreneurship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Edward Elgar Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Ayyagari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Demirgüç-Kunt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Maksimovic, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Firm innovation in emerging markets: The role of finance, governance, and competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Journal of Financial and Quantitative Analysis 46, 1545–1580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1017/S0022109011000378 Baron, Tang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The role of entrepreneurs in firm-level innovation: Joint effects of positive affect, creativity, and environmental dynamism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Journal of Business Venturing 26, 49– 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='jbusvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='002 26 Beckman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Burton, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=", O'Reilly, C." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Early teams: The impact of team demography on VC financing and going public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Journal of Business Venturing 22, 147–173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' BEEPS, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The business environment in the transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='ebrd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='com Biga-Diambeidou, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Bruna, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Dang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Houanti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Does gender diversity among new venture team matter for R&D intensity in technology-based new ventures?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Evidence Blau, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Inequality and heterogeneity: A primitive theory of social structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Free Press New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Boudreaux, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Institutional quality and innovation: some cross-country evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Journal of Entrepreneurship and Public Policy 6, 26–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1108/JEPP-04- 2016-0015 Boudreaux, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Nikolaev, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Capital is not enough: opportunity entrepreneurship and formal institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Small Business Economics 53, 709–738.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1007/s11187-018-0068-7 Boudreaux, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Nikolaev, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Klein, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Socio-cognitive traits and entrepreneurship: The moderating role of economic institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Journal of Business Venturing 34, 178– 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Bradley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', McMullen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Artz, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Simiyu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Capital is not enough: Innovation in developing economies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Journal of Management Studies 49, 684–717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1467-6486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='x Brixy, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Brunow, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=", D'Ambrosio, A." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The unlikely encounter: Is ethnic diversity in start- ups associated with innovation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Research Policy 49, 103950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='respol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='103950 Brush, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Research on women business owners: past trends, a new perspective and future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Entrepreneurship: Theory and Practice 16, 5–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Cohen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Levinthal, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Absorptive capacity: A new perspective on learning and innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Administrative Science Quarterly 35, 128–152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Cropley, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' and Cropley, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Innovation capacity, organisational culture and gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' European Journal of Innovation Management, 20, 493-510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1108/EJIM- 12-2016-0120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2307/2393553.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Dai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Byun, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Ding, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The direct and indirect impact of gender diversity in new venture teams on innovation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Entrepreneurship Theory and Practice 43, 505–528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1177/1042258718807696 Dezsö, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Ross, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Does female representation in top management improve firm performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' A panel data investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Strategic Management Journal 33, 1072–1089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1002/smj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1955 Díaz-García, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', González-Moreno, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Sáez-Martínez, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Gender diversity within R&D teams: Its impact on radicalness of innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Innovation 15, 149–160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='5172/impp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='149 Eagly, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' When passionate advocates meet research on diversity, does the honest broker stand a chance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Journal of Social Issues 72, 199–222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1111/josi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12163 Eagly, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Sex differences in social behavior: A social-role interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Psychology Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Faems, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Subramanian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' R&D manpower and technological performance: The impact of demographic and task-related diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Research Policy 42, 1624–1633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='respol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='001 27 Fama, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Jensen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Agency problems and residual claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The Journal of Law and Economics 26, 327–349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1086/467038 Garcia-Martinez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Zouaghi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Garcia-Marco, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Diversity is strategy: the effect of R&D team diversity on innovative performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' R&D Management 47, 311–329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1111/radm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12244 Godwin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Stevens, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', & Brenner, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Forced to play by the rules?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Theorizing how mixed–sex founding teams benefit women entrepreneurs in male–dominated contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Entrepreneurship Theory and Practice, 30, 623-642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1540-6520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Gorodnichenko, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Schnitzer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" Financial constraints and innovation: Why poor countries don't catch up." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Journal of the European Economic Association 11, 1115–1152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1111/jeea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12033 Hargadon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' How breakthroughs happen: The surprising truth about how companies innovate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Harvard Business Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Heckman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Sample selection bias as a specification error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Econometrica 47, 153–161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2307/1912352 Hoogendoorn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', van Praag, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Ethnic diversity and team performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' A field experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Tinbergen Institute Discussion Paper 068-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Hoogendoorn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Oosterbeek, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', van Praag, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The impact of gender diversity on the performance of business teams: Evidence from a field experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Management Science, 59, 1514-1528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Horbach, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Jacob, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The relevance of personal characteristics and gender diversity for (eco-)innovation activities at the firm-level: Results from a linked employer–employee database in Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Business Strategy and the Environment 27, 924–934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1002/bse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2042 Jennings, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Greenwood, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Lounsbury, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Suddaby, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Institutions, entrepreneurs, and communities: A special issue on entrepreneurship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Journal of Business Venturing, 28, 1-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Jennings, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Brush, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Research on women entrepreneurs: challenges to (and from) the broader entrepreneurship literature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Academy of Management Annals 7, 663–715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Jeong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Harrison, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Glass breaking, strategy making, and value creating: Meta- analytic outcomes of women as CEOs and TMT members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' AMJ 60, 1219–1252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='5465/amj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='0716 Kanter, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Some effects of proportion on group life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' American Journal of Sociology, 82, 965–90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Kim, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Do equally-owned small businesses have equal access to credit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Small Business Economics 27, 369–386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1007/s11187-005-2558-7 Kohler, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Karlson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Holm, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Comparing coefficients of nested nonlinear probability models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The Stata Journal 11, 420–438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Beckman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Unpacking board diversity: Women director experience and corporate social responsibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Paper presented at the Academy of Management Meeting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Manolova, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='., Manev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Carter, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Gyoshev, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" Breaking the family and friends' circle: Predictors of external financing usage among men and women entrepreneurs in a transitional economy." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Venture Capital, 8, 109-132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 28 McCann, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Bahl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The influence of competition from informal firms on new product development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Strategic Management Journal 38, 1518–1535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1002/smj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2585 Miller, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Triana, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Demographic diversity in the boardroom: Mediators of the board diversity–firm performance relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Journal of Management Studies 46, 755– 786.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1467-6486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='x Na, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Shin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" The gender effect on a firm's innovative activities in the emerging economies." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Sustainability 11, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='3390/su11071992 OECD, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Education at a Glance, OECD Indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' OECD, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Oslo Manual: Guidelines for Collecting and Interpreting Innovation Data, 3rd edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Østergaard, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Timmermans, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Kristinsson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Does a different view create something new?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The effect of employee diversity on innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Research Policy 40, 500–509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='respol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='004 Parker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The Economics of Entrepreneurship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Cambridge: Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Pelled, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Demographic diversity, conflict, and work group outcomes: An intervening process theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Organization Science, 7:615-631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1287/orsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='615 Post, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Byron, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Women on boards and firm financial performance: A meta-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Academy of Management Journal 58, 1546–1571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='5465/amj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='0319 Raudenbush, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Bryk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Hierarchical linear models: Applications and data analysis methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Sage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Ritter-Hayashi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Vermeulen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Knoben, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=" Is this a man's world?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The effect of gender diversity and gender equality on firm innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' PLoS One 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1371/journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='pone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='0222443 Rosenbaum, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Rubin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The central role of the propensity score in observational studies for causal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Biometrika 70, 41–55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1093/biomet/70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='41 Rosenbusch, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Brinckmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Bausch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Is innovation always beneficial?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' A meta- analysis of the relationship between innovation and performance in SMEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Journal of Business Venturing 26, 441–457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='jbusvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='002 Ruef, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The entrepreneurial group: Social identities, relations, and collective action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Princeton University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Ruiz-Jiménez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Fuentes-Fuentes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' del M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Ruiz-Arroyo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Knowledge combination capability and innovation: The effects of gender diversity on Top Management teams in technology-based Firms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Journal of Business Ethics 135, 503–515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1007/s10551-014-2462-7 Schilling, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Green, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Recombinant search and breakthrough idea generation: An analysis of high impact papers in the social sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Research Policy 40, 1321–1331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='respol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='009 Schumpeter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 1934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The Theory of Economic Development: An Inquiry Into Profits, Capital, Credit, Interest, and the Business Cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Transaction Publishers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Shane, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Technology strategy for managers and entrepreneurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Pearson/Prentice Hall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Strohmeyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Tonoyan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Bridging the gender gap in employment growth: On the role of innovativeness and occupational segregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The International Journal of Entrepreneurship and Innovation 6, 259-273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' httpsdoi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='5367/000000005775179865 29 Strohmeyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Tonoyan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Jennings, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Jacks-(and Jills)-of-all-trades: On whether, how and why gender influences firm innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Journal of Business Venturing 32, 498–518.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='jbusvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='001 Talke, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Salomo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Rost, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' How top management team diversity affects innovativeness and performance via the strategic choice to focus on innovation fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Research Policy 39, 907–918.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='respol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='001 Terjesen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Sealy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Singh, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Women directors on corporate boards: A review and research agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Corporate Governance: An International Review 17, 320–337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1467-8683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='x Teruel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Segarra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The link between gender diversity and innovation: What is the role of firm size?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' International Review of Entrepreneurship 15, 319–340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Thébaud, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2015a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Business as plan B: Institutional foundations of gender inequality in entrepreneurship across 24 industrialized countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Administrative Science Quarterly 60, 671–711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1177/0001839215591627 Thébaud, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2015b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Status beliefs and the spirit of capitalism: Accounting for gender biases in entrepreneurship and innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Social Forces, 94, 61-86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Tonoyan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Strohmeyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Jennings, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Gender gaps in perceived start-up ease: Implications of sex-based labor market segregation for entrepreneurship across 22 European countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Administrative Science Quarterly 65, 181–225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Triana, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' del C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Richard, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Su, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Gender diversity in senior management, strategic change, and firm performance: Examining the mediating nature of strategic change in high tech firms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Research Policy 48, 1681–1693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='respol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='013 Tyrowicz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Terjesen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Mazurek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' All on board?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' New evidence on board gender diversity from a large panel of European firms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' European Management Journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='emj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='001 Vogel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Puhan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Shehu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Kliger, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Beese, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Funding decisions and entrepreneurial team diversity: A field study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Journal of Economic Behavior & Organization 107, 595–613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='jebo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='021 Xie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Zong, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', Lu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Gender diversity in R&D teams and innovation efficiency: Role of the innovation context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Research Policy 49, 103885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='respol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='103885 Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' An institutional approach to gender diversity and firm performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Organization Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1287/orsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1297 30 Table 1: Descriptive statistics and correlations Gender diverse ownership Mean SD Firms with Firms without Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 1 Over’ll firm innov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='81 t=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='66** 1 2 Prod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' & proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' innov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='41 t=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='15*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='88 1 3 Org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' & markt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' innov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='41 t=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='48*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='56 1 4 Product innov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='23 t=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='18** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='45 1 5 Process innov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='18 t=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='80** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='49 1 6 Organizational innov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='19 t=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='70*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='49 1 7 Marketing innovation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='21 t=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='76*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='57 1 8 Gender-diverse owner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 n/a n/a n/a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 1 9 R&D activity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='10 t=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='91*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 1 10 Breadth of extern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' capital 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='67 t=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='10*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='10 1 11 SME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='96 t= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 1 12 Startup 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='17 t= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='19** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 1 13 Top Manager: Female 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='15 t=14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='16*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='17 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 1 14 Years of industry exper experience 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='44 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='54 t=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='82*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='27 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07 1 15 Foreign ownership 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 t=-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='46** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 1 16 State ownership 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 1 17 Exporting activity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='19 t=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='13 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='16 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='14 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 1 18 International quality certification 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='25 t=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='36*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='13 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='18 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='10 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='10 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='28 1 19 Technology licenses 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='17 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='08 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='21 1 20 Private firm/JV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='90 t=-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='80*** -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='27 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 1 21 Part of larger business 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='08 t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='11 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 1 22 Public company 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='84 t=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='51*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 1 23 Metropolitan area 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='35 t=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='91 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='10 1 24 Cntry’s business dens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='68 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='17 t=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='11*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='16 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='10 1 25 Cntry’s p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' GDP -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='15 t=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='24*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='27 1 26 Cntry’s R&D exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='15 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='33 t=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='30*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='11 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='85 1 Notes: correlations above |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01| are statistically significant, p ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' a 16 industry dummies are exclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Hence, correlations are not reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The industry distribution of innovation is statistically significant (overall χ2= 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='80***) with details as follows: IT (45%), machinery and equipment (38%), electronics (37%), plastics and rubber (36%), other manufacturing (31%), food (31%), basic metals and furniture (30%), wholesale (26%), non-metallic minerals (25%), chemicals (24%), textiles and garment (23%), construction (22%), retail (20%), other services (19%), transport (18%), and hotel and restaurants (17%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 31 Table 2: Results of multilevel regression analyses for Hypothesis 1 through 3 Multilevel model: GLS GLS GLS Dependent variable: Overall Firm Innovativeness Product and Process Innovation Organizational and Marketing Innovation Model: 1a 1b 1c 2a 2b 2c 3a 3b 3c Independent variable Gender diversity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='171** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='144** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='076** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='057* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='094** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='084** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='055 in firm ownership (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) Mediators R&D investments 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='362*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='677*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='680*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) Breadth of external capital 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='160*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='067*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='093*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01) Control variables SME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='338 *** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='273 *** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='265 *** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='230 *** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) Start up 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='023 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) Top Manager: Female 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='008 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) Years of industry experience experience 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00) Foreign ownership 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='057 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) State ownership 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='075 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='13) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='12) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='07) Exporting activity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='322** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='195*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='185** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='125*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='136** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='071*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) Intern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' quality certification 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='252** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='146*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='129** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='078*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='123** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='069*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) Technology license 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='430** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='302*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='230** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='167*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='198** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='133*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) Private firm/JV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='021 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) Part of larger business 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='107** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='093*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03) Public company 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) 16 Industry dummies No Yes Yes No Yes Yes No Yes Yes Overall model statistics Constant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='874** 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='129** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='871** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='436* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='546* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='428* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='437* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='577** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='438*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='18) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='16) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='08) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='10) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09) ICC1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='126*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='086* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='110*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='03) Snijders/Bosker R2 level (1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='0038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='0719 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='0026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='0729 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1811 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='0032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='0558 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1653 Snijders/Bosker R2 level (2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='0124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='0317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2624 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='0130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='0361 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2743 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='0103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='0502 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2491 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' of cases level 1 7614 7614 7614 7636 7637 7637 7639 7639 7639 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' of cases level 2 29 29 29 29 29 29 29 29 29 Notes: standard errors in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' *** p ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='001, ** p ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01, * p ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 (two-tailed tests) 1ICC=intra-class correlation 32 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' KHB Mediation Analysis Results for Hypotheses 2 and 3 Dependent Variable: Overall Firm Innovativeness Product and Process Innovation Organizational and Marketing Innovation Model: (1) (2) (3) Summary of effects for specified predictor X on firm innovation Total direct and indirect effect 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='16*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='063*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='095*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) Direct effect 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='10* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='063*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) Combined indirect effect 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='059* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='029* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='031*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01) Total amount mediated 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='43% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02% Indirect effects of specified predictor X on firm innovation through proposed mediator R&D investments 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='022* (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='84% 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09% Breadth of external capital 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='016** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='007** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='009*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='00) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='23% 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='42% 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='94% Notes: unstandardized coefficients are displayed in the first row, standard errors in parentheses in the second row, and percentage reduced due to mediation in the third row;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' all control variables from table 2 are included in the mediation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' *** p ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='001, ** p ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01, * p ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 (two-tailed tests) 33 Table 4: HLM Robustness Checks and Mediation for Separate Types of Innovation New product development New processes Organizational innovation Marketing innovation Independent Variable: Gender diversity in firm ownership 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='280* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='240** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='165 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='08) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='08) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09) Mediator 1: R&D investments 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='707*** 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='669*** 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='712*** 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='707*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='09) Mediator 2: Breadth of external capital 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='226*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='248*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='302*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='325*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='04) Control variables all included all included all included all included all included all included all included all included Total amount of mediation in KHB n/a 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='83% n/a 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='85% n/a 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='21% n/a 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content="38% McKelvey & Zavoina's R2 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1681 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='0916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1535 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='0894 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1563 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='0557 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1269 N of cases level1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' level 2 7,651;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 29 7,651;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 29 7,651;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 29 7,651;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 29 7,653;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 29 7,653;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 29 7,651;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 29 7,651;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 29 Notes: Full results available upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' *** p ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='001, ** p ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01, * p ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 (two-tailed tests) 34 Table 5: Additional HLM Robustness Check and Mediation Results for Hypotheses 1-3 Model 1: Alternate IV2 Model 2: Alternate for M13 Model 3a: Alternate (1) for M24a Model 3b: Alternate (2) for M24b Model 4: Without solo owned firms5 Model 5: Without case outliers6 Model 6: Higher innovation industries7 Model 7: Lower innovation industries8 Model 8: Heckman two part model with HLM9 IV: Gender diversity in firm ownership1 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='142** ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='071 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='176** ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='103* [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='171** ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='087 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='164** ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='093* [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='161** ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='089 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='171** ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='089* [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='224** ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='116 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='139* ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='090* n/a M1: R&D 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='363*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='989*** 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='361*** 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='371*** 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='360*** 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='362*** 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='374*** 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='359*** n/a M2:External capital 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='160*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='127*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='235*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='289*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='158*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='160*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='175*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='145*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='111*** Control variables All included All included All included All included All included All included All included All included All included Total amount mediated 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='80% 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='39% 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='35% 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='96% 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='95% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='06% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='39% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='85% n/a Snijders/Bosker R2 level (1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2473 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2154 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2483 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='1737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2612 Snijders/Bosker R2 level (2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2609 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='4141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2663 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2632 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2716 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2624 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='3737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='2001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='5226 N of cases level1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' level 2 7614;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 29 7544;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 29 7617;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 29 7850;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 29 6873;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 29 7614;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 29 3025;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 29 4589;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 29 7279;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='28 *** p ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='001, ** p ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='01, * p ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='05 (two-tailed tests) 1 DV=overall innovativeness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Regression coefficients for the “gender diversity in firm ownership” without standard errors are displayed in the table (full results are available upon request): values reported with brackets are the regression coefficients based on the analysis that does not include any mediators or controls (analogous to models 1a, 2a, and 3a in Table 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' values without square brackets are the regression coefficients based on the analysis with all mediators and controls included (analogous to model 1c, 2c, and 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 2 Alternate gender diversity index= coded 1 if the firm has a minimum of 59% gender diversity in the ownership structure (else 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 3 Alternate measurement for “R&D investments”=HR-pertinent practices enhancing firm innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Coded 1 if the firm has given its “employees some time to develop or try out a new approach or new idea about products or services, business process, firm management, or marketing” (else 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 4a Alternate measure (1) for external capital=a two-item formative index derived from questions about whether the company had used any of the following external sources for financing its working capital: (1) “borrowed from banks (private and state-owned)”, (2) “borrowed from non-bank financial institutions which include microfinance institutions, credit cooperatives, credit unions, or finance companies”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Each was coded dichotomously as 1 = yes, 0 = no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' Our alternate index for financial capital is thus a count variable ranging from 0 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 4b Alternate measure (2) for external capital= Coded 1 if a firm received any “subsidies from the national, regional, or local governments or European Union sources” over the last three years (else 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 5 Without solo-owned firms = analysis excludes all firms with sole proprietorship (N=839).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 6 Without case outliers = The “dfbeta” analysis indicated that there were no individual case outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 7 Higher-innovation industries =Industries with above-population-means for innovation intensity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='8680 (machinery and equipment, plastics and rubber, basic metals, furniture, non-metallic mineral products, other manufacturing, electronics and IT, wholesale, and food).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 8 Lower-innovation industries = Industries with below-population-means for innovation intensity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='8680 (retail, hotel and restaurants, other services, construction, transport, textiles and garments, chemicals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' 9 Heckman two-part model with hierarchical linear modeling (HLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We used the Heckman (1979) model to correct for a potential self-selection bias into R&D activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' The Heckman correction model is a two-step process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' In our case, the first step involved estimating a model for explaining an owner-manager’s self-selection into R&D activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' We used firm size, firm age, firm industries, location in a metropolitan area, country-fixed effects, firm subsidies, HR-pertinent innovation enhancement practices, foreign ownership, and state ownership, as the predictor variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' In the second step, we estimated a multilevel model for the focal dependent variable, including the inverse Mill’s ratio or Rho (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=', the predicted probability of self-selection into R&D activities) from the selection equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} +page_content=' See the online appendix for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfMPso/content/2301.01127v1.pdf'} diff --git a/g9E5T4oBgHgl3EQfFA61/content/tmp_files/2301.05419v1.pdf.txt b/g9E5T4oBgHgl3EQfFA61/content/tmp_files/2301.05419v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e792d554f0f4fef4ec409eac0188584f8a7a8bda --- /dev/null +++ b/g9E5T4oBgHgl3EQfFA61/content/tmp_files/2301.05419v1.pdf.txt @@ -0,0 +1,918 @@ +Gamow Shell Model description of 40Ca(d,p) transfer reaction +A. Mercenne,1 N. Michel,2, ∗ J.P. Linares Fern´andez,3 and M. P�loszajczak3 +1Center for Theoretical Physics, Sloane Physics Laboratory, +Yale University, New Haven, Connecticut 06520, USA +2Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China +3Grand Acc´el´erateur National d’Ions Lourds (GANIL), +CEA/DSM - CNRS/IN2P3, BP 55027, F-14076 Caen Cedex, France +(Dated: January 16, 2023) +Transfer reactions are essential to determine spectroscopic factors and astrophysical reaction rates. +However, their theoretical evaluation is typically effected using standard reaction theory, from which +structure degrees of freedom are absent. While reaction cross sections have been implemented in +the frame of the no-core shell model with continuum, this model can be applied in practice only +to the lightest nuclei. The use of the core + valence nucleon picture is then necessary to include +inter-nucleon correlations in reaction cross sections involving medium nuclei. For this, we will use +the recently developed coupled-channel Gamow Shell Model (GSM-CC) for direct reactions and +extend it to the evaluation of transfer cross sections. As an example, we will study the 40Ca(d,p) +transfer reaction with GSM-CC. Experimental data can be successfully reproduced, but at the price +of the use of a very phenomenological Hamiltonian. +I. +INTRODUCTION +Nuclear reaction rates are one of the most important +ingredients in describing many astrophysical phenomena. +However, direct measurements of the cross sections at +stellar energies are very challenging, especially when it +involves charged particles. Instead, transfer reactions can +be used as an indirect method to access the desired re- +action rates of astrophysical interest [1, 2]. More par- +ticularly, (d,p) reactions have been extensively used to +extract spectroscopic factors for astrophysically relevant +isotopes to constrain neutron-capture rates [3–7]. Indeed, +one-nucleon transfer reactions are the probe of choice to +obtain information about the nuclear response to nucleon +addition (single-particle strength) as a function of en- +ergy, angular momentum and parity. Traditionally, one- +nucleon transfer reactions to bound states can be used +to extract information regarding direct capture, where +those populating the continuum are used to extract the +resonant and/or compound capture. +To describe the thousands of reactions of astrophysi- +cal interest at the relevant energies, one has to rely on +theoretical approaches. +The properties of radioactive +nuclei, underpinning the nuclear mechanisms involved +in astrophysical processes, are strongly affected by cou- +plings to many-body continuum of scattering and decay +channels. Therefore, a unified theory of these nuclei in- +volves a comprehensive description of bound states, res- +onances and scattering many-body states within a sin- +gle theoretical framework, and this is one of the main +goals of the nuclear theory. A pioneer work in this di- +rection was initiated with the continuum shell model [8– +13], and has been extended to ab initio description of +structure and reactions of light nuclei within the no-core +∗ nicolas.michel@impcas.ac.cn +shell model coupled with the resonating-group method +(NCSM/RGM) [14, 15] and the no-core shell model with +continuum (NCSMC) [16, 17]. Deuteron induced trans- +fer reactions to s-shell and p-shell have been investigated +with NCSM/RGM [18, 19] in the context of primordial +and stellar nucleosynthesis. In the same effort to unify +nuclear structure and reactions, progress have been made +in the development of microscopic ab initio optical po- +tentials for deuteron induced transfer reactions [20]. +An alternative approach to describe radioactive nuclei +within a unifying framework has been proposed with the +open quantum system formulation of the shell model, +the Gamow Shell Model (GSM) [21, 22]. +GSM offers +the most general treatment of couplings between dis- +crete and scattering states, as it makes use of Slater +determinants defined in the Berggren ensemble [23] of +single-particle states. For the description of scattering +properties and reactions, it is convenient to formulate +GSM in the representation of reaction channels (GSM- +CC) [24]. +The Hamiltonian of GSM-CC is Hermitian +because matrix elements are calculated in the harmonic +oscillator basis. However, the calculation of resonances +using this Hamiltonian is done in the Berggren basis, so +that the Hamiltonian matrix in GSM-CC becomes com- +plex symmetric. +The cross sections are calculated by +coupling the real-energy incoming partial waves to the +target states given by the Hermitian Hamiltonian. Con- +sequently, the framework related to cross section calcu- +lation is fully Hermitian, whereas complex energies arise +for resonances because one diagonalizes the complex sym- +metric Hamiltonian matrix induced by Berggren basis +representation. GSM in the coupled-channel representa- +tion, which is based on the RGM, has been applied to the +description of 6Li via deuteron induced elastic scattering +on 4He [25]. +To benchmark GSM-CC for transfer reactions, we ap- +ply it to (d, p) reactions on the doubly magic stable 40Ca +nuclei. +The Letter is organized as follows. +The gen- +arXiv:2301.05419v1 [nucl-th] 13 Jan 2023 + +2 +eral formalism of the GSM-CC is briefly introduced in +Chap. II. The Hamiltonian and results of GSM-CC cal- +culations are presented in Chap. III. In particular, the +low-energy spectra and binding energies of 41Ca, 41Sc, +42Ca, 42Sc, 42Ti are discussed and the description of elas- +tic cross sections: +40Ca(n,n)40Ca, 40Ca(p,p)40Ca, and +transfer cross section 40Ca(d,p)41Ca is presented. +Fi- +nally, conclusions are summarized in Chap. IV. +II. +THEORETICAL FRAMEWORK +In this section, we will briefly outline the GSM-CC +formalism, as more details can be found in Refs. [22, 24, +25]. +We work in the cluster orbital shell model (COSM) +formalism [26], i.e. that all space coordinates are defined +with respect to that of a given inert core: +r = rlab − R(core) +CM +(1) +where rlab is the space coordinate in the laboratory +frame, R(core) +CM +is the center of mass coordinate of the +inert core in the laboratory frame, and r is the COSM +space coordinate. More precisely, rlab and r are respec- +tively the laboratory and COSM valence nucleon coor- +dinates in the case of one-nucleon systems, while they +correspond to the center of mass coordinate in the lab- +oratory and COSM frames, respectively, in the case of +a many-nucleon cluster projectile. The fundamental ad- +vantage of COSM is that it is translationally invariant, +as COSM space coordinates are clearly relative, so that +no spurious center of mass excitation can occur therein +[22, 26]. +The A-body state of the system is decomposed into +reaction channels : +|ΨJA +MA⟩ = +� +c +� +∞ +0 +|(c, r)JA +MA⟩ uJAMA +c +(r) +r +r2 dr , +(2) +where the radial amplitude uJAMA +c +(r), describing the rel- +ative motion of the projectile with respect to the core in +a channel c, is the solution to be determined for a given +total angular momentum JA and its projection MA. Note +that, in case of a cluster channel, the coordinates of the +cluster nucleons are not present in Eq. (2), as r is the +COSM radial coordinate of the cluster center of mass. +This is in contrast with usual formulations of channels +in other models and embodies the cluster approximation +used in GSM-CC. In fact, we do not need to explicitly +treat nucleonic coordinates inside the composite projec- +tile, as we restrict our GSM-CC basis to the two-cluster +mass partitioning case. However, it is possible to effec- +tively include deuteron break-up by including channels +that involve intrinsic scattering states of the deuteron +calculated using the Berggren basis. This was done in +Ref.[25] where the scattering reaction 4He(d,d) is consid- +ered. +The channel states are defined as: +|(c, r)⟩ = ˆ +A |{|ΨJT +T ⟩ ⊗ |r ℓ Jint JP⟩}JA +MA⟩ . +(3) +The channel index c stands for the partitions and quan- +tum numbers {A−a, JT; a, L, Jint, JP}, and ˆ +A is the inter- +cluster antisymmetrizer that acts among the nucleons +pertaining to different clusters. +The states |ΨJT +T ⟩ and +|r ℓ Jint JP⟩ are the target and projectile channel states +with their associated total angular momentum JT and +JP, respectively. The angular momentum couplings read +JP = Jint + ℓℓℓ and JA = JP + JT. +The coupled-channel equations can then be formally +derived from the Schr¨odinger equation: +H |ΨJA +MA⟩ = +E |ΨJA +MA⟩, as: +� +c +� ∞ +0 +r2 (Hcc′(r, r′) − ENcc′(r, r′)) uc(r) +r += 0 , +(4) +with E the scattering energy of the A-body system, and +where the kernels are defined as: +Hcc′(r, r′) = ⟨(c, r)| ˆH |(c′, r′)⟩ +(5) +Ncc′(r, r′) = ⟨(c, r)|(c′, r′)⟩ +(6) +For sake of clarity, we have dropped the total angular +momentum labels JA and MA, but one should keep in +mind that the resolution of Eq. +(4) is done for fixed +values of JA and MA. +Due to the decoupling of the target and projectile at +high energy, it is more convenient to express the Hamil- +tonian ˆH as simply: +ˆH = ˆHT + ˆHP + ˆHTP , +(7) +where ˆHT and ˆHP are the Hamiltonians of the target and +projectile, respectively, while the inter-cluster Hamilto- +nian ˆHTP is defined as: ˆHTP = ˆH − ˆHT − ˆHP, where ˆH +is considered here as a standard shell model Hamiltonian. +To be more specific, ˆHT is the center-of-mass free in- +trinsic Hamiltonian of the target, and its eigenvectors +are |ΨJT +T ⟩ with eigenvalues EJT +T . The projectile Hamilto- +nian is then given by ˆHP, and can be decomposed as fol- +low: ˆHP = ˆHint + ˆHCM, where ˆHint describes its intrinsic +properties and ˆHCM the movement of its center-of-mass, +defined in a single channel c, and where one radial coor- +dinate r occurs: +ˆHCM = +ℏ2 +2 ˜mP +� +− d2 +dr2 + ℓ(ℓ + 1) +r2 +� ++ U ℓ +CM(r) , +(8) +where ˜mP in this equation is the reduced mass of the pro- +jectile and U ℓ +CM(r) is the basis-generating Woods-Saxon +(WS) potential for nucleon projectile, while it is the +weighted sum of proton and neutron basis-generating WS +potentials for deuteron wave functions [22, 25]. Its cen- +tral and spin-orbit parts U ℓ +CM,C(r) and U ℓ +CM,SO(r) read + +3 +in the latter case: +U ℓ +CM,C(r) = U ℓ +p,C(r) + U ℓ +n,C(r) +(9) +U ℓ +CM,SO(r) = 1 +2 U ℓ +p,SO(r) + 1 +2 U ℓ +n,SO(r) , +(10) +where U ℓ +p,C(r), U ℓ +p,SO(r) and U ℓ +n,C(r), U ℓ +n,SO(r) are the +WS basis-generating central and spin-orbit potentials for +proton and neutron, respectively. The potential U ℓ +CM(r) +of Eq. (13) then reads: +U ℓ +CM(r) = U ℓ +CM,C(r) ++ 1 +2 U ℓ +CM,SO(r) (ℓℓℓ · Jint) , +(11) +where an additional 1/2 factor in the spin-orbit part +arises because the deuteron nucleons have an orbital an- +gular momentum approximately equal to ℓ/2 [22, 25]. +In order to calculate the kernels Eqs. (5) and (6), one +expands |(c, r)⟩ onto a one-body Berggren basis: +|(c, r)⟩ = +� +n +unℓ(r) +r +|(c, n)⟩ , +(12) +where +|(c, n)⟩ = ˆ +A |{|ΨJT +T ⟩ ⊗ |n ℓ Jint JP⟩}JA +MA⟩, +with +ˆHCM |nℓ⟩ = ECM |nℓ⟩, implying that n refers to the +projectile center-of-mass shell number in the Berggren +basis state. +Spin-dependence has not been added in +the unℓ(r) wave function notation for simplicity. +The +basis of |nℓ⟩ states is then generated by diagonalizing +ˆHCM. +Note that |Jint⟩ is the eigenvector of ˆHint with +the eigenvalue EJint +P +. +Consequently, we can expand Eq. (5) onto the basis of +|(c, n)⟩ using Eq. (12) and derive the following expression +for the Hamiltonian kernel: +Hcc′(r, r′) = +� +ˆHCM + EJT +T + EJint +P +� δ(r − r′) +rr′ +δcc′ ++ ˜Vcc′(r, r′) +(13) +where ˜Vcc′(r, r′) includes the remaining short-range po- +tential terms of the Hamiltonian kernels and is the inter- +cluster potential. Note that Hcc′(r, r′) reduces to its di- +agonal part at large distance as ˜Vcc′(r, r′) vanishes iden- +tically at a large but finite radius outside the target. +Hence, nucleon transfer, which is induced by ˜Vcc′(r, r′), +and consequently ˆHTP, can only occur in the vicinity of +the target and not in the asymptotic region. +Clearly, the determination of ˜Vcc′(r, r′) involves the +calculation of the matrix elements of ˆHTP, which contain +a shell model Hamiltonian. In order to compute ˆHTP, +one has to expand each |(c, n)⟩ onto a basis of Slater +determinants built upon single-particle (s.p.) states of +the Berggren ensemble. In practice, the intrinsic target +and projectile states, |ΨJT +T ⟩ and |Jint⟩ respectively, are +already calculated with that basis, as ˆHT and ˆHint are +solved using the GSM. Note that, in general, as we deal +with very light projectiles, ˆHint is solved within a no-core +framework, and this will be the case in the present study. +The remaining task consists in expanding |n ℓ Jint⟩ in a +basis of Slater determinants. In GSM-CC, this is done +by applying a center-of-mass excitation raising operator +onto |Jint⟩. More details can be found in [22, 25]. +The many-body matrix elements of the norm kernel +Eq. (6) are calculated using the Slater determinant ex- +pansion of the cluster wave functions |(c, n)⟩. The treat- +ment of the non-orthogonality of channels is the same as +in the one-nucleon projectile case [24]. Note that the an- +tisymmetry of channels, enforced by the antisymmetrizer +in Eq. (3), is exactly taken into account through the ex- +pansion of many-body targets and projectiles with Slater +determinants. +Once the kernels are computed, the coupled-channel +equations (4) can be solved using a numerical method +based on a Berggren basis expansion of the Green’s func- +tion (H − E)−1, that takes advantage of GSM complex +energies. Details of this method can be found in Refs. +[22, 25]. +One remarks that a formulation of coupled-channel +equations based on Berggren basis expansions has been +formulated in Ref. [27] and applied therein to the calcu- +lation of the deuteron bound state and phase shifts. +III. +RESULTS AND DISCUSSIONS +We consider a 40Ca core with one or two valence nu- +cleons to study the 40Ca(d,p) reaction. All partial waves +up to ℓ = 4 are included in the model space of GSM +and GSM-CC. As all considered target nuclei are well +bound, it is sufficient to define model spaces with HO +wave functions in GSM, i.e. GSM reduces to standard +shell model for target structure. For this, one includes +all HO wave functions bearing n ≤ 5 in s, p, d, f, g partial +waves above the 40Ca core. The HO length used is 1.88 +fm. No truncation is imposed. As described in Sec. II, +the many-body resonant and scattering wave functions +in GSM-CC are expanded in a Berggren basis of reaction +channels. +The core of the Hamiltonian in both GSM and GSM- +CC is mimicked by a WS potential and the residual +interaction between nucleons is the Furutani-Horiuchi- +Tamagaki (FHT) interaction [28]. Basis-generating po- +tentials in GSM-CC are also of WS type. +All WS potentials possess a diffuseness d = 0.65 fm +and a radius R0 = 1.27A1/3 = 4.34 fm, so that they +differ only by the central and spin-orbit potential depths, +denoted respectively by Vo and Vso. +The single-particle states of 41Ca and 41Sc, that is the +7/2− +1 , 3/2− +1 , 5/2− +1 and 1/2− +1 states, correspond to the +one-body states of the fp shell. Thus, the core WS po- +tentials for partial waves ℓ = 1, 3 have been fitted to +reproduce the single-particle states of 41Sc and 41Ca, re- +spectively. Core potentials for ℓ = 0, 2, 4 partial waves + +4 +play a very small role in the structure of A = 40 − 42 +nuclei, hence they cannot be determined on experimen- +tal energies. Therefore, their values have been fitted to +reproduce the reaction cross sections. WS core potential +depths are listed in Tab. I. +TABLE I. The central (Vo) and spin-orbit (Vso) potential +depths of the core WS potentials in proton (p) and neutron +parts (n) for partial waves ℓ = 0, . . . , 4 (in MeV). +Depth ℓ = 0 ℓ = 1 ℓ = 2 ℓ = 3 ℓ = 4 +Vo(p) +50 +55.142 +62 +56.601 56.5 +Vso(p) +— +6.012 +5 +2.884 +7 +Vo(n) +60.5 54.302 59.5 54.204 +40 +Vso(n) +— +5.007 +2 +2.850 +2 +The parameters of the FHT interaction have been fit- +ted to reproduce the low-lying spectra of A = 42 nuclei +in GSM and GSM-CC and are listed in Tab. II. +TABLE II. The optimized parameters of the FHT interaction +consist of central (V ST +c +), spin-orbit, (V ST +LS ) and tensor (V ST +T +) +coupling constants [30]. S = 0, 1 and T = 0, 1 are the spin +and isospin of the two nucleons, respectively. Parameters are +given in MeV for the central and spin-orbit parts, and in MeV +fm−2 for the tensor part. +V 11 +c +V 10 +c +V 00 +c +V 01 +c +V 10 +LS +V 11 +LS +V 10 +T +V 11 +T +17.745 −4.516 −0.210 −7.386 −2509 1000 −4.102 −0.257 +TABLE III. Same as Tab. I, but for the basis-generating WS +potentials +Depth ℓ = 0 ℓ = 1 ℓ = 2 ℓ = 3 ℓ = 4 +Vo(p) +61 +60 +43 +56 +59 +Vso(p) +— +6.681 +5 +2.854 +5 +Vo(n) +65 +54.302 65.13 54.204 +65 +Vso(n) +— +5.007 +3.9 +2.850 +5 +The deuteron projectile is issued from a GSM calcula- +tion using a Berggren basis defined with two-body rela- +tive coordinates, whereby the N3LO interaction is diag- +onalized (see also Ref. [22] for calculations of diproton, +dineutron and deuteron observables in that framework). +One has to generate antisymmetric composite states by +adding deuteron wave functions to target states after- +wards. However, as the latter are defined in HO model +spaces with laboratory coordinates, deuteron wave func- +tions must be expanded in the same basis. For this, the +bound and scattering deuteron eigenstates issued from +Berggren basis diagonalization are firstly expanded in a +basis of HO states defined with two-body relative coordi- +nates and then in a basis of HO states in laboratory co- +ordinates using Talmi-Brody-Moshinsky coefficients. For +the latter operation, one uses an HO basis defined in a +10 ℏω space. +The transformation from laboratory co- +ordinates to COSM coordinates is neglected, as it can +be shown that its effect is minimal compared to the +TABLE IV. The comparison of GSM-CC separation energies +Sn, Sp, S2n, S2p, and Sd with experimental ones [29]. Energies +are given in units of MeV. +Nucleus S(th) +n +S(exp) +n +S(th) +2n +S(exp) +2n +S(th) +d +S(exp) +d +41Ca +8.38 8.363 +- +- +- +- +42Ca +11.36 11.48 19.74 19.8 +- +- +42Sc +11.42 11.55 +- +- +12.53 12.48 +Nucleus S(th) +p +S(exp) +p +S(th) +2p +S(exp) +2p +S(th) +d +S(exp) +d +41Sc +1.11 +1.09 +- +- +- +- +42Ti +3.57 +3.75 +4.68 +4.83 +- +- +42Sc +4.15 +4.27 +- +- +12.53 12.48 +other theoretical approximations present in our model +[22]. +This allows to recapture the overall structure of +deuteron eigenstates. The energy of the deuteron ground +state is fixed at its experimental energy of -2.2 MeV, +which is close to its theoretical value, equal to -2.1 MeV. +Deuteron break-up can be taken into account by includ- +ing the scattering deuteron eigenstates arising from the +Berggren basis diagonalization, as was done in Ref.[25]. +However, as we only consider a small deuteron projectile +energy in the following, of about 1.8 MeV, along with an +inert 40Ca target core, deuteron break-up cannot occur +in our present calculations due to energy conservation, so +that only the deuteron ground state will be included in +deuteron channels. +As only nucleon projectiles are present in 40Ca(p,p) +and 40Ca(n,n) reactions, the GSM-CC reaction channels +[40Ca(0+ +1 ) ⊗ p(Lj)]Jπ, [40Ca(0+ +1 ) ⊗ n(Lj)]Jπ are directly +defined and solved in the GSM-CC Berggren basis. How- +ever, this is not possible when considering the 40Ca(d,p) +reaction, where one has to use the HO basis to build all +composite states, as we saw for 40Ca+d channels. Thus, +in this case, proton and neutron wave functions are also +expanded with the HO basis so as to form the composite +basis HO states of 41Ca + p and 41Sc + n. +The HO composite basis states are used only for the +generation of the potentials entering Eq. (13), where HO +energy truncation is effected at E(HO) +max = 8 ℏω. The GSM- +CC Hamiltonian of Eq. (13) is solved afterwards using +the GSM-CC Berggren basis (see Sec. II D of Ref.[25]). +The GSM-CC Berggren basis of proton, neutron and +deuteron projectiles is generated by WS potentials whose +parameters are listed in Tab. III (see Sec. II for formu- +las). +The Berggren basis of protons and neutrons consists +of all partial waves up to ℓ = 4. Those for deuteron con- +sist of 3S1, 3P0, 3P1, 3P2, 3D1, 3D2, 3D3 channels. Conse- +quently, the composite channels are those of 40Ca(0+ +1 ) ++ d, 41Ca(Kπ) + p and 41Sc(Kπ) + n, where Kπ = +1/2−, 3/2−, 5/2− and 7/2−. +For the calculation of cross sections, the Jπ quan- +tum numbers of composites are restricted to 0−, 1−, +2−, 1+, 2+ and 3+. +Contours are discretized with 21 +Gauss-Legendre points for protons and neutrons and 30 +Gauss-Legendre points for deuterons. Contours are de- + +5 +FIG. 1. The low-energy positive-parity states of 42Ca, 42Sc and 42Ti nuclei calculated using the GSM-CC are compared to the +experimental data [29]. +103 +106 +109 +40Ca(p,p) +0 +25 +50 +75 +100 +125 +150 +175 +C.M. (deg) +0 +500 +1000 +1500 +2000 +40Ca(n,n) +d /d (mb/sr) +FIG. 2. Cross sections 40Ca(p,p) at CM energy 9.61 MeV and +40Ca(n,n) at CM energy 2.69 MeV. Cross section and angle +are given in CM coordinates. +Experimental data for cross +sections 40Ca(p,p) and 40Ca(n,n) are taken from Refs.[31, 32], +respectively. +fined with kpeak = 0.2 − 0.01i, kmiddle = 0.4 − 0.01i and +kmax = 2 fm−1. Corrective factors have been added in +positive-parity channels, which are equal to 1.14, 0.99, +1.08, 1.04, 1.1 and 1.12 in 0+, 1+, 2+, 3+, 5+ and 7+ +channels, respectively. They allow to reproduce the low- +energy, positive-parity spectra of 42Ca, 42Sc, 42Ti nuclei +(see Fig.(1)). Note that their influence on the 40Ca(d,p) +cross section is minimal. The larger values of the correc- +tive factors for 0+, 5+, and 7+ channels might be due to +the absence of 0+ +2 , 3− +1 , and 2+ +1 low-energy core excitation +in 40Ca. +Results for one- and two-nucleon separation energies +Sn, Sp, S2n, S2p, and Sd calculated in GSM-CC for +A = 41 and 42 nuclei are compared with experimental +separation energies in Tab. +IV. The excited states of +A = 42 nuclei are reproduced in GSM-CC with a typical +precision of ∼50 keV. However, their impact on consid- +ered cross sections is minimal. +The calculated cross sections are compared with the +data in Figs. 2 and 3. The GSM-CC transfer cross sec- +tion 40Ca(d,p) is very well reproduced both in the form +and the magnitude. Also the calculated proton and neu- +tron elastic cross sections reproduce experimental data +very well, except for 40Ca(n,n) at large angles θCM where +the GSM-CC cross section exhibits small oscillations. It +is possible to determine a WS neutron core potential re- +producing correctly the 40Ca(n,n) cross section in the full +range of θCM. However, in this case, the 40Ca(d,p) cross +section cannot be reproduced satisfactorily, so that we +preferred not to optimally fit the 40Ca(n,n) cross section +for other cross sections to be well reproduced. +Note that it is possible to obtain a good reproduc- +tion of the 40Ca(d,p) cross section data only by fine tun- +ing both the Hamiltonian parameters and the WS basis- +generating potential describing the continuum wave func- +tions and their asymptotes, which is essential for a cor- +rect description of the cross sections. Indeed, 40Ca(d,p) +cross sections vary by large factors in an energy interval +centered on 1.853 MeV of a few hundreds of keV [33]. +Thus, important cancellations occur between all partial +waves. +Hence, in order to compensate for these intri- +cate phenomena, the parameters of WS basis-generating +potential had to be fitted as well to reproduce the exper- +imental data. +IV. +CONCLUSIONS +The microscopic description of reaction cross sections +demands to use models where structure and reaction de- +grees of freedom are present. This is the case in GSM-CC, +where target and projectile wave functions are calculated +with shell model. Scattering wave functions can be evalu- +ated afterwards from coupled-channel equations, defined +with the microscopically calculated reaction potentials. +However, contrary to direct reactions, transfer reactions +are very rarely studied at microscopic level. +Thus, we studied the 40Ca(d,p) transfer reaction cross +section in GSM-CC, along with associated direct nucleon +scattering reactions 40Ca(p,p) and 40Ca(n,n). +This is, + +42Ca +42Ti +42c +-18- +-3 - +-11- +2 +-11.049 +-11.033 ++-18.268 +2 + -3.218 + +-18.319 +2 + -3.281 +-11.16 +M +e +2-19- +7 +-12.019 +1+ -12.0 +E[Me +-4 +-12 +1 +-12.024 +7+-12.005 +0 +-12.635 +0 +-12.632 +-19.834 +Exp +GSM-CC +Exp +GSM-CC +-20- +GSM-CC +Exp +-13 +-5 J6 +0 +25 +50 +75 +100 +125 +150 +175 +lab (deg) +0.04 +0.06 +0.08 +0.10 +40Ca(d,p) +d /d (mb/sr) +FIG. 3. Cross section 40Ca(d,p) at CM energy 1.853 MeV. +Besides projectile energy, cross section and angle are given +in laboratory coordinates. Experimental data are taken from +Ref.[34]. +up to our knowledge, the first calculation of this type +in heavier nuclei combining shell model and coupled- +channel equation approaches. We could obtain a good +overall reproduction of experimental cross sections, ex- +cept for 40Ca(n,n) at large angle. The 40Ca(d,p) transfer +reaction cross section is particularly well described. How- +ever, this came at the price of having to very precisely +fine tune Hamiltonian and Berggren basis parameters. +Consequently, while the first GSM-CC calculation ap- +plied to transfer reaction cross section is satisfactory from +a phenomenological point of view, it still remains very +difficult in practice to systematically study transfer re- +action cross sections with the GSM-CC. The renormal- +ization of neglected reaction channels using complex cou- +pling channel-channel potentials seems to be necessary in +the future for that matter. +ACKNOWLEDGMENTS +This work has been supported by the National Nat- +ural Science Foundation of China under Grant Nos. +12175281 and 11975282; the Strategic Priority Research +Program of Chinese Academy of Sciences under Grant +No. +XDB34000000; the State Key Laboratory of Nu- +clear Physics and Technology, Peking University under +Grant No. NPT2020KFY13. +[1] D. W. Bardayan, J. Phys. G: Nucl. Part. Phys. 43, +043001 (2016). +[2] F. M. Nunes, G. Potel, T. Poxon-Pearson, +and J. A. +Cizewski, Annu. Rev. Nucl. Part. Sci. 70, 147 (2020). +[3] J. S. Thomas, D. W. Bardayan, J. C. Blackmon, J. A. +Cizewski, U. Greife, C. J. Gross, M. S. Johnson, K. L. +Jones, R. L. Kozub, J. F. Liang, R. J. Livesay, Z. Ma, +B. H. Moazen, C. D. Nesaraja, D. Shapira, +and M. S. +Smith, Phys. Rev. C 71, 021302(R) (2005). +[4] R. L. Kozub, D. W. Bardayan, J. C. Batchelder, J. C. +Blackmon, +C. R. Brune, +A. E. Champagne, +J. A. +Cizewski, T. Davinson, U. Greife, C. J. Gross, C. C. Jew- +ett, R. J. Livesay, Z. Ma, B. H. Moazen, C. D. Nesaraja, +L. Sahin, J. P. Scott, D. Shapira, M. S. Smith, J. S. +Thomas, and P. J. Woods, Phys. Rev. C 71, 032801(R) +(2005). +[5] J. S. Thomas, G. Arbanas, D. W. Bardayan, J. C. Black- +mon, J. A. Cizewski, D. J. Dean, R. P. Fitzgerald, +U. Greife, C. J. Gross, M. S. Johnson, K. L. Jones, R. L. +Kozub, J. F. Liang, R. J. Livesay, Z. Ma, B. H. Moazen, +C. D. Nesaraja, D. Shapira, M. S. Smith, +and D. W. +Visser, Phys. Rev. C 76, 044302 (2007). +[6] R. L. Kozub, G. Arbanas, A. S. Adekola, D. W. Bar- +dayan, J. C. Blackmon, K. Y. Chae, K. A. Chipps, J. A. +Cizewski, L. Erikson, R. Hatarik, W. R. Hix, K. L. Jones, +W. Krolas, J. F. Liang, Z. Ma, C. Matei, B. H. Moazen, +C. D. Nesaraja, S. D. Pain, D. Shapira, J. F. Shriner, +M. S. Smith, +and T. P. Swan, Phys. Rev. Lett. 109, +172501 (2012). +[7] B. Manning, G. Arbanas, J. A. Cizewski, R. L. Kozub, +S. Ahn, J. M. Allmond, D. W. Bardayan, K. Y. Chae, +K. A. Chipps, M. E. Howard, K. L. Jones, J. F. Liang, +M. Matos, C. D. Nesaraja, F. M. Nunes, P. D. O’Malley, +S. D. Pain, W. A. Peters, S. T. Pittman, A. Ratkiewicz, +K. T. Schmitt, D. Shapira, M. S. Smith, and L. Titus, +Phys. Rev. C 99, 041302(R) (2019). +[8] I. Rotter, Rep. Prog. Phys 54, 635 (1991). +[9] J. Oko�lowicz, M. P�loszajczak, and I. Rotter, Phys. Rep. +374, 271 (2003). +[10] K. +Bennaceur, +F. +Nowacki, +J. +Oko�lowicz, +and +M. P�loszajczak, Nucl. Phys. A 671, 203 (2000). +[11] J. Rotureau, J. Oko�lowicz, +and M. P�loszajczak, Phys. +Rev. Lett. 95, 042503 (2005). +[12] J. Rotureau, J. Oko�lowicz, +and M. P�loszajczak, Nucl. +Phys. A 767, 13 (2006). +[13] A. Volya and V. Zelevinsky, Phys. Rev. Lett. 94, 052501 +(2005). +[14] S. Quaglioni and P. Navr´atil, Phys. Rev. Lett. 101, +092501 (2008). +[15] S. Quaglioni and P. Navr´atil, Phys. Rev. C 79, 044606 +(2009). +[16] S. Baroni, P. Navr´atil, and S. Quaglioni, Phys. Rev. Lett. +110, 022505 (2013). +[17] S.Baroni, P.Navr´atil, and S.Quaglioni, Phys. Rev. C 87, +034326 (2013). +[18] P. Navr´atil and S. Quaglioni, Phys. Rev. Lett. 108, +042503 (2012). +[19] F. Raimondi, G. Hupin, P. Navr´atil, and S. Quaglioni, +Phys. Rev. C 93, 054606 (2016). +[20] J. Rotureau, G. Potel, W. Li, and F. M. Nunes, J. Phys. +G: Nucl. Part. Phys. 47, 065103 (2020). +[21] N. Michel, +W. Nazarewicz, +M. P�loszajczak, +and +T. Vertse, J. Phys. G: Nucl. Part. Phys. 36, 013110 +(2009). + +7 +[22] N. Michel and M. P�loszajczak, Gamow Shell Model, The +Unified Theory of Nuclear Structure and Reactions, Vol. +983 (Springer, 2021). +[23] T. Berggren, Nucl. Phys. A 109, 265 (1968). +[24] Y. Jaganathen, N. Michel, +and M. P�loszajczak, Phys. +Rev. C 89, 034624 (2014). +[25] A. Mercenne, N. Michel, and M. P�loszajczak, Phys. Rev. +C 99, 044606 (2019). +[26] Y. Suzuki and K. Ikeda, Phys. Rev. C 38, 410 (1988). +[27] R. Id Betan, Phys. Lett. B 730, 18 (2014). +[28] H. Furutani, H. Horiuchi, and R. Tamagaki, Prog. Theor. +Phys. 62, 981 (1979). +[29] http://www.nndc.bnl.gov/ensdf (2015). +[30] Y. +Jaganathen, +R. +M. +I. +Betan, +N. +Michel, +W. +Nazarewicz, +and +M. +P�loszajczak, +Phys. Rev. +C 96, 054316 (2017). +[31] J. F. Dicello, G. Igo, W. T. Leland, +and F. G. Perey, +Phys. Rev. C 4, 1130 (1971). +[32] D. Winterhalter, Z. Phys. 200, 487 (1971). +[33] G. Brown, A. Denning, and A. Macgregor, Nucl. Phys. +A 153, 145 (1970). +[34] I. Fodor, I. Szentp´etery, and J. Zim´anyi, Nucl. Phys 73, +155 (1965). + diff --git a/g9E5T4oBgHgl3EQfFA61/content/tmp_files/load_file.txt b/g9E5T4oBgHgl3EQfFA61/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..15fe7c0100de2400c3b78761c1de0ff8afc141d5 --- /dev/null +++ b/g9E5T4oBgHgl3EQfFA61/content/tmp_files/load_file.txt @@ -0,0 +1,716 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf,len=715 +page_content='Gamow Shell Model description of 40Ca(d,p) transfer reaction A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Mercenne,1 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Michel,2, ∗ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Linares Fern´andez,3 and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' P�loszajczak3 1Center for Theoretical Physics, Sloane Physics Laboratory, Yale University, New Haven, Connecticut 06520, USA 2Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China 3Grand Acc´el´erateur National d’Ions Lourds (GANIL), CEA/DSM - CNRS/IN2P3, BP 55027, F-14076 Caen Cedex, France (Dated: January 16, 2023) Transfer reactions are essential to determine spectroscopic factors and astrophysical reaction rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' However, their theoretical evaluation is typically effected using standard reaction theory, from which structure degrees of freedom are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' While reaction cross sections have been implemented in the frame of the no-core shell model with continuum, this model can be applied in practice only to the lightest nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The use of the core + valence nucleon picture is then necessary to include inter-nucleon correlations in reaction cross sections involving medium nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' For this, we will use the recently developed coupled-channel Gamow Shell Model (GSM-CC) for direct reactions and extend it to the evaluation of transfer cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' As an example, we will study the 40Ca(d,p) transfer reaction with GSM-CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Experimental data can be successfully reproduced, but at the price of the use of a very phenomenological Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' INTRODUCTION Nuclear reaction rates are one of the most important ingredients in describing many astrophysical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' However, direct measurements of the cross sections at stellar energies are very challenging, especially when it involves charged particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Instead, transfer reactions can be used as an indirect method to access the desired re- action rates of astrophysical interest [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' More par- ticularly, (d,p) reactions have been extensively used to extract spectroscopic factors for astrophysically relevant isotopes to constrain neutron-capture rates [3–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Indeed, one-nucleon transfer reactions are the probe of choice to obtain information about the nuclear response to nucleon addition (single-particle strength) as a function of en- ergy, angular momentum and parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Traditionally, one- nucleon transfer reactions to bound states can be used to extract information regarding direct capture, where those populating the continuum are used to extract the resonant and/or compound capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' To describe the thousands of reactions of astrophysi- cal interest at the relevant energies, one has to rely on theoretical approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The properties of radioactive nuclei, underpinning the nuclear mechanisms involved in astrophysical processes, are strongly affected by cou- plings to many-body continuum of scattering and decay channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Therefore, a unified theory of these nuclei in- volves a comprehensive description of bound states, res- onances and scattering many-body states within a sin- gle theoretical framework, and this is one of the main goals of the nuclear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' A pioneer work in this di- rection was initiated with the continuum shell model [8– 13], and has been extended to ab initio description of structure and reactions of light nuclei within the no-core ∗ nicolas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='michel@impcas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='cn shell model coupled with the resonating-group method (NCSM/RGM) [14, 15] and the no-core shell model with continuum (NCSMC) [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Deuteron induced trans- fer reactions to s-shell and p-shell have been investigated with NCSM/RGM [18, 19] in the context of primordial and stellar nucleosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' In the same effort to unify nuclear structure and reactions, progress have been made in the development of microscopic ab initio optical po- tentials for deuteron induced transfer reactions [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' An alternative approach to describe radioactive nuclei within a unifying framework has been proposed with the open quantum system formulation of the shell model, the Gamow Shell Model (GSM) [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' GSM offers the most general treatment of couplings between dis- crete and scattering states, as it makes use of Slater determinants defined in the Berggren ensemble [23] of single-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' For the description of scattering properties and reactions, it is convenient to formulate GSM in the representation of reaction channels (GSM- CC) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The Hamiltonian of GSM-CC is Hermitian because matrix elements are calculated in the harmonic oscillator basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' However, the calculation of resonances using this Hamiltonian is done in the Berggren basis, so that the Hamiltonian matrix in GSM-CC becomes com- plex symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The cross sections are calculated by coupling the real-energy incoming partial waves to the target states given by the Hermitian Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Con- sequently, the framework related to cross section calcu- lation is fully Hermitian, whereas complex energies arise for resonances because one diagonalizes the complex sym- metric Hamiltonian matrix induced by Berggren basis representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' GSM in the coupled-channel representa- tion, which is based on the RGM, has been applied to the description of 6Li via deuteron induced elastic scattering on 4He [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' To benchmark GSM-CC for transfer reactions, we ap- ply it to (d, p) reactions on the doubly magic stable 40Ca nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The Letter is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The gen- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='05419v1 [nucl-th] 13 Jan 2023 2 eral formalism of the GSM-CC is briefly introduced in Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The Hamiltonian and results of GSM-CC cal- culations are presented in Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' In particular, the low-energy spectra and binding energies of 41Ca, 41Sc, 42Ca, 42Sc, 42Ti are discussed and the description of elas- tic cross sections: 40Ca(n,n)40Ca, 40Ca(p,p)40Ca, and transfer cross section 40Ca(d,p)41Ca is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Fi- nally, conclusions are summarized in Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' THEORETICAL FRAMEWORK In this section, we will briefly outline the GSM-CC formalism, as more details can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [22, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' We work in the cluster orbital shell model (COSM) formalism [26], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' that all space coordinates are defined with respect to that of a given inert core: r = rlab − R(core) CM (1) where rlab is the space coordinate in the laboratory frame, R(core) CM is the center of mass coordinate of the inert core in the laboratory frame, and r is the COSM space coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' More precisely, rlab and r are respec- tively the laboratory and COSM valence nucleon coor- dinates in the case of one-nucleon systems, while they correspond to the center of mass coordinate in the lab- oratory and COSM frames, respectively, in the case of a many-nucleon cluster projectile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The fundamental ad- vantage of COSM is that it is translationally invariant, as COSM space coordinates are clearly relative, so that no spurious center of mass excitation can occur therein [22, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The A-body state of the system is decomposed into reaction channels : |ΨJA MA⟩ = � c � +∞ 0 |(c, r)JA MA⟩ uJAMA c (r) r r2 dr , (2) where the radial amplitude uJAMA c (r), describing the rel- ative motion of the projectile with respect to the core in a channel c, is the solution to be determined for a given total angular momentum JA and its projection MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Note that, in case of a cluster channel, the coordinates of the cluster nucleons are not present in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' (2), as r is the COSM radial coordinate of the cluster center of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' This is in contrast with usual formulations of channels in other models and embodies the cluster approximation used in GSM-CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' In fact, we do not need to explicitly treat nucleonic coordinates inside the composite projec- tile, as we restrict our GSM-CC basis to the two-cluster mass partitioning case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' However, it is possible to effec- tively include deuteron break-up by including channels that involve intrinsic scattering states of the deuteron calculated using the Berggren basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' This was done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [25] where the scattering reaction 4He(d,d) is consid- ered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The channel states are defined as: |(c, r)⟩ = ˆ A |{|ΨJT T ⟩ ⊗ |r ℓ Jint JP⟩}JA MA⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' (3) The channel index c stands for the partitions and quan- tum numbers {A−a, JT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' a, L, Jint, JP}, and ˆ A is the inter- cluster antisymmetrizer that acts among the nucleons pertaining to different clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The states |ΨJT T ⟩ and |r ℓ Jint JP⟩ are the target and projectile channel states with their associated total angular momentum JT and JP, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The angular momentum couplings read JP = Jint + ℓℓℓ and JA = JP + JT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The coupled-channel equations can then be formally derived from the Schr¨odinger equation: H |ΨJA MA⟩ = E |ΨJA MA⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' as: � c � ∞ 0 r2 (Hcc′(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' r′) − ENcc′(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' r′)) uc(r) r = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' (4) with E the scattering energy of the A-body system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' and where the kernels are defined as: Hcc′(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' r′) = ⟨(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' r)| ˆH |(c′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' r′)⟩ (5) Ncc′(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' r′) = ⟨(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' r)|(c′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' r′)⟩ (6) For sake of clarity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' we have dropped the total angular momentum labels JA and MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' but one should keep in mind that the resolution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' (4) is done for fixed values of JA and MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Due to the decoupling of the target and projectile at high energy, it is more convenient to express the Hamil- tonian ˆH as simply: ˆH = ˆHT + ˆHP + ˆHTP , (7) where ˆHT and ˆHP are the Hamiltonians of the target and projectile, respectively, while the inter-cluster Hamilto- nian ˆHTP is defined as: ˆHTP = ˆH − ˆHT − ˆHP, where ˆH is considered here as a standard shell model Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' To be more specific, ˆHT is the center-of-mass free in- trinsic Hamiltonian of the target, and its eigenvectors are |ΨJT T ⟩ with eigenvalues EJT T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The projectile Hamilto- nian is then given by ˆHP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' and can be decomposed as fol- low: ˆHP = ˆHint + ˆHCM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' where ˆHint describes its intrinsic properties and ˆHCM the movement of its center-of-mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' defined in a single channel c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' and where one radial coor- dinate r occurs: ˆHCM = ℏ2 2 ˜mP � − d2 dr2 + ℓ(ℓ + 1) r2 � + U ℓ CM(r) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' (8) where ˜mP in this equation is the reduced mass of the pro- jectile and U ℓ CM(r) is the basis-generating Woods-Saxon (WS) potential for nucleon projectile,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' while it is the weighted sum of proton and neutron basis-generating WS potentials for deuteron wave functions [22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Its cen- tral and spin-orbit parts U ℓ CM,C(r) and U ℓ CM,SO(r) read 3 in the latter case: U ℓ CM,C(r) = U ℓ p,C(r) + U ℓ n,C(r) (9) U ℓ CM,SO(r) = 1 2 U ℓ p,SO(r) + 1 2 U ℓ n,SO(r) , (10) where U ℓ p,C(r), U ℓ p,SO(r) and U ℓ n,C(r), U ℓ n,SO(r) are the WS basis-generating central and spin-orbit potentials for proton and neutron, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The potential U ℓ CM(r) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' (13) then reads: U ℓ CM(r) = U ℓ CM,C(r) + 1 2 U ℓ CM,SO(r) (ℓℓℓ · Jint) , (11) where an additional 1/2 factor in the spin-orbit part arises because the deuteron nucleons have an orbital an- gular momentum approximately equal to ℓ/2 [22, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' In order to calculate the kernels Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' (5) and (6), one expands |(c, r)⟩ onto a one-body Berggren basis: |(c, r)⟩ = � n unℓ(r) r |(c, n)⟩ , (12) where |(c, n)⟩ = ˆ A |{|ΨJT T ⟩ ⊗ |n ℓ Jint JP⟩}JA MA⟩, with ˆHCM |nℓ⟩ = ECM |nℓ⟩, implying that n refers to the projectile center-of-mass shell number in the Berggren basis state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Spin-dependence has not been added in the unℓ(r) wave function notation for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The basis of |nℓ⟩ states is then generated by diagonalizing ˆHCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Note that |Jint⟩ is the eigenvector of ˆHint with the eigenvalue EJint P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Consequently, we can expand Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' (5) onto the basis of |(c, n)⟩ using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' (12) and derive the following expression for the Hamiltonian kernel: Hcc′(r, r′) = � ˆHCM + EJT T + EJint P � δ(r − r′) rr′ δcc′ + ˜Vcc′(r, r′) (13) where ˜Vcc′(r, r′) includes the remaining short-range po- tential terms of the Hamiltonian kernels and is the inter- cluster potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Note that Hcc′(r, r′) reduces to its di- agonal part at large distance as ˜Vcc′(r, r′) vanishes iden- tically at a large but finite radius outside the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Hence, nucleon transfer, which is induced by ˜Vcc′(r, r′), and consequently ˆHTP, can only occur in the vicinity of the target and not in the asymptotic region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Clearly, the determination of ˜Vcc′(r, r′) involves the calculation of the matrix elements of ˆHTP, which contain a shell model Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' In order to compute ˆHTP, one has to expand each |(c, n)⟩ onto a basis of Slater determinants built upon single-particle (s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=') states of the Berggren ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' In practice, the intrinsic target and projectile states, |ΨJT T ⟩ and |Jint⟩ respectively, are already calculated with that basis, as ˆHT and ˆHint are solved using the GSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Note that, in general, as we deal with very light projectiles, ˆHint is solved within a no-core framework, and this will be the case in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The remaining task consists in expanding |n ℓ Jint⟩ in a basis of Slater determinants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' In GSM-CC, this is done by applying a center-of-mass excitation raising operator onto |Jint⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' More details can be found in [22, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The many-body matrix elements of the norm kernel Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' (6) are calculated using the Slater determinant ex- pansion of the cluster wave functions |(c, n)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The treat- ment of the non-orthogonality of channels is the same as in the one-nucleon projectile case [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Note that the an- tisymmetry of channels, enforced by the antisymmetrizer in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' (3), is exactly taken into account through the ex- pansion of many-body targets and projectiles with Slater determinants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Once the kernels are computed, the coupled-channel equations (4) can be solved using a numerical method based on a Berggren basis expansion of the Green’s func- tion (H − E)−1, that takes advantage of GSM complex energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Details of this method can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [22, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' One remarks that a formulation of coupled-channel equations based on Berggren basis expansions has been formulated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [27] and applied therein to the calcu- lation of the deuteron bound state and phase shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' RESULTS AND DISCUSSIONS We consider a 40Ca core with one or two valence nu- cleons to study the 40Ca(d,p) reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' All partial waves up to ℓ = 4 are included in the model space of GSM and GSM-CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' As all considered target nuclei are well bound, it is sufficient to define model spaces with HO wave functions in GSM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' GSM reduces to standard shell model for target structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' For this, one includes all HO wave functions bearing n ≤ 5 in s, p, d, f, g partial waves above the 40Ca core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The HO length used is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='88 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' No truncation is imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' II, the many-body resonant and scattering wave functions in GSM-CC are expanded in a Berggren basis of reaction channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The core of the Hamiltonian in both GSM and GSM- CC is mimicked by a WS potential and the residual interaction between nucleons is the Furutani-Horiuchi- Tamagaki (FHT) interaction [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Basis-generating po- tentials in GSM-CC are also of WS type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' All WS potentials possess a diffuseness d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='65 fm and a radius R0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='27A1/3 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='34 fm, so that they differ only by the central and spin-orbit potential depths, denoted respectively by Vo and Vso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The single-particle states of 41Ca and 41Sc, that is the 7/2− 1 , 3/2− 1 , 5/2− 1 and 1/2− 1 states, correspond to the one-body states of the fp shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Thus, the core WS po- tentials for partial waves ℓ = 1, 3 have been fitted to reproduce the single-particle states of 41Sc and 41Ca, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Core potentials for ℓ = 0, 2, 4 partial waves 4 play a very small role in the structure of A = 40 − 42 nuclei, hence they cannot be determined on experimen- tal energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Therefore, their values have been fitted to reproduce the reaction cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' WS core potential depths are listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The central (Vo) and spin-orbit (Vso) potential depths of the core WS potentials in proton (p) and neutron parts (n) for partial waves ℓ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' , 4 (in MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Depth ℓ = 0 ℓ = 1 ℓ = 2 ℓ = 3 ℓ = 4 Vo(p) 50 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='142 62 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='601 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='5 Vso(p) — 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='012 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='884 7 Vo(n) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='5 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='302 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='5 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='204 40 Vso(n) — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='007 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='850 2 The parameters of the FHT interaction have been fit- ted to reproduce the low-lying spectra of A = 42 nuclei in GSM and GSM-CC and are listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The optimized parameters of the FHT interaction consist of central (V ST c ), spin-orbit, (V ST LS ) and tensor (V ST T ) coupling constants [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' S = 0, 1 and T = 0, 1 are the spin and isospin of the two nucleons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Parameters are given in MeV for the central and spin-orbit parts, and in MeV fm−2 for the tensor part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' V 11 c V 10 c V 00 c V 01 c V 10 LS V 11 LS V 10 T V 11 T 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='745 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='516 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='210 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='386 −2509 1000 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='102 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='257 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Same as Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' I, but for the basis-generating WS potentials Depth ℓ = 0 ℓ = 1 ℓ = 2 ℓ = 3 ℓ = 4 Vo(p) 61 60 43 56 59 Vso(p) — 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='681 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='854 5 Vo(n) 65 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='302 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='13 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='204 65 Vso(n) — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='007 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='850 5 The deuteron projectile is issued from a GSM calcula- tion using a Berggren basis defined with two-body rela- tive coordinates, whereby the N3LO interaction is diag- onalized (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [22] for calculations of diproton, dineutron and deuteron observables in that framework).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' One has to generate antisymmetric composite states by adding deuteron wave functions to target states after- wards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' However, as the latter are defined in HO model spaces with laboratory coordinates, deuteron wave func- tions must be expanded in the same basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' For this, the bound and scattering deuteron eigenstates issued from Berggren basis diagonalization are firstly expanded in a basis of HO states defined with two-body relative coordi- nates and then in a basis of HO states in laboratory co- ordinates using Talmi-Brody-Moshinsky coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' For the latter operation, one uses an HO basis defined in a 10 ℏω space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The transformation from laboratory co- ordinates to COSM coordinates is neglected, as it can be shown that its effect is minimal compared to the TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The comparison of GSM-CC separation energies Sn, Sp, S2n, S2p, and Sd with experimental ones [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Energies are given in units of MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Nucleus S(th) n S(exp) n S(th) 2n S(exp) 2n S(th) d S(exp) d 41Ca 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='38 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='363 42Ca 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='36 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='48 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='74 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='8 42Sc 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='42 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='55 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='53 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='48 Nucleus S(th) p S(exp) p S(th) 2p S(exp) 2p S(th) d S(exp) d 41Sc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='09 42Ti 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='57 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='68 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='83 42Sc 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='27 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='53 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='48 other theoretical approximations present in our model [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' This allows to recapture the overall structure of deuteron eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The energy of the deuteron ground state is fixed at its experimental energy of -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='2 MeV, which is close to its theoretical value, equal to -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='1 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Deuteron break-up can be taken into account by includ- ing the scattering deuteron eigenstates arising from the Berggren basis diagonalization, as was done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' However, as we only consider a small deuteron projectile energy in the following, of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='8 MeV, along with an inert 40Ca target core, deuteron break-up cannot occur in our present calculations due to energy conservation, so that only the deuteron ground state will be included in deuteron channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' As only nucleon projectiles are present in 40Ca(p,p) and 40Ca(n,n) reactions, the GSM-CC reaction channels [40Ca(0+ 1 ) ⊗ p(Lj)]Jπ, [40Ca(0+ 1 ) ⊗ n(Lj)]Jπ are directly defined and solved in the GSM-CC Berggren basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' How- ever, this is not possible when considering the 40Ca(d,p) reaction, where one has to use the HO basis to build all composite states, as we saw for 40Ca+d channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Thus, in this case, proton and neutron wave functions are also expanded with the HO basis so as to form the composite basis HO states of 41Ca + p and 41Sc + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The HO composite basis states are used only for the generation of the potentials entering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' (13), where HO energy truncation is effected at E(HO) max = 8 ℏω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The GSM- CC Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' (13) is solved afterwards using the GSM-CC Berggren basis (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' II D of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The GSM-CC Berggren basis of proton, neutron and deuteron projectiles is generated by WS potentials whose parameters are listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' III (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' II for formu- las).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The Berggren basis of protons and neutrons consists of all partial waves up to ℓ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Those for deuteron con- sist of 3S1, 3P0, 3P1, 3P2, 3D1, 3D2, 3D3 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Conse- quently, the composite channels are those of 40Ca(0+ 1 ) + d, 41Ca(Kπ) + p and 41Sc(Kπ) + n, where Kπ = 1/2−, 3/2−, 5/2− and 7/2−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' For the calculation of cross sections, the Jπ quan- tum numbers of composites are restricted to 0−, 1−, 2−, 1+, 2+ and 3+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Contours are discretized with 21 Gauss-Legendre points for protons and neutrons and 30 Gauss-Legendre points for deuterons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Contours are de- 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The low-energy positive-parity states of 42Ca, 42Sc and 42Ti nuclei calculated using the GSM-CC are compared to the experimental data [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 103 106 109 40Ca(p,p) 0 25 50 75 100 125 150 175 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' (deg) 0 500 1000 1500 2000 40Ca(n,n) d /d (mb/sr) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Cross sections 40Ca(p,p) at CM energy 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='61 MeV and 40Ca(n,n) at CM energy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='69 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Cross section and angle are given in CM coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Experimental data for cross sections 40Ca(p,p) and 40Ca(n,n) are taken from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [31, 32], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' fined with kpeak = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='01i, kmiddle = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='01i and kmax = 2 fm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Corrective factors have been added in positive-parity channels, which are equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='14, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='99, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='08, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='04, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='12 in 0+, 1+, 2+, 3+, 5+ and 7+ channels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' They allow to reproduce the low- energy, positive-parity spectra of 42Ca, 42Sc, 42Ti nuclei (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Note that their influence on the 40Ca(d,p) cross section is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The larger values of the correc- tive factors for 0+, 5+, and 7+ channels might be due to the absence of 0+ 2 , 3− 1 , and 2+ 1 low-energy core excitation in 40Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Results for one- and two-nucleon separation energies Sn, Sp, S2n, S2p, and Sd calculated in GSM-CC for A = 41 and 42 nuclei are compared with experimental separation energies in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The excited states of A = 42 nuclei are reproduced in GSM-CC with a typical precision of ∼50 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' However, their impact on consid- ered cross sections is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The calculated cross sections are compared with the data in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The GSM-CC transfer cross sec- tion 40Ca(d,p) is very well reproduced both in the form and the magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Also the calculated proton and neu- tron elastic cross sections reproduce experimental data very well, except for 40Ca(n,n) at large angles θCM where the GSM-CC cross section exhibits small oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' It is possible to determine a WS neutron core potential re- producing correctly the 40Ca(n,n) cross section in the full range of θCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' However, in this case, the 40Ca(d,p) cross section cannot be reproduced satisfactorily, so that we preferred not to optimally fit the 40Ca(n,n) cross section for other cross sections to be well reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Note that it is possible to obtain a good reproduc- tion of the 40Ca(d,p) cross section data only by fine tun- ing both the Hamiltonian parameters and the WS basis- generating potential describing the continuum wave func- tions and their asymptotes, which is essential for a cor- rect description of the cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Indeed, 40Ca(d,p) cross sections vary by large factors in an energy interval centered on 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='853 MeV of a few hundreds of keV [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Thus, important cancellations occur between all partial waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Hence, in order to compensate for these intri- cate phenomena, the parameters of WS basis-generating potential had to be fitted as well to reproduce the exper- imental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' CONCLUSIONS The microscopic description of reaction cross sections demands to use models where structure and reaction de- grees of freedom are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' This is the case in GSM-CC, where target and projectile wave functions are calculated with shell model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Scattering wave functions can be evalu- ated afterwards from coupled-channel equations, defined with the microscopically calculated reaction potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' However, contrary to direct reactions, transfer reactions are very rarely studied at microscopic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Thus, we studied the 40Ca(d,p) transfer reaction cross section in GSM-CC, along with associated direct nucleon scattering reactions 40Ca(p,p) and 40Ca(n,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' This is, 42Ca 42Ti 42c 18- 3 - 11- 2 +-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='049 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='033 +-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='268 2 + -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='218 +-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='319 2 + -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='281 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='16 M e 2-19- 7 +-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='019 1+ -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='0 E[Me 4 12 1 +-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='024 7+-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='005 0 +-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='635 0 +-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='632 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='834 Exp GSM-CC Exp GSM-CC 20- GSM-CC Exp 13 5 J6 0 25 50 75 100 125 150 175 lab (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='10 40Ca(d,p) d /d (mb/sr) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Cross section 40Ca(d,p) at CM energy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='853 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Besides projectile energy, cross section and angle are given in laboratory coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Experimental data are taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' up to our knowledge, the first calculation of this type in heavier nuclei combining shell model and coupled- channel equation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' We could obtain a good overall reproduction of experimental cross sections, ex- cept for 40Ca(n,n) at large angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The 40Ca(d,p) transfer reaction cross section is particularly well described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' How- ever, this came at the price of having to very precisely fine tune Hamiltonian and Berggren basis parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Consequently, while the first GSM-CC calculation ap- plied to transfer reaction cross section is satisfactory from a phenomenological point of view, it still remains very difficult in practice to systematically study transfer re- action cross sections with the GSM-CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' The renormal- ization of neglected reaction channels using complex cou- pling channel-channel potentials seems to be necessary in the future for that matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work has been supported by the National Nat- ural Science Foundation of China under Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 12175281 and 11975282;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' the Strategic Priority Research Program of Chinese Academy of Sciences under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' XDB34000000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' the State Key Laboratory of Nu- clear Physics and Technology, Peking University under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' NPT2020KFY13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Bardayan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' G: Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 43, 043001 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [2] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Nunes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Potel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Poxon-Pearson, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Cizewski, Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 70, 147 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Thomas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Bardayan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Blackmon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Cizewski, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Greife, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Gross, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Johnson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Jones, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Kozub, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Liang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Livesay, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Ma, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Moazen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Nesaraja, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Shapira, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Smith, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' C 71, 021302(R) (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [4] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Kozub, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Bardayan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Batchelder, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Blackmon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Brune, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Champagne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Cizewski, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Davinson, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Greife, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Gross, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Jew- ett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Livesay, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Ma, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Moazen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Nesaraja, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Sahin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Scott, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Shapira, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Smith, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Thomas, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Woods, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' C 71, 032801(R) (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Thomas, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Arbanas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Bardayan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Black- mon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Cizewski, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Dean, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Fitzgerald, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Greife, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Gross, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Johnson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Jones, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Kozub, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Liang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Livesay, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Ma, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Moazen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Nesaraja, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Shapira, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Smith, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Visser, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' C 76, 044302 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Kozub, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Arbanas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Adekola, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Bar- dayan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Blackmon, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Chae, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Chipps, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Cizewski, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Erikson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Hatarik, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Hix, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Jones, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Krolas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Liang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Ma, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Matei, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Moazen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Nesaraja, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Pain, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Shapira, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Shriner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Smith, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Swan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 109, 172501 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [7] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Manning, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Arbanas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Cizewski, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Kozub, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Ahn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Allmond, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Bardayan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Chae, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Chipps, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Howard, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Jones, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Liang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Matos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Nesaraja, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Nunes, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' O’Malley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Pain, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Peters, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Pittman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Ratkiewicz, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Schmitt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Shapira, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Smith, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Titus, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' C 99, 041302(R) (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [8] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rotter, Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Phys 54, 635 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Oko�lowicz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' P�loszajczak, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rotter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 374, 271 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [10] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Bennaceur, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Nowacki, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Oko�lowicz, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' P�loszajczak, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' A 671, 203 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rotureau, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Oko�lowicz, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' P�loszajczak, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 95, 042503 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rotureau, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Oko�lowicz, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' P�loszajczak, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' A 767, 13 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Volya and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Zelevinsky, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 94, 052501 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Quaglioni and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Navr´atil, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 101, 092501 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Quaglioni and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Navr´atil, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' C 79, 044606 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Baroni, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Navr´atil, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Quaglioni, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 110, 022505 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [17] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='Baroni, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='Navr´atil, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='Quaglioni, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' C 87, 034326 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [18] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Navr´atil and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Quaglioni, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 108, 042503 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [19] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Raimondi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Hupin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Navr´atil, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Quaglioni, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' C 93, 054606 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rotureau, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Potel, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Li, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Nunes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' G: Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 47, 065103 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [21] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Michel, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Nazarewicz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' P�loszajczak, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Vertse, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' G: Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 36, 013110 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 7 [22] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Michel and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' P�loszajczak, Gamow Shell Model, The Unified Theory of Nuclear Structure and Reactions, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 983 (Springer, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [23] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Berggren, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' A 109, 265 (1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [24] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Jaganathen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Michel, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' P�loszajczak, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' C 89, 034624 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [25] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Mercenne, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Michel, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' P�loszajczak, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' C 99, 044606 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [26] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Suzuki and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Ikeda, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' C 38, 410 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [27] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Id Betan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' B 730, 18 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [28] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Furutani, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Horiuchi, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Tamagaki, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 62, 981 (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [29] http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='nndc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='bnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content='gov/ensdf (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [30] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Jaganathen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Betan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Michel, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Nazarewicz, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' P�loszajczak, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' C 96, 054316 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Dicello, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Igo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Leland, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Perey, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' C 4, 1130 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [32] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Winterhalter, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' 200, 487 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [33] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Brown, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Denning, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Macgregor, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' A 153, 145 (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' [34] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Fodor, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Szentp´etery, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Zim´anyi, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} +page_content=' Phys 73, 155 (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfFA61/content/2301.05419v1.pdf'} diff --git a/g9FKT4oBgHgl3EQfty7v/vector_store/index.pkl b/g9FKT4oBgHgl3EQfty7v/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..1fe227dab1d5c197f4a81ef08c80c051854dec6b --- /dev/null +++ b/g9FKT4oBgHgl3EQfty7v/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a267a73f838be2e61feeac28e59502bc6cd34cae28fd5ebaec5cc60b694d438c +size 123175 diff --git a/hNE1T4oBgHgl3EQffQSM/content/tmp_files/2301.03216v1.pdf.txt b/hNE1T4oBgHgl3EQffQSM/content/tmp_files/2301.03216v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..61c3c01a8ece652a10ce381fba3e3ca5ed0924fa --- /dev/null +++ b/hNE1T4oBgHgl3EQffQSM/content/tmp_files/2301.03216v1.pdf.txt @@ -0,0 +1,635 @@ +Reservoir Prediction by Machine Learning Methods on The Well Data and Seismic +Attributes for Complex Coastal Conditions. +Dmitry Ivlev +Petroleum Overseas ME JSC +dm.ivlev@gmail.com + + + + +The aim of this work was to predict the probability of the spread of rock formations with hydrocarbon- +collecting properties in the studied coastal area using a stack of machine learning algorithms and data +augmentation and modification methods. This research develops the direction of machine learning where +training is conducted on well data and spatial attributes. Methods for overcoming the limitations of this +direction are shown, two methods - augmentation and modification of the well data sample: Spindle and Revers- +Calibration. The implementation and effectiveness of these methods are demonstrated on real data. +Materials and methods. Data from 15 drilled wells, 930 seismic field attributes and 22 additional space +attributes were used in this work. The proposed approach is based on binary classification algorithms with +training on well data and includes the following sequence of applying methods: creating training datasets, +selecting the features, creating a population of classification models, assessing the quality of the forecast, +Reverse-Calibrating the seismic field position, creating additional datasets using Spindle method, assessing the +contribution of the features in the forecast, combining models into an ensemble, the final forecast. +Results. A prediction of the spatial development of reservoirs was made - a three-dimensional cube of +calibrated probabilities of belonging of the studied space to the class of reservoirs was obtained. The estimation +of the classification quality for the classification algorithms and changes of the forecast quality from application +of the Reversal-Calibration and Spindle methods were done. +Conclusion. Considering the difficulties in interpreting seismic data in coastal conditions, the proposed +approach is a tool that is capable of working with the entire set of geophysical and geological data, extracting +knowledge from a 159-dimensional space of spatial attributes, and making a forecast of the spread of facies +with acceptable quality - the F1 measure for the collector class is 0.798 on average for the evaluation of drilling +results under different geological conditions. It has been shown that the sequential application of the proposed +augmentation methods in the implemented technology stack increases the quality of the collector forecast by +more than 1.5 times relative to the original sample. +Keywords: machine learning, downhole data learning, seismic attributes, facies prediction, rock properties +prediction, classification, augmentation methods, ensemble learning, feature selection, evaluation of +contribution of features in prediction. +Conflict of interest: Due to a possible conflict of interest, the work lacks geographic referencing, the name of +the wells has been changed, and the map does not show the geographic location of the north pole direction. + + +Introduction. +The task of forecasting the spatial distribution of rock formations that are reservoirs for +hydrocarbons is both relevant and non-trivial. Machine learning algorithms are one of the tools for +such forecasting, and there are different approaches to predicting the geophysical properties of rock +formations using them. +This approach has the advantage of training and adjusting models using actual data on +environmental properties obtained from geophysical surveys. The complexity of implementing this +approach is determined by two constraints: a small sample of well data and uncertainty in the position +of the seismic wave field relative to well trajectories. To overcome these constraints, the Spindle +method of data augmentation was proposed in [1]: a method of expanding the sample by copying the +full sequence of the target geophysical survey or its interpretation with a small shift in the values of +spatial data at a new position. This method is based on the assumption that within a given shift in the +near-well space, the lithological section does not significantly change, and the sample will include +multiple interpretations of the same geological interval. +However, in coastal areas with significant spatial variability of the geological section, large +deviations of the well trajectory from the vertical and reduced quality of seismic observations over +the area, there are difficulties in predicting changes in environmental properties by machine learning +algorithms, including using the proposed data augmentation method. +Based on empirical studies of spatial data in these conditions, the development of the Spindle +method - the method of Reverse-Calibrating (RC) the position of the seismic wave field relative to +well trajectories - is proposed in this work. This is achieved by searching for the best response of the +machine learning model with the target function of reconstructing a continuous sequence of lithology +classes along the wellbore using surrounding attribute values in space. +The aim of this work is to predict the probability of the presence of rock formations with +collector properties in the studied area by implementing a technological stack of machine learning +algorithms and demonstrating the implementation of Reverse-Calibration and Spindle methods on +actual data, and evaluate the change in forecast quality. The research includes the following sequence +of actions: creation of datasets for training, feature selection, creation of a population of classification +models, evaluation of forecast quality, creation of an additional dataset using the Spindle method, +evaluation of the contribution of features to the forecast, combining models into an ensemble, final +forecast, and analysis of the obtained results. + +The study area is located in the coastal part of a marine basin. A hydrocarbon deposit has been +discovered and is being exploited within the study area. The main area of research is located on an +exploration license block. Due to the potential conflict of interests, the work does not include a +geographical reference to the location, the names of the wells have been changed, and the geographical +position of the direction towards the North Pole is not indicated on the map. The initial data consists +of a single 3D seismic survey of the area. The measurement overlap ratio decreases from 27 to 9 units +towards the coastline. 16 variants of wave field cubes at depths have been obtained as a result of +reprocessing using various transformation graphs. An area of study has been extracted from the 16 +cubes. The width of the study area is 4875 meters, the length is 11025 meters, and the thickness is 800 +meters. The cubes have been brought to a common seismic grid with a lateral step of 25 meters, a +vertical step of 4 meters. The total volume of the cube is 17.199 million points. The cubes have a size +of approximately 1.1 GB each. +Primary filtering of seismic data was carried out based on the Phik correlation coefficient +between the values of amplitudes between the cubes. If the coefficient exceeded the threshold of 0.95, +one of the cubes was excluded from further work. 5 cubes were left with a Phik coefficient of 0.64- +0.92 (Table 1). For each cube, 185 different attributes of the seismic wave field were generated. The +space features are standardized and a power transformation Yeo-Johnson is performed. +Table 1. Seismic cubes included in the study +# +Migration +Route +Stack +Filter +Name in study +1 +KPSDM +FWI +BWXT +no +KFRBSU +2 +KPSDM +FWI +FULL +yes +KFRFSU +3 +KPSDM +FWI +IDB +yes +KFRISF +4 +RTM +FWI +BWXT +no +RFRBSU +5 +KPSDM +Tomo +BWXT IDB +yes +KTRBISF + +An additional set of features was used in the work, including results from aeromagnetic +surveying and lidar imaging, as well as results from stratigraphic interpretation in six versions. The +results of the aeromagnetic survey and lidar imaging were projected onto a depth grid with a single +value. Discrete cubes with assigned categories between horizons were created based on tracing options +for stratigraphic horizons. The total number of additional data was 22 cubes with seismic grid +parameters. + +The study used 15 wells with the results of lithological log interpretation in the study interval: +11 wells from the field and 4 wells from the exploration area. The results of lithological interpretation +are coded into two classes - reservoir and non-reservoir. The reservoir is coded as 1, non- +reservoir as 0. +The base dataset includes results of classification of well logging data projected onto a grid +based on the dominant frequency of the class in the seismic cube element intersecting the well +trajectory. Additionally, vectors were created from spatial data, where each class definition along the +well trajectory was assigned 952 spatial features. +Feature selection. Feature selection for further training was carried out in two stages on the +base dataset. In the first stage, the BoostARoota algorithm [2] was used. The CatBoost [3] algorithm +from the gradient boosting decision tree (GBDT) family was used as an evaluator for its work. In the +second stage, the importance of features was evaluated by training the same algorithm, but with the +addition of noise to the dataset and calculating the Shapley index for the features. Random data (noise) +forms a threshold value for the Shapley index. All features with values of the index below the +threshold are excluded from the dataset. After two stages of selection, 159 features of the studied +space were left from the original 952 for training. All features from the additional dataset showed low +significance for classification and were excluded. +Augmentation - expansion and modification of the dataset. Reverse-Calibration was +implemented by copying well trajectories along with classification well logging. The copying of wells +was done with a large discrete step of lateral displacement of 0, 10, 20 meters to the sides of the world, +and up/down in the vertical direction - 0, 5, 10 meters. The total number of trajectories for one well +was 125, including the original position. For each trajectory, the lithological well logging +classification was projected onto the grid, and vectors were created from spatial data in the position +of the shifted trajectory. +The Reverse-Calibration method was developed based on two assumptions: +▪ first, the position of the seismic wave field relative to the well trajectory may be shifted +due to a more complex physical model of the environment of the studied space compared +to the synthetic one; +▪ second, the seismic wave field carries information about the real environment, which +implies the existence of an area in the field space whose attributes best correspond to the +lithology sequence along the wellbore. + +During the exploratory data analysis, the CatBoost algorithm on the test sample showed +relatively low-quality classification of collectors on the base dataset (F1 0.64) and even lower quality +on the dataset created by the wreath method (F1 0.61). At the same time, experiments with samples +showed that when using only part of the data from the wreath method, the quality of the forecast +improved (F1 0.66), surpassing the quality obtained on the base dataset. +The search for the position of the seismic wave field relative to the original is carried out by +iterating over the positions of the wells in the dataset created using the method of Spindle and +optimizing the classification error function of the machine learning algorithm on cross-validation for +each combination of trajectories. The positions can be shifted by simultaneously optimizing the error +for all datasets, or by optimizing the error for each well independently. The option of simultaneous +optimization of all groups of well positions produces a unique combination of well positions +𝑋𝑑,𝑙 = 𝑉 (𝑋𝐷,𝐿), 𝑑 ∈ 𝐷, 𝑙 ∈ 𝐿, +where 𝑋𝐷,𝐿 is an array of all well positions D generated using the Spindle method, with spatial +data vectors L, the function 𝑉 (⋅) generates 𝑋𝑑,𝑙 - an array of unique combinations of well d with an +array of vectors l for that combination, either randomly or according to a search grid. +𝑋𝑑,𝑙 +𝑚𝑖𝑛 = 𝑎𝑟𝑔𝑚𝑖𝑛𝑋𝑑,𝑙 [ 𝐸 (С (𝑉 (𝑋𝐷,𝐿) +𝑛) )], 𝑛 ∈ 𝑁, +where each combination n from the number of well position combinations N is given to the +classification function С (⋅), the result of the classification is evaluated by the error function 𝐸 (⋅). +𝑋𝑑,𝑙 +𝑚𝑖𝑛 is an array of well sets and their vectors with the minimum classification error value. +This paper implements a version that optimizes the error by shifting all trajectories +simultaneously. Given the significant number of combinations of discrete well positions, grid or +random search may take a long time, so a search optimizer, the tree-structured Parzen estimator +(TPESampler), was used to select combinations [4, 5]. The Random Forest algorithm was used to +evaluate the quality of classification. +The quality of classification was evaluated using the ROC AUC metric on cross-validation. +Cross-validation was performed on wells, in which data from one well were extracted from the dataset, +and the classifier was trained on the remaining data. The quality metric of the trained classifier was +assessed on the extracted well, and then the data for the extracted well were returned to the general +sample, and the next well was selected from it, and the assessment process was repeated. To evaluate + +the overall combination of offset wells, the ROC AUC quality assessment results for the wells were +averaged. The ROC AUC estimate for the initial positions was 0.627, and for the best-found +position 0.810. The distances of the trajectory shifts (shifts seismic wave field) with the best response +of the model from the initial are given in Table 2. A trend of the overall trend of the field shift in the +area of wells related to the field, and a divergent trend in the search area are noted. +Table 2. Offsets of the seismic field relative to the well trajectories +Area +Wells +X +Y +Z +Field +P-4ST +10 +-20 +-5 +P-2ST +10 +-20 +-5 +P-10ST +10 +-20 +-5 +P-6 +20 +-20 +0 +P-3 +10 +-10 +-5 +P-9 +10 +-20 +-5 +P-7 +10 +-10 +-5 +P-8 +20 +-20 +-5 +P-5 +0 +-20 +0 +P-10 +10 +-10 +-10 +P-11 +10 +-10 +-5 +Sub mean +10.9 +-16.4 +-4.5 +Sub std +5.4 +5 +2.7 +Wild +J-1X +20 +10 +5 +SEP-1X +-10 +-10 +10 +SP-1 +10 +-10 +-5 +SP-1ST +10 +-20 +-5 +Sub mean +7.5 +-7.5 +1.3 +Sub std +12.6 +12.6 +7.5 +Total mean + +10 +-14 +-3 +Total std + +7.6 +8.3 +4.9 + +Two datasets were formed for further studies: the base dataset (Base) and the Reverse-Calibrated +dataset (RC). The number of labels and the ratio of classes are given in Table 3. + + +Table 3. Quantity of datasets for training +Class +Base +RC +Spindle RC +0 +934 +937 +57091 +1 +193 +192 +13816 +Sum +1127 +1129 +70907 + +Protocol for data handling was developed to prevent data leakage - accidental exchange of +any information between the test and training sets. The protocol included standardization and Yeo- +Johnson power transformation applied to each attribute cube, followed by the transformation +parameters being applied to the well datasets (Base and RC) from the same combinations of wells. +Eight variations of splits (batches) were formed: training, validation, and test parts for the datasets +(Base and RC) from the same combinations of wells. The test parts were formed from data from a +combination of two wells: one from the search block and one from the field. These parts were isolated +and used only for quality classification assessment. For fine-tuning of models and control of learning, +validation parts were formed in the same way - one well from the search part and one from the field. +The combination of the eight variations of splits ensured that the data from the test wells would not +be included in the training set in any individual variant. At the same time, the eight models of a single +machine learning algorithm, in total, saw the entire dataset in various combinations of training and +were tuned to the entire range of geological conditions revealed by the wells. +The design of training on well data consists of two stages. In the first stage, basic models are +trained based on the data processing protocol, after which the models are evaluated for quality on two +sets of data for 8 batches. In the second stage, a meta-model is trained on an ensemble of basic models. +Basic models. The type of machine learning algorithms currently determines the type of dataset. +Data types are divided into unstructured (image, sound, etc.) and structured (tabular). Deep neural +network architectures are used for unstructured data. It is believed that the main advantage of deep +learning in working with unstructured data lies in the ability to study the feature extraction pipeline +(Raschka 2022). +In this work, sets of tabular data were used, where the features of the space - attributes of the +seismic field were already extracted and class labels were assigned to them. For such data, gradient +boosting decision trees (GBDT) in its specific implementations CatBoost [3], LightGBM [6] and +XGboost [7] have proven to be effective in practice. + +At the same time, deep learning algorithms for tabular data types are emerging. One such +algorithm that showed high classification quality on the datasets used in the work was TabPFN (Prior- +data Fitted Network) [8], described as "a trained Transformer that can perform classification". This +method is well suited for subsequent ensemble with the results of GBDT algorithms, as its errors are +not correlated with the errors of these methods. The current limitation for this algorithm is a maximum +of 100 features and a dataset limited to 1000 instances. Therefore, the 100 best features determined +using the Shepley index on GBDT family algorithms were input to it. It worked only with non- +extended datasets. +Hyperparameter tuning of models. For all algorithms, hyperparameter optimization was +carried out by maximizing the ROC AUC metric on the cross-validation of the training and validation +sets for each batch separately, with balancing of the class ratio and subsequent return of the validation +instances to the training set. For gradient boosting decision tree models (GBTD), the metric was +maximized using the TPESampler optimizer. For TabPFN, by sequentially iterating from 2 to 200 of +the “N_ensemble_configurations” parameter, and the maximum ROC AUC value was reached with +this parameter set to 12. +Model training. The training of GBDT algorithms was monitored on the validation set for each +of the 8 combinations of dataset splits. The TabPFN algorithm, due to its unique characteristics, was +trained simultaneously on the validation and training sets. As a result of training four algorithms on +the Base and Reverse-Calibrated datasets, 64 models were obtained. +Evaluating classification quality. In accordance with the data handling protocol, the quality of +classification was assessed on the isolated test set for each batch individually. The F1 metric was used +to evaluate the quality of the classes. The metrics were grouped according to the groups being studied +and their values were reduced to the mean. +Classification quality for algorithms. Table 4 presents the results of evaluating the +performance of the algorithms. The average values of the F1 metric for 16 models of each algorithm +are shown. The best forecast quality for collectors was demonstrated by the CatBoost algorithm from +the GBDT family. The second in quality was the transformer architecture-based algorithm TabPFN. +Comparing GBDT algorithms for forecast quality is often a demonstration of the author's expertise in +selecting numerous hyperparameters for their settings. This is what distinguishes the TabPFN +algorithm, which is trained in one pass on the entire available dataset and currently has one + +hyperparameter, and at the same time has comparable quality to the GBDT family's forecast. For +further research, based on the experimental design, all algorithms used have good forecast quality. +Table 4. Average classification quality (F1 measure) by algorithms +Parameters +CatBoost +XGBost +LightGBM TabPFN +class 0 +0.892 +0.887 +0.883 +0.880 +class 1 +0.766 +0.727 +0.723 +0.736 + +Quality prediction on datasets. Table 5 shows the averaged prediction metrics grouped by +datasets: a dataset with vectors from the original positions of the seismic wave field (Base) and a +dataset shifted using the Reverse-Calibration method (RC). It follows that, as a result of using the +shifted dataset, the quality of the collector prediction based on the F1 metric increased on average +from 0.691 to 0.786. This increase is significant and means that the quality of the collector prediction +increased 1.49 times, taking into account the value for a random prediction equal to 0.5. Given that +the field calibration by finding the best response to lithology was carried out within the standard error +limits of the processing and interpretation results of 3D seismic exploration work, it can be assumed +that the new field position better reflects the real lithological sequences exposed by the wells. Thus, +it is shown that the method proposed in the work allows to eliminate one of the difficulties of +lithological prediction based on seismic data. This can be achieved by separating the field parameters +that affect the lithology from the field parameters that affect the physical state of the rock and the +properties of the fluid in it. +Table 5. Average classification quality (F1 measure) by datasets +Parameters +Base +RC +Spindle RC +class 0 +0.875 +0.898 +0.906 +class 1 +0.691 +0.786 +0.798 + +New dataset. From further research, models and datasets obtained from the original wave field +position relative to the well were excluded. From the new positions obtained by Reverse-Calibration, +an additional set (Table 3, Spindle RC) was formed using the Spindle method. The discrete copy steps +were smaller than for Reverse-Calibration and amounted to 0, 5, 10 meters to the sides of the earth +and 0, 2, 4 meters vertically. The number and ratio of the obtained label vectors are given in Table 3. +For the new set, according to the data work protocol, a population of models was created and the + +quality of classification was assessed (Table 5, RC and Spindle RC). The Spindle augmentation +method from the new position increased the quality of classification of both classes. The quality +increase was 1.56 times for the collector forecast, relative to the original set. +Feature Importance Assessment. Using the GBDT model population, the importance of +features for forecasting was evaluated (Table 6). The evaluation was carried out for each batch +separately using the SHAP library [9] by calculating the Shapley index. The obtained results were +scaled (min-max) within the batch, and then the sum was taken for each feature. Table 6 shows 20 out +of 159 features with the highest total contribution. The first column of the table shows the name of +the seismic cube based on the processing graph used in Table 1, the second column shows the attribute +name, and the third column shows the rank by total contribution. The greatest contribution to +forecasting was made by attributes of the spectral decomposition with the Morlet wavelet, where the +number in the name is the decomposition frequency in Hz. The second most significant for forecasting +was the acoustic impedance. The most frequently significant features were obtained from seismic +cubes using the Image Domain Beam (IDB) in the processing graph. + + + +Table 6. The twenty most important features for GBDT algorithms +Seis.cub +Features +Rank +KFRISF +spectral_dec.-morlet-1 +1.000 +KFRFSU +spectral_dec.-morlet-1 +0.882 +KTRBISF +acoustic_impedance +0.750 +KFRISF +acoustic_impedance +0.667 +KFRISF +spectral_dec.-morlet-3 +0.590 +KFRISF +spectral_dec.-ricker-3 +0.562 +KFRISF +spectral_dec.-morlet-4 +0.495 +KTRBISF +dominant_frequency +0.477 +KFRFSU +amplitude_spectrum +0.475 +KTRBISF +sweetness +0.462 +KTRBISF +spectral_dec.-morlet-2 +0.402 +KTRBISF +spectral_dec.-morlet-1 +0.391 +KFRFSU +spectral_dec.-morlet-4 +0.361 +RFRBSU +sweetness +0.349 +KTRBISF +amplitude_spectrum +0.321 +KTRBISF +spectral_dec.-morlet-13 +0.316 +KFRISF +instantaneous_frequency +0.292 +KTRBSU +spectral_dec.-morlet-1 +0.288 +KTRBISF +instantaneous_frequency +0.286 +KFRFSU +spectral_dec.-ricker-4 +0.276 + +Ensemble learning. For the final prediction of the probability of collectors spread, the obtained +model population on the datasets (RC and Spindle RC) was ensembled using the stacking method. +This ensemble learning method uses a meta-model that is trained and makes a prediction based on the +predictions of the base models. The logistic regression algorithm was used as the meta-model. Before +training the ensemble, the GBDT models from the population were further trained on the full dataset +- a fixed number of epochs. And for the TabPFN algorithm, an additional model was trained on the +full dataset of Reverse-Calibration. The total number of models used for ensemble learning was 57. +Result of ensemble learning. As a result of ensemble learning, a prediction was made and a +3D cube of calibrated probabilities of the studied space belonging to the class of collectors was +obtained with a seismic data grid resolution (probabilistic space). The space was modified by the +volume-interpolated values of the displacement tensor from Table 2 with the opposite sign. + + +Figure 1. Map of predicted rock thicknesses with reservoir properties within the study area, with white rectangles in the direction of the section. +Number – beginning of the section, dashed number – end. The brown lines show the vertiacal projections of the wells. Above and below from +the map, vertical sections of the probability space are shown, corresponding to the section numbers from the map + + +reservoir thickness, m +250 +150 +50 +reservoir probability +0.5 +500 +500 +1000 +1500 +2000Description of map and sections. Figure 1 shows a map of the thickness of rock formations +with collector properties within the study area. The map is obtained by vertically summing the voxels +of the cube that have a probability of belonging to the collector class greater than 0.5, and multiplying +the resulting value by the vertical resolution (4 meters). The sections of the probabilistic space along +7 wells and one prospective area are shown. The sections have a length of 1500 meters and a thickness +of 800 meters. Along the well sections, their trajectory and lithological logs are shown with three +discrete categories highlighted in colors along the trajectory: green - reservoir, red – non-reservoir, +blue - no data. +Description of the obtained results. The sections of the probability space 2, 3, 4, 5 show that +the model has learned to predict the probability of collectors for well data sufficiently well. The map +shows the complex spatial structure of the geological body associated with collectors. The body's +presence in the section in the form of steps is well correlated with the geological conception of the +development of the research area. +The final model is a geology exploration model with a pragmatic task - to find rocks capable +of containing hydrocarbons. It was trained on all available data, and its predictive power can only be +tested according to the Popper criteria. As an example, to evaluate the model's quality, it is possible +to confirm one of the following forecasts using the listed methods: +▪ drilling exploration well can confirm that there are large traps on the left side of the field +with a thickness and area comparable to the open field; +▪ drilling sidetrack in the lower direction can confirm the presence of part of one of these traps +shown on the section in Figure 1; +▪ drilling sidetrack in the lower direction from the well shown on the third section of Figure 1 +can uncover a massive collector; +▪ drilling exploration well from the shore to an area intersecting section 5 on Figure 1 and not +encountering a massive reservoir in the studied interval; +▪ drilling exploration well from the shore through section 8 on Figure 1 in the area with the +maximum predicted probability and encountering deposits related to collectors. + + + +Conclusions +1. The complexity of traditional approaches to seismic data interpretation in coastal marine areas is +demonstrated on cross-sections 1, 3, 7 in Figure 1. The situation of drilling "blind" wells is +characteristic of these conditions, where the historical success rate of even exploitation drilling at the +field is 0.5. The proposed approach for these conditions is a tool that, using a technological stack of +machine learning algorithms, dataset expansion and modification mechanisms, can work with the +entire set of geophysical data, extract knowledge from a 159-dimensional space of seismic attributes, +and make a forecast of facies distribution with acceptable quality. +2. It is shown that the average forecast quality for facies belonging to hydrocarbon collectors was F1 +0.798 for newly "drilled" wells, always in the test sample, including one well from the field and one +well from the search area. The final ensemble of algorithms improved prediction quality. +3. The work develops the direction of machine learning, where algorithms are trained on well data and +attribute space. In order to overcome the limitations of this direction, the article develops two methods +of well data augmentation: +▪ the Spindle method, which enriches the set of well data by information in the near-well space +and can be used individually or to implement the Reverse-Calibration method; +▪ the Reverse-Calibration method, which searches for a new position of the seismic wave field +relative to well trajectories by finding the best response of the machine learning model with the +target function of restoring a continuous sequence of lithology classes along the wellbore using +surrounding attribute space values. +During the implementation of the entire technological stack of machine learning algorithms, it was +shown that the sequential application of the augmentation methods - Spindle and Reverse-Calibration, +proposed in the work, increases the forecast quality of collectors for this research area by 1.56 times +relative to the original dataset. + + +List of references +1 +Ivlev, D. Prediction of geophysical properties of rocks on rare well data and attributes of +seismic waves by machine learning methods on the example of the Achimov formation. +arXiv preprint arXiv:2106.13274v2 2021. +2 +BoostARoota GitHub repository. https://github.com/chasedehan/BoostARoota. +3 +Prokhorenkova, G. Gusev, A. Vorobev, A. Dorogush, and A. Gulin. CatBoost: unbiased +boosting with categorical features. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. +Cesa-Bianchi, and R. Garnett, editors, Proceedings of the 31st International Conference on +Advances in Neural Information Processing Systems (NeurIPS’18). Curran Associates, +2018. +4 +Bergstra, James S., et al. “Algorithms for hyper-parameter optimization.” Advances in +Neural Information Processing Systems. 2011. +5 +Bergstra, James, Daniel Yamins, and David Cox. “Making a science of model search: +Hyperparameter optimization in hundreds of dimensions for vision architectures.” +Proceedings of The 30th International Conference on Machine Learning. 2013. +6 +Ke, Q. Meng, T. Finley, T. Wang, W. Chen, W. Ma, Q. Ye, and T.-Y. Liu. Lightgbm: A +highly efficient gradient boosting decision tree. In I. Guyon, U. von Luxburg, S. Bengio, H. +Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Proceedings of the 30th +International Conference on Advances in Neural Information Processing Systems +(NeurIPS’17). Curran Associates, 2017. +7 +Chen and C. Guestrin. Xgboost: A scalable tree boosting system. In B. Krishnapuram, M. +Shah, A. Smola, C. Aggarwal, D. Shen, and R. Rastogi, editors, Proceedings of the 22nd +ACM SIGKDD International Conference on Knowledge Discovery and Data Mining +(KDD’16), pages 785–794. ACM Press, 2016. +8 +Hollmann, N.; Müller, S.; Eggensperger, K.; Hutter, F. TabPFN: A Transformer That Solves +Small Tabular Classification Problems in a Second. arXiv preprint arXiv:2207.01848v4 +2022. +9 +Scott Lundberg, Su-In Lee. A Unified Approach to Interpreting Model Predictions arXiv +arXiv:1705.07874v2 2017. + + diff --git a/hNE1T4oBgHgl3EQffQSM/content/tmp_files/load_file.txt b/hNE1T4oBgHgl3EQffQSM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..222f494101666a2056a0f4648bb6612a648a83a3 --- /dev/null +++ b/hNE1T4oBgHgl3EQffQSM/content/tmp_files/load_file.txt @@ -0,0 +1,371 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf,len=370 +page_content='Reservoir Prediction by Machine Learning Methods on The Well Data and Seismic Attributes for Complex Coastal Conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Dmitry Ivlev Petroleum Overseas ME JSC dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='ivlev@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='com The aim of this work was to predict the probability of the spread of rock formations with hydrocarbon- collecting properties in the studied coastal area using a stack of machine learning algorithms and data augmentation and modification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' This research develops the direction of machine learning where training is conducted on well data and spatial attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Methods for overcoming the limitations of this direction are shown, two methods - augmentation and modification of the well data sample: Spindle and Revers- Calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The implementation and effectiveness of these methods are demonstrated on real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Materials and methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Data from 15 drilled wells, 930 seismic field attributes and 22 additional space attributes were used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The proposed approach is based on binary classification algorithms with training on well data and includes the following sequence of applying methods: creating training datasets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' selecting the features,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' creating a population of classification models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' assessing the quality of the forecast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Reverse-Calibrating the seismic field position,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' creating additional datasets using Spindle method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' assessing the contribution of the features in the forecast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' combining models into an ensemble,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' the final forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' A prediction of the spatial development of reservoirs was made - a three-dimensional cube of calibrated probabilities of belonging of the studied space to the class of reservoirs was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The estimation of the classification quality for the classification algorithms and changes of the forecast quality from application of the Reversal-Calibration and Spindle methods were done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Considering the difficulties in interpreting seismic data in coastal conditions, the proposed approach is a tool that is capable of working with the entire set of geophysical and geological data, extracting knowledge from a 159-dimensional space of spatial attributes, and making a forecast of the spread of facies with acceptable quality - the F1 measure for the collector class is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='798 on average for the evaluation of drilling results under different geological conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' It has been shown that the sequential application of the proposed augmentation methods in the implemented technology stack increases the quality of the collector forecast by more than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='5 times relative to the original sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Keywords: machine learning, downhole data learning, seismic attributes, facies prediction, rock properties prediction, classification, augmentation methods, ensemble learning, feature selection, evaluation of contribution of features in prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Conflict of interest: Due to a possible conflict of interest, the work lacks geographic referencing, the name of the wells has been changed, and the map does not show the geographic location of the north pole direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The task of forecasting the spatial distribution of rock formations that are reservoirs for hydrocarbons is both relevant and non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Machine learning algorithms are one of the tools for such forecasting, and there are different approaches to predicting the geophysical properties of rock formations using them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' This approach has the advantage of training and adjusting models using actual data on environmental properties obtained from geophysical surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The complexity of implementing this approach is determined by two constraints: a small sample of well data and uncertainty in the position of the seismic wave field relative to well trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' To overcome these constraints, the Spindle method of data augmentation was proposed in [1]: a method of expanding the sample by copying the full sequence of the target geophysical survey or its interpretation with a small shift in the values of spatial data at a new position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' This method is based on the assumption that within a given shift in the near-well space, the lithological section does not significantly change, and the sample will include multiple interpretations of the same geological interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' However, in coastal areas with significant spatial variability of the geological section, large deviations of the well trajectory from the vertical and reduced quality of seismic observations over the area, there are difficulties in predicting changes in environmental properties by machine learning algorithms, including using the proposed data augmentation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Based on empirical studies of spatial data in these conditions, the development of the Spindle method - the method of Reverse-Calibrating (RC) the position of the seismic wave field relative to well trajectories - is proposed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' This is achieved by searching for the best response of the machine learning model with the target function of reconstructing a continuous sequence of lithology classes along the wellbore using surrounding attribute values in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The aim of this work is to predict the probability of the presence of rock formations with collector properties in the studied area by implementing a technological stack of machine learning algorithms and demonstrating the implementation of Reverse-Calibration and Spindle methods on actual data, and evaluate the change in forecast quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The research includes the following sequence of actions: creation of datasets for training, feature selection, creation of a population of classification models, evaluation of forecast quality, creation of an additional dataset using the Spindle method, evaluation of the contribution of features to the forecast, combining models into an ensemble, final forecast, and analysis of the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The study area is located in the coastal part of a marine basin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' A hydrocarbon deposit has been discovered and is being exploited within the study area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The main area of research is located on an exploration license block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Due to the potential conflict of interests, the work does not include a geographical reference to the location, the names of the wells have been changed, and the geographical position of the direction towards the North Pole is not indicated on the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The initial data consists of a single 3D seismic survey of the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The measurement overlap ratio decreases from 27 to 9 units towards the coastline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' 16 variants of wave field cubes at depths have been obtained as a result of reprocessing using various transformation graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' An area of study has been extracted from the 16 cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The width of the study area is 4875 meters, the length is 11025 meters, and the thickness is 800 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The cubes have been brought to a common seismic grid with a lateral step of 25 meters, a vertical step of 4 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The total volume of the cube is 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='199 million points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The cubes have a size of approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='1 GB each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Primary filtering of seismic data was carried out based on the Phik correlation coefficient between the values of amplitudes between the cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' If the coefficient exceeded the threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='95, one of the cubes was excluded from further work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' 5 cubes were left with a Phik coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='64- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='92 (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' For each cube, 185 different attributes of the seismic wave field were generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The space features are standardized and a power transformation Yeo-Johnson is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Seismic cubes included in the study # Migration Route Stack Filter Name in study 1 KPSDM FWI BWXT no KFRBSU 2 KPSDM FWI FULL yes KFRFSU 3 KPSDM FWI IDB yes KFRISF 4 RTM FWI BWXT no RFRBSU 5 KPSDM Tomo BWXT IDB yes KTRBISF An additional set of features was used in the work, including results from aeromagnetic surveying and lidar imaging, as well as results from stratigraphic interpretation in six versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The results of the aeromagnetic survey and lidar imaging were projected onto a depth grid with a single value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Discrete cubes with assigned categories between horizons were created based on tracing options for stratigraphic horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The total number of additional data was 22 cubes with seismic grid parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The study used 15 wells with the results of lithological log interpretation in the study interval: 11 wells from the field and 4 wells from the exploration area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The results of lithological interpretation are coded into two classes - reservoir and non-reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The reservoir is coded as 1, non- reservoir as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The base dataset includes results of classification of well logging data projected onto a grid based on the dominant frequency of the class in the seismic cube element intersecting the well trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Additionally, vectors were created from spatial data, where each class definition along the well trajectory was assigned 952 spatial features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Feature selection for further training was carried out in two stages on the base dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' In the first stage, the BoostARoota algorithm [2] was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The CatBoost [3] algorithm from the gradient boosting decision tree (GBDT) family was used as an evaluator for its work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' In the second stage, the importance of features was evaluated by training the same algorithm, but with the addition of noise to the dataset and calculating the Shapley index for the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Random data (noise) forms a threshold value for the Shapley index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' All features with values of the index below the threshold are excluded from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' After two stages of selection, 159 features of the studied space were left from the original 952 for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' All features from the additional dataset showed low significance for classification and were excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Augmentation - expansion and modification of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Reverse-Calibration was implemented by copying well trajectories along with classification well logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The copying of wells was done with a large discrete step of lateral displacement of 0, 10, 20 meters to the sides of the world, and up/down in the vertical direction - 0, 5, 10 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The total number of trajectories for one well was 125, including the original position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' For each trajectory, the lithological well logging classification was projected onto the grid, and vectors were created from spatial data in the position of the shifted trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The Reverse-Calibration method was developed based on two assumptions: ▪ first, the position of the seismic wave field relative to the well trajectory may be shifted due to a more complex physical model of the environment of the studied space compared to the synthetic one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' ▪ second, the seismic wave field carries information about the real environment, which implies the existence of an area in the field space whose attributes best correspond to the lithology sequence along the wellbore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' During the exploratory data analysis, the CatBoost algorithm on the test sample showed relatively low-quality classification of collectors on the base dataset (F1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='64) and even lower quality on the dataset created by the wreath method (F1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' At the same time, experiments with samples showed that when using only part of the data from the wreath method, the quality of the forecast improved (F1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='66), surpassing the quality obtained on the base dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The search for the position of the seismic wave field relative to the original is carried out by iterating over the positions of the wells in the dataset created using the method of Spindle and optimizing the classification error function of the machine learning algorithm on cross-validation for each combination of trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The positions can be shifted by simultaneously optimizing the error for all datasets, or by optimizing the error for each well independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The option of simultaneous optimization of all groups of well positions produces a unique combination of well positions 𝑋𝑑,𝑙 = 𝑉 (𝑋𝐷,𝐿), 𝑑 ∈ 𝐷, 𝑙 ∈ 𝐿, where 𝑋𝐷,𝐿 is an array of all well positions D generated using the Spindle method, with spatial data vectors L, the function 𝑉 (⋅) generates 𝑋𝑑,𝑙 - an array of unique combinations of well d with an array of vectors l for that combination, either randomly or according to a search grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' 𝑋𝑑,𝑙 𝑚𝑖𝑛 = 𝑎𝑟𝑔𝑚𝑖𝑛𝑋𝑑,𝑙 [ 𝐸 (С (𝑉 (𝑋𝐷,𝐿) 𝑛) )], 𝑛 ∈ 𝑁, where each combination n from the number of well position combinations N is given to the classification function С (⋅), the result of the classification is evaluated by the error function 𝐸 (⋅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' 𝑋𝑑,𝑙 𝑚𝑖𝑛 is an array of well sets and their vectors with the minimum classification error value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' This paper implements a version that optimizes the error by shifting all trajectories simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Given the significant number of combinations of discrete well positions, grid or random search may take a long time, so a search optimizer, the tree-structured Parzen estimator (TPESampler), was used to select combinations [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The Random Forest algorithm was used to evaluate the quality of classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The quality of classification was evaluated using the ROC AUC metric on cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Cross-validation was performed on wells, in which data from one well were extracted from the dataset, and the classifier was trained on the remaining data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The quality metric of the trained classifier was assessed on the extracted well, and then the data for the extracted well were returned to the general sample, and the next well was selected from it, and the assessment process was repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' To evaluate the overall combination of offset wells, the ROC AUC quality assessment results for the wells were averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The ROC AUC estimate for the initial positions was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='627, and for the best-found position 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The distances of the trajectory shifts (shifts seismic wave field) with the best response of the model from the initial are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' A trend of the overall trend of the field shift in the area of wells related to the field, and a divergent trend in the search area are noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Offsets of the seismic field relative to the well trajectories Area Wells X Y Z Field P-4ST 10 20 5 P-2ST 10 20 5 P-10ST 10 20 5 P-6 20 20 0 P-3 10 10 5 P-9 10 20 5 P-7 10 10 5 P-8 20 20 5 P-5 0 20 0 P-10 10 10 10 P-11 10 10 5 Sub mean 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='9 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='5 Sub std 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='4 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='7 Wild J-1X 20 10 5 SEP-1X 10 10 10 SP-1 10 10 5 SP-1ST 10 20 5 Sub mean 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='3 Sub std 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='5 Total mean 10 14 3 Total std 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='9 Two datasets were formed for further studies: the base dataset (Base) and the Reverse-Calibrated dataset (RC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The number of labels and the ratio of classes are given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Quantity of datasets for training Class Base RC Spindle RC 0 934 937 57091 1 193 192 13816 Sum 1127 1129 70907 Protocol for data handling was developed to prevent data leakage - accidental exchange of any information between the test and training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The protocol included standardization and Yeo- Johnson power transformation applied to each attribute cube, followed by the transformation parameters being applied to the well datasets (Base and RC) from the same combinations of wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Eight variations of splits (batches) were formed: training, validation, and test parts for the datasets (Base and RC) from the same combinations of wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The test parts were formed from data from a combination of two wells: one from the search block and one from the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' These parts were isolated and used only for quality classification assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' For fine-tuning of models and control of learning, validation parts were formed in the same way - one well from the search part and one from the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The combination of the eight variations of splits ensured that the data from the test wells would not be included in the training set in any individual variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' At the same time, the eight models of a single machine learning algorithm, in total, saw the entire dataset in various combinations of training and were tuned to the entire range of geological conditions revealed by the wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The design of training on well data consists of two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' In the first stage, basic models are trained based on the data processing protocol, after which the models are evaluated for quality on two sets of data for 8 batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' In the second stage, a meta-model is trained on an ensemble of basic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Basic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The type of machine learning algorithms currently determines the type of dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Data types are divided into unstructured (image, sound, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=') and structured (tabular).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Deep neural network architectures are used for unstructured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' It is believed that the main advantage of deep learning in working with unstructured data lies in the ability to study the feature extraction pipeline (Raschka 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' In this work, sets of tabular data were used, where the features of the space - attributes of the seismic field were already extracted and class labels were assigned to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' For such data, gradient boosting decision trees (GBDT) in its specific implementations CatBoost [3], LightGBM [6] and XGboost [7] have proven to be effective in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' At the same time, deep learning algorithms for tabular data types are emerging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' One such algorithm that showed high classification quality on the datasets used in the work was TabPFN (Prior- data Fitted Network) [8], described as "a trained Transformer that can perform classification".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' This method is well suited for subsequent ensemble with the results of GBDT algorithms, as its errors are not correlated with the errors of these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The current limitation for this algorithm is a maximum of 100 features and a dataset limited to 1000 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Therefore, the 100 best features determined using the Shepley index on GBDT family algorithms were input to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' It worked only with non- extended datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Hyperparameter tuning of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' For all algorithms, hyperparameter optimization was carried out by maximizing the ROC AUC metric on the cross-validation of the training and validation sets for each batch separately, with balancing of the class ratio and subsequent return of the validation instances to the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' For gradient boosting decision tree models (GBTD), the metric was maximized using the TPESampler optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' For TabPFN, by sequentially iterating from 2 to 200 of the “N_ensemble_configurations” parameter, and the maximum ROC AUC value was reached with this parameter set to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The training of GBDT algorithms was monitored on the validation set for each of the 8 combinations of dataset splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The TabPFN algorithm, due to its unique characteristics, was trained simultaneously on the validation and training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' As a result of training four algorithms on the Base and Reverse-Calibrated datasets, 64 models were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Evaluating classification quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' In accordance with the data handling protocol, the quality of classification was assessed on the isolated test set for each batch individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The F1 metric was used to evaluate the quality of the classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The metrics were grouped according to the groups being studied and their values were reduced to the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Classification quality for algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Table 4 presents the results of evaluating the performance of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The average values of the F1 metric for 16 models of each algorithm are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The best forecast quality for collectors was demonstrated by the CatBoost algorithm from the GBDT family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The second in quality was the transformer architecture-based algorithm TabPFN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=" Comparing GBDT algorithms for forecast quality is often a demonstration of the author's expertise in selecting numerous hyperparameters for their settings." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=" This is what distinguishes the TabPFN algorithm, which is trained in one pass on the entire available dataset and currently has one hyperparameter, and at the same time has comparable quality to the GBDT family's forecast." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' For further research, based on the experimental design, all algorithms used have good forecast quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Average classification quality (F1 measure) by algorithms Parameters CatBoost XGBost LightGBM TabPFN class 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='887 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='883 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='880 class 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='766 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='727 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='723 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='736 Quality prediction on datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Table 5 shows the averaged prediction metrics grouped by datasets: a dataset with vectors from the original positions of the seismic wave field (Base) and a dataset shifted using the Reverse-Calibration method (RC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' It follows that, as a result of using the shifted dataset, the quality of the collector prediction based on the F1 metric increased on average from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='691 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='786.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' This increase is significant and means that the quality of the collector prediction increased 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='49 times, taking into account the value for a random prediction equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Given that the field calibration by finding the best response to lithology was carried out within the standard error limits of the processing and interpretation results of 3D seismic exploration work, it can be assumed that the new field position better reflects the real lithological sequences exposed by the wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Thus, it is shown that the method proposed in the work allows to eliminate one of the difficulties of lithological prediction based on seismic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' This can be achieved by separating the field parameters that affect the lithology from the field parameters that affect the physical state of the rock and the properties of the fluid in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Average classification quality (F1 measure) by datasets Parameters Base RC Spindle RC class 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='898 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='906 class 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='691 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='786 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='798 New dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' From further research, models and datasets obtained from the original wave field position relative to the well were excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' From the new positions obtained by Reverse-Calibration, an additional set (Table 3, Spindle RC) was formed using the Spindle method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The discrete copy steps were smaller than for Reverse-Calibration and amounted to 0, 5, 10 meters to the sides of the earth and 0, 2, 4 meters vertically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The number and ratio of the obtained label vectors are given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' For the new set, according to the data work protocol, a population of models was created and the quality of classification was assessed (Table 5, RC and Spindle RC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The Spindle augmentation method from the new position increased the quality of classification of both classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The quality increase was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='56 times for the collector forecast, relative to the original set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Feature Importance Assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Using the GBDT model population, the importance of features for forecasting was evaluated (Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The evaluation was carried out for each batch separately using the SHAP library [9] by calculating the Shapley index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The obtained results were scaled (min-max) within the batch, and then the sum was taken for each feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Table 6 shows 20 out of 159 features with the highest total contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The first column of the table shows the name of the seismic cube based on the processing graph used in Table 1, the second column shows the attribute name, and the third column shows the rank by total contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The greatest contribution to forecasting was made by attributes of the spectral decomposition with the Morlet wavelet, where the number in the name is the decomposition frequency in Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The second most significant for forecasting was the acoustic impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The most frequently significant features were obtained from seismic cubes using the Image Domain Beam (IDB) in the processing graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The twenty most important features for GBDT algorithms Seis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='cub Features Rank KFRISF spectral_dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='-morlet-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='000 KFRFSU spectral_dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='-morlet-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='882 KTRBISF acoustic_impedance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='750 KFRISF acoustic_impedance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='667 KFRISF spectral_dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='-morlet-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='590 KFRISF spectral_dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='-ricker-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='562 KFRISF spectral_dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='-morlet-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='495 KTRBISF dominant_frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='477 KFRFSU amplitude_spectrum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='475 KTRBISF sweetness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='462 KTRBISF spectral_dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='-morlet-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='402 KTRBISF spectral_dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='-morlet-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='391 KFRFSU spectral_dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='-morlet-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='361 RFRBSU sweetness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='349 KTRBISF amplitude_spectrum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='321 KTRBISF spectral_dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='-morlet-13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='316 KFRISF instantaneous_frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='292 KTRBSU spectral_dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='-morlet-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='288 KTRBISF instantaneous_frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='286 KFRFSU spectral_dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='-ricker-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='276 Ensemble learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' For the final prediction of the probability of collectors spread, the obtained model population on the datasets (RC and Spindle RC) was ensembled using the stacking method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' This ensemble learning method uses a meta-model that is trained and makes a prediction based on the predictions of the base models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The logistic regression algorithm was used as the meta-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Before training the ensemble, the GBDT models from the population were further trained on the full dataset a fixed number of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' And for the TabPFN algorithm, an additional model was trained on the full dataset of Reverse-Calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The total number of models used for ensemble learning was 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Result of ensemble learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' As a result of ensemble learning, a prediction was made and a 3D cube of calibrated probabilities of the studied space belonging to the class of collectors was obtained with a seismic data grid resolution (probabilistic space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The space was modified by the volume-interpolated values of the displacement tensor from Table 2 with the opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Map of predicted rock thicknesses with reservoir properties within the study area, with white rectangles in the direction of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Number – beginning of the section, dashed number – end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The brown lines show the vertiacal projections of the wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Above and below from the map, vertical sections of the probability space are shown, corresponding to the section numbers from the map reservoir thickness, m 250 150 50 reservoir probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='5 500 500 1000 1500 2000Description of map and sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Figure 1 shows a map of the thickness of rock formations with collector properties within the study area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The map is obtained by vertically summing the voxels of the cube that have a probability of belonging to the collector class greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='5, and multiplying the resulting value by the vertical resolution (4 meters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The sections of the probabilistic space along 7 wells and one prospective area are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The sections have a length of 1500 meters and a thickness of 800 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Along the well sections, their trajectory and lithological logs are shown with three discrete categories highlighted in colors along the trajectory: green - reservoir, red – non-reservoir, blue - no data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Description of the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The sections of the probability space 2, 3, 4, 5 show that the model has learned to predict the probability of collectors for well data sufficiently well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The map shows the complex spatial structure of the geological body associated with collectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=" The body's presence in the section in the form of steps is well correlated with the geological conception of the development of the research area." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The final model is a geology exploration model with a pragmatic task - to find rocks capable of containing hydrocarbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' It was trained on all available data, and its predictive power can only be tested according to the Popper criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=" As an example, to evaluate the model's quality, it is possible to confirm one of the following forecasts using the listed methods: ▪ drilling exploration well can confirm that there are large traps on the left side of the field with a thickness and area comparable to the open field;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' ▪ drilling sidetrack in the lower direction can confirm the presence of part of one of these traps shown on the section in Figure 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' ▪ drilling sidetrack in the lower direction from the well shown on the third section of Figure 1 can uncover a massive collector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' ▪ drilling exploration well from the shore to an area intersecting section 5 on Figure 1 and not encountering a massive reservoir in the studied interval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' ▪ drilling exploration well from the shore through section 8 on Figure 1 in the area with the maximum predicted probability and encountering deposits related to collectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Conclusions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The complexity of traditional approaches to seismic data interpretation in coastal marine areas is demonstrated on cross-sections 1, 3, 7 in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The situation of drilling "blind" wells is characteristic of these conditions, where the historical success rate of even exploitation drilling at the field is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The proposed approach for these conditions is a tool that, using a technological stack of machine learning algorithms, dataset expansion and modification mechanisms, can work with the entire set of geophysical data, extract knowledge from a 159-dimensional space of seismic attributes, and make a forecast of facies distribution with acceptable quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' It is shown that the average forecast quality for facies belonging to hydrocarbon collectors was F1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='798 for newly "drilled" wells, always in the test sample, including one well from the field and one well from the search area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The final ensemble of algorithms improved prediction quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' The work develops the direction of machine learning, where algorithms are trained on well data and attribute space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' In order to overcome the limitations of this direction, the article develops two methods of well data augmentation: ▪ the Spindle method, which enriches the set of well data by information in the near-well space and can be used individually or to implement the Reverse-Calibration method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' ▪ the Reverse-Calibration method, which searches for a new position of the seismic wave field relative to well trajectories by finding the best response of the machine learning model with the target function of restoring a continuous sequence of lithology classes along the wellbore using surrounding attribute space values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' During the implementation of the entire technological stack of machine learning algorithms, it was shown that the sequential application of the augmentation methods - Spindle and Reverse-Calibration, proposed in the work, increases the forecast quality of collectors for this research area by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='56 times relative to the original dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' List of references 1 Ivlev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Prediction of geophysical properties of rocks on rare well data and attributes of seismic waves by machine learning methods on the example of the Achimov formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='13274v2 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' 2 BoostARoota GitHub repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='com/chasedehan/BoostARoota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' 3 Prokhorenkova, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Gusev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Vorobev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Dorogush, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Gulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' CatBoost: unbiased boosting with categorical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' In S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Bengio, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Wallach, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Larochelle, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Grauman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Cesa-Bianchi, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Garnett, editors, Proceedings of the 31st International Conference on Advances in Neural Information Processing Systems (NeurIPS’18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Curran Associates, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' 4 Bergstra, James S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' “Algorithms for hyper-parameter optimization.” Advances in Neural Information Processing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' 5 Bergstra, James, Daniel Yamins, and David Cox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' “Making a science of model search: Hyperparameter optimization in hundreds of dimensions for vision architectures.” Proceedings of The 30th International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' 6 Ke, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Meng, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Finley, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Ma, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Ye, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Lightgbm: A highly efficient gradient boosting decision tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' In I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Guyon, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' von Luxburg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Bengio, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Wallach, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Fergus, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Vishwanathan, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Garnett, editors, Proceedings of the 30th International Conference on Advances in Neural Information Processing Systems (NeurIPS’17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Curran Associates, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' 7 Chen and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Guestrin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Xgboost: A scalable tree boosting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' In B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Krishnapuram, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Shah, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Smola, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Aggarwal, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Shen, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Rastogi, editors, Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD’16), pages 785–794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' ACM Press, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' 8 Hollmann, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Müller, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Eggensperger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' Hutter, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' TabPFN: A Transformer That Solves Small Tabular Classification Problems in a Second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='01848v4 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' 9 Scott Lundberg, Su-In Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content=' A Unified Approach to Interpreting Model Predictions arXiv arXiv:1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} +page_content='07874v2 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE1T4oBgHgl3EQffQSM/content/2301.03216v1.pdf'} diff --git a/i9FQT4oBgHgl3EQflDYi/content/2301.13360v1.pdf b/i9FQT4oBgHgl3EQflDYi/content/2301.13360v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..459ef7f718b43e919f51e5778de4087c375715bf --- /dev/null +++ b/i9FQT4oBgHgl3EQflDYi/content/2301.13360v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:58b463bc44b131b1e4fa86b3e41b65f1bf28905ea75761fdfd564903ca616459 +size 1189056 diff --git a/i9FQT4oBgHgl3EQflDYi/vector_store/index.faiss b/i9FQT4oBgHgl3EQflDYi/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..75330d975f5a1c79986a3ce718720297e8885053 --- /dev/null +++ b/i9FQT4oBgHgl3EQflDYi/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:53cc38f01d557f9cd7335d2a01d7a1f130751204e3a1a9181bd0f3abbc95d441 +size 2490413 diff --git a/i9FQT4oBgHgl3EQflDYi/vector_store/index.pkl b/i9FQT4oBgHgl3EQflDYi/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..8de4ac56851f7594087489c242da46308e67a82a --- /dev/null +++ b/i9FQT4oBgHgl3EQflDYi/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2800e14f6b5d4fecab6f98989348156e197b2dd4e680877e5d035a2d7e558f8f +size 107322 diff --git a/l9E4T4oBgHgl3EQftw0Q/vector_store/index.faiss b/l9E4T4oBgHgl3EQftw0Q/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..3aefe9a539a6b86a35e26cd7a10a381f74ae8867 --- /dev/null +++ b/l9E4T4oBgHgl3EQftw0Q/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ba41577e1b0c4a8740103020d24bb4277e38aa8d3732cb7986e3b1cf506c6322 +size 3538989 diff --git a/l9E4T4oBgHgl3EQftw0Q/vector_store/index.pkl b/l9E4T4oBgHgl3EQftw0Q/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..f752bf3c16cdd86ca393e54903795dad85c80f5b --- /dev/null +++ b/l9E4T4oBgHgl3EQftw0Q/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8afff416f0c2c6f37acecdb240c56c01d3fcdd58c561b78fa85fc8e940f2afbb +size 147175 diff --git a/mNE2T4oBgHgl3EQfeQdZ/vector_store/index.faiss b/mNE2T4oBgHgl3EQfeQdZ/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..e7649f2deba763c71b3a4c7e5e12ddaf7f1527b2 --- /dev/null +++ b/mNE2T4oBgHgl3EQfeQdZ/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bcdf904ea227b162894ae318b9cf97b3b0e36aafe2b9eafb1bcf5d514ccc61dd +size 2162733 diff --git a/mdE2T4oBgHgl3EQfJQZR/vector_store/index.faiss b/mdE2T4oBgHgl3EQfJQZR/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..4a12d4e1704698f791a174c5b1413259f708c8c1 --- /dev/null +++ b/mdE2T4oBgHgl3EQfJQZR/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:817ff8858e347fd8e8eb272fb1fdc4e147e14ece18fa5d4a442ec249c32f90cf +size 8519725 diff --git a/mtFRT4oBgHgl3EQfaTey/content/tmp_files/2301.13556v1.pdf.txt b/mtFRT4oBgHgl3EQfaTey/content/tmp_files/2301.13556v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1401ba3f3f55483a1d509d8151afba5ed0b7b2f8 --- /dev/null +++ b/mtFRT4oBgHgl3EQfaTey/content/tmp_files/2301.13556v1.pdf.txt @@ -0,0 +1,1025 @@ +Purposeful and Operation-based Cognitive +System for AGI +February 1, 2023 +Abstract +This paper proposes a new cognitive model, acting as the main com- +ponent of an AGI agent. The model is introduced in its mature state, +and as an extension of previous models, DENN, and especially AKREM, +by including operational models (frames/classes) and will. In addition, +it is mainly based on the duality principle in every known intelligent as- +pect, such as exhibiting both top-down and bottom-up model learning, +generalization verse specialization, and more. Furthermore, a holistic ap- +proach is advocated for AGI designing and cognition under constraints or +efficiency is proposed, in the form of reusability and simplicity. Finally, +reaching this mature state is described via a cognitive evolution from in- +fancy to adulthood, utilizing a consolidation principle. The final product +of this cognitive model is a dynamic operational memory of models and +instances. +1 +Introduction +Our consistent goal is to construct a basic realistic model for AGI (Artificial +General Intelligence). +It is a gradual process with many versions along the +way. Hence, this paper presents MOM (Model Of Models), the next version of +AKREM (Associative Knowledge Representation) [Komarovsky, 2022b]. +AKREM is a mature-state knowledge representation model, based mainly +on the assumption that communication is about encoding the sender’s will into +a sequence of words (a message), and then decoding it by the recipient. The +model proposes a representation of any message, in a hierarchical form based +on grouping, by generating some essence in a given level, from details in the +lower level. The lowest level details are founded upon some DNN (Deep Neural +Network), generating the basic concepts and actions (from which the details +are made of) from unstructured input. Finally, while the will concept exists in +AKREM, it will be expanded upon in this paper. +Following this, new additions to the presented model, including new associa- +tions, operationability, modeling, consolidation, and reusability, are introduced. +First, while AKREM assumes that the learned elements are either objects or +1 +arXiv:2301.13556v1 [cs.AI] 31 Jan 2023 + +static actions (verbs), new associations are introduced: object’s attributes and +relations. Next, these connections are all static representations of knowledge, +i.e., the hierarchies cannot be changed. Therefore, operationability introduces a +new type of association to objects: actions that act upon them, thus producing +new knowledge elements. This makes the connections in AKREM ’s hierarchies +dynamic, hence it allows the freedom to update and create new hierarchies. +Next, modeling introduces some basic cognitive operations, e.g. +abstraction +and grouping. Both gather many details into fewer. Grouping specifically is +about connecting elements via some common property. It could be for exam- +ple a chronology in a plot or other common properties/actions grouped into +classes. Finally, consolidation is a process in time, that collapses a huge amount +of possibilities into small set of patterns, of any kind. +All the operations above are considered to be bidirectional, i.e. everything +lies in some range between extremes, i.e. everything has its inverse (dualism). +In grouping, it is from the whole to its parts and vice versa, and in abstraction, +it is from instances to classes and vice versa. Consolidation is an operation +in time, creating models and memory, while its inverse is forgetting, which is +also an operation in time. Will also lies in the dichotomy of determinism and +randomness. +The product of these cognitive operations is a dynamic memory of models, +formed as a semantic network of elements. It encourages a holistic approach +for AGI designing: one simple system for multiple functions, such as short-term +and long-term memories, problem-solving, communication, learning, and any +cognitive function. Moreover, if in the early epoch of AI, symbolic reasoning +was dominant, and nowadays connectionism dominates, then we come to a new +era, where we should combine and include many conflict perspectives, in co- +operation and competition. Our holistic perspective embrace this duality and +other dualities also. +Lastly, operational modeling presents AGI agent’s intelligence in its mature +state, which is the state of how its knowledge should be represented. However, to +accomplish this state, a cognitive evolution over time is required, which utilizes +the consolidation principle. +2 +Will +In this section, the importance of will as an essential element in human in- +telligence is elaborated upon, starting from the previously presented model, +AKREM. +Will in AKREM is represented in the levels of any specific hierarchy. Start- +ing from the most detailed aspects of will, at the lowest level, and finishing at +the most abstract will or its essence, at the top. The top level represents some +kind of experience uniqueness to differentiate it from other memories that use +the same low-level structures. +This hierarchical will is especially demonstrated in a constrained environ- +ment, as our reality, for topics like problem-solving and communication. +In +2 + +problem-solving, the main will produces sub-wills in lower levels, till it reaches +the final solution at the bottom level (Fig 1(a)). The final result is a plan or a +sequence of actions. See more in Appendix A.2. Similarly, in communication, +the sender encodes/converts his will into a sequence of actions (in a language +form), while the recipient on the other side decodes the intention/will from this +sequence (Fig 1(b)). In both cases, evaluation is necessary, hence this top-down +process is cyclic and non-linear. +(a) 3-level problem-solving in state space +(b) Communication model +Figure 1: Two cases of will in a constraint environment +3 + +The AGI +agent's +or +human's +models +Proposed Solution: +Detailed Solution: +Reality +(details)the +Sequence +form +Lowest will +ofAll the above are specific cases of will, but there is also a more general +will. As recipients of reality, our main cognitive will is to find the most ap- +propriate/simple model to fit all the pieces/details in the right place or to +make the most sense of them1. This is similar to decoding a will from a mes- +sage/mystery/riddle, and it can be rephrased as a general problem-solving task +to comprehend reality. First, it is done internally, by reorganizing our mod- +els (mostly during a sleep phase), and later it is done externally, in any kind of +problem-solving, or in understanding a message/story/riddle/situation/phenomena. +The best model will allow us to move from place to place in it easily, perform +new actions, and produce conclusions/solutions easily. This results with an un- +derstanding, or the ability to control any aspect in the complex model. So in a +sense, we have two wills governing our cognition: controlling and subsequently +making sense. Obviously, these wills are enforcing each other. +Additionally, there are different categories of will, such as chronology, cau- +sation, and purposefulness. In stories, they are very intertwined/mixed. It is +because purposefulness is a higher manifestation of will (usually applied in hu- +mans), while causation is a lower one, usually applied to animals/objects (e.g. +”A causes B”), and chronology is simply the way will is implemented: in a delay. +You first want, and then you try to accomplish it. Or in the case of causation, +there is a law, as a fixed kind of will (e.g. gravity), and then it is realized. See +more in Fig 2. +Figure 2: Types of will +In conclusion, purpose is everywhere, in all of our daily interactions. We try +to figure out animals’ intentions or people’s hidden will, to make sense, or in our +case to make our model complete, i.e. enable prediction. Since a specific human +has his own will, he tries to figure out other people’s will to enforce its own +will over theirs. Sometimes it is via competition, while sometimes it is through +cooperation. +However, how is will actually included in MOM ? It is discussed in 5 and +Appendix A.4. +1Making sense is also important for explainability, especially storytelling modeling can +provide this. +Hence, explainability should be generative and flexible, in its most general +conception +4 + +Humans +To who: +Animals +- Objects +Random +Free +Physical +What: +Desires +-ness +Will +Laws +Characteristics +Impossible to Model +Hard to Model (as stochastic +Easy to Model (as +variable) +deterministic variable) +Totally: +Uncertain/Stochastic +Uncertain/stochastic +Certain/Deterministic +Totally not: +Predictable/Controllable Unpredictable/not Controllable +Predictable/Controllable3 +New Associations +Inspired by semantic nets, some general connections are proposed, such as in- +heritance (is-a relation), instance property (is-instance-of relation), part-whole +property (part-of relation), an attribute of an object property (has relation), +assigning a value to attribute (value relation), synonyms, antonyms/opposites +and more, see [Wiktionary, 2022]. All these connections are static and can rep- +resent stories as separate items as it is in AKREM. But to connect these details, +as a sequence of applied actions, operationability is introduced. +4 +Operationability +It is hypothesized that thinking is operational. Meaning it is a process, gener- +ating new facts along the way, via some set of actions. Hence, the first principle +added to AKREM is operationability, which turns it from static to dynamic +knowledge representation. That is, unlike static connections as in AKREM and +knowledge/scene-graphs (representing facts or a scene shot) - action connec- +tions in MOM can also be productive (produce new elements). It adds degrees +of freedom to the current cognitive model, to move in new directions along +the hierarchy, i.e. to create new hierarchies on the fly or update old ones, via +admissible actions. +Subsequently, a minimal set of primitive operations is proposed, to func- +tion as basic operations, which can be the building blocks for more complex +and composite operations/actions. Operations such as logical relations (AND, +OR, NOT, all, each, (in)equalities, exists, count), flow operations like loop +operations (while, for) and if-else conditionals, mathematical operations (+,- +,*,/,min,max,norm,log), and other relations. This set of tools can replace DNN +units and DNN’s fixed structure, in a program-search process. It can be imple- +mented for example via Reservoir network, a random mix of basic rule compo- +nents/blocks, yielding an algorithm best describing an operation. +At the same time, prior knowledge is needed to be inserted, within all ele- +ments, by including: number of visits, uncertainty, rate of update, and measure +of consolidation. The measure of consolidation is to prioritize different options +associated to some element, to separate the relevant from the irrelevant, e.g. +admissible actions, which can be used inversely in creativity mode - by picking +the less expected directions to follow. +Moreover, action’s admissibility is needed for two reasons. Firstly, due to +the elimination of entry conditions necessary for an action to be performed, +e.g. on which types (integer, string, etc). And secondly, it is due to the ability +to use High-Order Logic, as in λ-calculus, which removes any restrictions on +an object’s slot or action’s argument. Hence, relevancy is needed to constrain +action’s admissible space. +5 + +5 +Modeling +If operationability is considered, as the addition of freedom to move in a 2D +knowledge representation, then modeling is an addition of a new dimension, i.e. +converting it to 3D. It is about extending the ”is-a” operation into a program- +ming abstraction, as in OOP (Object-Oriented Programming), or in abstract +mathematics, such as algebra or category theory2. Meaning, that while usually +semantic networks represent this operation in a 2D graph, here the instances are +totally separated from classes, and from classes of classes, and so on. Resulting +with a multi-level of abstractions, while for simplicity two types of levels can +be distinguished, in the final LTM (Long-Term Memory), see Fig. 3. In sum- +mary, at first all different associations including operationability are describing +objects, and then abstraction extends objects as instances into classes, which +represent models. Consequently, unlike AKREM, these full models are impor- +tant for natural communication, e.g. +for context-based conversations, where +undelivered missing information (common sense) is needed to be filled. +This extension has several implications. First, in perception from senses: +from the basic recognition of instances in AKREM, to a multi-level recognition of +instances and classes. Next, every element is learned and can be abstracted (into +class), i.e. objects, actions, relations, and attributes. For example, action with +attributes, relation with attributes such as strength, numeric attribute/values as +class (integer, real), group types (sets, lists, arrays) with their group operations +(slicing, union, sorting), and more. +Next, this extension enables answering +the question about how will is implemented in MOM, see 2. Alternatively to +AKREM ’s hierarchy by will, which it is not clear how it can be implemented, +it could be generated by abstraction, while the will/intention will is serving +as an additional and independent variable in the models that construct the +hierarchy. Additionally, feeling measures could be included, as influencing will. +Moreover, since there are several levels of will, correspondingly there are the +main variable and secondary variables representing these wills, perhaps with +different significance intensities, depending on the abstraction level. +Finally, the novelty here, is that unlike DL which performs program-search in +an un-interpretable way, here however, additional inductive bias is introduced: +separating of models and performing program-search to relevant actions, in +consistency with other models and actions. This makes MOM both usable and +interpretable. +5.1 +Learning the modeling +Here, model learning mechanism is proposed, where two contrary but complet- +ing learning approaches in AI are combined [Day, 2022]: empirical, i.e. from +examples (induction), and expertise (rule-based). This is a learnable symbolic +manipulation, or can also be referred to as a hybrid approach or Neuro-Symbolic, +see [Marcus, 2003]. Empirical is bottom-up (from examples to rules), e.g. via +2Group properties include identity object to use compositionality of an operation and its +inverse operation. This implies duality. +6 + +observation or passive interaction. Rule-based is straight from the top, via rules +in abstract language, e.g. via conversation or observation. It can then descend +to examples of these rules (deduction). +These approaches might contain models of concepts that do not belong to +them both, but only to one of them. +For example, concepts that are hard +to define, like love, God, beauty, and tacit/unconscious knowledge like walk- +ing and breathing - all can be modeled simply by examples. +Similarly are +the sub-symbolic features, like audio and visual inputs - they do not have +logic/linguistic/symbolic meaning, hence should be modeled by examples, as +it is done in DL nowadays. Hence, these non-symbolic concepts can be learned +via usual non-interpretable DL. On the other hand, abstract concepts, like those +in math and sciences, that appear less in the physical reality, can be learned +solely in the top levels of LTM. +Moreover, in this hybrid approach, DL (Deep Learning) is used twice. On the +one hand, DL is extended from its too constraint program-search to be much +more flexible, if more operations are added as building blocks, see 4. Hence, +symbolism is learned and adaptive just like in DL, differently from expert/rule- +based AI. On the other hand, different input sensors are fused to represent +specific symbols/concepts, i.e., the uninterpreted features in DL become sym- +bolic tokens (Fig. 3). +Figure 3: Proposed cognitive model basic diagram +The reason the hybrid approach is preferred over DL-only, is that DL usu- +ally does not implement compositionality, modularity and abstraction [Dickson, +2022]. One of the effects of this shortcoming is the shortcut effect, where the DL +categorizes something due to the wrong reasons. It can be solved in a system, +that learns a model with alignment and consistency-check with other models +learned so far, which MOM supplies (continual learning). +7 + +4 +Temporal abstractions +Symbolicmodels +(Interpretable) +LTM +Instances +symbols +Sub-Symbolio +External +models +Interactions +(Uninterpretable) +World +Input +OutputAdditionally, the topmost level is actually temporal and used for creativity +and problem-solving. In this level, temporal new abstractions are created, by +stripping off attributes/actions/relations, thus connecting distant or different +abstractions to perform analogy or transfer learning between different domains. +For example, the abstraction is via the number of edges in the polygon classes +(Fig. 3). +Furthermore, learning can be divided into online and offline modes. In wak- +ing periods, i.e. when sensors are active, the learning is online and minimal +since most resources are dedicated to fast response (e.g. fast optimization into +local optimums). However, during sleeping periods, sensors are inactive, and +previous memories can be used for improving models to make overall sense, e.g. +larger time scale is used to generate causal relations in models. In such a case, +it is a slow processing, implemented e.g. as a Neural Architecture Search or a +Genetic Algorithm, to get out of local minima and search for a global one. +Finally, another issue precedes learning: how to obtain separate models at +all. One way, is like the DENN (Dynamic and evolving NN) idea [Komarovsky, +2022a], i.e. always learning the ”model of everything” while refining it more and +more with every new experience, such that new sub-models are produced. The +idea here, is like Jeff’s hierarchy [Hawkins and Blakeslee, 2007], where the top +model is always reached, at perception from senses, and it decides which lower +model will handle the situation. For example, in recognizing a specific type of +problem, it chooses the most appropriate model for solving this problem. Or, +when encountering new knowledge, it selects the most appropriate model to +handle its assimilation in the current memory of models. +Another way to facilitate modeling evolution is via consolidation. +6 +Consolidation +So far, the cognitive model has been presented in its mature state. Now, the +discussion is about how to reach it. This is a process in time, which is mainly +based on consolidation. +Consolidation is about transforming from chaos to some stable order of pat- +terns, or from a continuous realm to a discrete one, as in quantum mechanics. +An infinite amount of details is hard to handle (i.e. to understand and then +to control), therefore consolidation to fewer patterns is required. Consolidation +also allows for fuzzy logic and categories [Wyler, 1995]. +Consolidation can be expressed in many forms, such as: +• in the conversion of sub-symbolic to symbolic, for any type of element +• in cognitive evolution: from flexible (at infancy) to less flexible (at adult- +hood) +• in modeling, at program search, from huge hypothesis space for possible +programs to a small set of hypotheses (as in DL). It is both in the micro +(within models) and in the macro (between models) +8 + +• in testing multiple versions of an unknown model, and finally converging +into less/one version(s) that are/is consistent with evidence +• and in grouping/abstraction, where some separate elements become con- +nected +Note that causality is a special case of modeling, a spatio-temporal one, +where re-occurrence is consolidated. More generally, re-occurrence help in learn- +ing both static objects and dynamic basic/composite events (equivalent to sce- +narios/scripts in OOP). +Additionally, MOM enables multiple parallel versions of the same thing, +since any specific topic or subject can have multiple theories/models, sometimes +in conflict. Hence, like in the quantum superposition realm, multiple-version +combinations could be tryout, and consolidation can help in collapsing them +into fewer versions. +Those versions should make the most sense, i.e. +to be +consistent on different occasions or supporting the majority of evidence. Thus, +it just maybe, that at infancy, a highly uncertain period, there are many versions +created, and with time - only the most consistent ones survive (consolidate). +Lastly, two operations help in producing consolidation. On the one hand, to +deal with a stochastic environment and ambiguous signals, repetition provides +memory prioritized by relevancy. +Repetition is never exactly over the same +thing, but rather over many different examples of a thing. Repetition is needed +also with guided tutoring of an AGI agent.Conversely, sparsification is about +reducing irrelevant signals. +6.1 +Reusability +An additional form of consolidation is reusability, since the more learning pro- +gresses, the fewer new models are proposed in favor of using existing ones. +Hence, reusability is expressed via exploration (mostly at early stages) verse +exploitation (in stable or mature stages), as in Reinforcement Learning. In the +beginning, many possible codes are generated for models, but as time goes by +the process is less exploratory and more exploitative, i.e. there is more emphasis +on retrieving known codes, while testing fewer new codes in parallel. In addi- +tion, reusability aligns perfectly with abstraction/grouping, in a constrained +environment and limited resources. They are both needed to hold control of as +much as possible, with minimum effort, i.e. without generating many models of +each thing. +In practice, reusability is about using less of the initial available tools, as the +learning evolves. Meaning, while regular DL tools (if, sum, activation function) +or the primitive tools 4 can be used for program search of basic action methods +or relation methods, the new methods apply reusability. +In such methods, +less primitive tools are used while the current methods are used more, thus +encouraging more connectivity in the network. +Moreover, Functional Programming can be applied to assist reusability. On +the one hand, the general/outer structure is OOP, i.e. elements are grouped in +an OOP fashion. On the other hand, methods are kept in a pure operational +9 + +immutable form [Chen, 2019, Van Roy et al., 2009]. Meaning, having small +and simple methods, which maximally reuse other functions, and without in- +ner variables, due to objects-memory-only assumption. Meaning, methods that +are comprised of other methods, as much as possible. This is compositional- +ity/grouping applied in actions. See Fig. 4. +Figure 4: Actions, objects, compositionality, and reusability +7 +Cognitive model comparison +Here all models developed so far are compared, in Fig. 5. +Figure 5: Comparison table of models discussed so far +In summary, DENN stores each new combination of events, while AKREM +stores dynamically in episodic memory any newly encountered combination of +basic events. Events which are stored separately in two types of memory. MOM +unites those separate memories into one dynamic operational memory, consisting +of concepts, actions, relations and any instance of those. +10 + +object +object +object +function +object +function +function +object +object +object +function +function +object +object +function +function +functionAssociative knowledge +Model +Dynamic evolving DNN +Model of models +representation +Plus +Growing/adapting and +Separate basic instances to +Stores relational and +applies continuous and +fixed and small memory, +operational info, for +most importantly gradual +while new combinations of +both instances and +learning +basic events stored in +classes. Allow thinking +episodic memory +creating new objects. +Minus +Instances +Instances +Consolidation (e.g. from +Only event +Only representation (no +sub-symbolic data to +discrimination +learning) +symbolic) +Stage in +Infancy +Adulthood +Childhood until +development +adulthood +Take-aways +-Constantly refining +Generating and perceiving- A full cognitive +events on the one hand, +messages, and uniquely +model, producing +while grouping them on +storing them in episodic +dynamic operational +the other hand +memory +memory of models +and instances +Implementability +High +Low +Medium +level8 +Conclusion +The following is the summary of the paper including some key takeaways. +First, a new cognitive model is introduced to join the existing cognitive archi- +tectures, in the form of dynamic operational memory of models and instances. +In it, a holistic approach is embraced, assuming that intelligence should be +highly versatile and diverse, instead of ”picking one side”. +Next, our model includes will as an essential part of the modeling process. +Thus, in case of its absence, it turns most of the learned models to be very +partial. In the OOP formulation, the will is an additional variable, and it is +mostly significant in the top level while it is least significant in the lowest one. +Next, operationability turns static knowledge representation into a dynamic +one, thus enabling cognitive processes. +The actions are learned via regular +program search, with either DL or other tools, in a self-supervised manner. +Next, one way to ensure that the continual learning is consistent, is by imple- +menting local learning, i.e., concentrating on updating only one/some model(s), +while confirming compatibility with other models. A model can be learned ei- +ther from examples or directly by using existing operations (logically). Another +way to ensure that learning is consistent, is by a slow process of consolidation. +It is ensured by maintaining a high level of flexibility over a long period, while +pursuing more and more consistency within and between models. +Next, reusability is utilized to enhance connectivity between models, instead +of learning them as separate entities. +Finally, the cognitive model is designed via inverse engineering. Meaning, +starting from our highly aware and mature cognitive state of mind, and then +tracking back in time to study its evolution. +9 +Future Work +The main problem is how to implement a cognitive system, that produces the +appropriate models, i.e. how grouping/clustering occur, to generate the right +models. Also, how these models produce new ones by the correct compositions. +In addition, there is the issue of how sub-symbolic become symbolic. Per- +haps the models produce objects and actions directly upon sub-symbolic data. +Furthermore, if continuing this line of thought, then models may be removed at +all, which converts this problem to pure DL-based approach. +Additionally, relevant to the last issue, is model evolving. Is it model re- +finement of some main model to sub-models, which controls how models of +knowledge are used, or is it that all the models are separate. And if it is by +refinement, is it one large DL-based model, and all the rest are knowledge mod- +els, or if we continue this line of thought - again, end up with pure DL-based +one huge model, containing implicitly all the different models, their actions and +attributes. But then how elements and abstraction are implemented in such a +model? More about the suggestion above see Appendix A.3.1. +11 + +Finally, though hierarchy by abstraction can be implemented, but how can +it be implemented by will, i.e. how to decide when and what to group in such +a hierarchy? +These are all open questions to deal with. +P.S.: Neuro-Symbolic AI for me is merely taking inspiration of class-based +structure, to act as the final stage of learning, while DL is the main tool to +reach it. So it is all about flexibility of DNNs, only that we use consolidation +to finally reach symbols. Another implication of this, is memory. First, since it +is vague and mostly reconstructed. It means it is not recorded accurately. And +second, since it probably used in sleeping periods similarly in a vague form. +A +Appendix +A.1 +Examples of MOM in action +Fig. 6 contains examples of cognition processes. As seen, actions denoted as +arrows, can perform changes in several objects to their different attributes. High +admissibility expressed via salient color. Fig. 6(b) is a MOM representation of +evolving state, representing a story. +A state is consisting of several objects +(joining/leaving as the story evolves), including their attributes and actions +that change the state. The story: ”David entered his room. He searched for +something on the floor. Then he searched in the basket. Then he searched under +his bed, and was thrilled to find the ball there. In the meanwhile, his mother +entered home. She put her keys on the desk. Then she removed her shoes and put +her sunglasses on the desk. Then she searched for David, found him, and they +sat to eat lunch together”. The same story is represented via AKREM, without +the evolution of details, see the video link in [Komarovsky, 2022b]. Note, that +if repeated often, the sequence of David or anybody searching in some place +can be grouped/abstracted as an event class, also referred to as Trans-Frame +[Minsky, 1988]. +Next, Fig. 7 contains examples of model learning, based on Fig. 3. The as- +sumption here, is that similarly to DNN pattern matching and then executing +some task - here it is performed explicitly, via if condition as pattern match- +ing, and execution following. Each sub-figure contains the type of interaction +(passive or active) and the different modalities involved: A=audio, V=vision, +P=physical act. +As seen, operation in MOM involves time, i.e. it can be either immediate +or include past/future of any scale, which enable causality modeling. +Also, +naming, or the inclusion of language to describe objects is not necessary in +model learning. The learning still occurs, even before its name is introduced, or +if it is forgotten for some reason. +Finally, the conditions in Fig. 7 could be alternatively grouped/abstracted +as event classes instead of being learned as an action. Meaning, the pattern +matching can be replaced by event class to be recognized, and the possible +reaction to this pattern can be formed as an admissible action in such a class. +12 + +For example, ”OR” can assign the same action to different objects, and ”AND” +assigns an action to a group of objects. +(a) Developing equations (grouped chronologically) +(b) A part of a story +Figure 6: Examples of cognition processes +13 + +X=y-3 +substitute +2x+4=10 +X2 +Equation +object: +2(y-3)+4=10 +X+2=5Mom +David +state +state +room +Human +object: +Eating +David's +David +lunch +rom +Enter a room +together +has a +Sitting +room +David +Mom +David +near +in a +state +state +mom, at +floor in +Searching +room +a table +the room + something +floor +room +on the floor +David +basket +searching +Mom +Searching +without +basket in + something +Keys on +keys +the room +in a basket +desk +room +Put her +David +keys on +searching +a desk +Searching +something +Keys at +mom +Mom in +under the bed +home +room +David +searching +"in the +Entering +meantime" +home +Finding it there +chronological +bed +David +connection +finish +David's +searching +mom(a) Multi-modal fusion +(b) Causality relation1 +(c) Causality relation2 +(d) Interaction +(e) Reinforcement Learning (Pavlov experiment) +Figure 7: Examples of model learning +14 + +If A2=thunder and +later(V2,A2) and +audio +V2=... +thunder +A2 → +t +(t=10sec) +V2 +vision +Observation +(t=Osec)If V3= +andlater(V3,V4) and +V4=. +V3 +V4 +Observation +vision +vision +(t=Osec) +(t=10sec)If V5="ball thrown to +me" then output P6 +(kick), with the proper +intense, angle, .. +V5 +P6 +vision +Interaction +Physical +act +(t=Osec) +(t=10sec)If A7="bell ring" then +expect food coming, +which triggers body +preparing itself +A7 +P8 +audio +Interaction +Physical +act +(t=Osec) +(t=3sec) +Bell +SalivationIf A1="Horse" and +"Horse" +audio +A1 一→ +V1=.. +V1 +vision +ObservationA.2 +Problem-solving and Designing Appendix +A.2.1 +Problem-solving +Problem-solving is a broad topic, which is about handling any given situation, +and not only solving puzzles/mysteries/science. In any such situation, we can +either recognize a previous similar pattern (System 1), and apply automatic +reaction, i.e. immediate resolution, or if it is not the case, try to generate a new +solution (System 2). In this section, the latter option is discussed. +In this context, it is represented within some state space, where a problem +is situated at some point or a region in the space. Additionally, a problem is +expressing the current (problematic) situation, involving a general will to get +out of this situation. Hence, a will is not yet formulated at this stage (Fig. 8(a)). +Next, it is about deciding upon some goal states to be reached, to gain a desired +resolution. When goal states are defined, the will become purposeful. Purpose +is a more definite will, because it gives some “direction”, either a vague direction +or a strong one, to specific goal states (Fig. 8(b)). Since will derives action(s), it +is represented similarly to an action in the state space - as a vector, transmitting +one situation to another. Meaning, will is defining the direction the agent wants +to move, before it found the admissible/legitimate/allowable way to realize it, +in the given environment. Finally, the agent starts to plan how to solve the +given problem, under given constraints, i.e. +where one cannot fulfill its will +directly, but instead look for some legitimate way to accomplish this, in the +given circumstances. +(a) First: the problem +(b) Next: will turns into a purpose +Figure 8: Phases of will refinement in problem-solving +After the will is refined and transferred to a purpose, the search for a solution +is initiated, and it is depicted in Fig 1(a). This is the next phase of problem- +solving: realization. The figure demonstrates a coarse-to-fine hierarchy, where +the will along with its refinement, is placed at a top level. This level is vague, +since nothing is perceived clearly about the ground level. However, descending +the levels reveal more and more details, and get the will more closely to realiza- +tion. It is similar to the process of zooming in on a geographical map. Higher +15 + +In Problem-Solving: +problem +goal states +initial state:In Problem-Solving: +purpose +problem +goal states +initial state!levels propose general models as potential stations in a possible trajectory from +a problem state to a goal state. Then at descending, finer models are proposed, +consistent with the upper levels, to move from a problem to a goal state. The +search at any level can be performed by any heuristic/learned model, such as +back/forward chaining, Depth-First Search/Breadth-First Search techniques, or +any combination of those. Note the mismatch between our model level (pro- +posed solution) and data level (detailed solution), in Fig 1(a). It is due to our +inclination towards abstracting, i.e. memorizing the essences and less the de- +tails, which is essential for efficient learning, as described also in 6, where it is +better to learn several patterns than to lose yourself in a non-pattern realm, +where all we see are details. +In summary, this approach is non-local, i.e. similar to Means-Ends Analysis, +it is looking simultaneously at the whole region, only within different resolutions. +It is also cyclic and non-linear, both in the will-refining stage and in the real- +ization stage. At will refining, it is since sometimes the goal states cannot be +reached, so other states are needed to be generated, sometimes as a compromise. +At the realization stage, it is since descending in levels might result in conflicts +or failures, due to misalignment between the lowest models of reality and the +actual reality. Hence, returning to higher levels for trying different solutions is +needed. +A.2.2 +Designing +While in problem-solving, the will was generated from a problem, i.e. growing +from an initial state, in designing it is the opposite. Here, instead, it is growing +from the final state(s), searching for the best state to start the full solution +from. It is like creating a story backward: starting from the end, to reach some +beginning (Fig 9). +In designing there is a goal and a will to go there, but no specification of +the problem or the initial states. So it is an iterative process, starting from +searching for a problem to reach the goal, then continuing with a specific will +connecting the problem with the goal, resulting with a problem to solve. +16 + +Figure 9: Designing approach +A.2.3 +Summary of Problem-Solving and Designing +One can spot a duality here also. +On the one hand, problem-solving is an +analytical perspective, looking for a resolution or finishing a problem, by usually +a systemic view, breaking it to parts and then looking for some appropriate +solution, that serves as a better state than the one we started with (problem +state). On the other hand, designing is based on a holistic perspective, where +instead of finding a fast/analytical resolution to a problem (“to make everyone +happy and go on with our lives”) it is about empathy/consideration, i.e. +it +is about looking for the roots of the problem, and not just shutting it down +quickly. It takes the opposite approach: instead of reducing the problem, it +tries to track its sources and thus solving the causes that generated this problem +state/situation. By doing so, it searches for a better problem to solve, which +solves multiple other problems. Either way, will is either getting somewhere +(goal) or getting out of something (problem). +17 + +In Designing: +will +problem? +goal statesA.3 +Implementation +A.3.1 +Combining Will and Modeling +The final question, is how all the discussed above, i.e. in A.2, and in 2, i.e. +will-related topics, are combined with the operational model. There are a few +options, as discussed in 9, but we emphasize one of them here. One option, is to +have one main supervising model, that depending on the category of the situa- +tion, assigns the proper model to handle it. E.g. model for problem-solving, for +learning, and for story message (where it connects separate events sequentially). +Conversation for example is about taking turns, waiting till me/other side fin- +ished, recognizing our models, etc. Perceiving fictional information is treated +differently than factual information, and so on. In conclusion, this option de- +rived since problem-solving and alike are very complex models, which is why the +suggestion to separate them from the knowledge models. But it extends further +- perhaps there is separation of model representation. May be some models can +be represented as operational classes, but others cannot. These others could +be not interpretable nor can be explained by the agent, since they are in the +background of thinking itself, thus they are “hidden” or implicit. See Fig. 10. +Figure 10: Separating cognitive and knowledge models +18 + +I learning +problem solving +conversation +model l +object +object +object +Cognitive +function +object +function +Knowledge +function +object +models +object +models +object +function +function +object +object +function +function +function +symbols +Sub-Symbolic models +Input +OutputA.3.2 +Multi-scale Consolidation in Model Learning +Next question, is how models can be learned separately? One solution is by +assuming some initial network of unlearned models, see Fig. 11(b). But before +that we assume consolidation in multiple scales, as if there is consolidation also +in scaling (discrete amount of them). It can be encountered through many phe- +nomenon in nature, e.g. in the universe (consolidation into stars/solar-systems +and galaxies), in fractals (such as snow-flakes), and in other recursive structures. +See for example the nested structure in Fig. 11(a), and in the transition from +Fig. 11(b) to Fig. 11(c), consolidation in multiple levels, both in micro (within +models), and in macro (between models). We see that modeling or reorganiza- +tion of inner elements, is occuring at many levels of models, i.e. from the basic +models to the most complex ones. +(a) Nested DNN structure +(b) Initial Nested DNN +(c) Learned Nested DNN +Figure 11: Nested DNNs for model learning +19 + +..A.4 +Attention +After assuming a bunch of unlearned models, Fig. 11(b), we can assume that +will, acting like a flashlight or a beacon, produce consistent attention (over time) +to learn/attend each model (or several of them) separately. +This explains why an infant is usually very focused over his toys (e.g. a ball), +and tracking them is essential for this process. This effect stick till adulthood, +also in the process of using the cognitive model (i.e. after the learning stage). It +is the need to be attentive only to a limited set of models (7 ± 2 items in WM). +See examples in Fig. 12. +(a) Attending specific model1 +(b) Attending specific model2 +Figure 12: Local model learning via attention +Next question is how this will is applied in story telling/hearing and in prob- +lem solving? For example, in my presentation, the problem-solving 3-layered +slide actually showing these beams, searching for solutions! +We can see it in the following Fig. 13. +Figure 13: Learning reframed as problem-solving task +20 + +Knowledge +models +Cognitive +models +symbols +Sub-Symbolic +models +Inputs +OutputKnowledge +models +Cognitive +models +symbols +Sub-Symbolic +models +Inputs +OutputThe AGI +agent's or +human's +models +Reality +(details)We can say that learning, is making-sense type of problem-solving. So again, +there is will, coming from the top, like a projector, focusing on one/few models, +while tracking it in the real world. More accurately, we should have changed +the top will, not as a purpose to reach but as the point-wise will at the start of +a problem. +Moreover, we could say that the hierarchy is abstraction, as claimed earlier, +and will is actually only on the top but ”shining” directly on the models. Then, +we could say that during waking hours, an infant is gathering instances of its +current models, e.g. a ball, and at sleeping, he uses these instances to train his +models, for the purpose of making sense. The waking hours do not do it, they +can only perform cognitive operations, which is actions in these models. So first, +he tries to figure out different models, then he tries to model them also in time, +thus able eventually to track them, which is a validation of the correctness of his +”ball” model for example. Because the final test of his model is prediction, hence +temporal modeling is what enables prediction, or more specifically forecasting +(prediction in time). +Note, that attention to a few models also implies that just as humans, AGI +agent need not to understand and model everything, but only what it is focused +on or interested with. Also, there is the idea of bidirectional attention, which +is bottom-up (external) verse top-down (its own will), and describes the com- +petition between having a (strong) will to be highly influenced by the outside. +In AGI’s case, it should be mostly navigated by external guidance, if will is not +engineered into it. +In addition, attention can have different ”focal length”, like the theory of +vision, having small pinhole perception at lower levels, and a bigger one at +higher ones. Meaning, the ability to sometimes see small details and sometimes +see the big picture. In model attention it is the same: we can both have low- +level more detailed attention on smaller models, upto a high-level attention for +more general or composite models. In comparison to classical object detection +in computer vision, high-level concepts use only the higher-level features for the +classification task, but more generally there is no reason not to be attentive to +low-level features whenever is needed. +Finally, attention in our perspective is very similar to the attention in DL, +only without regulation. Meaning, without considering the ideal state of consol- +idating into symbolic reasoning of models and operations and most importantly +allowing for dynamic abstraction. However, DL’s attention is similar in that it +too allow for multiple implicit functions in a given learning NN, since it react +differently depending on the input. In other words, the DNN can be regarded +as a group of undeclared models/functions, generated by attention units, thus +implicitly implement compositionality and reusability. +21 + +References +Jing +Chen. +A +brief +survey +of +“programming +paradigms”, +2019. +URL +https://medium.com/@jingchenjc2019/ +a-brief-survey-of-programming-paradigms-207543a84e2b. +Taylor Day. Knowledge-based artificial neural network modeling assessment: +integrating heterogeneous genomics data to uncover lifespan regulation. 2022. +Ben Dickson. 4 deep thoughts on deep learning in 2022, 2022. URL https:// +venturebeat.com/ai/four-thoughts-on-ai-deep-learning-in-2022/. +Jeff Hawkins and Sandra Blakeslee. On intelligence: How a new understanding +of the brain will lead to the creation of truly intelligent machines. Macmillan, +2007. +Shimon Komarovsky. Dynamic and evolving neural network for event discrimi- +nation. In Ben Goertzel, Matt Ikl´e, Alexey Potapov, and Denis Ponomaryov, +editors, Artificial General Intelligence, pages 40–50. Springer International +Publishing, 2022a. ISBN 978-3-031-19907-3. +Shimon Komarovsky. Hierarchical temporal dnn and associative knowledge rep- +resentation. In Ben Goertzel, Matt Ikl´e, Alexey Potapov, and Denis Pono- +maryov, editors, Artificial General Intelligence, pages 51–61. Springer Inter- +national Publishing, 2022b. ISBN 978-3-031-19907-3. +Gary F Marcus. The algebraic mind: Integrating connectionism and cognitive +science. MIT press, 2003. +Marvin Minsky. Society of mind. Simon and Schuster, 1988. +Peter Van Roy et al. Programming paradigms for dummies: What every pro- +grammer should know. New computational paradigms for computer music, +104:616–621, 2009. +Wiktionary. +Semantic relations, 2022. +URL https://en.wiktionary.org/ +wiki/Wiktionary:Semantic_relations. +Oswald Wyler. Fuzzy logic and categories of fuzzy sets. In Non-Classical Logics +and Their Applications to Fuzzy Subsets, pages 235–268. Springer, 1995. +22 + diff --git a/mtFRT4oBgHgl3EQfaTey/content/tmp_files/load_file.txt b/mtFRT4oBgHgl3EQfaTey/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3f76e6e8f5f59b07d382f8d7fa40285cdc6f0b7b --- /dev/null +++ b/mtFRT4oBgHgl3EQfaTey/content/tmp_files/load_file.txt @@ -0,0 +1,577 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf,len=576 +page_content='Purposeful and Operation-based Cognitive System for AGI February 1, 2023 Abstract This paper proposes a new cognitive model, acting as the main com- ponent of an AGI agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' The model is introduced in its mature state, and as an extension of previous models, DENN, and especially AKREM, by including operational models (frames/classes) and will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In addition, it is mainly based on the duality principle in every known intelligent as- pect, such as exhibiting both top-down and bottom-up model learning, generalization verse specialization, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Furthermore, a holistic ap- proach is advocated for AGI designing and cognition under constraints or efficiency is proposed, in the form of reusability and simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Finally, reaching this mature state is described via a cognitive evolution from in- fancy to adulthood, utilizing a consolidation principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' The final product of this cognitive model is a dynamic operational memory of models and instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 1 Introduction Our consistent goal is to construct a basic realistic model for AGI (Artificial General Intelligence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It is a gradual process with many versions along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Hence, this paper presents MOM (Model Of Models), the next version of AKREM (Associative Knowledge Representation) [Komarovsky, 2022b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' AKREM is a mature-state knowledge representation model, based mainly on the assumption that communication is about encoding the sender’s will into a sequence of words (a message), and then decoding it by the recipient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' The model proposes a representation of any message, in a hierarchical form based on grouping, by generating some essence in a given level, from details in the lower level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' The lowest level details are founded upon some DNN (Deep Neural Network), generating the basic concepts and actions (from which the details are made of) from unstructured input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Finally, while the will concept exists in AKREM, it will be expanded upon in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Following this, new additions to the presented model, including new associa- tions, operationability, modeling, consolidation, and reusability, are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' First, while AKREM assumes that the learned elements are either objects or 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='13556v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='AI] 31 Jan 2023 static actions (verbs), new associations are introduced: object’s attributes and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Next, these connections are all static representations of knowledge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=', the hierarchies cannot be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Therefore, operationability introduces a new type of association to objects: actions that act upon them, thus producing new knowledge elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' This makes the connections in AKREM ’s hierarchies dynamic, hence it allows the freedom to update and create new hierarchies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Next, modeling introduces some basic cognitive operations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' abstraction and grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Both gather many details into fewer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Grouping specifically is about connecting elements via some common property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It could be for exam- ple a chronology in a plot or other common properties/actions grouped into classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Finally, consolidation is a process in time, that collapses a huge amount of possibilities into small set of patterns, of any kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' All the operations above are considered to be bidirectional, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' everything lies in some range between extremes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' everything has its inverse (dualism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In grouping, it is from the whole to its parts and vice versa, and in abstraction, it is from instances to classes and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Consolidation is an operation in time, creating models and memory, while its inverse is forgetting, which is also an operation in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Will also lies in the dichotomy of determinism and randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' The product of these cognitive operations is a dynamic memory of models, formed as a semantic network of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It encourages a holistic approach for AGI designing: one simple system for multiple functions, such as short-term and long-term memories, problem-solving, communication, learning, and any cognitive function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Moreover, if in the early epoch of AI, symbolic reasoning was dominant, and nowadays connectionism dominates, then we come to a new era, where we should combine and include many conflict perspectives, in co- operation and competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Our holistic perspective embrace this duality and other dualities also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Lastly, operational modeling presents AGI agent’s intelligence in its mature state, which is the state of how its knowledge should be represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' However, to accomplish this state, a cognitive evolution over time is required, which utilizes the consolidation principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 2 Will In this section, the importance of will as an essential element in human in- telligence is elaborated upon, starting from the previously presented model, AKREM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Will in AKREM is represented in the levels of any specific hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Start- ing from the most detailed aspects of will, at the lowest level, and finishing at the most abstract will or its essence, at the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' The top level represents some kind of experience uniqueness to differentiate it from other memories that use the same low-level structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' This hierarchical will is especially demonstrated in a constrained environ- ment, as our reality, for topics like problem-solving and communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In 2 problem-solving, the main will produces sub-wills in lower levels, till it reaches the final solution at the bottom level (Fig 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' The final result is a plan or a sequence of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' See more in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Similarly, in communication, the sender encodes/converts his will into a sequence of actions (in a language form), while the recipient on the other side decodes the intention/will from this sequence (Fig 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In both cases, evaluation is necessary, hence this top-down process is cyclic and non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=" (a) 3-level problem-solving in state space (b) Communication model Figure 1: Two cases of will in a constraint environment 3 The AGI agent's or human's models Proposed Solution: Detailed Solution: Reality (details)the Sequence form Lowest will ofAll the above are specific cases of will, but there is also a more general will." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' As recipients of reality, our main cognitive will is to find the most ap- propriate/simple model to fit all the pieces/details in the right place or to make the most sense of them1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' This is similar to decoding a will from a mes- sage/mystery/riddle, and it can be rephrased as a general problem-solving task to comprehend reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' First, it is done internally, by reorganizing our mod- els (mostly during a sleep phase), and later it is done externally, in any kind of problem-solving, or in understanding a message/story/riddle/situation/phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' The best model will allow us to move from place to place in it easily, perform new actions, and produce conclusions/solutions easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' This results with an un- derstanding, or the ability to control any aspect in the complex model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' So in a sense, we have two wills governing our cognition: controlling and subsequently making sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Obviously, these wills are enforcing each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Additionally, there are different categories of will, such as chronology, cau- sation, and purposefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In stories, they are very intertwined/mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It is because purposefulness is a higher manifestation of will (usually applied in hu- mans), while causation is a lower one, usually applied to animals/objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' ”A causes B”), and chronology is simply the way will is implemented: in a delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' You first want, and then you try to accomplish it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Or in the case of causation, there is a law, as a fixed kind of will (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' gravity), and then it is realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' See more in Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Figure 2: Types of will In conclusion, purpose is everywhere, in all of our daily interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' We try to figure out animals’ intentions or people’s hidden will, to make sense, or in our case to make our model complete, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' enable prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Since a specific human has his own will, he tries to figure out other people’s will to enforce its own will over theirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Sometimes it is via competition, while sometimes it is through cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' However, how is will actually included in MOM ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It is discussed in 5 and Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 1Making sense is also important for explainability, especially storytelling modeling can provide this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' explainability should be generative and flexible,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' in its most general conception 4 Humans To who: Animals Objects Random Free Physical What: Desires ness Will Laws Characteristics Impossible to Model Hard to Model (as stochastic Easy to Model (as variable) deterministic variable) Totally: Uncertain/Stochastic Uncertain/stochastic Certain/Deterministic Totally not: Predictable/Controllable Unpredictable/not Controllable Predictable/Controllable3 New Associations Inspired by semantic nets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' some general connections are proposed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' such as in- heritance (is-a relation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' instance property (is-instance-of relation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' part-whole property (part-of relation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' an attribute of an object property (has relation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' assigning a value to attribute (value relation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' synonyms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' antonyms/opposites and more,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' see [Wiktionary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' All these connections are static and can rep- resent stories as separate items as it is in AKREM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' But to connect these details, as a sequence of applied actions, operationability is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 4 Operationability It is hypothesized that thinking is operational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Meaning it is a process, gener- ating new facts along the way, via some set of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Hence, the first principle added to AKREM is operationability, which turns it from static to dynamic knowledge representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' That is, unlike static connections as in AKREM and knowledge/scene-graphs (representing facts or a scene shot) - action connec- tions in MOM can also be productive (produce new elements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It adds degrees of freedom to the current cognitive model, to move in new directions along the hierarchy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' to create new hierarchies on the fly or update old ones, via admissible actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Subsequently, a minimal set of primitive operations is proposed, to func- tion as basic operations, which can be the building blocks for more complex and composite operations/actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Operations such as logical relations (AND, OR, NOT, all, each, (in)equalities, exists, count), flow operations like loop operations (while, for) and if-else conditionals, mathematical operations (+,- ,*,/,min,max,norm,log), and other relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' This set of tools can replace DNN units and DNN’s fixed structure, in a program-search process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It can be imple- mented for example via Reservoir network, a random mix of basic rule compo- nents/blocks, yielding an algorithm best describing an operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' At the same time, prior knowledge is needed to be inserted, within all ele- ments, by including: number of visits, uncertainty, rate of update, and measure of consolidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' The measure of consolidation is to prioritize different options associated to some element, to separate the relevant from the irrelevant, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' admissible actions, which can be used inversely in creativity mode - by picking the less expected directions to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Moreover, action’s admissibility is needed for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Firstly, due to the elimination of entry conditions necessary for an action to be performed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' on which types (integer, string, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' And secondly, it is due to the ability to use High-Order Logic, as in λ-calculus, which removes any restrictions on an object’s slot or action’s argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Hence, relevancy is needed to constrain action’s admissible space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 5 5 Modeling If operationability is considered, as the addition of freedom to move in a 2D knowledge representation, then modeling is an addition of a new dimension, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' converting it to 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It is about extending the ”is-a” operation into a program- ming abstraction, as in OOP (Object-Oriented Programming), or in abstract mathematics, such as algebra or category theory2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Meaning, that while usually semantic networks represent this operation in a 2D graph, here the instances are totally separated from classes, and from classes of classes, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Resulting with a multi-level of abstractions, while for simplicity two types of levels can be distinguished, in the final LTM (Long-Term Memory), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In sum- mary, at first all different associations including operationability are describing objects, and then abstraction extends objects as instances into classes, which represent models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Consequently, unlike AKREM, these full models are impor- tant for natural communication, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' for context-based conversations, where undelivered missing information (common sense) is needed to be filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' This extension has several implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' First, in perception from senses: from the basic recognition of instances in AKREM, to a multi-level recognition of instances and classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Next, every element is learned and can be abstracted (into class), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' objects, actions, relations, and attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' For example, action with attributes, relation with attributes such as strength, numeric attribute/values as class (integer, real), group types (sets, lists, arrays) with their group operations (slicing, union, sorting), and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Next, this extension enables answering the question about how will is implemented in MOM, see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Alternatively to AKREM ’s hierarchy by will, which it is not clear how it can be implemented, it could be generated by abstraction, while the will/intention will is serving as an additional and independent variable in the models that construct the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Additionally, feeling measures could be included, as influencing will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Moreover, since there are several levels of will, correspondingly there are the main variable and secondary variables representing these wills, perhaps with different significance intensities, depending on the abstraction level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Finally, the novelty here, is that unlike DL which performs program-search in an un-interpretable way, here however, additional inductive bias is introduced: separating of models and performing program-search to relevant actions, in consistency with other models and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' This makes MOM both usable and interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='1 Learning the modeling Here, model learning mechanism is proposed, where two contrary but complet- ing learning approaches in AI are combined [Day, 2022]: empirical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' from examples (induction), and expertise (rule-based).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' This is a learnable symbolic manipulation, or can also be referred to as a hybrid approach or Neuro-Symbolic, see [Marcus, 2003].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Empirical is bottom-up (from examples to rules), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' via 2Group properties include identity object to use compositionality of an operation and its inverse operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' This implies duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 6 observation or passive interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Rule-based is straight from the top, via rules in abstract language, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' via conversation or observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It can then descend to examples of these rules (deduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' These approaches might contain models of concepts that do not belong to them both, but only to one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' For example, concepts that are hard to define, like love, God, beauty, and tacit/unconscious knowledge like walk- ing and breathing - all can be modeled simply by examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Similarly are the sub-symbolic features, like audio and visual inputs - they do not have logic/linguistic/symbolic meaning, hence should be modeled by examples, as it is done in DL nowadays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Hence, these non-symbolic concepts can be learned via usual non-interpretable DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' On the other hand, abstract concepts, like those in math and sciences, that appear less in the physical reality, can be learned solely in the top levels of LTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Moreover, in this hybrid approach, DL (Deep Learning) is used twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' On the one hand, DL is extended from its too constraint program-search to be much more flexible, if more operations are added as building blocks, see 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Hence, symbolism is learned and adaptive just like in DL, differently from expert/rule- based AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' On the other hand, different input sensors are fused to represent specific symbols/concepts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=', the uninterpreted features in DL become sym- bolic tokens (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Figure 3: Proposed cognitive model basic diagram The reason the hybrid approach is preferred over DL-only, is that DL usu- ally does not implement compositionality, modularity and abstraction [Dickson, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' One of the effects of this shortcoming is the shortcut effect, where the DL categorizes something due to the wrong reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It can be solved in a system, that learns a model with alignment and consistency-check with other models learned so far, which MOM supplies (continual learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 7 4 Temporal abstractions Symbolicmodels (Interpretable) LTM Instances symbols Sub-Symbolio External models Interactions (Uninterpretable) World Input OutputAdditionally, the topmost level is actually temporal and used for creativity and problem-solving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In this level, temporal new abstractions are created, by stripping off attributes/actions/relations, thus connecting distant or different abstractions to perform analogy or transfer learning between different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' For example, the abstraction is via the number of edges in the polygon classes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Furthermore, learning can be divided into online and offline modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In wak- ing periods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' when sensors are active, the learning is online and minimal since most resources are dedicated to fast response (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' fast optimization into local optimums).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' However, during sleeping periods, sensors are inactive, and previous memories can be used for improving models to make overall sense, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' larger time scale is used to generate causal relations in models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In such a case, it is a slow processing, implemented e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' as a Neural Architecture Search or a Genetic Algorithm, to get out of local minima and search for a global one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Finally, another issue precedes learning: how to obtain separate models at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' One way, is like the DENN (Dynamic and evolving NN) idea [Komarovsky, 2022a], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' always learning the ”model of everything” while refining it more and more with every new experience, such that new sub-models are produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' The idea here, is like Jeff’s hierarchy [Hawkins and Blakeslee, 2007], where the top model is always reached, at perception from senses, and it decides which lower model will handle the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' For example, in recognizing a specific type of problem, it chooses the most appropriate model for solving this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Or, when encountering new knowledge, it selects the most appropriate model to handle its assimilation in the current memory of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Another way to facilitate modeling evolution is via consolidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 6 Consolidation So far, the cognitive model has been presented in its mature state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Now, the discussion is about how to reach it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' This is a process in time, which is mainly based on consolidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Consolidation is about transforming from chaos to some stable order of pat- terns, or from a continuous realm to a discrete one, as in quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' An infinite amount of details is hard to handle (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' to understand and then to control), therefore consolidation to fewer patterns is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Consolidation also allows for fuzzy logic and categories [Wyler, 1995].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Consolidation can be expressed in many forms, such as: in the conversion of sub-symbolic to symbolic, for any type of element in cognitive evolution: from flexible (at infancy) to less flexible (at adult- hood) in modeling, at program search, from huge hypothesis space for possible programs to a small set of hypotheses (as in DL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It is both in the micro (within models) and in the macro (between models) 8 in testing multiple versions of an unknown model, and finally converging into less/one version(s) that are/is consistent with evidence and in grouping/abstraction, where some separate elements become con- nected Note that causality is a special case of modeling, a spatio-temporal one, where re-occurrence is consolidated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' More generally, re-occurrence help in learn- ing both static objects and dynamic basic/composite events (equivalent to sce- narios/scripts in OOP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Additionally, MOM enables multiple parallel versions of the same thing, since any specific topic or subject can have multiple theories/models, sometimes in conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Hence, like in the quantum superposition realm, multiple-version combinations could be tryout, and consolidation can help in collapsing them into fewer versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Those versions should make the most sense, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' to be consistent on different occasions or supporting the majority of evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Thus, it just maybe, that at infancy, a highly uncertain period, there are many versions created, and with time - only the most consistent ones survive (consolidate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Lastly, two operations help in producing consolidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' On the one hand, to deal with a stochastic environment and ambiguous signals, repetition provides memory prioritized by relevancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Repetition is never exactly over the same thing, but rather over many different examples of a thing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Repetition is needed also with guided tutoring of an AGI agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='Conversely, sparsification is about reducing irrelevant signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='1 Reusability An additional form of consolidation is reusability, since the more learning pro- gresses, the fewer new models are proposed in favor of using existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Hence, reusability is expressed via exploration (mostly at early stages) verse exploitation (in stable or mature stages), as in Reinforcement Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In the beginning, many possible codes are generated for models, but as time goes by the process is less exploratory and more exploitative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' there is more emphasis on retrieving known codes, while testing fewer new codes in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In addi- tion, reusability aligns perfectly with abstraction/grouping, in a constrained environment and limited resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' They are both needed to hold control of as much as possible, with minimum effort, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' without generating many models of each thing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In practice, reusability is about using less of the initial available tools, as the learning evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Meaning, while regular DL tools (if, sum, activation function) or the primitive tools 4 can be used for program search of basic action methods or relation methods, the new methods apply reusability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In such methods, less primitive tools are used while the current methods are used more, thus encouraging more connectivity in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Moreover, Functional Programming can be applied to assist reusability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' On the one hand, the general/outer structure is OOP, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' elements are grouped in an OOP fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' On the other hand, methods are kept in a pure operational 9 immutable form [Chen, 2019, Van Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=', 2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Meaning, having small and simple methods, which maximally reuse other functions, and without in- ner variables, due to objects-memory-only assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Meaning, methods that are comprised of other methods, as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' This is compositional- ity/grouping applied in actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Figure 4: Actions, objects, compositionality, and reusability 7 Cognitive model comparison Here all models developed so far are compared, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Figure 5: Comparison table of models discussed so far In summary, DENN stores each new combination of events, while AKREM stores dynamically in episodic memory any newly encountered combination of basic events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Events which are stored separately in two types of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' MOM unites those separate memories into one dynamic operational memory, consisting of concepts, actions, relations and any instance of those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 10 object object object function object function function object object object function function object object function function functionAssociative knowledge Model Dynamic evolving DNN Model of models representation Plus Growing/adapting and Separate basic instances to Stores relational and applies continuous and fixed and small memory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' operational info,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' for most importantly gradual while new combinations of both instances and learning basic events stored in classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Allow thinking episodic memory creating new objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Minus Instances Instances Consolidation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' from Only event Only representation (no sub-symbolic data to discrimination learning) symbolic) Stage in Infancy Adulthood Childhood until development adulthood Take-aways Constantly refining Generating and perceiving- A full cognitive events on the one hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' messages,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' and uniquely model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' producing while grouping them on storing them in episodic dynamic operational the other hand memory memory of models and instances Implementability High Low Medium level8 Conclusion The following is the summary of the paper including some key takeaways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' First, a new cognitive model is introduced to join the existing cognitive archi- tectures, in the form of dynamic operational memory of models and instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In it, a holistic approach is embraced, assuming that intelligence should be highly versatile and diverse, instead of ”picking one side”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Next, our model includes will as an essential part of the modeling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Thus, in case of its absence, it turns most of the learned models to be very partial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In the OOP formulation, the will is an additional variable, and it is mostly significant in the top level while it is least significant in the lowest one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Next, operationability turns static knowledge representation into a dynamic one, thus enabling cognitive processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' The actions are learned via regular program search, with either DL or other tools, in a self-supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Next, one way to ensure that the continual learning is consistent, is by imple- menting local learning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=', concentrating on updating only one/some model(s), while confirming compatibility with other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' A model can be learned ei- ther from examples or directly by using existing operations (logically).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Another way to ensure that learning is consistent, is by a slow process of consolidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It is ensured by maintaining a high level of flexibility over a long period, while pursuing more and more consistency within and between models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Next, reusability is utilized to enhance connectivity between models, instead of learning them as separate entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Finally, the cognitive model is designed via inverse engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Meaning, starting from our highly aware and mature cognitive state of mind, and then tracking back in time to study its evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 9 Future Work The main problem is how to implement a cognitive system, that produces the appropriate models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' how grouping/clustering occur, to generate the right models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Also, how these models produce new ones by the correct compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In addition, there is the issue of how sub-symbolic become symbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Per- haps the models produce objects and actions directly upon sub-symbolic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Furthermore, if continuing this line of thought, then models may be removed at all, which converts this problem to pure DL-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Additionally, relevant to the last issue, is model evolving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Is it model re- finement of some main model to sub-models, which controls how models of knowledge are used, or is it that all the models are separate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' And if it is by refinement, is it one large DL-based model, and all the rest are knowledge mod- els, or if we continue this line of thought - again, end up with pure DL-based one huge model, containing implicitly all the different models, their actions and attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' But then how elements and abstraction are implemented in such a model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' More about the suggestion above see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 11 Finally, though hierarchy by abstraction can be implemented, but how can it be implemented by will, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' how to decide when and what to group in such a hierarchy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' These are all open questions to deal with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' : Neuro-Symbolic AI for me is merely taking inspiration of class-based structure, to act as the final stage of learning, while DL is the main tool to reach it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' So it is all about flexibility of DNNs, only that we use consolidation to finally reach symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Another implication of this, is memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' First, since it is vague and mostly reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It means it is not recorded accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' And second, since it probably used in sleeping periods similarly in a vague form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' A Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='1 Examples of MOM in action Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 6 contains examples of cognition processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' As seen, actions denoted as arrows, can perform changes in several objects to their different attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' High admissibility expressed via salient color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 6(b) is a MOM representation of evolving state, representing a story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' A state is consisting of several objects (joining/leaving as the story evolves), including their attributes and actions that change the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' The story: ”David entered his room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' He searched for something on the floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Then he searched in the basket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Then he searched under his bed, and was thrilled to find the ball there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In the meanwhile, his mother entered home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' She put her keys on the desk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Then she removed her shoes and put her sunglasses on the desk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Then she searched for David, found him, and they sat to eat lunch together”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' The same story is represented via AKREM, without the evolution of details, see the video link in [Komarovsky, 2022b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Note, that if repeated often, the sequence of David or anybody searching in some place can be grouped/abstracted as an event class, also referred to as Trans-Frame [Minsky, 1988].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Next, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 7 contains examples of model learning, based on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' The as- sumption here, is that similarly to DNN pattern matching and then executing some task - here it is performed explicitly, via if condition as pattern match- ing, and execution following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Each sub-figure contains the type of interaction (passive or active) and the different modalities involved: A=audio, V=vision, P=physical act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' As seen, operation in MOM involves time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' it can be either immediate or include past/future of any scale, which enable causality modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Also, naming, or the inclusion of language to describe objects is not necessary in model learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' The learning still occurs, even before its name is introduced, or if it is forgotten for some reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Finally, the conditions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 7 could be alternatively grouped/abstracted as event classes instead of being learned as an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Meaning, the pattern matching can be replaced by event class to be recognized, and the possible reaction to this pattern can be formed as an admissible action in such a class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 12 For example, ”OR” can assign the same action to different objects, and ”AND” assigns an action to a group of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=" (a) Developing equations (grouped chronologically) (b) A part of a story Figure 6: Examples of cognition processes 13 X=y-3 substitute 2x+4=10 X2 Equation object: 2(y-3)+4=10 X+2=5Mom David state state room Human object: Eating David's David lunch rom Enter a room together has a Sitting room David Mom David near in a state state mom," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' at ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='floor in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='Searching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='room ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='a table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='the room ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='something ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='floor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='room ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='on the floor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='David ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='basket ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='searching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='Mom ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='Searching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='without ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='basket in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='something ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='Keys on ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='keys ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='the room ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='in a basket ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='desk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='room ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='Put her ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='David ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='keys on ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='searching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='a desk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='Searching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='something ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='Keys at ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='mom ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='Mom in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='under the bed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='home ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='room ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='David ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='searching "' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='in the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='Entering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='meantime" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='home ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='Finding it there ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='chronological ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='bed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='David ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='connection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='finish ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content="David's " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='searching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='mom(a) Multi-modal fusion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='(b) Causality relation1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='(c) Causality relation2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='(d) Interaction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='(e) Reinforcement Learning (Pavlov experiment) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='Figure 7: Examples of model learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='If A2=thunder and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='later(V2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='A2) and audio V2=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' thunder A2 → t (t=10sec) V2 vision Observation (t=Osec)If V3= andlater(V3,V4) and V4=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' V3 V4 Observation vision vision (t=Osec) (t=10sec)If V5="ball thrown to me" then output P6 (kick), with the proper intense, angle, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='. V5 P6 vision Interaction Physical act (t=Osec) (t=10sec)If A7="bell ring" then expect food coming, which triggers body preparing itself A7 P8 audio Interaction Physical act (t=Osec) (t=3sec) Bell SalivationIf A1="Horse" and "Horse" audio A1 一→ V1=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='. V1 vision ObservationA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='2 Problem-solving and Designing Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='1 Problem-solving Problem-solving is a broad topic, which is about handling any given situation, and not only solving puzzles/mysteries/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In any such situation, we can either recognize a previous similar pattern (System 1), and apply automatic reaction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' immediate resolution, or if it is not the case, try to generate a new solution (System 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In this section, the latter option is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In this context, it is represented within some state space, where a problem is situated at some point or a region in the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Additionally, a problem is expressing the current (problematic) situation, involving a general will to get out of this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Hence, a will is not yet formulated at this stage (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 8(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Next, it is about deciding upon some goal states to be reached, to gain a desired resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' When goal states are defined, the will become purposeful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Purpose is a more definite will, because it gives some “direction”, either a vague direction or a strong one, to specific goal states (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 8(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Since will derives action(s), it is represented similarly to an action in the state space - as a vector, transmitting one situation to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Meaning, will is defining the direction the agent wants to move, before it found the admissible/legitimate/allowable way to realize it, in the given environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Finally, the agent starts to plan how to solve the given problem, under given constraints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' where one cannot fulfill its will directly, but instead look for some legitimate way to accomplish this, in the given circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' (a) First: the problem (b) Next: will turns into a purpose Figure 8: Phases of will refinement in problem-solving After the will is refined and transferred to a purpose, the search for a solution is initiated, and it is depicted in Fig 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' This is the next phase of problem- solving: realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' The figure demonstrates a coarse-to-fine hierarchy, where the will along with its refinement, is placed at a top level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' This level is vague, since nothing is perceived clearly about the ground level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' However, descending the levels reveal more and more details, and get the will more closely to realiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It is similar to the process of zooming in on a geographical map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Higher 15 In Problem-Solving: problem goal states initial state:In Problem-Solving: purpose problem goal states initial state!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='levels propose general models as potential stations in a possible trajectory from a problem state to a goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Then at descending, finer models are proposed, consistent with the upper levels, to move from a problem to a goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' The search at any level can be performed by any heuristic/learned model, such as back/forward chaining, Depth-First Search/Breadth-First Search techniques, or any combination of those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Note the mismatch between our model level (pro- posed solution) and data level (detailed solution), in Fig 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It is due to our inclination towards abstracting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' memorizing the essences and less the de- tails, which is essential for efficient learning, as described also in 6, where it is better to learn several patterns than to lose yourself in a non-pattern realm, where all we see are details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In summary, this approach is non-local, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' similar to Means-Ends Analysis, it is looking simultaneously at the whole region, only within different resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It is also cyclic and non-linear, both in the will-refining stage and in the real- ization stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' At will refining, it is since sometimes the goal states cannot be reached, so other states are needed to be generated, sometimes as a compromise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' At the realization stage, it is since descending in levels might result in conflicts or failures, due to misalignment between the lowest models of reality and the actual reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Hence, returning to higher levels for trying different solutions is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='2 Designing While in problem-solving, the will was generated from a problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' growing from an initial state, in designing it is the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Here, instead, it is growing from the final state(s), searching for the best state to start the full solution from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It is like creating a story backward: starting from the end, to reach some beginning (Fig 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In designing there is a goal and a will to go there, but no specification of the problem or the initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' So it is an iterative process, starting from searching for a problem to reach the goal, then continuing with a specific will connecting the problem with the goal, resulting with a problem to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 16 Figure 9: Designing approach A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='3 Summary of Problem-Solving and Designing One can spot a duality here also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' On the one hand, problem-solving is an analytical perspective, looking for a resolution or finishing a problem, by usually a systemic view, breaking it to parts and then looking for some appropriate solution, that serves as a better state than the one we started with (problem state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' On the other hand, designing is based on a holistic perspective, where instead of finding a fast/analytical resolution to a problem (“to make everyone happy and go on with our lives”) it is about empathy/consideration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' it is about looking for the roots of the problem, and not just shutting it down quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It takes the opposite approach: instead of reducing the problem, it tries to track its sources and thus solving the causes that generated this problem state/situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' By doing so, it searches for a better problem to solve, which solves multiple other problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Either way, will is either getting somewhere (goal) or getting out of something (problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 17 In Designing: will problem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' goal statesA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='3 Implementation A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='1 Combining Will and Modeling The final question, is how all the discussed above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='2, and in 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' will-related topics, are combined with the operational model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' There are a few options, as discussed in 9, but we emphasize one of them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' One option, is to have one main supervising model, that depending on the category of the situa- tion, assigns the proper model to handle it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' model for problem-solving, for learning, and for story message (where it connects separate events sequentially).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Conversation for example is about taking turns, waiting till me/other side fin- ished, recognizing our models, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Perceiving fictional information is treated differently than factual information, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In conclusion, this option de- rived since problem-solving and alike are very complex models, which is why the suggestion to separate them from the knowledge models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' But it extends further perhaps there is separation of model representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' May be some models can be represented as operational classes, but others cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' These others could be not interpretable nor can be explained by the agent, since they are in the background of thinking itself, thus they are “hidden” or implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Figure 10: Separating cognitive and knowledge models 18 I learning problem solving conversation model l object object object Cognitive function object function Knowledge function object models object models object function function object object function function function symbols Sub-Symbolic models Input OutputA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='2 Multi-scale Consolidation in Model Learning Next question, is how models can be learned separately?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' One solution is by assuming some initial network of unlearned models, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 11(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' But before that we assume consolidation in multiple scales, as if there is consolidation also in scaling (discrete amount of them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It can be encountered through many phe- nomenon in nature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' in the universe (consolidation into stars/solar-systems and galaxies), in fractals (such as snow-flakes), and in other recursive structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' See for example the nested structure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 11(a), and in the transition from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 11(b) to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 11(c), consolidation in multiple levels, both in micro (within models), and in macro (between models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' We see that modeling or reorganiza- tion of inner elements, is occuring at many levels of models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' from the basic models to the most complex ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' (a) Nested DNN structure (b) Initial Nested DNN (c) Learned Nested DNN Figure 11: Nested DNNs for model learning 19 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='.A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='4 Attention After assuming a bunch of unlearned models, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 11(b), we can assume that will, acting like a flashlight or a beacon, produce consistent attention (over time) to learn/attend each model (or several of them) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' This explains why an infant is usually very focused over his toys (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' a ball), and tracking them is essential for this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' This effect stick till adulthood, also in the process of using the cognitive model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' after the learning stage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' It is the need to be attentive only to a limited set of models (7 ± 2 items in WM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' See examples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' (a) Attending specific model1 (b) Attending specific model2 Figure 12: Local model learning via attention Next question is how this will is applied in story telling/hearing and in prob- lem solving?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' For example, in my presentation, the problem-solving 3-layered slide actually showing these beams, searching for solutions!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' We can see it in the following Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=" Figure 13: Learning reframed as problem-solving task 20 Knowledge models Cognitive models symbols Sub-Symbolic models Inputs OutputKnowledge models Cognitive models symbols Sub-Symbolic models Inputs OutputThe AGI agent's or human's models Reality (details)We can say that learning, is making-sense type of problem-solving." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' So again, there is will, coming from the top, like a projector, focusing on one/few models, while tracking it in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' More accurately, we should have changed the top will, not as a purpose to reach but as the point-wise will at the start of a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Moreover, we could say that the hierarchy is abstraction, as claimed earlier, and will is actually only on the top but ”shining” directly on the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Then, we could say that during waking hours, an infant is gathering instances of its current models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' a ball, and at sleeping, he uses these instances to train his models, for the purpose of making sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' The waking hours do not do it, they can only perform cognitive operations, which is actions in these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' So first, he tries to figure out different models, then he tries to model them also in time, thus able eventually to track them, which is a validation of the correctness of his ”ball” model for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Because the final test of his model is prediction, hence temporal modeling is what enables prediction, or more specifically forecasting (prediction in time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Note, that attention to a few models also implies that just as humans, AGI agent need not to understand and model everything, but only what it is focused on or interested with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Also, there is the idea of bidirectional attention, which is bottom-up (external) verse top-down (its own will), and describes the com- petition between having a (strong) will to be highly influenced by the outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In AGI’s case, it should be mostly navigated by external guidance, if will is not engineered into it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In addition, attention can have different ”focal length”, like the theory of vision, having small pinhole perception at lower levels, and a bigger one at higher ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Meaning, the ability to sometimes see small details and sometimes see the big picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In model attention it is the same: we can both have low- level more detailed attention on smaller models, upto a high-level attention for more general or composite models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In comparison to classical object detection in computer vision, high-level concepts use only the higher-level features for the classification task, but more generally there is no reason not to be attentive to low-level features whenever is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Finally, attention in our perspective is very similar to the attention in DL, only without regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Meaning, without considering the ideal state of consol- idating into symbolic reasoning of models and operations and most importantly allowing for dynamic abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' However, DL’s attention is similar in that it too allow for multiple implicit functions in a given learning NN, since it react differently depending on the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In other words, the DNN can be regarded as a group of undeclared models/functions, generated by attention units, thus implicitly implement compositionality and reusability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 21 References Jing Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' A brief survey of “programming paradigms”, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' URL https://medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='com/@jingchenjc2019/ a-brief-survey-of-programming-paradigms-207543a84e2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Taylor Day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Knowledge-based artificial neural network modeling assessment: integrating heterogeneous genomics data to uncover lifespan regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Ben Dickson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 4 deep thoughts on deep learning in 2022, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' URL https:// venturebeat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='com/ai/four-thoughts-on-ai-deep-learning-in-2022/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Jeff Hawkins and Sandra Blakeslee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' On intelligence: How a new understanding of the brain will lead to the creation of truly intelligent machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Macmillan, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Shimon Komarovsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Dynamic and evolving neural network for event discrimi- nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In Ben Goertzel, Matt Ikl´e, Alexey Potapov, and Denis Ponomaryov, editors, Artificial General Intelligence, pages 40–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Springer International Publishing, 2022a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' ISBN 978-3-031-19907-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Shimon Komarovsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Hierarchical temporal dnn and associative knowledge rep- resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In Ben Goertzel, Matt Ikl´e, Alexey Potapov, and Denis Pono- maryov, editors, Artificial General Intelligence, pages 51–61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Springer Inter- national Publishing, 2022b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' ISBN 978-3-031-19907-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Gary F Marcus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' The algebraic mind: Integrating connectionism and cognitive science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' MIT press, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Marvin Minsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Society of mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Simon and Schuster, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Peter Van Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Programming paradigms for dummies: What every pro- grammer should know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' New computational paradigms for computer music, 104:616–621, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Wiktionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Semantic relations, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' URL https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='wiktionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content='org/ wiki/Wiktionary:Semantic_relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Oswald Wyler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Fuzzy logic and categories of fuzzy sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' In Non-Classical Logics and Their Applications to Fuzzy Subsets, pages 235–268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' Springer, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} +page_content=' 22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFRT4oBgHgl3EQfaTey/content/2301.13556v1.pdf'} diff --git a/n9FKT4oBgHgl3EQfFy0J/content/2301.11721v1.pdf b/n9FKT4oBgHgl3EQfFy0J/content/2301.11721v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a237c6869d62cc220a29703ce3f36fb5e142387e --- /dev/null +++ b/n9FKT4oBgHgl3EQfFy0J/content/2301.11721v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ad3bd3c550e22f7dccb900f5cb6aed303c2fada04b5ea09985d5355d30c00115 +size 733927 diff --git a/n9FKT4oBgHgl3EQfFy0J/vector_store/index.faiss b/n9FKT4oBgHgl3EQfFy0J/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..151329e1c048aeba119a5ffd9d365e1760ef9ec9 --- /dev/null +++ b/n9FKT4oBgHgl3EQfFy0J/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:478e2a3710bdd28710fbe298153fa789c03593eb2396d4543bddbe66bca2d434 +size 5373997 diff --git a/ntE_T4oBgHgl3EQf7xzu/content/tmp_files/2301.08372v1.pdf.txt b/ntE_T4oBgHgl3EQf7xzu/content/tmp_files/2301.08372v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8edf9cc9e0f4fbb47c23d27a2d3bb4da094db136 --- /dev/null +++ b/ntE_T4oBgHgl3EQf7xzu/content/tmp_files/2301.08372v1.pdf.txt @@ -0,0 +1,1742 @@ +Screen Correspondence: Mapping Interchangeable Elements +between UIs +Jason Wu, Amanda Swearngin, Xiaoyi Zhang, Jeffrey Nichols, Jeffrey Bigham +{jsonwu,aswearngin,xiaoyiz,jwnichols,jbigham}@apple.com +Login Page +Sign in with your email address to access your +account. +Email +Test App +Don’t have an account or forgot it? +Your account is used to access data across all your +devices. +Your email address is used to enable services when you sign in, +including backup which automatically backs up the data on your +devices. Your device’s identifier may be used to check eligibility for +special offers. For more information, see our terms of service. +Login Page +Don’t have an account or forgot your password? +Your email address is used to enable services when you sign in, including +backup which automatically backs up the data on your devices. Your +device’s identifier may be used to check eligibility for special offers. For +more information, see our terms of service. +Required +Required +Email +Password +Sign in with your email address to access +your account. +Input Screen +Exemplar +Inferred Semantics +Text_Instruction +Text_Title +Button_Back +Button_Disabled +Input_Email +Input_Passwd +Button_Enabled +Text_Footer +Text_Footer +Button_Enabled +Input_Email +Text_Instruction +Text_Title +Button_Back +Button_Disabled +Login Page +Sign in with your email address to access your +account. +Email +Test App +Don’t have an account or forgot it? +Your account is used to access data across all your +devices. +Your email address is used to enable services when you sign in, +including backup which automatically backs up the data on your +devices. Your device’s identifier may be used to check eligibility for +special offers. For more information, see our terms of service. +Image +Icon +Icon +Image +Unmatched +Figure 1: Screen correspondence produces a mapping of similar UI elements across two UIs that have related elements. Screen- +shots are encoded using a multi-modal model that segments and featurizes UI elements. Mappings are generated that link +element pairs that have the same or similar functionality across UI screens. +ABSTRACT +Understanding user interface (UI) functionality is a useful yet chal- +lenging task for both machines and people. In this paper, we in- +vestigate a machine learning approach for screen correspondence, +which allows reasoning about UIs by mapping their elements onto +previously encountered examples with known functionality and +properties. We describe and implement a model that incorporates +element semantics, appearance, and text to support correspondence +computation without requiring any labeled examples. Through a +comprehensive performance evaluation, we show that our approach +improves upon baselines by incorporating multi-modal properties +of UIs. Finally, we show three example applications where screen +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than ACM +must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, +to post on servers or to redistribute to lists, requires prior specific permission and/or a +fee. Request permissions from permissions@acm.org. +, , +© 2022 Association for Computing Machinery. +ACM ISBN 978-1-4503-XXXX-X/18/06...$15.00 +https://doi.org/10.1145/1122445.1122456 +correspondence facilitates better UI understanding for humans and +machines: (i) instructional overlay generation, (ii) semantic UI ele- +ment search, and (iii) automated interface testing. +KEYWORDS +user interface modeling, ui semantics, element correspondence +ACM Reference Format: +Jason Wu, Amanda Swearngin, Xiaoyi Zhang, Jeffrey Nichols, Jeffrey Bigham. +2022. Screen Correspondence: Mapping Interchangeable Elements between +UIs. In . ACM, New York, NY, USA, 13 pages. https://doi.org/10.1145/1122445. +1122456 +1 +INTRODUCTION +Understanding how user interfaces (UIs) can be operated to achieve +some goal can be challenging for both machines and humans, es- +pecially those who are less tech-savvy. While automated systems +in the right circumstances can provide useful assistance [62, 63] or +automatically complete the task themselves [34, 36], people can be +hindered or completely blocked by apps that do not provide neces- +sary metadata, such as the view hierarchy. A promising approach +involves inferring UI functionality solely from the pixels rendered +arXiv:2301.08372v1 [cs.HC] 20 Jan 2023 + +10:14 +Dubsmash +Cancel +Next +Apple ID +Sign in with your Apple ID to use iCloud and other +Apple services. +Apple ID +Email +Don't have an Apple ID or forgot it? +Your Apple ID is the account you use to access all +Apple services. +Your Apple ID information is used to enable Apple services when you +sign in,includingiCloud Backupwhichautomatically backs upthedata +on your device in case you need to replace or restore it.Your device +serialnumbermaybeusedtocheckeligibilityforserviceoffers. +See how your data is managed...口9:41 +三 +< Back +Next +Apple ID +SigninwithyourAppleIDtouseiCloud,the +AppStore,andotherAppleservices +AppleID +j.appleseed@icloud.com +Password +Required +Forgotpasswordordon'thaveanAppleID? +Your Apple ID information is used to enable Apple services when +you sign in,including iCloud Backup which automatically backs up +the data on vour device in case vou need to replace or restore it. +Your device serial number may be used to check eligibility for +service offers.Seehow yourdataismanaged.. +UsedifferentAppleIDsforiCloud&other, , +Wu et al. +to the screen, but to date this method has primarily been useful +for identifying the location and type of typical UI elements [65] +and not higher-level semantics. For example, these algorithms can +identify that a screen has a button that contains the text “Login,” but +are unaware of the higher-level concept of logging in to a service, +and they cannot infer what the role of this button would be in that +process. +There are many higher-level semantics in user interfaces (e.g., lo- +gin, account registration, shopping carts), which would correspond +to an enormous number of classes if we attempted to use a classifier +to predict their occurrence. Instead of making class predictions, an +alternate approach to data inference involves directly comparing +inputs to previously encountered examples with known properties. +Studies on human [55] and machine [56] learning suggest that di- +rect comparison is a useful tool, especially when relevant examples +are available in the form of analogies [4, 24] or templates [20, 22]. +This concept can be highly effective for UIs, as many belong to the +same app or are constructed to serve a similar purpose. For example, +knowledge of how an app screen was previously interacted with by +an app crawler or automated UI tester could aid in producing more +robust and consistent results when visited again. Similar inferences +can also be made for related screens from different apps, such as +by determining that a button with the label “Login” in a new app is +likely used to submit a login request because that is how a similar +button is used in a known app. +In this paper, we pose the problem of screen correspondence to +map interchangeable elements between two UI screenshots (Figure +1). We introduce a multi-modal transformer model for detecting, fea- +turizing, and matching UI elements. Our approach is unsupervised, +which allows it to work without a large dataset of labeled examples, +which could be costly and time-consuming to collect. In a perfor- +mance evaluation with strong baselines, we compare our approach +to existing correspondence algorithms used in computer vision +(CV) and heuristics such as schema-matching. Our results indicate +that our multi-modal model outperforms all existing baselines. +We describe and implement three example applications that show +the utility of screen correspondence for humans and machines to +understand UI functionality. We create an application to gener- +ate instructional overlays by transferring high-quality human- +authored coach marks (a type of instructional label) from one screen +to another of the same category (e.g., two registration screens). To +support UI design search and exemplar-based exploration, we used +our model to index a large dataset of UI elements and screens. +Finally, we built a system to aid an automated app crawler by +identifying mappings between the elements of screens from differ- +ent runs. +To summarize, we make the following contributions: +• We introduce screen correspondence as a method of mapping +interchangeable elements between UI screens from their +screenshots. +• We describe a machine learning approach to generating cor- +respondence between two UI screenshots, and we show it +outperforms existing baselines. +• We show the utility of screen correspondence in three ex- +ample applications that improve both human and machine +understanding of UI functionality. +2 +RELATED WORK +Our work is related to recent work in understanding user interfaces +from their pixels, and also a variety of methods for understanding +applications in terms of their many screens. We also overview +machine learning solutions to correspondence problems in other +domains, such as computer vision and natural language processing. +2.1 +Predicting Screen Semantics +Computational representation of user interfaces are useful for many +downstream tasks, such as design assistance [32, 39], accessibility +improvement [65], and task-oriented systems. Screen Recognition +[65] generates accessibility metadata of a UI from screenshots using +an object detection model and heuristics. Screen Parsing [60] gener- +ates structured UI models from screenshots of UIs. Several models +[6, 15, 41, 64] have also been trained to predict the semantics of +unlabeled icons found in mobile apps. These models can be applied +to improve the accessibility of mobile apps, either as a tool during +design time or as an automated system that repairs existing apps +at runtime. Most of these models map UI elements to a pre-defined +set of classes (e.g. UI element and icon type), which may exclude +less common components [7]. +An alternative is to train models using self-supervision [9, 18, 35], +which allows them to take advantage of larger unlabeled datasets. +Screen2Vec [35] and other pixel-based autoencoders [9, 39] map UIs +to fixed-length embedding vectors which can be used to represent +semantic properties. The Pixel-words model [18] employs a trans- +former model architecture and masked training objective based on +prior work in NLP [11]. Our work builds upon these approaches to +train a model for identifying UI element correspondences between +screens. +2.2 +Multi-screen Understanding +While many automated UI systems can benefit from understanding +the semantics of a single screen, screens are rarely used in isolation. +Any task or interaction trace requires reasoning about multiple app +screens and how they are related to each other. +StoryDroid is a system that extracts a storyboard of Android +apps from APK files as an “App Transition Graph” [8]. ActionBert +[23] models the relationship between two consecutive UI screens +by predicting, among other things, which UI element was tapped on +the first screen to reach the second (i.e., link component prediction). +Longer sequences of touch interactions have also been modeled to +better understand user behavior and app usage [31, 69]. +A particular problem that many multi-screen systems aim to ad- +dress is identifying whether two screens are instances of the same +UI, a problem which we refer to as “screen fingerprinting.” NEAR +[61] detects near-duplicate pages on the web using a combination +of visual and DOM-based features. Prior work [14] used super- +vised learning to predict the relationship and transitions between +screenshots by, among other things, classifying whether inputs +were different instances of the same screen (e.g., a news app with +dynamic content). +Screen fingerprinting is useful for comparing screens to known +examples; however, finer grain mappings (e.g., element-level finger- +printing) can result in higher fidelity comparisons and additional +benefits. Bricolage [30] is a system that renders the content of one + +Screen Correspondence: Mapping Interchangeable Elements between UIs +, , +web page using the style and layout of another. It employs a su- +pervised element matching model that featurizes web elements +based on their DOM representation and was trained on a dataset of +human-generated mappings. Interaction proxies [67] rely on a set +of equivalency heuristics to identify UI components and structures +found in Android view hierarchies to facilitate accessibility repair. +Our work is related to these approaches in multi-screen un- +derstanding and specifically element fingerprinting. While many +previous examples relied heavily on the availability of a structured +UI representation (e.g., DOM, view hierarchy) and were trained on +labeled data, our approach requires only screenshots of related apps +with optional labels. +2.3 +Machine Learning of Correspondence +Machine learning has been used to learn correspondences in other +domains, such as computer vision (CV) and natural language pro- +cessing (NLP), which are closely related to our approach. +A longstanding problem in CV is inferring accurate correspon- +dence of objects from different images. Homography estimation +[21] involves finding a mapping, either sparse or dense, or a trans- +formation matrix that describes perspective changes in two images +of the same object or scene. Optical flow [25] applies a similar +concept to finding mappings between consecutively taken images. +A common approach involves computing appearance descriptors +(keypoints), then creating a mapping that optimizes the global corre- +spondence [16, 38]. Recent work [1] has extended these approaches +using learned semantic features to infer correspondence between +images of inter-class or inter-domain objects. +Correspondence learning has also been useful for many tasks in +NLP such as pronoun co-reference resolution and commonsense rea- +soning, both of which rely on modeling correspondences between +words to resolve ambiguities [28, 33, 59]. Language translation +and understanding, particularly for low-resource languages, ben- +efit from learning word alignments to higher-resource languages +[12, 47]. Finally, other types of conditional natural language gen- +eration have benefited from learning alignments between words +with similar meanings [2, 48]. +Screen correspondence is related to many machine-learning +driven approaches to identifying correspondence. Our transformer +model builds upon many of these approaches by combining visual +information and word-alignment techniques to produce screen cor- +respondence. In our evaluation, we compared our system to several +baseline techniques from the CV and NLP literature. We show that +by incorporating multiple sources of information, our model gen- +erates better representations for UI elements, which leads to more +accurate correspondence predictions. +3 +SCREEN CORRESPONDENCE +We define screen correspondence as the task of mapping interchange- +able UI elements between two UI screens. While matching UI ele- +ments between screens may seem simple, it is a complex problem +(especially from pixels alone) with many practical use-cases. Pre- +vious work relied on mappings to retarget UIs [30], provide help +[62, 63], assist design [3], and test GUIs [5] and specifically called +for more robust matching to improve performance. +We primarily consider cases where two UI screens are of the same +category (e.g., Login or Registration) but from different apps (i.e., +intra-class examples). This is challenging because UI element pairs +across such screen type pairs may not share similar appearance, text, +or position. Instead, the model must reason about the purpose of +each element in the context of its screen. To give intuition why even +a seemingly simple example is hard, consider two Login screens +(Figure 1): one contains Username and Password fields, while another +contains Email and Password fields. Position and element type +information alone is unreliable for matching, since the text fields +may have different sizes or appear at different locations on each +screen. Appearance information alone is also noisy for matching, +since the text fields may have different visual themes. State-of-the- +art text encoders, even those trained on phrases, are unreliable +(e.g., most text models would produce a higher similarity score for +Username and Password than Email). +To detect UI element correspondences between different UI +screens, we built a system that (i) automatically detects UI elements +and text from screenshots, (ii) generates multi-modal embeddings +for each element, and (iii) establishes mappings between individ- +ual UI elements with high similarity. Figure 2 shows a high-level +overview of our approach. +3.1 +UI Element Detection +The first stage of our system identifies semantically relevant pieces +of information from a UI screenshot, such as UI elements and text. +The input is a bitmap and the output is a list of detected UI elements +and pieces of text. +We use a pre-trained object detection model from previous work +[60] that was trained to recognize UI elements in iOS app screens. +The pre-trained model uses the Faster-RCNN architecture [45] and +was trained on the AMP dataset [65], which is separate from the +main dataset used in this paper. It achieved a class-weighted mAP +score of 0.8. We use post-processing procedures, such as score- +based thresholding and inter-class non-max suppression (NMS), to +improve the quality of the output. Optical character recognition +(OCR) is performed using Tesseract [53], an open-source, off-the- +shelf OCR software package. We run OCR on regions of the screen +that correspond to text elements as detected by our element detec- +tor. +3.2 +UI Element Encoder +Using the elements segmented from the screenshot, we generate +representations that encode properties useful for comparison. In +our work, we consider relative positioning, element/icon category, +visual appearance, and text, properties which we hypothesize to be +relevant to element semantics. We used pre-trained models to pre- +dict these properties (element detection [60] and icon type (“com- +mon icon classifier" from previous work [7])) from screenshots. +Note that the pre-trained models were trained on different datasets +(i.e., no sample overlap) than the ones used in our paper. +3.2.1 +Modality Representations. We use off-the-shelf models to +generate modality-specific features for each element, then feed +their output into a screen transformer model, which combines and +learns further associations between them. + +, , +Wu et al. +UI Element Detection +e1 +Screen Transformer +e2 +e3 +e4 +en +UI Element Detection +e1 +Screen Transformer +e2 +e3 +e4 +en +Edges denote embedding similarity +Edges w/ high similarity form correspondences +Correspondence Matching +Figure 2: Overview of our screen correspondence approach. Elements and text from two screenshots are first extracted using +UI element detection then featurized using a screen transformer model. Finally, a correspondence between UI elements are +generated from element pairs with highly similar embeddings relative to other candidates. +Screen Transformer +Element Label Embedding +Appearance Embedding +Text Embedding +Pos Embedding +e1 +Element 1 +Visual +Element 2 +Visual +Element 3 +Visual +Element 1 +Label +Element 2 +Label +Element 3 +Label +el1 +Element 1 +Text +Element 3 +Text +el2 +el3 +ea1 +ea2 +ea3 +et1 +et3 +e2 +e3 +Modality Pooling +Contextual Embedding +One embedding +per UI element +One embedding +per modality +Figure 3: Architecture diagram of our screen transformer model. Each modality-specific input is treated as separate inputs +to our transformer model, which implicitly aligns them based on their positional information. Note that elements may be +missing modalities (Element 2 in this example). After the per-modality inputs are processed by our transformer, we generate +element embeddings (i.e., one per element) by pooling together outputs corresponding to the same original element. +Positional information: Previous work [14, 18] encoded element +position as a simple concatenation of bounding box coordinates. We +hypothesized that relative position may be more effective, since UI +interactions such as scrolling, text flow, and dynamic content could +cause changes in absolute position but have less effect on relative +ordering. We adopted a relative positional encoding scheme used to +improve the performance of language models [49] that incorporates +pairwise distance when calculating the attention score between +two elements. +Element Category: We categorized elements based on their UI and +icon type. Our pre-trained element detector classifies elements into +12 categories, as defined by previous work [65]. Three of these can +be further delineated into sub-categories. We separate the Toggle +and Checkbox classes based on their selection state (e.g., Toggle on +and Toggle off). We also classified common icon types using a sep- +arate pre-trained CNN model [65]. In total, we consider 83 unique +categories of elements and represent them as one-hot vectors. +Visual Appearance: We featurized regions using the intermediate +representations of a proposal-based object detector. Similar ap- +proaches have also been used by visual question-answering models, +which also need to take into account multiple visual information [? +] Since our UI element detector is based off of a similar proposal- +based architecture, we retrieve the activations of the object pro- +posals corresponding to detected elements using the fc6 layer [52]. + +22:29 +News +January7 +Top Stories +Apple Al Maryah lsland +opens Friday in the heart of +Abu Dhabi +The new store creates a direct connection from +The Galleria Al Maryah lsland to the water's +edge, delivering the best of Apple with shoreline +views. +More coverage > +Today +News+ +Audio +Following +Search22:29 +'News +April 6 +Top Stories +Apple's Worldwide +Developers Conference +returns in its all online +format +Apple today announced it will host its annual +Worldwide Developers Conference WWDC) in an +developers to attend +More coverage > +Today +News+ +Audio +Following +SearchScreen Correspondence: Mapping Interchangeable Elements between UIs +, , +This approach to featurizing appearance is beneficial since it results +in a fixed-size representation for image regions without the need +to explicitly resize or crop them. +Text: Numerous embedding methods have been developed for +representing words, sentences, and documents. Sentence transform- +ers are transformer-based models for encoding variable-length text +into an embedding space representative of semantic meaning [44]. +Since much of the text on UI screens is relatively short, we use a +variant specifically trained to encode phrases [58]. +3.2.2 +Transformer Model. To further enrich and learn associa- +tions between the modality-specific element representations, we +designed a model that generates a fixed-size embedding for each +detected UI element. Our model is based on the transformer architec- +ture (Figure 3), which has been used for UI representation learning +[18, 23]. The modifications we described (e.g., relative positioning +and appearance features) are aimed at improving performance on +the correspondence task. +Because not all elements have the same attributes, e.g., not all UI +elements have text, we rely on the transformer’s attention mecha- +nism to implicitly align information. Each modality-specific repre- +sentation with the exception of position is first embedded with a +separate linear layer to a common size. Instead of creating one input +vector for each element by concatenating features from each modal- +ity, we create an input vector for each modality of each element. For +example, a login button could result in three input vectors for ele- +ment category, visual appearance, and text. All inputs are fed into a +series of stacked self-attention blocks, which results in one output +embedding for each original input vector. Finally, we use pooling +to recover the output embeddings associated with each original UI +element and compute their mean to incorporate information from +all of the modalities. +3.2.3 +Unsupervised Training. We did not have access to labeled +data during the development of our model, so we used unsupervised +training to learn its parameters. Masked element prediction is a +training objective that requires the model to reconstruct an input +that has been corrupted by randomized masking (i.e., replacing a +portion of the input with 0’s). Previous work [54] has shown that +this training objective encourages the model to learn semantically +relevant representations since it must learn to associate masked +information with other sources of information. +The reconstruction loss was measured separately for all modal- +ities (element category, visual appearance, and text) then added +together to obtain the model’s total loss. L2-loss was used for re- +construction of visual and text features, and cross-entropy loss was +used for reconstruction of element category. +3.3 +Correspondence Matching +After we used our screen transformer model to featurize UI ele- +ments on two screens, we perform a matching procedure to predict +correspondences between them. For a pair consisting of a source +screen with 𝑀 elements and a target screen with 𝑁, we construct a +𝑀 × 𝑁 cost matrix 𝐶 ∈ R𝑀𝑥𝑁 to represent correspondence scores. +The matching cost 𝐶𝑖,𝑗 is computed using cosine similarity. +Several approaches have been used to generate correspondences +from cost matrices [47]. A simple approach of matching based solely +on highest cosine similarity may make suboptimal decisions when +one element has more than one likely match. In our final implemen- +tation, we formulated correspondence mapping as an optimization +problem that finds the assignment between two sets that results in +the lowest overall cost [29]. We employ this approach for match- +ing elements between screens, since elements are more likely to +be dissimilar. To reduce false positives, we employ additional pre- +processing and post-processing steps. Before running the best-cost +optimization, we prune unlikely matches from the cost-matrix so +that each element only considers its 𝑘 closest neighbors. After- +wards, we ignore matches where 𝐶𝑖,𝑗 < 𝑐. We tuned the values of +𝑘 = 5,𝑐 = 0.4 based on manual examination of a small number of +examples. This approach is similar to approaches in text decoding +models that consider only the top 𝑘 most likely tokens, which have +been shown to generate higher quality output by reducing the effect +from low-probability outputs. +4 +DATASET +We developed and trained our transfer model on two datasets of +app screens that were generated by manual crawling of popular +mobile apps: Crawls and Rico. The Crawls dataset, which was +used by prior UI modeling work [7], consists of 750,000 iOS app +screens from 6,000 apps and was collected by crowdworkers who +were instructed to manually explore mobile applications through a +remote interface that periodically captured screenshots and addi- +tional metadata of the current app screen. The Rico dataset [9] is +a publicly available dataset of 72,000 Android screens from 9,700 +apps that was also collected by crowdworkers remotely operating +devices. We divided each dataset into training (70%), validation +(15%), and testing (15%) splits by their crawl ID, which corresponds +to which app was crawled. +4.1 +Evaluation Dataset +While our training algorithm does not depend on labeled data +(i.e., unsupervised), we manually collected a small set of labeled +examples (900 pairs across 90 screens) from each dataset to evaluate +our system. +4.1.1 +Data Collection. Our evaluation dataset consists of data from +9 types of screens that we hypothesized could have correspon- +dences: Media Player, In-App Purchases, Login, Permission Request, +Register, Pre-Login, Pop-up, Search, and Web Views. We initially +asked crowdworkers to categorize a set of screenshots outside of +the training split based on a criteria for each category. Unlike app +categories, which might be used to categorize apps (e.g., finance, +health, social media), we focused on screen categories, since both +a health app and a banking app might both contain a login screen +that could contain correspondences. For each of our two datasets, +we sampled a small number of screens from each category for cor- +respondence labeling (9 categories x 10 screens = 90 screens total). +The 9 categories that we chose do not cover all possibilities, but we +believe they constitute a reasonable subset. More detailed descrip- +tions and criteria of each category is available in the appendix of +this paper. +4.1.2 +Data Labeling. We created a labeling interface to annotate +our small evaluation split. First, a randomly selected element was + +, , +Wu et al. +shown on a screen, and the interface displayed a prompt asking if +the element was likely to appear on other screens of the same type: +“Are elements of similar functionality likely to appear on other Login +screens?” If the user responded “Yes,” the application displayed a +prompt for a label: “What is the role of this element in the current +screen?” We built our interface to include auto-complete function- +ality to encourage labelers to identify correspondence categories +that could generalize across screens, e.g., “login button” instead +of “button to log into my credit card account.” The autocomplete +list was pre-populated with 5 choices for each category and was +auto-updated with novel descriptions. If a label was provided on +the first step, then the user was shown other screens from the same +category and asked to select elements with a similar role, if they +were present on the screens. +A drawback of this approach is that it is slow, since it requires +providing a role description before elements are matched. However, +we found the additional consideration of element role is useful for +reasoning about correspondences. +5 +EVALUATION +We evaluate our model against several baselines and ablated ver- +sions of our model. Our results show that compared to heuristic +and traditional key-point methods used in CV, multi-modal trans- +former encodings lead to better correspondences. Furthermore, our +ablation experiments show that the architectural improvements we +made lead to modest performance gains. +5.1 +Baselines +In this section, we describe the baselines used in our performance +evaluation. We focus primarily on other unsupervised approaches, +since our constraint was that we didn’t have any labeled data avail- +able for training. Similar supervised approaches exist [30], but +they depend on element-level annotations and access to underlying +source code (i.e., HTML). +For comparison, we chose a variety of baselines that include +keypoint-based methods used for image matching and heuristics +such as schema-matching. Our main constraint was that we did +not have large quantities of labeled data for supervised machine +learning methods, so we selected unsupervised techniques for com- +parison. +ORB: As a review, image correspondence relies on the compu- +tation of semantic features from regions of the image. Semantic +features, usually invariant to surface-level changes such as trans- +lation and scale, are first calculated for small, localized regions of +the image. When this process is repeated recursively, the receptive +field increases, and globally-aware features can be learned. ORB +[46] is a traditional CV approach to generating descriptor features. +We first computed ORB descriptors for each screenshot which re- +sulted in numerous keypoints at salient points of the image. Using +brute-force matching, keypoints from one image were matched +onto keypoints from another image based on descriptor similarity. +Finally, to translate keypoint similarity to UI element similarity, we +used an object detector to compute the boundaries of UI elements +and matched elements based on the number of matching keypoints +contained within them. +Neural Best Buddies: Neural Best Buddies (NBB) uses the internal +representations of a deep CNN to featurize and match image regions. +Like ORB, it also generates keypoint descriptors but uses activations +from a convolutional neural network (CNN). One advantage that +CNN features have over traditional methods is that learned features +can better correspond to semantic properties that the network was +trained on (e.g., image classification). To run our experiments, we +used the code released by the authors of the paper 1. The original +paper focuses on finding correspondences between “natural images” +and use a VGG-19 model [51] that was pretrained on ImageNet [10]. +Since UI screenshots have different properties than images found +in ImageNet, we initially tried to train a CNN model better repre- +sentative of UIs using an unsupervised autoencoder objective due +to the lack of labels in our training set. However, we found the +autoencoder model did not produce good outputs, so we report +results using the pre-trained ImageNet model. +Schema-matching Heuristic: One drawback of keypoint-based +methods that we explored is that keypoints are generated using +the entire image as input and without knowledge of UI element +locations. Schema-matching is an approach that first considers each +predicted element as a discrete object, then uses its attributes (i.e., +schema) to compare similarity to other candidates. We implemented +a heuristic that uses schema matching through incorporating the +predicted UI element/icon type by concatenating their one-hot class +predictions into a single vector and applying the same best-cost +matching algorithm [29]. More sophisticated schema-matching may +incorporate additional information, such as UI hierarchy (e.g., an +element that belongs in a list should be matched to another element +in a list). While possible to predict [60], we did not incorporate +hierarchical information since it requires complex techniques for +tree matching but expect it would perform similarly to [30], which +uses hierarchical information. +Screen Transformer Ablations: Our performance evaluation in- +cludes ablated variations of our main transformer model. Trans- +former models allow learning more sophisticated representations of +elements through data, which provides advantages over manually- +defined schemas. We evaluate several ablated versions of our model +to understand the performance impact of our architectural changes. +Specifically, the ablated versions of our transformers removes cer- +tain components that we hypothesized to improve correspondence +matching, such as relative positional embedding, visual appearance +information, and text. In addition, we evaluated the Pixel-words +transformer [18], which our model is based on, but we adjust the +number of element classes, layers, attention heads, and hidden di- +mensions to be the same as our other models. The Pixel-words +transformer also includes a “layout embedding" network which +featurizes the layout of UI using a semantic map which is fed into +an autoencoder. To summarize, the Pixel-words configuration (i) +considers categorical and text information, (ii) uses absolute posi- +tional encodings, and (iii) includes an additional layout embedding +component. +5.2 +Results +5.2.1 +Baseline Comparison. Our evaluation results (Table 1) shows +the benefit of our multi-modal model over simpler baselines. We +1https://github.com/kfiraberman/neural_best_buddies + +Screen Correspondence: Mapping Interchangeable Elements between UIs +, , +Table 1: Performance of our approach and other baselines +for screen correspondence. Our approach leads to the best +performance, reaching an F1 score 0.61. We also included ab- +lated versions of our model without relative positional em- +beddings, appearance features, and text features. +CRAWLS +RICO +Model Configuration +P +R +F1 +P +R +F1 +Screen Trans. +0.66 +0.57 +0.61 +0.83 +0.41 +0.55 +Screen Tran. (w/o Relative) +0.58 +0.53 +0.56 +0.74 +0.37 +0.49 +Screen Trans. (w/o Appearance) +0.74 +0.44 +0.55 +0.77 +0.38 +0.51 +Screen Trans. (w/o Text) +0.66 +0.63 +0.59 +0.77 +0.37 +0.50 +Screen Trans (Pixel-words) +0.70 +0.49 +0.58 +0.83 +0.22 +0.35 +Heuristic +0.48 +0.59 +0.53 +0.80 +0.32 +0.45 +ORB +0.25 +0.17 +0.20 +0.63 +0.21 +0.31 +NBB +0.22 +0.15 +0.18 +0.58 +0.17 +0.26 +Figure 4: Performance across different categories in the +Crawls (Top) and Rico (bottom) datasets using the full +Screen Transformer model configuration. The average clas- +sification performance was F1=0.61 on Crawls and F1=0.55 +on Rico. +employ standard classification metrics to measure the accuracy +of element-to-element correspondences generated by our model. +Since elements in our evaluation dataset are labeled using their +ground truth bounding boxes instead of our element detector’s +predictions, we first match predicted detections to ground truth +elements using the best Intersection-over-Union (IoU) score. Due +to our labeling procedure where one element is highlighted at a +time, the examples in our evaluation set were only partially labeled, +meaning that screens contained only a randomly sampled subset +of all possible corresponding pairs. Our best model configuration +reaches an F1 score of 0.61. Screens in our dataset contained an +average of around 20 elements, so correct correspondence required +finding the best out of match out of all possible candidates. The +ORB and NBB baselines are based on keypoint-based matching, +which is commonly used in CV to compare images. Among them, +ORB performs the best, achieving higher correspondence accuracy +but performed poorly due to low recall. One possible reason is +that keypoints are generated at visually salient locations of the +image, such as edges and corners, and without any knowledge of +where UI elements are. Thus, some UI elements may not contain +many keypoints within them, reducing the quality of matches. The +schema-matching heuristic performed substantially better than +keypoint-based methods and reached high recall by directly using +the outputs of pre-existing models (i.e., element detection, icon +classification). Precision was lower, possibly due to the difficulty of +accurately matching ambiguous elements without knowledge of +additional context. +Our ablation experiments revealed that our modifications to the +base transformer architecture led to modest improvements in terms +of F1 score but also had other consequences for precision and recall. +For example, our model trained without appearance information +was the lowest performing variation but reached the highest preci- +sion score. We attribute these variations to the information encoded +in each modality and may warrant different configurations based +on intended use-case. +5.2.2 +Performance across UI Categories. Figure 4 provides a more +in-depth breakdown correspondence by UI category. Our model +achieved the best performance on the Website View and In-App +Purchase categories and the worst performance on the Media Player +and Pre-Login categories. +One major source of error for our model was the presence of +sub-categories within our dataset. For example, we manually ex- +amined examples from the Pre-Login and Login categories, which +received relatively low performance. We discovered a consider- +able difference between apps that used different authentication +providers, such as OAuth and Single-Sign-On (SSO). For example, +a “traditional” login screen might include text fields for entering a +username and password, but an app using a SSO provide (e.g., Sign +in with Apple) might only contain a button without any text fields. +We found that there was also variance within media player screens +– video players and music players had significant differences and +some media players were full screen while others were not. Since +our correspondence model uses contextual information (i.e., in- +formation from other elements on the same screen) and relative +positional encoding, this could significantly affect the computed +representation. One strategy to address this is the formation of +sub-categories with a more consistent set of elements e.g., creating +separate categories for traditional login screens and those with +other types of authentication. +5.2.3 +Performance across Datasets. We evaluated all models and +baselines on both the Crawls and Rico dataset. Overall perfor- +mance between the two datasets were similar, although the Rico +models performed slightly worse (F1=0.55) than ones trained on +Crawls (F1=0.61). One possible reason for the performance dis- +crepancy is that Crawls is an order of magnitude larger and the +model was exposed to more variation during training time, which +is beneficial for unsupervised training techniques. While the full +transformer model is the best-performing configuration for both + +Performance by Category (Crawls) +0.75 +0.5 +0.25 +Media +In-App +Login Permission Register Pre-Login Pop-up +Search +Website +Player +Purchase +Request +View +Precision +Recall +F1 +Performance by Category (Rico) +0.75 +0.5 +0.25 +0 +Media +In-App +Login Permission Register Pre-Login Pop-up +Search +Website +Player +Purchase +Request +View +Precision +Recall +F1, , +Wu et al. +Table 2: Performance of our approach and other baselines +screen correspondence for same-screen pairs in the Crawls +dataset. Many configurations, including our model, reach a +maximum F1 score of 0.76. We attribute labeling noise and +the IoU element matching process used to assign predicted +element locations to ground-truth boxes. +Model Configuration +P +R +F1 +Screen Transformer +0.85 +0.68 +0.76 +Screen Transformer (w/o Relative) +0.86 +0.68 +0.76 +Screen Transformer (w/o Appearance) +0.87 +0.66 +0.75 +Screen Transformer (w/o Text) +0.85 +0.68 +0.76 +Screen Transformer (Pixel-words) +0.88 +0.66 +0.76 +Heuristic +0.87 +0.66 +0.75 +ORB +0.78 +0.48 +0.59 +NBB +0.53 +0.25 +0.34 +datasets, the relative performance ablated models were affected dif- +ferently. Notably, the Rico models without text and the Pixel-Words +model performed much worse, suggesting that its evaluation set +may have contained more text-heavy screens. +5.2.4 +Correspondence between Same-screen Pairs. In addition to +evaluating our models on screens from different related apps (i.e., +intra-class pairs), we also investigated performance on same-screen +pairs. Same-screen correspondence is useful for identifying the +same UI element across multiple versions of the same screen. For +example, an app’s appearance may change following an update or +from dynamically updated content (e.g., a news page loads content +from a remote source). Following prior work [14], we consider two +screenshots to be the “same” if they represent different instances +of the same underlying implementation, possibly with significantly +different appearance. Correspondence mapping can help guide au- +tomated systems such as crawlers to behave more consistently +in these situations. We randomly selected screen groups with the +same app ID as those in the testing split of our Crawls dataset, then +randomly sampled two screenshots from each group, resulting in +888 total pairs. Upon manual inspection, we found that some of +the sampled pairs had only minimal visual changes. To filter out +“easy pairs,” we constructed a heuristic that attempted to match +elements based only on bounding box location. If all elements in a +pair were successfully matched, we discarded the example, since it +meant that no significant dynamic change (e.g., scrolling, dynamic +content) occurred. After this process, the final dataset contains 607 +examples. We did not repeat this for the Rico dataset because the +authors applied a heuristic to filter out repeated views of the same +screen [9]. +Our observations and performance results (Table 2) show that +same-screen correspondence is generally higher. Since same-screen +pairs are usually more visually similar, the model can rely more +heavily on surface-level features and in many cases perform direct +matching, such as looking for recurring text. Many configurations, +including our model, reached a maximum F1 score of 0.76. Er- +rors from labeling noise and IoU element matching (e.g., matching +ground truth bounding boxes to predictions) may have established +an effective ceiling, since our element detection model introduced +errors (has a class-weighted mAP score of 0.8). +6 +EXAMPLE APPLICATIONS +We describe three example applications that show the utility of +screen correspondence to human and machine understanding of UI +functionality. Generating and transferring a type of instructional +overlay called coach marks can help users navigate unfamiliar UIs +by mapping them to previously encountered ones of the same class. +UI search is useful for app designers to find how concepts are +expressed across apps (e.g., what are different ways of expressing a +search intent?). Finally, automated GUI testing can be made more +robust by accounting for variations in visual presentation between +different app versions without requiring platform-specific APIs or +metadata. These example applications are not meant to be novel, but +we believe they show that accurate screen correspondence allows +many existing systems to work under a wider range of conditions, +e.g., using pixel data alone or improved robustness to dynamic +visual changes. +6.1 +Instructional Overlays +We used our model to improve users’ understanding of complex or +newly installed apps by creating an infrastructure that could be used +to crowdsource coach marks for apps. Coach marks are instructional +overlays that are sometimes shown to provide assistance to users +when an app is first launched, and can be helpful for exposing UI +functionality. While it is possible to automatically generate natural +language for describing screens [57] and widgets [37] using deep +models, they are often affected by surface-level appearance and may +be prone to producing generic outputs [37]. Building a model that +produces natural language also introduces significant complexity +that can be similar achieved with a correspondence mapping. A +better approach might be to crowdsource users [43] or developers +to write coach marks for screens in a subset of apps, and then +apply our screen correspondence technology to map these coach +marks onto a much larger set of screens with similar purposes. This +idea builds on the template-based matching scheme of Yeh et al. +[62] for generating contextual help, and expands their idea beyond +same-screen applications to also intra-class usage. +We applied our model’s intra-class correspondence capabilities +to automatically transfer annotations from one screen to another +related app of the same category (Figure 5). We first populated a +small database of instructional text for elements from app screens +in one of the categories from our evaluation data. In a real imple- +mentation of this system, an interface would be created to allow +users to author new instructional text for screenshots that they up- +load. Each screen in the database was associated with its featurized +elements as a key, and each instruction in the database was associ- +ated with its element. Our current prototype is a proof-of-concept +implementation where the user can upload a screenshot image file +through a web interface. On the uploaded screen, we perform a +nearest-neighbor search to retrieve the screens in our database that +are most similar. If the distance is sufficiently close, we run our +screen correspondence matching, which also returns a “matching +cost.” If enough matches are discovered and the matching cost is + +Screen Correspondence: Mapping Interchangeable Elements between UIs +, , +Exemplar Labeling +Generated Help Overlay +Figure 5: Coach marks are useful for uncovering function- +ality in apps. High-quality natural language descriptions of +UI components can be difficult to generate, so we curated a +small number of labeled examples from different app cate- +gories. Element descriptions from this labeled set are trans- +ferred onto unseen app screens of the same type using the +correspondence mapping. Depending on the use-case, the +exemplar can be manually provided (e.g., developer wishes +to label many similar screens at once) or automatically re- +trieved (e.g., a help-generation app uses a separate classifier +to find an exemplar from a database of labeled screens.) +below a heuristically set threshold, we directly render the annota- +tions to the screenshot using image drawing APIs and display the +annotated image. +In a complete implementation, the matching and rendering algo- +rithms would be built into a mobile operating system and run on +the user’s mobile device so that it would not require the user to exit +their current app to use our tool. In the future, we plan to improve +the user experience and investigate ways that these overlays could +be surfaced contextually. +In this example, we show that accurate intra-class screen corre- +spondence can facilitate transferring coach marks, which can help +users discover new app functionality and documentation. Other pos- +sible applications of screen correspondence to improving end-user +usage include transferring more types of accessiblity meta-data, +example-based re-targeting of UIs [30] and using input redirection +techniques to improve the accessibility of UI components [66]. +6.2 +UI Element Search Engine +UI search can help app designers find how concepts are expressed +across apps and provide example starting points when designing +a new app. Previous work indexed databases of UI screens using +visual properties [3], structural properties [60], sketches [26]. We +focus specifically on returning relevant UI elements instead of +screens, and leverage our model’s intraclass matching abilities to +improve the search process. +We integrated our screen correspondence model into a UI search +engine to support tag-based search and exemplar-based refinement. +The implementation of our UI search engine is a web app that +indexed elements from 130,000 UI screens using a variety of meta- +data, including detected element classes, icon types, and text, which +are stored in a database. Our app features a search page, where +users can first perform an initial search by entering text or tags +in a search bar. Results are returned based on matching attributes +found in the property database. Matching elements are shown in +the context of their app screen and highlighted with a bounding box. +When a result is selected, users are brought to the element inspector +page, where users can examine the properties of all elements on +the screen. +One limitation of tag-based search is that it is difficult to specify +target properties that do not belong to the pre-defined set of tags. +For example, a “plus” icon displayed on the top or bottom of a list +may indicate adding to the list while a “plus” icon displayed next +to a list item is more likely to representing adding the item from +the list. It would be difficult to disambiguate between these cases as +they share the same tag. Thus, we used our correspondence model +to enable exemplar-based search refinement, which allows users to +“narrow in” on more specific results. To enable this functionality, +we computed embeddings for UI elements in our database and +stored this information into a vector data store which supports +fast approximate nearest-neighbor search. We added a “search for +similar items” button on the element inspector page, which finds +results with a high similarity to the target element according to the +cosine similarity metric. Figure 6 shows an example flow of our UI +element search engine. +6.3 +Automated GUI Testing +Finally, we used our model to improve the robustness of automated +GUI tests using our model’s same-screen matching capability. Auto- +mated testing is useful for ensuring the quality of GUIs. Specifically, +visual-based methods can be employed in these systems to search +for targets based on their rendered appearance, which allows for +easier authoring of testing scripts and reduces the dependency of +testing frameworks on specific UI toolkits [5]. However, strong +reliance on visual similarity may lead to failures caused by change +in visual style, such as updated application theme or icons [5]. +In such applications, the quality of UI element matching is impor- +tant for automated GUI testing because poor matching capability + +Not Secure - )9-0620-25-srv.mr3.simcloud.apple.com ++ +UI Helper +Upload Exemplar Label Screen +Screen +Descriptions +Consider the elements on the right. Provide a short +5:13 +description for each. +4 +The verification code is usually sent to your email or text m +Verify code +Element 1 +Element 2 +Complete the login +Cancel +5 +Enter code herel +Berify巴 +Not Secure - )9-0620-25-srv.mr3.simcloud.apple.com ++ +UI Helper +Upload Exemplar Label Screen +Generated +Correspondences +5:13 +3:58 +K +Verity code +WOODWILD +Enter your code +EVERGREEN +WOODWILD +- The verification code is usually +sentte yoyr email or text messages. +Verify +- Enter the code here +cGomplete the login +I didn't get a code +By continuing, 1 agree to reoeive emails from Evergreen. I understand that 1 am +free to withdraw consent at any time., , +Wu et al. +Tag-based Search +Element Inspector +Refine by Exemplar +Figure 6: An example usage flow of our UI element search engine. The user first searches for icon elements that contain the +“add” tag. The results page shows UI screens with a matching element highlighted (Left). The user selects a result screen +where an add button is placed on the top right of the screen. The inspection page provides details about element info and +allows searching for similar elements (Center). Another search query is run using the embedded element of interest. The new +results are similar to the query in that they are all located at the top right of the screen and they appear to be used for adding +items to a gallery (Right). This example shows how designers can start a search using natural language or tag-based queries +then refine the results based on exemplars. +Recorded Action +Replayed Action +Figure 7: Automated UI testing techniques execute an inter- +action trace (either manually pre-defined or automatically +generated) to detect functional regressions, visual regres- +sions, and other unexpected behavior. Updated versions of +apps may lead to small changes in layout and visual appear- +ance and knowledge of same-screen correspondence can im- +prove the consistency and robustness of tests. This exam- +ple shows a automated application performing a previously +recorded action, despite the target’s appearance change. +can lead to a failure to replicate recorded interaction traces in a +scripted testing scenario, and repeated visits to the same screens in +a random crawler stress test example. As shown by previous work +on screen similarity [14], methods that rely heavily on surface-level +appearance may have high precision but low recall due to possi- +ble variations between UIs. We applied our screen correspondence +model to improve the robustness of these matches. +We built a prototype system that interacts with remotely con- +nected smartphone devices through a VNC interface. Our software +sends commands through this interface to simulate interactions, +such as clicking and swiping. We also include a “recording” mode +that allows a tester to record an interaction trace, during which all +of the screenshots and interactions are saved. When replaying the +interaction trace, the saved screenshots and interacted elements are +used to match the current state of the VNC output. Specifically, for +each step in the saved trace, we identify the UI element with which +the tester interacted, such as the button that was pressed. Then, on +the live VNC view, we find the corresponding element and apply +the recorded interaction to it, similar to previous work on tutorial +consumption [68]. Figure 7 illustrates how our automated tester nav- +igates an app where the appearance of a target element has changed. +Used in conjunction with traditional template-matching techniques, +which offer high precision but low recall, correspondence matching +can help improve the overall performance of automated testers. +7 +LIMITATIONS & FUTURE WORK +Our evaluation shows that correspondences can be automatically +identified through machine learning and matching approaches. +Some types of screens are more likely to have correspondences de- +tectable by our system (e.g., Website Views and In-App purchases) +than others (e.g., media players). The required accuracy level de- +pends largely on the final application, since different use-case since +different performance attributes. For example, using correspon- +dences to generate contextual help (instructional overlays) may +result in a better experience if only very confident matches are used, +as incorrect instructions can lead to confusion and frustration from +the user. GUI testing and crawling is less tolerant to mistakes, since +an incorrect action can make it impossible to access the rest of an +application. On the other hand, UI design search is more forgiving, +since it can provide value if most of the returned elements are cor- +rect (does not need to be the top choice). Our current evaluation +does not account for the requirements of down-stream applications, +although based on the example applications we implemented, we +found them to provide acceptable performance. We plan to further +evaluate our system in down-stream applications. + +0 +十 +< +UI Search Engine +Search Results +Search +add +O All +xel O +Text Field +o Icon + Segmented Control +Search +314 +315 +22:01 +22:17 +22:14 +田 +Favorites +田 +< Shortcuts +Select +田 +Projects +All Shortcuts +V Default Room +Kitchen +[Q bearch +Cancel +二楼 +8 +卧室 +Find Photos +New Shortcut 1 +洗衣房 +1actior +Home Settings +大 +Room Settings +Accessibility＋ +< +UI Search Engine +Screen 9359 +Q Search Similar Screens +@ Search Similar Elements +> array [14] +>0 [3] +>1 [3] +Projects +口 ++ +>2 [3] +3 [3] +口 +4 [3] +> data [6] +id : 184000 +screen_id : 9359 +>5 [3] +6 [3] +>7 [3] +8 [3] +Prolec +Projeci +9 [3] +JUn, 15 2021 2:30 AN +J00.15 20212-28 AN +10 [3] +>11 [3] +>12 [3] +dataPointType : annotation0 +< +UI Search Engine +Similar Screen Search Results +22:01 +三3 +4:55 +22:017 +田 +Favorites +TestName +? +Edit +田 +Projects +口田 +Activity +Alarm +SleepIWakeUp +NoAlarm +SET UP +Other +00:30 +taned +NoFavorites +09:19 +Alarm +10:03 +Alarn +16:07 +16:09 +Q +1710_736635_1623724210395.json +4513_1629564_1624926702762.json +1530_682461_1623646544824.json +6747_2295141_1625682246038.json +α +a +α +α4:27 +Collections +What's New +9 tips +14 +Welcome to iPhone +Get to know your iPhone +8 tips +Essentials +Must know features you'll love +8 tips +Genius Picks +Favorites from our experts +13 tips +Photography +Take and perfect your best shot +12 tips5:48 +Collections +What'sNew +11 tips +15 +Welcome to iPhone +Get to know your iPhone +9 tips +Essentials +Must-know features you'll love +10 tips +Genius Picks +Favorites from our experts +9 tips +Accessibility +大 +Make iPhone work for you +10 tips5:48 +< Collections +Welcome to +iPhoneScreen Correspondence: Mapping Interchangeable Elements between UIs +, , +A limitation of our current experiments is that they focus only +on mobile UIs that belong to a set of 9 categories that we identified. +These 9 categories do not cover all possibilities of app screens, but +they cover a considerable subset. Our model is likely to perform +better for complex app screens if given a small amount of annota- +tions to fine-tune on. Moreover, since we only use pixel information +as input to our model, we believe that our approach is likely to +generalize well to other types of graphical UIs that also represent +their output as pixels. In the future, we aim to replicate our exper- +iments on other types of graphical UIs with varying screen sizes +and shapes. +We see several opportunities to improve the performance of our +system. Since our system relies on several individual components, +it may be useful to quantify the performance of each separately. +We used a pre-trained element detector model that produced noisy +output for the correspondence matching. Previous work [60] has +shown that element detectors perform poorly on more complex +screens due to the increased number of elements and sometimes +miss smaller elements. Future work could investigate a screen cor- +respondence system that uses a more accurate element detector +model or accepts manual annotations as input. More advanced +matching techniques can also be employed, such those that consider +multi-scale correspondence, which first process smaller sub-regions +before merging their predictions globally. Separately, prior work +on image correspondence [27] has shown improved performance +by scaling images during training and inference. A similar idea +could be applied to UIs by first predicting their UI hierarchy [60] +and generating mappings for groups of elements. Our model could +also use different unsupervised pre-training objectives to help it +build better representations of UI elements for our matching task +[28, 50]. +Our work focuses on mapping interchangeable elements with +similar functionality between UI screens, however there are other +relationships that can be modeled. Categorization of different rela- +tions in language analogies [19, 40] show that antonym, categorical, +and functional connections can enrich the expressiveness of lan- +guage and rhetoric. We plan to focus future modeling efforts on +identifying and inferring a wider range of similar relationships that +exist in UIs. +Finally, our work explores inferring UI functionality from a sin- +gle previously encountered example, yet we believe our approach +may extend to multiple examples [17]. For example, non-parametric +machine learning methods such as the k-nearest neighbors algo- +rithm often benefit from considering more than one example at a +time. +8 +CONCLUSION +In this paper, we explore screen correspondence as a machine learn- +ing technique for inferring UI functionality by directly leveraging +previously encountered examples. We describe our model architec- +ture and training procedure that incorporates information about UI +semantics, appearance, and text when generating correspondence +mappings between screenshots. In a comprehensive evaluation with +strong baselines, we show that our approach outperforms corre- +spondence algorithms by leveraging multiple information sources +found in UIs. Finally, we show how three example applications of +screen correspondence: (i) transferring coach marks from related +apps, (ii) UI element search, and (iii) automated GUI testing. Broadly, +our work demonstrates the feasibility of learning UI semantics by +mapping to prior examples. +A +MODEL HYPERPARAMETERS +Model +Hyperparameter +Value +Screen Transformer +optimizer +Adam +learning rate +1e-4 +weight decay +1e-5 +dropout +0.25 +hidden size +256 +num layers +4 +num heads +4 +We trained our models with early stopping and stopped training +when validation loss did not improve for 10 epochs. We imple- +mented our model using the PyTorch [42] and PyTorch Lightning +[13] frameworks. +B +UI CATEGORY CRITERIA +We collected a small dataset of 9 screen categories for evaluation +of our model’s intra-class correspondence capabilities. We used the +following guidelines to categorize apps. +• Media Player - A screen that allows users to play media +content such as music or video. Usually contains controls +for adjusting playback, volume, and sharing. +• In-App Purchase - A screen that asks users to make a purchase +for a subscription or to access some part of an app. Usually +contains buttons for making the purchase, dismissing the +screen, or signing up for a trial. +• Login - A screen that asks users to log into an app or service. +It may contain fields for entering username and password +or buttons for third party authentication services. +• Permission Request - A screen that asks users to enable some +permission, which are usually associated with security set- +tings such as location or camera access. +• Register - A screen that asks the user to create an account. +May contain a form to register or buttons for third party +authentication providers. +• Pre-Login - A screen that contains controls to access other +parts of the app either by logging in or registering for an +account. This usually comes before the login page. +• Pop-up - A screen with a pop-up or dialog model that is +displayed over other app content. Pop-ups may contain con- +trols for accepting or dismissing it. For pop-ups that ask for +permission or purchases, see other categories. +• Search - A screen for entering and submitting a search query. +May include a search bar and filtering controls. +• Website View - A screen where an app opens an external +website. May contain a URL bar and forward/backward con- +trols. + +, , +Wu et al. +REFERENCES +[1] Kfir Aberman, Jing Liao, Mingyi Shi, Dani Lischinski, Baoquan Chen, and Daniel +Cohen-Or. 2018. Neural best-buddies: Sparse cross-domain correspondence. ACM +Transactions on Graphics (TOG) 37, 4 (2018), 1–14. +[2] Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. 2014. Neural ma- +chine translation by jointly learning to align and translate. +arXiv preprint +arXiv:1409.0473 (2014). +[3] Sara Bunian, Kai Li, Chaima Jemmali, Casper Harteveld, Yun Fu, and Magy Seif +Seif El-Nasr. 2021. VINS: Visual Search for Mobile User Interface Design. In +Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems. +1–14. +[4] Joel Chan, Joseph Chee Chang, Tom Hope, Dafna Shahaf, and Aniket Kittur. 2018. +Solvent: A mixed initiative system for finding analogies between research papers. +Proceedings of the ACM on Human-Computer Interaction 2, CSCW (2018), 1–21. +[5] Tsung-Hsiang Chang, Tom Yeh, and Robert C Miller. 2010. GUI testing using +computer vision. In Proceedings of the SIGCHI Conference on Human Factors in +Computing Systems. 1535–1544. +[6] Jieshan Chen, Chunyang Chen, Zhenchang Xing, Xiwei Xu, Liming Zhu, Guo- +qiang Li, and Jinshui Wang. 2020. Unblind your apps: Predicting natural-language +labels for mobile gui components by deep learning. In Proceedings of the ACM/IEEE +42nd International Conference on Software Engineering. 322–334. +[7] Jieshan Chen, Amanda Swearngin, Jason Wu, Titus Barik, Jeffrey Nichols, and +Xiaoyi Zhang. 2022. Towards Complete Icon Labeling in Mobile Applications. In +Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems. +1–14. +[8] Sen Chen, Lingling Fan, Chunyang Chen, Ting Su, Wenhe Li, Yang Liu, and Lihua +Xu. 2019. Storydroid: Automated generation of storyboard for Android apps. +In 2019 IEEE/ACM 41st International Conference on Software Engineering (ICSE). +IEEE, 596–607. +[9] Biplab Deka, Zifeng Huang, Chad Franzen, Joshua Hibschman, Daniel Afergan, +Yang Li, Jeffrey Nichols, and Ranjitha Kumar. 2017. Rico: A mobile app dataset +for building data-driven design applications. In Proceedings of the 30th Annual +ACM Symposium on User Interface Software and Technology. 845–854. +[10] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. 2009. Imagenet: +A large-scale hierarchical image database. In 2009 IEEE conference on computer +vision and pattern recognition. Ieee, 248–255. +[11] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018. Bert: +Pre-training of deep bidirectional transformers for language understanding. arXiv +preprint arXiv:1810.04805 (2018). +[12] Chris Dyer, Victor Chahuneau, and Noah A Smith. 2013. A simple, fast, and +effective reparameterization of ibm model 2. In Proceedings of the 2013 Conference +of the North American Chapter of the Association for Computational Linguistics: +Human Language Technologies. 644–648. +[13] WA +Falcon +and +.al. +2019. +PyTorch +Lightning. +GitHub. +Note: +https://github.com/PyTorchLightning/pytorch-lightning 3 (2019). +[14] Shirin Feiz, Jason Wu, Xiaoyi Zhang, Amanda Swearngin, Titus Barik, and Jeffrey +Nichols. 2022. Understanding Screen Relationships from Screenshots of Smart- +phone Applications. In Proceedings of the 27th Annual Conference on Intelligent +User Interfaces. 1–12. +[15] Sidong Feng, Suyu Ma, Jinzhong Yu, Chunyang Chen, TingTing Zhou, and Yankun +Zhen. 2021. Auto-icon: An automated code generation tool for icon designs +assisting in ui development. In 26th International Conference on Intelligent User +Interfaces. 59–69. +[16] Martin A Fischler and Robert C Bolles. 1981. Random sample consensus: a +paradigm for model fitting with applications to image analysis and automated +cartography. Commun. ACM 24, 6 (1981), 381–395. +[17] Huazhu Fu, Xiaochun Cao, and Zhuowen Tu. 2013. Cluster-based co-saliency +detection. IEEE Transactions on Image Processing 22, 10 (2013), 3766–3778. +[18] Jingwen Fu, Xiaoyi Zhang, Yuwang Wang, Wenjun Zeng, Sam Yang, and Grayson +Hilliard. 2021. Understanding Mobile GUI: from Pixel-Words to Screen-Sentences. +arXiv preprint arXiv:2105.11941 (2021). +[19] Anna Gladkova, Aleksandr Drozd, and Satoshi Matsuoka. 2016. Analogy-based +detection of morphological and semantic relations with word embeddings: what +works and what doesn’t.. In Proceedings of the NAACL Student Research Workshop. +8–15. +[20] Kelvin Guu, Tatsunori B Hashimoto, Yonatan Oren, and Percy Liang. 2018. Gen- +erating sentences by editing prototypes. Transactions of the Association for +Computational Linguistics 6 (2018), 437–450. +[21] Richard Hartley and Andrew Zisserman. 2003. Multiple view geometry in computer +vision. Cambridge university press. +[22] Junxian He, Taylor Berg-Kirkpatrick, and Graham Neubig. 2020. Learning sparse +prototypes for text generation. Advances in Neural Information Processing Systems +33 (2020), 14724–14735. +[23] Zecheng He, Srinivas Sunkara, Xiaoxue Zang, Ying Xu, Lijuan Liu, Nevan Wich- +ers, Gabriel Schubiner, Ruby Lee, Jindong Chen, and Blaise Aguera y Arcas. +2020. ActionBert: Leveraging User Actions for Semantic Understanding of User +Interfaces. arXiv preprint arXiv:2012.12350 (2020). +[24] Aaron Hertzmann, Charles E Jacobs, Nuria Oliver, Brian Curless, and David H +Salesin. 2001. Image analogies. In Proceedings of the 28th annual conference on +Computer graphics and interactive techniques. 327–340. +[25] Berthold KP Horn and Brian G Schunck. 1981. Determining optical flow. Artificial +intelligence 17, 1-3 (1981), 185–203. +[26] Forrest Huang, John F Canny, and Jeffrey Nichols. 2019. Swire: Sketch-based user +interface retrieval. In Proceedings of the 2019 CHI Conference on Human Factors in +Computing Systems. 1–10. +[27] Wei Jiang, Eduard Trulls, Jan Hosang, Andrea Tagliasacchi, and Kwang Moo +Yi. 2021. Cotr: Correspondence transformer for matching across images. In +Proceedings of the IEEE/CVF International Conference on Computer Vision. 6207– +6217. +[28] Vid Kocijan, Ana-Maria Cretu, Oana-Maria Camburu, Yordan Yordanov, and +Thomas Lukasiewicz. 2019. A Surprisingly Robust Trick for the Winograd Schema +Challenge. In ACL. +[29] Harold W Kuhn. 1955. The Hungarian method for the assignment problem. Naval +research logistics quarterly 2, 1-2 (1955), 83–97. +[30] Ranjitha Kumar, Jerry O Talton, Salman Ahmad, and Scott R Klemmer. 2011. +Bricolage: example-based retargeting for web design. In Proceedings of the SIGCHI +Conference on Human Factors in Computing Systems. 2197–2206. +[31] Seokjun Lee, Rhan Ha, and Hojung Cha. 2018. Click sequence prediction in +Android mobile applications. IEEE Transactions on Human-Machine Systems 49, 3 +(2018), 278–289. +[32] Luis A Leiva, Asutosh Hota, and Antti Oulasvirta. 2020. Enrico: A dataset for +topic modeling of mobile ui designs. In 22nd International Conference on Human- +Computer Interaction with Mobile Devices and Services. 1–4. +[33] Hector Levesque, Ernest Davis, and Leora Morgenstern. 2012. The winograd +schema challenge. In Thirteenth international conference on the principles of knowl- +edge representation and reasoning. +[34] Toby Jia-Jun Li, Amos Azaria, and Brad A Myers. 2017. SUGILITE: creating +multimodal smartphone automation by demonstration. In Proceedings of the 2017 +CHI conference on human factors in computing systems. 6038–6049. +[35] Toby Jia-Jun Li, Lindsay Popowski, Tom Mitchell, and Brad A Myers. 2021. +Screen2Vec: Semantic Embedding of GUI Screens and GUI Components. In Pro- +ceedings of the 2021 CHI Conference on Human Factors in Computing Systems. +1–15. +[36] Yang Li, Jiacong He, Xin Zhou, Yuan Zhang, and Jason Baldridge. 2020. Mapping +natural language instructions to mobile UI action sequences. arXiv preprint +arXiv:2005.03776 (2020). +[37] Yang Li, Gang Li, Luheng He, Jingjie Zheng, Hong Li, and Zhiwei Guan. 2020. +Widget captioning: Generating natural language description for mobile user +interface elements. arXiv preprint arXiv:2010.04295 (2020). +[38] Ce Liu, Jenny Yuen, and Antonio Torralba. 2010. Sift flow: Dense correspondence +across scenes and its applications. IEEE transactions on pattern analysis and +machine intelligence 33, 5 (2010), 978–994. +[39] Thomas F Liu, Mark Craft, Jason Situ, Ersin Yumer, Radomir Mech, and Ranjitha +Kumar. 2018. Learning design semantics for mobile apps. In Proceedings of the +31st Annual ACM Symposium on User Interface Software and Technology. 569–579. +[40] Hongjing Lu, Ying Nian Wu, and Keith J Holyoak. 2019. Emergence of analogy +from relation learning. Proceedings of the National Academy of Sciences 116, 10 +(2019), 4176–4181. +[41] Forough Mehralian, Navid Salehnamadi, and Sam Malek. 2021. Data-driven +accessibility repair revisited: on the effectiveness of generating labels for icons in +Android apps. In Proceedings of the 29th ACM Joint Meeting on European Software +Engineering Conference and Symposium on the Foundations of Software Engineering. +107–118. +[42] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory +Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. 2019. +Pytorch: An imperative style, high-performance deep learning library. Advances +in neural information processing systems 32 (2019). +[43] Vidya Ramesh, Charlie Hsu, Maneesh Agrawala, and Björn Hartmann. 2011. +ShowMeHow: translating user interface instructions between applications. In +Proceedings of the 24th annual ACM symposium on User interface software and +technology. 127–134. +[44] Nils Reimers and Iryna Gurevych. 2019. Sentence-bert: Sentence embeddings +using siamese bert-networks. arXiv preprint arXiv:1908.10084 (2019). +[45] Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. 2015. Faster r-cnn: +Towards real-time object detection with region proposal networks. Advances in +neural information processing systems 28 (2015), 91–99. +[46] Ethan Rublee, Vincent Rabaud, Kurt Konolige, and Gary Bradski. 2011. ORB: An +efficient alternative to SIFT or SURF. In 2011 International conference on computer +vision. Ieee, 2564–2571. +[47] Masoud Jalili Sabet, Philipp Dufter, François Yvon, and Hinrich Schütze. 2020. +SimAlign: High quality word alignments without parallel training data using +static and contextualized embeddings. arXiv preprint arXiv:2004.08728 (2020). +[48] Abigail See, Peter J Liu, and Christopher D Manning. 2017. Get to the point: +Summarization with pointer-generator networks. arXiv preprint arXiv:1704.04368 +(2017). + +Screen Correspondence: Mapping Interchangeable Elements between UIs +, , +[49] Peter Shaw, Jakob Uszkoreit, and Ashish Vaswani. 2018. Self-attention with +relative position representations. arXiv preprint arXiv:1803.02155 (2018). +[50] Ming Shen, Pratyay Banerjee, and Chitta Baral. 2021. Unsupervised Pronoun +Resolution via Masked Noun-Phrase Prediction. arXiv preprint arXiv:2105.12392 +(2021). +[51] Karen Simonyan and Andrew Zisserman. 2014. Very deep convolutional networks +for large-scale image recognition. arXiv preprint arXiv:1409.1556 (2014). +[52] Amanpreet Singh, Vivek Natarajan, Meet Shah, Yu Jiang, Xinlei Chen, Dhruv +Batra, Devi Parikh, and Marcus Rohrbach. 2019. Towards vqa models that can +read. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern +Recognition. 8317–8326. +[53] Ray Smith. 2007. An overview of the Tesseract OCR engine. In Ninth international +conference on document analysis and recognition (ICDAR 2007), Vol. 2. IEEE, 629– +633. +[54] Hao Tan and Mohit Bansal. 2019. Lxmert: Learning cross-modality encoder +representations from transformers. arXiv preprint arXiv:1908.07490 (2019). +[55] Michael Tomasello. 2005. +Constructing a language: A usage-based theory of +language acquisition. Harvard university press. +[56] Oriol Vinyals, Charles Blundell, Timothy Lillicrap, Daan Wierstra, et al. 2016. +Matching networks for one shot learning. Advances in neural information pro- +cessing systems 29 (2016). +[57] Bryan Wang, Gang Li, Xin Zhou, Zhourong Chen, Tovi Grossman, and Yang +Li. 2021. Screen2words: Automatic mobile UI summarization with multimodal +learning. In The 34th Annual ACM Symposium on User Interface Software and +Technology. 498–510. +[58] Shufan Wang, Laure Thompson, and Mohit Iyyer. 2021. Phrase-bert: Improved +phrase embeddings from bert with an application to corpus exploration. arXiv +preprint arXiv:2109.06304 (2021). +[59] Jason Wu, Karan Ahuja, Richard Li, Victor Chen, and Jeffrey Bigham. 2019. +ScratchThat: Supporting Command-Agnostic Speech Repair in Voice-Driven +Assistants. Proceedings of the ACM on Interactive, Mobile, Wearable and Ubiquitous +Technologies 3, 2 (2019), 1–17. +[60] Jason Wu, Xiaoyi Zhang, Jeff Nichols, and Jeffrey P Bigham. 2021. Screen Parsing: +Towards Reverse Engineering of UI Models from Screenshots. In The 34th Annual +ACM Symposium on User Interface Software and Technology. 470–483. +[61] Rahulkrishna Yandrapally, Andrea Stocco, and Ali Mesbah. 2020. Near-duplicate +detection in web app model inference. In Proceedings of the ACM/IEEE 42nd +International Conference on Software Engineering. 186–197. +[62] Tom Yeh, Tsung-Hsiang Chang, Bo Xie, Greg Walsh, Ivan Watkins, Krist Wong- +suphasawat, Man Huang, Larry S Davis, and Benjamin B Bederson. 2011. Creating +contextual help for GUIs using screenshots. In Proceedings of the 24th annual +ACM symposium on User interface software and technology. 145–154. +[63] Ja Eun Yu and Debaleena Chattopadhyay. 2020. “Maps are hard for me”: Identify- +ing How Older Adults Struggle with Mobile Maps. In the 22nd international ACM +SIGACCESS conference on computers and accessibility. 1–8. +[64] Xiaoxue Zang, Ying Xu, and Jindong Chen. 2021. Multimodal Icon Annotation +For Mobile Applications. In Proceedings of the 23rd International Conference on +Mobile Human-Computer Interaction. 1–11. +[65] Xiaoyi Zhang, Lilian de Greef, Amanda Swearngin, Samuel White, Kyle Murray, +Lisa Yu, Qi Shan, Jeffrey Nichols, Jason Wu, Chris Fleizach, et al. 2021. Screen +Recognition: Creating Accessibility Metadata for Mobile Applications from Pixels. +In Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems. +1–15. +[66] Xiaoyi Zhang, Anne Spencer Ross, Anat Caspi, James Fogarty, and Jacob O +Wobbrock. 2017. Interaction proxies for runtime repair and enhancement of +mobile application accessibility. In Proceedings of the 2017 CHI conference on +human factors in computing systems. 6024–6037. +[67] Xiaoyi Zhang, Anne Spencer Ross, and James Fogarty. 2018. Robust annotation of +mobile application interfaces in methods for accessibility repair and enhancement. +In Proceedings of the 31st Annual ACM Symposium on User Interface Software and +Technology. 609–621. +[68] Mingyuan Zhong, Gang Li, Peggy Chi, and Yang Li. 2021. HelpViz: Automatic +Generation of Contextual Visual Mobile Tutorials from Text-Based Instructions. +In The 34th Annual ACM Symposium on User Interface Software and Technology. +1144–1153. +[69] Xin Zhou and Yang Li. 2021. Large-Scale Modeling of Mobile User Click Behaviors +Using Deep Learning. In Fifteenth ACM Conference on Recommender Systems. 473– +483. + diff --git a/ntE_T4oBgHgl3EQf7xzu/content/tmp_files/load_file.txt b/ntE_T4oBgHgl3EQf7xzu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2cc86df6cd3b4f0ad9cdef7f7a2c05f3fa5fc11a --- /dev/null +++ b/ntE_T4oBgHgl3EQf7xzu/content/tmp_files/load_file.txt @@ -0,0 +1,1103 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf,len=1102 +page_content='Screen Correspondence: Mapping Interchangeable Elements between UIs Jason Wu, Amanda Swearngin, Xiaoyi Zhang, Jeffrey Nichols, Jeffrey Bigham {jsonwu,aswearngin,xiaoyiz,jwnichols,jbigham}@apple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='com Login Page Sign in with your email address to access your account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Email Test App Don’t have an account or forgot it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Your account is used to access data across all your devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Your email address is used to enable services when you sign in, including backup which automatically backs up the data on your devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Your device’s identifier may be used to check eligibility for special offers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' For more information, see our terms of service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Login Page Don’t have an account or forgot your password?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Your email address is used to enable services when you sign in, including backup which automatically backs up the data on your devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Your device’s identifier may be used to check eligibility for special offers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' For more information, see our terms of service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Required Required Email Password Sign in with your email address to access your account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Input Screen Exemplar Inferred Semantics Text_Instruction Text_Title Button_Back Button_Disabled Input_Email Input_Passwd Button_Enabled Text_Footer Text_Footer Button_Enabled Input_Email Text_Instruction Text_Title Button_Back Button_Disabled Login Page Sign in with your email address to access your account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Email Test App Don’t have an account or forgot it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Your account is used to access data across all your devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Your email address is used to enable services when you sign in, including backup which automatically backs up the data on your devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Your device’s identifier may be used to check eligibility for special offers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' For more information, see our terms of service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Image Icon Icon Image Unmatched Figure 1: Screen correspondence produces a mapping of similar UI elements across two UIs that have related elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Screen- shots are encoded using a multi-modal model that segments and featurizes UI elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Mappings are generated that link element pairs that have the same or similar functionality across UI screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' ABSTRACT Understanding user interface (UI) functionality is a useful yet chal- lenging task for both machines and people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In this paper, we in- vestigate a machine learning approach for screen correspondence, which allows reasoning about UIs by mapping their elements onto previously encountered examples with known functionality and properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We describe and implement a model that incorporates element semantics, appearance, and text to support correspondence computation without requiring any labeled examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Through a comprehensive performance evaluation, we show that our approach improves upon baselines by incorporating multi-modal properties of UIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Finally, we show three example applications where screen Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' , , © 2022 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' ACM ISBN 978-1-4503-XXXX-X/18/06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='1145/1122445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='1122456 correspondence facilitates better UI understanding for humans and machines: (i) instructional overlay generation, (ii) semantic UI ele- ment search, and (iii) automated interface testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' KEYWORDS user interface modeling, ui semantics, element correspondence ACM Reference Format: Jason Wu, Amanda Swearngin, Xiaoyi Zhang, Jeffrey Nichols, Jeffrey Bigham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Screen Correspondence: Mapping Interchangeable Elements between UIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' ACM, New York, NY, USA, 13 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='1145/1122445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 1122456 1 INTRODUCTION Understanding how user interfaces (UIs) can be operated to achieve some goal can be challenging for both machines and humans, es- pecially those who are less tech-savvy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' While automated systems in the right circumstances can provide useful assistance [62, 63] or automatically complete the task themselves [34, 36], people can be hindered or completely blocked by apps that do not provide neces- sary metadata, such as the view hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' A promising approach involves inferring UI functionality solely from the pixels rendered arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='08372v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='HC] 20 Jan 2023 10:14 Dubsmash Cancel Next Apple ID Sign in with your Apple ID to use iCloud and other Apple services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=" Apple ID Email Don't have an Apple ID or forgot it?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Your Apple ID is the account you use to access all Apple services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Your Apple ID information is used to enable Apple services when you sign in,includingiCloud Backupwhichautomatically backs upthedata on your device in case you need to replace or restore it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Your device serialnumbermaybeusedtocheckeligibilityforserviceoffers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' See how your data is managed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='口9:41 三 < Back Next Apple ID SigninwithyourAppleIDtouseiCloud,the AppStore,andotherAppleservices AppleID j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='appleseed@icloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content="com Password Required Forgotpasswordordon'thaveanAppleID?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Your Apple ID information is used to enable Apple services when you sign in,including iCloud Backup which automatically backs up the data on vour device in case vou need to replace or restore it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Your device serial number may be used to check eligibility for service offers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Seehow yourdataismanaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='. UsedifferentAppleIDsforiCloud&other, , Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' to the screen, but to date this method has primarily been useful for identifying the location and type of typical UI elements [65] and not higher-level semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' For example, these algorithms can identify that a screen has a button that contains the text “Login,” but are unaware of the higher-level concept of logging in to a service, and they cannot infer what the role of this button would be in that process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' There are many higher-level semantics in user interfaces (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', lo- gin, account registration, shopping carts), which would correspond to an enormous number of classes if we attempted to use a classifier to predict their occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Instead of making class predictions, an alternate approach to data inference involves directly comparing inputs to previously encountered examples with known properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Studies on human [55] and machine [56] learning suggest that di- rect comparison is a useful tool, especially when relevant examples are available in the form of analogies [4, 24] or templates [20, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' This concept can be highly effective for UIs, as many belong to the same app or are constructed to serve a similar purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' For example, knowledge of how an app screen was previously interacted with by an app crawler or automated UI tester could aid in producing more robust and consistent results when visited again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Similar inferences can also be made for related screens from different apps, such as by determining that a button with the label “Login” in a new app is likely used to submit a login request because that is how a similar button is used in a known app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In this paper, we pose the problem of screen correspondence to map interchangeable elements between two UI screenshots (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We introduce a multi-modal transformer model for detecting, fea- turizing, and matching UI elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our approach is unsupervised, which allows it to work without a large dataset of labeled examples, which could be costly and time-consuming to collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In a perfor- mance evaluation with strong baselines, we compare our approach to existing correspondence algorithms used in computer vision (CV) and heuristics such as schema-matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our results indicate that our multi-modal model outperforms all existing baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We describe and implement three example applications that show the utility of screen correspondence for humans and machines to understand UI functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We create an application to gener- ate instructional overlays by transferring high-quality human- authored coach marks (a type of instructional label) from one screen to another of the same category (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', two registration screens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' To support UI design search and exemplar-based exploration, we used our model to index a large dataset of UI elements and screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Finally, we built a system to aid an automated app crawler by identifying mappings between the elements of screens from differ- ent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' To summarize, we make the following contributions: We introduce screen correspondence as a method of mapping interchangeable elements between UI screens from their screenshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We describe a machine learning approach to generating cor- respondence between two UI screenshots, and we show it outperforms existing baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We show the utility of screen correspondence in three ex- ample applications that improve both human and machine understanding of UI functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2 RELATED WORK Our work is related to recent work in understanding user interfaces from their pixels, and also a variety of methods for understanding applications in terms of their many screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We also overview machine learning solutions to correspondence problems in other domains, such as computer vision and natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='1 Predicting Screen Semantics Computational representation of user interfaces are useful for many downstream tasks, such as design assistance [32, 39], accessibility improvement [65], and task-oriented systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Screen Recognition [65] generates accessibility metadata of a UI from screenshots using an object detection model and heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Screen Parsing [60] gener- ates structured UI models from screenshots of UIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Several models [6, 15, 41, 64] have also been trained to predict the semantics of unlabeled icons found in mobile apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' These models can be applied to improve the accessibility of mobile apps, either as a tool during design time or as an automated system that repairs existing apps at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Most of these models map UI elements to a pre-defined set of classes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' UI element and icon type), which may exclude less common components [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' An alternative is to train models using self-supervision [9, 18, 35], which allows them to take advantage of larger unlabeled datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Screen2Vec [35] and other pixel-based autoencoders [9, 39] map UIs to fixed-length embedding vectors which can be used to represent semantic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The Pixel-words model [18] employs a trans- former model architecture and masked training objective based on prior work in NLP [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our work builds upon these approaches to train a model for identifying UI element correspondences between screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='2 Multi-screen Understanding While many automated UI systems can benefit from understanding the semantics of a single screen, screens are rarely used in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Any task or interaction trace requires reasoning about multiple app screens and how they are related to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' StoryDroid is a system that extracts a storyboard of Android apps from APK files as an “App Transition Graph” [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' ActionBert [23] models the relationship between two consecutive UI screens by predicting, among other things, which UI element was tapped on the first screen to reach the second (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', link component prediction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Longer sequences of touch interactions have also been modeled to better understand user behavior and app usage [31, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' A particular problem that many multi-screen systems aim to ad- dress is identifying whether two screens are instances of the same UI, a problem which we refer to as “screen fingerprinting.” NEAR [61] detects near-duplicate pages on the web using a combination of visual and DOM-based features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Prior work [14] used super- vised learning to predict the relationship and transitions between screenshots by, among other things, classifying whether inputs were different instances of the same screen (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', a news app with dynamic content).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Screen fingerprinting is useful for comparing screens to known examples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' however, finer grain mappings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', element-level finger- printing) can result in higher fidelity comparisons and additional benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Bricolage [30] is a system that renders the content of one Screen Correspondence: Mapping Interchangeable Elements between UIs , , web page using the style and layout of another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' It employs a su- pervised element matching model that featurizes web elements based on their DOM representation and was trained on a dataset of human-generated mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Interaction proxies [67] rely on a set of equivalency heuristics to identify UI components and structures found in Android view hierarchies to facilitate accessibility repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our work is related to these approaches in multi-screen un- derstanding and specifically element fingerprinting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' While many previous examples relied heavily on the availability of a structured UI representation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', DOM, view hierarchy) and were trained on labeled data, our approach requires only screenshots of related apps with optional labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='3 Machine Learning of Correspondence Machine learning has been used to learn correspondences in other domains, such as computer vision (CV) and natural language pro- cessing (NLP), which are closely related to our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' A longstanding problem in CV is inferring accurate correspon- dence of objects from different images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Homography estimation [21] involves finding a mapping, either sparse or dense, or a trans- formation matrix that describes perspective changes in two images of the same object or scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Optical flow [25] applies a similar concept to finding mappings between consecutively taken images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' A common approach involves computing appearance descriptors (keypoints), then creating a mapping that optimizes the global corre- spondence [16, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Recent work [1] has extended these approaches using learned semantic features to infer correspondence between images of inter-class or inter-domain objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Correspondence learning has also been useful for many tasks in NLP such as pronoun co-reference resolution and commonsense rea- soning, both of which rely on modeling correspondences between words to resolve ambiguities [28, 33, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Language translation and understanding, particularly for low-resource languages, ben- efit from learning word alignments to higher-resource languages [12, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Finally, other types of conditional natural language gen- eration have benefited from learning alignments between words with similar meanings [2, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Screen correspondence is related to many machine-learning driven approaches to identifying correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our transformer model builds upon many of these approaches by combining visual information and word-alignment techniques to produce screen cor- respondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In our evaluation, we compared our system to several baseline techniques from the CV and NLP literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We show that by incorporating multiple sources of information, our model gen- erates better representations for UI elements, which leads to more accurate correspondence predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 3 SCREEN CORRESPONDENCE We define screen correspondence as the task of mapping interchange- able UI elements between two UI screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' While matching UI ele- ments between screens may seem simple, it is a complex problem (especially from pixels alone) with many practical use-cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Pre- vious work relied on mappings to retarget UIs [30], provide help [62, 63], assist design [3], and test GUIs [5] and specifically called for more robust matching to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We primarily consider cases where two UI screens are of the same category (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', Login or Registration) but from different apps (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', intra-class examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' This is challenging because UI element pairs across such screen type pairs may not share similar appearance, text, or position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Instead, the model must reason about the purpose of each element in the context of its screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' To give intuition why even a seemingly simple example is hard, consider two Login screens (Figure 1): one contains Username and Password fields, while another contains Email and Password fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Position and element type information alone is unreliable for matching, since the text fields may have different sizes or appear at different locations on each screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Appearance information alone is also noisy for matching, since the text fields may have different visual themes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' State-of-the- art text encoders, even those trained on phrases, are unreliable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', most text models would produce a higher similarity score for Username and Password than Email).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' To detect UI element correspondences between different UI screens, we built a system that (i) automatically detects UI elements and text from screenshots, (ii) generates multi-modal embeddings for each element, and (iii) establishes mappings between individ- ual UI elements with high similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Figure 2 shows a high-level overview of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='1 UI Element Detection The first stage of our system identifies semantically relevant pieces of information from a UI screenshot, such as UI elements and text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The input is a bitmap and the output is a list of detected UI elements and pieces of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We use a pre-trained object detection model from previous work [60] that was trained to recognize UI elements in iOS app screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The pre-trained model uses the Faster-RCNN architecture [45] and was trained on the AMP dataset [65], which is separate from the main dataset used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' It achieved a class-weighted mAP score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We use post-processing procedures, such as score- based thresholding and inter-class non-max suppression (NMS), to improve the quality of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Optical character recognition (OCR) is performed using Tesseract [53], an open-source, off-the- shelf OCR software package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We run OCR on regions of the screen that correspond to text elements as detected by our element detec- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='2 UI Element Encoder Using the elements segmented from the screenshot, we generate representations that encode properties useful for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In our work, we consider relative positioning, element/icon category, visual appearance, and text, properties which we hypothesize to be relevant to element semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We used pre-trained models to pre- dict these properties (element detection [60] and icon type (“com- mon icon classifier" from previous work [7])) from screenshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Note that the pre-trained models were trained on different datasets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', no sample overlap) than the ones used in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='1 Modality Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We use off-the-shelf models to generate modality-specific features for each element, then feed their output into a screen transformer model, which combines and learns further associations between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' , , Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' UI Element Detection e1 Screen Transformer e2 e3 e4 en UI Element Detection e1 Screen Transformer e2 e3 e4 en Edges denote embedding similarity Edges w/ high similarity form correspondences Correspondence Matching Figure 2: Overview of our screen correspondence approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Elements and text from two screenshots are first extracted using UI element detection then featurized using a screen transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Finally, a correspondence between UI elements are generated from element pairs with highly similar embeddings relative to other candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Screen Transformer Element Label Embedding Appearance Embedding Text Embedding Pos Embedding e1 Element 1 Visual Element 2 Visual Element 3 Visual Element 1 Label Element 2 Label Element 3 Label el1 Element 1 Text Element 3 Text el2 el3 ea1 ea2 ea3 et1 et3 e2 e3 Modality Pooling Contextual Embedding One embedding per UI element One embedding per modality Figure 3: Architecture diagram of our screen transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Each modality-specific input is treated as separate inputs to our transformer model, which implicitly aligns them based on their positional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Note that elements may be missing modalities (Element 2 in this example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' After the per-modality inputs are processed by our transformer, we generate element embeddings (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', one per element) by pooling together outputs corresponding to the same original element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Positional information: Previous work [14, 18] encoded element position as a simple concatenation of bounding box coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We hypothesized that relative position may be more effective, since UI interactions such as scrolling, text flow, and dynamic content could cause changes in absolute position but have less effect on relative ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We adopted a relative positional encoding scheme used to improve the performance of language models [49] that incorporates pairwise distance when calculating the attention score between two elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Element Category: We categorized elements based on their UI and icon type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our pre-trained element detector classifies elements into 12 categories, as defined by previous work [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Three of these can be further delineated into sub-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We separate the Toggle and Checkbox classes based on their selection state (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', Toggle on and Toggle off).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We also classified common icon types using a sep- arate pre-trained CNN model [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In total, we consider 83 unique categories of elements and represent them as one-hot vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Visual Appearance: We featurized regions using the intermediate representations of a proposal-based object detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Similar ap- proaches have also been used by visual question-answering models, which also need to take into account multiple visual information [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' ] Since our UI element detector is based off of a similar proposal- based architecture, we retrieve the activations of the object pro- posals corresponding to detected elements using the fc6 layer [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=" 22:29 News January7 Top Stories Apple Al Maryah lsland opens Friday in the heart of Abu Dhabi The new store creates a direct connection from The Galleria Al Maryah lsland to the water's edge, delivering the best of Apple with shoreline views." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=" More coverage > Today News+ Audio Following Search22:29 'News April 6 Top Stories Apple's Worldwide Developers Conference returns in its all online format Apple today announced it will host its annual Worldwide Developers Conference WWDC) in an developers to attend More coverage > Today News+ Audio Following SearchScreen Correspondence: Mapping Interchangeable Elements between UIs ," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' This approach to featurizing appearance is beneficial since it results in a fixed-size representation for image regions without the need to explicitly resize or crop them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Text: Numerous embedding methods have been developed for representing words, sentences, and documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Sentence transform- ers are transformer-based models for encoding variable-length text into an embedding space representative of semantic meaning [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Since much of the text on UI screens is relatively short, we use a variant specifically trained to encode phrases [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='2 Transformer Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' To further enrich and learn associa- tions between the modality-specific element representations, we designed a model that generates a fixed-size embedding for each detected UI element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our model is based on the transformer architec- ture (Figure 3), which has been used for UI representation learning [18, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The modifications we described (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', relative positioning and appearance features) are aimed at improving performance on the correspondence task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Because not all elements have the same attributes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', not all UI elements have text, we rely on the transformer’s attention mecha- nism to implicitly align information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Each modality-specific repre- sentation with the exception of position is first embedded with a separate linear layer to a common size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Instead of creating one input vector for each element by concatenating features from each modal- ity, we create an input vector for each modality of each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' For example, a login button could result in three input vectors for ele- ment category, visual appearance, and text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' All inputs are fed into a series of stacked self-attention blocks, which results in one output embedding for each original input vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Finally, we use pooling to recover the output embeddings associated with each original UI element and compute their mean to incorporate information from all of the modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='3 Unsupervised Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We did not have access to labeled data during the development of our model, so we used unsupervised training to learn its parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Masked element prediction is a training objective that requires the model to reconstruct an input that has been corrupted by randomized masking (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', replacing a portion of the input with 0’s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Previous work [54] has shown that this training objective encourages the model to learn semantically relevant representations since it must learn to associate masked information with other sources of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The reconstruction loss was measured separately for all modal- ities (element category, visual appearance, and text) then added together to obtain the model’s total loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' L2-loss was used for re- construction of visual and text features, and cross-entropy loss was used for reconstruction of element category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='3 Correspondence Matching After we used our screen transformer model to featurize UI ele- ments on two screens, we perform a matching procedure to predict correspondences between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' For a pair consisting of a source screen with 𝑀 elements and a target screen with 𝑁, we construct a 𝑀 × 𝑁 cost matrix 𝐶 ∈ R𝑀𝑥𝑁 to represent correspondence scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The matching cost 𝐶𝑖,𝑗 is computed using cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Several approaches have been used to generate correspondences from cost matrices [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' A simple approach of matching based solely on highest cosine similarity may make suboptimal decisions when one element has more than one likely match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In our final implemen- tation, we formulated correspondence mapping as an optimization problem that finds the assignment between two sets that results in the lowest overall cost [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We employ this approach for match- ing elements between screens, since elements are more likely to be dissimilar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' To reduce false positives, we employ additional pre- processing and post-processing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Before running the best-cost optimization, we prune unlikely matches from the cost-matrix so that each element only considers its 𝑘 closest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' After- wards, we ignore matches where 𝐶𝑖,𝑗 < 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We tuned the values of 𝑘 = 5,𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='4 based on manual examination of a small number of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' This approach is similar to approaches in text decoding models that consider only the top 𝑘 most likely tokens, which have been shown to generate higher quality output by reducing the effect from low-probability outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 4 DATASET We developed and trained our transfer model on two datasets of app screens that were generated by manual crawling of popular mobile apps: Crawls and Rico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The Crawls dataset, which was used by prior UI modeling work [7], consists of 750,000 iOS app screens from 6,000 apps and was collected by crowdworkers who were instructed to manually explore mobile applications through a remote interface that periodically captured screenshots and addi- tional metadata of the current app screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The Rico dataset [9] is a publicly available dataset of 72,000 Android screens from 9,700 apps that was also collected by crowdworkers remotely operating devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We divided each dataset into training (70%), validation (15%), and testing (15%) splits by their crawl ID, which corresponds to which app was crawled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='1 Evaluation Dataset While our training algorithm does not depend on labeled data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', unsupervised), we manually collected a small set of labeled examples (900 pairs across 90 screens) from each dataset to evaluate our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='1 Data Collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our evaluation dataset consists of data from 9 types of screens that we hypothesized could have correspon- dences: Media Player, In-App Purchases, Login, Permission Request, Register, Pre-Login, Pop-up, Search, and Web Views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We initially asked crowdworkers to categorize a set of screenshots outside of the training split based on a criteria for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Unlike app categories, which might be used to categorize apps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', finance, health, social media), we focused on screen categories, since both a health app and a banking app might both contain a login screen that could contain correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' For each of our two datasets, we sampled a small number of screens from each category for cor- respondence labeling (9 categories x 10 screens = 90 screens total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The 9 categories that we chose do not cover all possibilities, but we believe they constitute a reasonable subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' More detailed descrip- tions and criteria of each category is available in the appendix of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='2 Data Labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We created a labeling interface to annotate our small evaluation split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' First, a randomly selected element was , , Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' shown on a screen, and the interface displayed a prompt asking if the element was likely to appear on other screens of the same type: “Are elements of similar functionality likely to appear on other Login screens?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' If the user responded “Yes,” the application displayed a prompt for a label: “What is the role of this element in the current screen?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We built our interface to include auto-complete function- ality to encourage labelers to identify correspondence categories that could generalize across screens, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', “login button” instead of “button to log into my credit card account.” The autocomplete list was pre-populated with 5 choices for each category and was auto-updated with novel descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' If a label was provided on the first step, then the user was shown other screens from the same category and asked to select elements with a similar role, if they were present on the screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' A drawback of this approach is that it is slow, since it requires providing a role description before elements are matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' However, we found the additional consideration of element role is useful for reasoning about correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 5 EVALUATION We evaluate our model against several baselines and ablated ver- sions of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our results show that compared to heuristic and traditional key-point methods used in CV, multi-modal trans- former encodings lead to better correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Furthermore, our ablation experiments show that the architectural improvements we made lead to modest performance gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='1 Baselines In this section, we describe the baselines used in our performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We focus primarily on other unsupervised approaches, since our constraint was that we didn’t have any labeled data avail- able for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Similar supervised approaches exist [30], but they depend on element-level annotations and access to underlying source code (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', HTML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' For comparison, we chose a variety of baselines that include keypoint-based methods used for image matching and heuristics such as schema-matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our main constraint was that we did not have large quantities of labeled data for supervised machine learning methods, so we selected unsupervised techniques for com- parison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' ORB: As a review, image correspondence relies on the compu- tation of semantic features from regions of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Semantic features, usually invariant to surface-level changes such as trans- lation and scale, are first calculated for small, localized regions of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' When this process is repeated recursively, the receptive field increases, and globally-aware features can be learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' ORB [46] is a traditional CV approach to generating descriptor features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We first computed ORB descriptors for each screenshot which re- sulted in numerous keypoints at salient points of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Using brute-force matching, keypoints from one image were matched onto keypoints from another image based on descriptor similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Finally, to translate keypoint similarity to UI element similarity, we used an object detector to compute the boundaries of UI elements and matched elements based on the number of matching keypoints contained within them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Neural Best Buddies: Neural Best Buddies (NBB) uses the internal representations of a deep CNN to featurize and match image regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Like ORB, it also generates keypoint descriptors but uses activations from a convolutional neural network (CNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' One advantage that CNN features have over traditional methods is that learned features can better correspond to semantic properties that the network was trained on (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', image classification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' To run our experiments, we used the code released by the authors of the paper 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The original paper focuses on finding correspondences between “natural images” and use a VGG-19 model [51] that was pretrained on ImageNet [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Since UI screenshots have different properties than images found in ImageNet, we initially tried to train a CNN model better repre- sentative of UIs using an unsupervised autoencoder objective due to the lack of labels in our training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' However, we found the autoencoder model did not produce good outputs, so we report results using the pre-trained ImageNet model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Schema-matching Heuristic: One drawback of keypoint-based methods that we explored is that keypoints are generated using the entire image as input and without knowledge of UI element locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Schema-matching is an approach that first considers each predicted element as a discrete object, then uses its attributes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', schema) to compare similarity to other candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We implemented a heuristic that uses schema matching through incorporating the predicted UI element/icon type by concatenating their one-hot class predictions into a single vector and applying the same best-cost matching algorithm [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' More sophisticated schema-matching may incorporate additional information, such as UI hierarchy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', an element that belongs in a list should be matched to another element in a list).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' While possible to predict [60], we did not incorporate hierarchical information since it requires complex techniques for tree matching but expect it would perform similarly to [30], which uses hierarchical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Screen Transformer Ablations: Our performance evaluation in- cludes ablated variations of our main transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Trans- former models allow learning more sophisticated representations of elements through data, which provides advantages over manually- defined schemas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We evaluate several ablated versions of our model to understand the performance impact of our architectural changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Specifically, the ablated versions of our transformers removes cer- tain components that we hypothesized to improve correspondence matching, such as relative positional embedding, visual appearance information, and text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In addition, we evaluated the Pixel-words transformer [18], which our model is based on, but we adjust the number of element classes, layers, attention heads, and hidden di- mensions to be the same as our other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The Pixel-words transformer also includes a “layout embedding" network which featurizes the layout of UI using a semantic map which is fed into an autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' To summarize, the Pixel-words configuration (i) considers categorical and text information, (ii) uses absolute posi- tional encodings, and (iii) includes an additional layout embedding component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='2 Results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='1 Baseline Comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our evaluation results (Table 1) shows the benefit of our multi-modal model over simpler baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='com/kfiraberman/neural_best_buddies Screen Correspondence: Mapping Interchangeable Elements between UIs , , Table 1: Performance of our approach and other baselines for screen correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our approach leads to the best performance, reaching an F1 score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We also included ab- lated versions of our model without relative positional em- beddings, appearance features, and text features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' CRAWLS RICO Model Configuration P R F1 P R F1 Screen Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='55 Screen Tran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' (w/o Relative) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='49 Screen Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' (w/o Appearance) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='51 Screen Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' (w/o Text) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='50 Screen Trans (Pixel-words) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='35 Heuristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='45 ORB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='31 NBB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='26 Figure 4: Performance across different categories in the Crawls (Top) and Rico (bottom) datasets using the full Screen Transformer model configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The average clas- sification performance was F1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='61 on Crawls and F1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='55 on Rico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' employ standard classification metrics to measure the accuracy of element-to-element correspondences generated by our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Since elements in our evaluation dataset are labeled using their ground truth bounding boxes instead of our element detector’s predictions, we first match predicted detections to ground truth elements using the best Intersection-over-Union (IoU) score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Due to our labeling procedure where one element is highlighted at a time, the examples in our evaluation set were only partially labeled, meaning that screens contained only a randomly sampled subset of all possible corresponding pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our best model configuration reaches an F1 score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Screens in our dataset contained an average of around 20 elements, so correct correspondence required finding the best out of match out of all possible candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The ORB and NBB baselines are based on keypoint-based matching, which is commonly used in CV to compare images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Among them, ORB performs the best, achieving higher correspondence accuracy but performed poorly due to low recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' One possible reason is that keypoints are generated at visually salient locations of the image, such as edges and corners, and without any knowledge of where UI elements are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Thus, some UI elements may not contain many keypoints within them, reducing the quality of matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The schema-matching heuristic performed substantially better than keypoint-based methods and reached high recall by directly using the outputs of pre-existing models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', element detection, icon classification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Precision was lower, possibly due to the difficulty of accurately matching ambiguous elements without knowledge of additional context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our ablation experiments revealed that our modifications to the base transformer architecture led to modest improvements in terms of F1 score but also had other consequences for precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' For example, our model trained without appearance information was the lowest performing variation but reached the highest preci- sion score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We attribute these variations to the information encoded in each modality and may warrant different configurations based on intended use-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='2 Performance across UI Categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Figure 4 provides a more in-depth breakdown correspondence by UI category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our model achieved the best performance on the Website View and In-App Purchase categories and the worst performance on the Media Player and Pre-Login categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' One major source of error for our model was the presence of sub-categories within our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' For example, we manually ex- amined examples from the Pre-Login and Login categories, which received relatively low performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We discovered a consider- able difference between apps that used different authentication providers, such as OAuth and Single-Sign-On (SSO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' For example, a “traditional” login screen might include text fields for entering a username and password, but an app using a SSO provide (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', Sign in with Apple) might only contain a button without any text fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We found that there was also variance within media player screens – video players and music players had significant differences and some media players were full screen while others were not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Since our correspondence model uses contextual information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', in- formation from other elements on the same screen) and relative positional encoding, this could significantly affect the computed representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' One strategy to address this is the formation of sub-categories with a more consistent set of elements e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', creating separate categories for traditional login screens and those with other types of authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='3 Performance across Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We evaluated all models and baselines on both the Crawls and Rico dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Overall perfor- mance between the two datasets were similar, although the Rico models performed slightly worse (F1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='55) than ones trained on Crawls (F1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' One possible reason for the performance dis- crepancy is that Crawls is an order of magnitude larger and the model was exposed to more variation during training time, which is beneficial for unsupervised training techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' While the full transformer model is the best-performing configuration for both Performance by Category (Crawls) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='25 Media In-App Login Permission Register Pre-Login Pop-up Search Website Player Purchase Request View Precision Recall F1 Performance by Category (Rico) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='25 0 Media In-App Login Permission Register Pre-Login Pop-up Search Website Player Purchase Request View Precision Recall F1, , Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Table 2: Performance of our approach and other baselines screen correspondence for same-screen pairs in the Crawls dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Many configurations, including our model, reach a maximum F1 score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We attribute labeling noise and the IoU element matching process used to assign predicted element locations to ground-truth boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Model Configuration P R F1 Screen Transformer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='76 Screen Transformer (w/o Relative) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='76 Screen Transformer (w/o Appearance) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='75 Screen Transformer (w/o Text) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='76 Screen Transformer (Pixel-words) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='76 Heuristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='75 ORB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='59 NBB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='34 datasets, the relative performance ablated models were affected dif- ferently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Notably, the Rico models without text and the Pixel-Words model performed much worse, suggesting that its evaluation set may have contained more text-heavy screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='4 Correspondence between Same-screen Pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In addition to evaluating our models on screens from different related apps (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', intra-class pairs), we also investigated performance on same-screen pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Same-screen correspondence is useful for identifying the same UI element across multiple versions of the same screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' For example, an app’s appearance may change following an update or from dynamically updated content (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', a news page loads content from a remote source).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Following prior work [14], we consider two screenshots to be the “same” if they represent different instances of the same underlying implementation, possibly with significantly different appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Correspondence mapping can help guide au- tomated systems such as crawlers to behave more consistently in these situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We randomly selected screen groups with the same app ID as those in the testing split of our Crawls dataset, then randomly sampled two screenshots from each group, resulting in 888 total pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Upon manual inspection, we found that some of the sampled pairs had only minimal visual changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' To filter out “easy pairs,” we constructed a heuristic that attempted to match elements based only on bounding box location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' If all elements in a pair were successfully matched, we discarded the example, since it meant that no significant dynamic change (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', scrolling, dynamic content) occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' After this process, the final dataset contains 607 examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We did not repeat this for the Rico dataset because the authors applied a heuristic to filter out repeated views of the same screen [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our observations and performance results (Table 2) show that same-screen correspondence is generally higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Since same-screen pairs are usually more visually similar, the model can rely more heavily on surface-level features and in many cases perform direct matching, such as looking for recurring text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Many configurations, including our model, reached a maximum F1 score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Er- rors from labeling noise and IoU element matching (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', matching ground truth bounding boxes to predictions) may have established an effective ceiling, since our element detection model introduced errors (has a class-weighted mAP score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 6 EXAMPLE APPLICATIONS We describe three example applications that show the utility of screen correspondence to human and machine understanding of UI functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Generating and transferring a type of instructional overlay called coach marks can help users navigate unfamiliar UIs by mapping them to previously encountered ones of the same class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' UI search is useful for app designers to find how concepts are expressed across apps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', what are different ways of expressing a search intent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Finally, automated GUI testing can be made more robust by accounting for variations in visual presentation between different app versions without requiring platform-specific APIs or metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' These example applications are not meant to be novel, but we believe they show that accurate screen correspondence allows many existing systems to work under a wider range of conditions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', using pixel data alone or improved robustness to dynamic visual changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='1 Instructional Overlays We used our model to improve users’ understanding of complex or newly installed apps by creating an infrastructure that could be used to crowdsource coach marks for apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Coach marks are instructional overlays that are sometimes shown to provide assistance to users when an app is first launched, and can be helpful for exposing UI functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' While it is possible to automatically generate natural language for describing screens [57] and widgets [37] using deep models, they are often affected by surface-level appearance and may be prone to producing generic outputs [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Building a model that produces natural language also introduces significant complexity that can be similar achieved with a correspondence mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' A better approach might be to crowdsource users [43] or developers to write coach marks for screens in a subset of apps, and then apply our screen correspondence technology to map these coach marks onto a much larger set of screens with similar purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' This idea builds on the template-based matching scheme of Yeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [62] for generating contextual help, and expands their idea beyond same-screen applications to also intra-class usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We applied our model’s intra-class correspondence capabilities to automatically transfer annotations from one screen to another related app of the same category (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We first populated a small database of instructional text for elements from app screens in one of the categories from our evaluation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In a real imple- mentation of this system, an interface would be created to allow users to author new instructional text for screenshots that they up- load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Each screen in the database was associated with its featurized elements as a key, and each instruction in the database was associ- ated with its element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our current prototype is a proof-of-concept implementation where the user can upload a screenshot image file through a web interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' On the uploaded screen, we perform a nearest-neighbor search to retrieve the screens in our database that are most similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' If the distance is sufficiently close, we run our screen correspondence matching, which also returns a “matching cost.” If enough matches are discovered and the matching cost is Screen Correspondence: Mapping Interchangeable Elements between UIs , , Exemplar Labeling Generated Help Overlay Figure 5: Coach marks are useful for uncovering function- ality in apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' High-quality natural language descriptions of UI components can be difficult to generate, so we curated a small number of labeled examples from different app cate- gories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Element descriptions from this labeled set are trans- ferred onto unseen app screens of the same type using the correspondence mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Depending on the use-case, the exemplar can be manually provided (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', developer wishes to label many similar screens at once) or automatically re- trieved (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', a help-generation app uses a separate classifier to find an exemplar from a database of labeled screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=') below a heuristically set threshold, we directly render the annota- tions to the screenshot using image drawing APIs and display the annotated image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In a complete implementation, the matching and rendering algo- rithms would be built into a mobile operating system and run on the user’s mobile device so that it would not require the user to exit their current app to use our tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In the future, we plan to improve the user experience and investigate ways that these overlays could be surfaced contextually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In this example, we show that accurate intra-class screen corre- spondence can facilitate transferring coach marks, which can help users discover new app functionality and documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Other pos- sible applications of screen correspondence to improving end-user usage include transferring more types of accessiblity meta-data, example-based re-targeting of UIs [30] and using input redirection techniques to improve the accessibility of UI components [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='2 UI Element Search Engine UI search can help app designers find how concepts are expressed across apps and provide example starting points when designing a new app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Previous work indexed databases of UI screens using visual properties [3], structural properties [60], sketches [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We focus specifically on returning relevant UI elements instead of screens, and leverage our model’s intraclass matching abilities to improve the search process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We integrated our screen correspondence model into a UI search engine to support tag-based search and exemplar-based refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The implementation of our UI search engine is a web app that indexed elements from 130,000 UI screens using a variety of meta- data, including detected element classes, icon types, and text, which are stored in a database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our app features a search page, where users can first perform an initial search by entering text or tags in a search bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Results are returned based on matching attributes found in the property database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Matching elements are shown in the context of their app screen and highlighted with a bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' When a result is selected, users are brought to the element inspector page, where users can examine the properties of all elements on the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' One limitation of tag-based search is that it is difficult to specify target properties that do not belong to the pre-defined set of tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' For example, a “plus” icon displayed on the top or bottom of a list may indicate adding to the list while a “plus” icon displayed next to a list item is more likely to representing adding the item from the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' It would be difficult to disambiguate between these cases as they share the same tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Thus, we used our correspondence model to enable exemplar-based search refinement, which allows users to “narrow in” on more specific results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' To enable this functionality, we computed embeddings for UI elements in our database and stored this information into a vector data store which supports fast approximate nearest-neighbor search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We added a “search for similar items” button on the element inspector page, which finds results with a high similarity to the target element according to the cosine similarity metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Figure 6 shows an example flow of our UI element search engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='3 Automated GUI Testing Finally, we used our model to improve the robustness of automated GUI tests using our model’s same-screen matching capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Auto- mated testing is useful for ensuring the quality of GUIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Specifically, visual-based methods can be employed in these systems to search for targets based on their rendered appearance, which allows for easier authoring of testing scripts and reduces the dependency of testing frameworks on specific UI toolkits [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' However, strong reliance on visual similarity may lead to failures caused by change in visual style, such as updated application theme or icons [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In such applications, the quality of UI element matching is impor- tant for automated GUI testing because poor matching capability Not Secure - )9-0620-25-srv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='mr3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='simcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='apple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='com + UI Helper Upload Exemplar Label Screen Screen Descriptions Consider the elements on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Provide a short 5:13 description for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 4 The verification code is usually sent to your email or text m Verify code Element 1 Element 2 Complete the login Cancel 5 Enter code herel Berify巴 Not Secure - )9-0620-25-srv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='mr3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='simcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='apple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='com + UI Helper Upload Exemplar Label Screen Generated Correspondences 5:13 3:58 K Verity code WOODWILD Enter your code EVERGREEN WOODWILD The verification code is usually sentte yoyr email or text messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=" Verify Enter the code here cGomplete the login I didn't get a code By continuing, 1 agree to reoeive emails from Evergreen." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' I understand that 1 am free to withdraw consent at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', , Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Tag-based Search Element Inspector Refine by Exemplar Figure 6: An example usage flow of our UI element search engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The user first searches for icon elements that contain the “add” tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The results page shows UI screens with a matching element highlighted (Left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The user selects a result screen where an add button is placed on the top right of the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The inspection page provides details about element info and allows searching for similar elements (Center).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Another search query is run using the embedded element of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The new results are similar to the query in that they are all located at the top right of the screen and they appear to be used for adding items to a gallery (Right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' This example shows how designers can start a search using natural language or tag-based queries then refine the results based on exemplars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Recorded Action Replayed Action Figure 7: Automated UI testing techniques execute an inter- action trace (either manually pre-defined or automatically generated) to detect functional regressions, visual regres- sions, and other unexpected behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Updated versions of apps may lead to small changes in layout and visual appear- ance and knowledge of same-screen correspondence can im- prove the consistency and robustness of tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' This exam- ple shows a automated application performing a previously recorded action, despite the target’s appearance change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' can lead to a failure to replicate recorded interaction traces in a scripted testing scenario, and repeated visits to the same screens in a random crawler stress test example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' As shown by previous work on screen similarity [14], methods that rely heavily on surface-level appearance may have high precision but low recall due to possi- ble variations between UIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We applied our screen correspondence model to improve the robustness of these matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We built a prototype system that interacts with remotely con- nected smartphone devices through a VNC interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our software sends commands through this interface to simulate interactions, such as clicking and swiping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We also include a “recording” mode that allows a tester to record an interaction trace, during which all of the screenshots and interactions are saved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' When replaying the interaction trace, the saved screenshots and interacted elements are used to match the current state of the VNC output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Specifically, for each step in the saved trace, we identify the UI element with which the tester interacted, such as the button that was pressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Then, on the live VNC view, we find the corresponding element and apply the recorded interaction to it, similar to previous work on tutorial consumption [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Figure 7 illustrates how our automated tester nav- igates an app where the appearance of a target element has changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Used in conjunction with traditional template-matching techniques, which offer high precision but low recall, correspondence matching can help improve the overall performance of automated testers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 7 LIMITATIONS & FUTURE WORK Our evaluation shows that correspondences can be automatically identified through machine learning and matching approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Some types of screens are more likely to have correspondences de- tectable by our system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', Website Views and In-App purchases) than others (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=', media players).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The required accuracy level de- pends largely on the final application, since different use-case since different performance attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' For example, using correspon- dences to generate contextual help (instructional overlays) may result in a better experience if only very confident matches are used, as incorrect instructions can lead to confusion and frustration from the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' GUI testing and crawling is less tolerant to mistakes, since an incorrect action can make it impossible to access the rest of an application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' On the other hand, UI design search is more forgiving, since it can provide value if most of the returned elements are cor- rect (does not need to be the top choice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our current evaluation does not account for the requirements of down-stream applications, although based on the example applications we implemented, we found them to provide acceptable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We plan to further evaluate our system in down-stream applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='十 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='< ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='UI Search Engine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Search Results ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Search ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='add ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='O All ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='xel O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Text Field ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='o Icon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Segmented Control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Search ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='314 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='315 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='22:01 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='22:17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='22:14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='田 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Favorites ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='田 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='< Shortcuts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Select ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='田 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Projects ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='All Shortcuts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='V Default Room ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Kitchen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='[Q bearch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Cancel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='二楼 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='卧室 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Find Photos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='New Shortcut 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='洗衣房 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='1actior ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Home Settings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='大 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Room Settings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Accessibility＋ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='< ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='UI Search Engine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Screen 9359 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Q Search Similar Screens ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='@ Search Similar Elements ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='> array [14] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='>0 [3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='>1 [3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Projects ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='口 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='>2 [3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='3 [3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='口 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='4 [3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='> data [6] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='id : 184000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='screen_id : 9359 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='>5 [3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='6 [3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='>7 [3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='8 [3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Prolec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Projeci ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='9 [3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='JUn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 15 2021 2:30 AN J00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='15 20212-28 AN 10 [3] >11 [3] >12 [3] dataPointType : annotation0 < UI Search Engine Similar Screen Search Results 22:01 三3 4:55 22:017 田 Favorites TestName ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Edit 田 Projects 口田 Activity Alarm SleepIWakeUp NoAlarm SET UP Other 00:30 taned NoFavorites 09:19 Alarm 10:03 Alarn 16:07 16:09 Q 1710_736635_1623724210395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='json 4513_1629564_1624926702762.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='json 1530_682461_1623646544824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='json 6747_2295141_1625682246038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='json ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='α4:27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Collections ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content="What's New " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='9 tips ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Welcome to iPhone ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Get to know your iPhone ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='8 tips ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Essentials ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content="Must know features you'll love " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='8 tips ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Genius Picks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Favorites from our experts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='13 tips ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Photography ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Take and perfect your best shot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='12 tips5:48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Collections ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content="What'sNew " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='11 tips ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Welcome to iPhone ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Get to know your iPhone ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='9 tips ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Essentials ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content="Must-know features you'll love " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='10 tips ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Genius Picks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Favorites from our experts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='9 tips ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Accessibility ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='大 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Make iPhone work for you ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='10 tips5:48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='< Collections ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='Welcome to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='iPhoneScreen Correspondence: Mapping Interchangeable Elements between UIs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' A limitation of our current experiments is that they focus only on mobile UIs that belong to a set of 9 categories that we identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' These 9 categories do not cover all possibilities of app screens, but they cover a considerable subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our model is likely to perform better for complex app screens if given a small amount of annota- tions to fine-tune on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Moreover, since we only use pixel information as input to our model, we believe that our approach is likely to generalize well to other types of graphical UIs that also represent their output as pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In the future, we aim to replicate our exper- iments on other types of graphical UIs with varying screen sizes and shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We see several opportunities to improve the performance of our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Since our system relies on several individual components, it may be useful to quantify the performance of each separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We used a pre-trained element detector model that produced noisy output for the correspondence matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Previous work [60] has shown that element detectors perform poorly on more complex screens due to the increased number of elements and sometimes miss smaller elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Future work could investigate a screen cor- respondence system that uses a more accurate element detector model or accepts manual annotations as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' More advanced matching techniques can also be employed, such those that consider multi-scale correspondence, which first process smaller sub-regions before merging their predictions globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Separately, prior work on image correspondence [27] has shown improved performance by scaling images during training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' A similar idea could be applied to UIs by first predicting their UI hierarchy [60] and generating mappings for groups of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our model could also use different unsupervised pre-training objectives to help it build better representations of UI elements for our matching task [28, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Our work focuses on mapping interchangeable elements with similar functionality between UI screens, however there are other relationships that can be modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Categorization of different rela- tions in language analogies [19, 40] show that antonym, categorical, and functional connections can enrich the expressiveness of lan- guage and rhetoric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We plan to focus future modeling efforts on identifying and inferring a wider range of similar relationships that exist in UIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Finally, our work explores inferring UI functionality from a sin- gle previously encountered example, yet we believe our approach may extend to multiple examples [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' For example, non-parametric machine learning methods such as the k-nearest neighbors algo- rithm often benefit from considering more than one example at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 8 CONCLUSION In this paper, we explore screen correspondence as a machine learn- ing technique for inferring UI functionality by directly leveraging previously encountered examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We describe our model architec- ture and training procedure that incorporates information about UI semantics, appearance, and text when generating correspondence mappings between screenshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In a comprehensive evaluation with strong baselines, we show that our approach outperforms corre- spondence algorithms by leveraging multiple information sources found in UIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Finally, we show how three example applications of screen correspondence: (i) transferring coach marks from related apps, (ii) UI element search, and (iii) automated GUI testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Broadly, our work demonstrates the feasibility of learning UI semantics by mapping to prior examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' A MODEL HYPERPARAMETERS Model Hyperparameter Value Screen Transformer optimizer Adam learning rate 1e-4 weight decay 1e-5 dropout 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='25 hidden size 256 num layers 4 num heads 4 We trained our models with early stopping and stopped training when validation loss did not improve for 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We imple- mented our model using the PyTorch [42] and PyTorch Lightning [13] frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' B UI CATEGORY CRITERIA We collected a small dataset of 9 screen categories for evaluation of our model’s intra-class correspondence capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' We used the following guidelines to categorize apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Media Player - A screen that allows users to play media content such as music or video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Usually contains controls for adjusting playback, volume, and sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In-App Purchase - A screen that asks users to make a purchase for a subscription or to access some part of an app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Usually contains buttons for making the purchase, dismissing the screen, or signing up for a trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Login - A screen that asks users to log into an app or service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' It may contain fields for entering username and password or buttons for third party authentication services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Permission Request - A screen that asks users to enable some permission, which are usually associated with security set- tings such as location or camera access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Register - A screen that asks the user to create an account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' May contain a form to register or buttons for third party authentication providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Pre-Login - A screen that contains controls to access other parts of the app either by logging in or registering for an account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' This usually comes before the login page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Pop-up - A screen with a pop-up or dialog model that is displayed over other app content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Pop-ups may contain con- trols for accepting or dismissing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' For pop-ups that ask for permission or purchases, see other categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Search - A screen for entering and submitting a search query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' May include a search bar and filtering controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Website View - A screen where an app opens an external website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' May contain a URL bar and forward/backward con- trols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' , , Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' REFERENCES [1] Kfir Aberman, Jing Liao, Mingyi Shi, Dani Lischinski, Baoquan Chen, and Daniel Cohen-Or.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Neural best-buddies: Sparse cross-domain correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' ACM Transactions on Graphics (TOG) 37, 4 (2018), 1–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [2] Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Neural ma- chine translation by jointly learning to align and translate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' arXiv preprint arXiv:1409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='0473 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [3] Sara Bunian, Kai Li, Chaima Jemmali, Casper Harteveld, Yun Fu, and Magy Seif Seif El-Nasr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' VINS: Visual Search for Mobile User Interface Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 1–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [4] Joel Chan, Joseph Chee Chang, Tom Hope, Dafna Shahaf, and Aniket Kittur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Solvent: A mixed initiative system for finding analogies between research papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Proceedings of the ACM on Human-Computer Interaction 2, CSCW (2018), 1–21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [5] Tsung-Hsiang Chang, Tom Yeh, and Robert C Miller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' GUI testing using computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 1535–1544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [6] Jieshan Chen, Chunyang Chen, Zhenchang Xing, Xiwei Xu, Liming Zhu, Guo- qiang Li, and Jinshui Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Unblind your apps: Predicting natural-language labels for mobile gui components by deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the ACM/IEEE 42nd International Conference on Software Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 322–334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [7] Jieshan Chen, Amanda Swearngin, Jason Wu, Titus Barik, Jeffrey Nichols, and Xiaoyi Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Towards Complete Icon Labeling in Mobile Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 1–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [8] Sen Chen, Lingling Fan, Chunyang Chen, Ting Su, Wenhe Li, Yang Liu, and Lihua Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Storydroid: Automated generation of storyboard for Android apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In 2019 IEEE/ACM 41st International Conference on Software Engineering (ICSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' IEEE, 596–607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [9] Biplab Deka, Zifeng Huang, Chad Franzen, Joshua Hibschman, Daniel Afergan, Yang Li, Jeffrey Nichols, and Ranjitha Kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Rico: A mobile app dataset for building data-driven design applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the 30th Annual ACM Symposium on User Interface Software and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 845–854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [10] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Imagenet: A large-scale hierarchical image database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In 2009 IEEE conference on computer vision and pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Ieee, 248–255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [11] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Bert: Pre-training of deep bidirectional transformers for language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' arXiv preprint arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='04805 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [12] Chris Dyer, Victor Chahuneau, and Noah A Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' A simple, fast, and effective reparameterization of ibm model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the 2013 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 644–648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [13] WA Falcon and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' PyTorch Lightning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Note: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='com/PyTorchLightning/pytorch-lightning 3 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [14] Shirin Feiz, Jason Wu, Xiaoyi Zhang, Amanda Swearngin, Titus Barik, and Jeffrey Nichols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Understanding Screen Relationships from Screenshots of Smart- phone Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the 27th Annual Conference on Intelligent User Interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [15] Sidong Feng, Suyu Ma, Jinzhong Yu, Chunyang Chen, TingTing Zhou, and Yankun Zhen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Auto-icon: An automated code generation tool for icon designs assisting in ui development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In 26th International Conference on Intelligent User Interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 59–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [16] Martin A Fischler and Robert C Bolles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' ACM 24, 6 (1981), 381–395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [17] Huazhu Fu, Xiaochun Cao, and Zhuowen Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Cluster-based co-saliency detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' IEEE Transactions on Image Processing 22, 10 (2013), 3766–3778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [18] Jingwen Fu, Xiaoyi Zhang, Yuwang Wang, Wenjun Zeng, Sam Yang, and Grayson Hilliard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Understanding Mobile GUI: from Pixel-Words to Screen-Sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' arXiv preprint arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='11941 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [19] Anna Gladkova, Aleksandr Drozd, and Satoshi Matsuoka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Analogy-based detection of morphological and semantic relations with word embeddings: what works and what doesn’t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='. In Proceedings of the NAACL Student Research Workshop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 8–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [20] Kelvin Guu, Tatsunori B Hashimoto, Yonatan Oren, and Percy Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Gen- erating sentences by editing prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Transactions of the Association for Computational Linguistics 6 (2018), 437–450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [21] Richard Hartley and Andrew Zisserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Multiple view geometry in computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Cambridge university press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [22] Junxian He, Taylor Berg-Kirkpatrick, and Graham Neubig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Learning sparse prototypes for text generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 33 (2020), 14724–14735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [23] Zecheng He, Srinivas Sunkara, Xiaoxue Zang, Ying Xu, Lijuan Liu, Nevan Wich- ers, Gabriel Schubiner, Ruby Lee, Jindong Chen, and Blaise Aguera y Arcas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' ActionBert: Leveraging User Actions for Semantic Understanding of User Interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' arXiv preprint arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='12350 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [24] Aaron Hertzmann, Charles E Jacobs, Nuria Oliver, Brian Curless, and David H Salesin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Image analogies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the 28th annual conference on Computer graphics and interactive techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 327–340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [25] Berthold KP Horn and Brian G Schunck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Determining optical flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Artificial intelligence 17, 1-3 (1981), 185–203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [26] Forrest Huang, John F Canny, and Jeffrey Nichols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Swire: Sketch-based user interface retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 1–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [27] Wei Jiang, Eduard Trulls, Jan Hosang, Andrea Tagliasacchi, and Kwang Moo Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Cotr: Correspondence transformer for matching across images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 6207– 6217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [28] Vid Kocijan, Ana-Maria Cretu, Oana-Maria Camburu, Yordan Yordanov, and Thomas Lukasiewicz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' A Surprisingly Robust Trick for the Winograd Schema Challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In ACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [29] Harold W Kuhn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 1955.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The Hungarian method for the assignment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Naval research logistics quarterly 2, 1-2 (1955), 83–97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [30] Ranjitha Kumar, Jerry O Talton, Salman Ahmad, and Scott R Klemmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Bricolage: example-based retargeting for web design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2197–2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [31] Seokjun Lee, Rhan Ha, and Hojung Cha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Click sequence prediction in Android mobile applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' IEEE Transactions on Human-Machine Systems 49, 3 (2018), 278–289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [32] Luis A Leiva, Asutosh Hota, and Antti Oulasvirta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Enrico: A dataset for topic modeling of mobile ui designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In 22nd International Conference on Human- Computer Interaction with Mobile Devices and Services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [33] Hector Levesque, Ernest Davis, and Leora Morgenstern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' The winograd schema challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Thirteenth international conference on the principles of knowl- edge representation and reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [34] Toby Jia-Jun Li, Amos Azaria, and Brad A Myers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' SUGILITE: creating multimodal smartphone automation by demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the 2017 CHI conference on human factors in computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 6038–6049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [35] Toby Jia-Jun Li, Lindsay Popowski, Tom Mitchell, and Brad A Myers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Screen2Vec: Semantic Embedding of GUI Screens and GUI Components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Pro- ceedings of the 2021 CHI Conference on Human Factors in Computing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 1–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [36] Yang Li, Jiacong He, Xin Zhou, Yuan Zhang, and Jason Baldridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Mapping natural language instructions to mobile UI action sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' arXiv preprint arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='03776 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [37] Yang Li, Gang Li, Luheng He, Jingjie Zheng, Hong Li, and Zhiwei Guan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Widget captioning: Generating natural language description for mobile user interface elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='04295 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [38] Ce Liu, Jenny Yuen, and Antonio Torralba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Sift flow: Dense correspondence across scenes and its applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' IEEE transactions on pattern analysis and machine intelligence 33, 5 (2010), 978–994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [39] Thomas F Liu, Mark Craft, Jason Situ, Ersin Yumer, Radomir Mech, and Ranjitha Kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Learning design semantics for mobile apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the 31st Annual ACM Symposium on User Interface Software and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 569–579.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [40] Hongjing Lu, Ying Nian Wu, and Keith J Holyoak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Emergence of analogy from relation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences 116, 10 (2019), 4176–4181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [41] Forough Mehralian, Navid Salehnamadi, and Sam Malek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Data-driven accessibility repair revisited: on the effectiveness of generating labels for icons in Android apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the 29th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 107–118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [42] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Pytorch: An imperative style, high-performance deep learning library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Advances in neural information processing systems 32 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [43] Vidya Ramesh, Charlie Hsu, Maneesh Agrawala, and Björn Hartmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' ShowMeHow: translating user interface instructions between applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the 24th annual ACM symposium on User interface software and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 127–134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [44] Nils Reimers and Iryna Gurevych.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Sentence-bert: Sentence embeddings using siamese bert-networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' arXiv preprint arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='10084 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [45] Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Faster r-cnn: Towards real-time object detection with region proposal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Advances in neural information processing systems 28 (2015), 91–99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [46] Ethan Rublee, Vincent Rabaud, Kurt Konolige, and Gary Bradski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' ORB: An efficient alternative to SIFT or SURF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In 2011 International conference on computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Ieee, 2564–2571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [47] Masoud Jalili Sabet, Philipp Dufter, François Yvon, and Hinrich Schütze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' SimAlign: High quality word alignments without parallel training data using static and contextualized embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' arXiv preprint arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='08728 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [48] Abigail See, Peter J Liu, and Christopher D Manning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Get to the point: Summarization with pointer-generator networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' arXiv preprint arXiv:1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='04368 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Screen Correspondence: Mapping Interchangeable Elements between UIs , , [49] Peter Shaw, Jakob Uszkoreit, and Ashish Vaswani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Self-attention with relative position representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' arXiv preprint arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='02155 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [50] Ming Shen, Pratyay Banerjee, and Chitta Baral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Unsupervised Pronoun Resolution via Masked Noun-Phrase Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' arXiv preprint arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='12392 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [51] Karen Simonyan and Andrew Zisserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Very deep convolutional networks for large-scale image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' arXiv preprint arXiv:1409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='1556 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [52] Amanpreet Singh, Vivek Natarajan, Meet Shah, Yu Jiang, Xinlei Chen, Dhruv Batra, Devi Parikh, and Marcus Rohrbach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Towards vqa models that can read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 8317–8326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [53] Ray Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' An overview of the Tesseract OCR engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Ninth international conference on document analysis and recognition (ICDAR 2007), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' IEEE, 629– 633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [54] Hao Tan and Mohit Bansal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Lxmert: Learning cross-modality encoder representations from transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' arXiv preprint arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='07490 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [55] Michael Tomasello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Constructing a language: A usage-based theory of language acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Harvard university press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [56] Oriol Vinyals, Charles Blundell, Timothy Lillicrap, Daan Wierstra, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Matching networks for one shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Advances in neural information pro- cessing systems 29 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [57] Bryan Wang, Gang Li, Xin Zhou, Zhourong Chen, Tovi Grossman, and Yang Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Screen2words: Automatic mobile UI summarization with multimodal learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In The 34th Annual ACM Symposium on User Interface Software and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 498–510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [58] Shufan Wang, Laure Thompson, and Mohit Iyyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Phrase-bert: Improved phrase embeddings from bert with an application to corpus exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' arXiv preprint arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content='06304 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [59] Jason Wu, Karan Ahuja, Richard Li, Victor Chen, and Jeffrey Bigham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' ScratchThat: Supporting Command-Agnostic Speech Repair in Voice-Driven Assistants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Proceedings of the ACM on Interactive, Mobile, Wearable and Ubiquitous Technologies 3, 2 (2019), 1–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [60] Jason Wu, Xiaoyi Zhang, Jeff Nichols, and Jeffrey P Bigham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Screen Parsing: Towards Reverse Engineering of UI Models from Screenshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In The 34th Annual ACM Symposium on User Interface Software and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 470–483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [61] Rahulkrishna Yandrapally, Andrea Stocco, and Ali Mesbah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Near-duplicate detection in web app model inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the ACM/IEEE 42nd International Conference on Software Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 186–197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [62] Tom Yeh, Tsung-Hsiang Chang, Bo Xie, Greg Walsh, Ivan Watkins, Krist Wong- suphasawat, Man Huang, Larry S Davis, and Benjamin B Bederson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Creating contextual help for GUIs using screenshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the 24th annual ACM symposium on User interface software and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 145–154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [63] Ja Eun Yu and Debaleena Chattopadhyay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' “Maps are hard for me”: Identify- ing How Older Adults Struggle with Mobile Maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In the 22nd international ACM SIGACCESS conference on computers and accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [64] Xiaoxue Zang, Ying Xu, and Jindong Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Multimodal Icon Annotation For Mobile Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the 23rd International Conference on Mobile Human-Computer Interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [65] Xiaoyi Zhang, Lilian de Greef, Amanda Swearngin, Samuel White, Kyle Murray, Lisa Yu, Qi Shan, Jeffrey Nichols, Jason Wu, Chris Fleizach, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Screen Recognition: Creating Accessibility Metadata for Mobile Applications from Pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 1–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [66] Xiaoyi Zhang, Anne Spencer Ross, Anat Caspi, James Fogarty, and Jacob O Wobbrock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Interaction proxies for runtime repair and enhancement of mobile application accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the 2017 CHI conference on human factors in computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 6024–6037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [67] Xiaoyi Zhang, Anne Spencer Ross, and James Fogarty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Robust annotation of mobile application interfaces in methods for accessibility repair and enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Proceedings of the 31st Annual ACM Symposium on User Interface Software and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 609–621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [68] Mingyuan Zhong, Gang Li, Peggy Chi, and Yang Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' HelpViz: Automatic Generation of Contextual Visual Mobile Tutorials from Text-Based Instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In The 34th Annual ACM Symposium on User Interface Software and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 1144–1153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' [69] Xin Zhou and Yang Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' Large-Scale Modeling of Mobile User Click Behaviors Using Deep Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' In Fifteenth ACM Conference on Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} +page_content=' 473– 483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE_T4oBgHgl3EQf7xzu/content/2301.08372v1.pdf'} diff --git a/o9FKT4oBgHgl3EQfGS3Z/content/2301.11725v1.pdf b/o9FKT4oBgHgl3EQfGS3Z/content/2301.11725v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9ff92574fbb2b3c3e134647a064159ea364dfff6 --- /dev/null +++ b/o9FKT4oBgHgl3EQfGS3Z/content/2301.11725v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a01a6432818a3daedc94a2c749db674198795055c42d9f1e984fe85a46c54315 +size 585036 diff --git a/oNE0T4oBgHgl3EQf8wJq/content/tmp_files/2301.02792v1.pdf.txt b/oNE0T4oBgHgl3EQf8wJq/content/tmp_files/2301.02792v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1b9504a37fdaa2f2675afc6fb58307b26cf42f55 --- /dev/null +++ b/oNE0T4oBgHgl3EQf8wJq/content/tmp_files/2301.02792v1.pdf.txt @@ -0,0 +1,1150 @@ +Linguistic-style-aware Neural Networks for +Fake News Detection +Xinyi Zhou1*†, Jiayu Li2*, Qinzhou Li3†, Reza Zafarani2 +1Paul G. Allen School of Computer Science and Engineering, University of Washington, Seattle, WA 98195 +2Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244 +3Google, Durham, NC 27701 +Abstract—We propose the hierarchical recursive neural net- +work (HERO) to predict fake news by learning its linguistic style, +which is distinguishable from the truth, as psychological theories +reveal. We first generate the hierarchical linguistic tree of news +documents; by doing so, we translate each news document’s lin- +guistic style into its writer’s usage of words and how these words +are recursively structured as phrases, sentences, paragraphs, and, +ultimately, the document. By integrating the hierarchical linguis- +tic tree with the neural network, the proposed method learns +and classifies the representation of news documents by capturing +their locally sequential and globally recursive structures that +are linguistically meaningful. It is the first work offering the +hierarchical linguistic tree and the neural network preserving +the tree information to our best knowledge. Experimental results +based on public real-world datasets demonstrate the proposed +method’s effectiveness, which can outperform state-of-the-art +techniques in classifying short and long news documents. We +also examine the differential linguistic style of fake news and the +truth and observe some patterns of fake news.1 +Index Terms—fake news, neural network, linguistic style +I. INTRODUCTION +“Fake news,” as deceptive and misleading news articles (or +statements at times), has been broadly discussed along with its +influence on democracies and economies [1]. Public health has +also been negatively impacted, especially with the “infodemic” +that we face along with the pandemic [2]. Effective fake news +detection has thus become an urgent task to mitigate such +detrimental impacts. +Psychological theories, such as Undeutsch hypothesis [3], +have suggested that the linguistic style of fake news is +distinguishable from that of the truth. Therefore, effective +techniques can be designed to identify fake news by analyzing +the linguistic style of news articles [1]. Linguistic style can be +captured by looking at the writer’s usage of words (lexically +and semantically) and the way these words are further formed +into sentences (syntactic level) and the document (discourse +level) [1]. Within a machine learning framework, existing +studies have captured a news article’s linguistic style by +computing the frequencies of each word [4], [5], part of speech +(POS, at the syntactic level) [5]–[7], and rhetorical relationship +(RR, at the discourse level) [5], [8]. These frequencies form +a news article’s representation, which is further classified by, +*The authors are equally contributed. +†This work was completed while the authors were at Syracuse University. +1The code and data are available at https://github.com/Code4Graph/HERO. +e.g., support vector machines (SVM) and random forests to +predict the news as fake news or the truth. +These studies have advanced linguistic-style-aware fake +news +prediction. +However, +translating +a +news +article’s +linguistic style into the appearances of words, POSs, and RRs +overlooks the linguistic structure that reveals how the article’s +words, POSs, and RRs are assembled. Specifically, we can +form a hierarchical linguistic tree for each news article; see +Section III-A for the details and Figure 1 for an illustrated +tree +for +the +news +piece +“Vitamin D determines +severity in COVID-19, so government advice +needs to change, experts urge: Researchers +point to changes in government advice +in Wales, England, and Scotland.” +This +tree +explicitly presents the order of words used in the article, +syntactic structure revealing how these words are recursively +structured as the elementary discourse units (EDUs, which +are meaningful phrases, sentences, or paragraphs) through +POSs, and discourse structure exhibiting how these EDUs +are recursively structured as the entire article through RRs. +Previous approaches paid full attention to the tree’s node +information by looking if this news piece used a specific +word (e.g., “COVID-19”), POS (e.g., “NNP”), or RR (e.g., +“NS-elaboration”) in the corpus or how many times it appears +without considering the relational (edge) information among +the nodes. Although Zhou et al. [9] and P´erez-Rosas et +al. [4] computed the frequencies of production rules (at +the syntactic level only), each rule can merely show the +structure within a fundamental component of the tree (i.e., +parent-children, such as VP → VBZ NP). Each fundamental +component is investigated independently by overlooking +how components are connected to form the tree; the tree’s +structure is hence preserved locally rather than globally. In +addition, the representation of news articles obtained by the +frequencies of these local structures is often high-dimensional +and sparse, which can be adverse to the prediction task. +Present work. To address the above problems, we propose the +hierarchical recursive neural network (HERO) for fake news +prediction. The architecture of the proposed neural network +adaptively preserves the global structure of the hierarchical +linguistic tree of various news articles. To our best knowledge, +2Source: https://www.politifact.com/factchecks/2020/may/01/rudy-giuliani/ +rudy-giuliani-wrong-about-us-policy-grant-amount-w/ +arXiv:2301.02792v1 [cs.CL] 7 Jan 2023 + +3.7 +Why did the US in 2017 give $3.7m to the Wuhan Lab in China? +Such grants were prohibited in 2014. Did President Obama grant an exception? +Why did the US in 2017 give $3.7m to +the Wuhan Lab in China? +NS-Elaboration +Such grants were prohibited in 2014. +Did President Obama grant an exception? +WHADVP +SP +? +Why +did +VP +PP +VP +the +US +in +2017 +give +NP +PP +$ +m to +NP +NP +PP +the Wuhan +Lab +in +China +NP +VP +VP +Such +grants +were +VP +. +prohibited +PP +in +2014 +Did +NP +VP +? +president +Obama +grant +NP +an +exception +Fig. 1: The hierarchical linguistic tree for the news piece “Why did the US in 2017 give $3.7m to the Wuhan Lab in China? +Such grants were prohibited in 2014. Did President Obama grant an exception?” verified as a false statement by PolitiFact.2Blue +nodes: RRs. Green nodes: POSs. +this is the first work that develops hierarchical linguistic +trees. Leveraging the developed trees, the proposed neural +network can learn the linguistic-style-aware representations +of news articles by explicitly capturing the writers’ usage of +words and the linguistically meaningful ways in which these +words are structured as phrases, sentences, paragraphs, and, +ultimately, the documents. We conduct extensive experiments +on real-world datasets with well-established and state-of-the- +art approaches, which demonstrate the effectiveness of the +proposed neural network in predicting fake news. Additionally, +we examine the differential linguistic style of fake news and +the truth and identify statistically significant and consistent +patterns of fake news across datasets. +The rest of this paper is organized as follows. We review re- +lated work in Section II. We introduce the proposed method in +Section III and detail the experiments designed and conducted +to evaluate the proposed method in Section IV. We conclude +in Section V. +II. RELATED WORK +Fake news prediction methods can be categorized as +content-based or propagation-based depending on whether the +method focuses on investigating news content or its propaga- +tion on social media. +Propagation-based methods can utilize rich auxiliary social +media information, including news spreaders’ intent [2] or +profiles [10], relationships between news spreaders and their +posts [11], social feedback [12]–[14], social networks [15], and +propagation paths [16], [17]. Nevertheless, they can only be +deployed after news articles published on news outlets have +been disseminated on social media. In comparison, content- +based methods have the primary advantage of predicting fake +news early when news articles have been published online but +have not been spread [5]. Additionally, an effective content- +based method can be easily extended further by incorporating +social context information. With this consideration, we focus +on analyzing news content to predict fake news and review re- +lated work on content-based fake news prediction approaches. +As news articles are mainly text, content-based methods +start by manually extracting linguistic features and predicting +fake news using common classifiers such as SVM [4]. Such +linguistic features have been related to lexicons (e.g., bag-of- +words) [5], POSs [5], [6], context-free grammars (production +rules) [4], [5], RRs [5], [8], readability [6], [18], and n- +grams that preserve the sequences of words or POSs [7]. +Though news features can be easily interpreted within this +machine learning framework, features cannot be automatically +extracted, which can significantly impact the prediction per- +formance; hence, the performance heavily relies on experts’ +involvement and experience. More importantly, as detailed in +Section I, it is difficult to capture the global structure of news +text (language) at any of the syntactic and discourse levels +with these hand-crafted features. Compared to these methods, +the proposed neural network can learn the features of news +articles, which capture the global and hierarchical structures +that news linguistic styles carry. +Recently, neural networks (e.g., Bi-LSTM [7], [19] and +Text-CNN [20]) have been frequently employed to identify +fake news. These models can learn the features of news +text (sometimes, combined with other modalities in news +content, such as images [20], [21]). These neural networks + +News +document +D +D +EDU +RR +EDU +EDU +Discourse parsing +EDU +POS +W +W +W +W +POS +EDU +POS +W +W +POS +EDU +POS +W +W +POS +Constituency +parsing + T F +Softmax +Fake/true news +Hierarchical linguistic tree construction +Feature extraction via hierarchical recursive neural network +Fake news prediction +Output +Input +Document +EDUs +CoreNLP +Discourse +structure +w/o RRs +Discourse +structure +Level-specific classifiers +Transition-based +system +LayerNorm ++ +LayerNorm ++ +word +tag +position ++ +Input +Encoder +Decoder +Output +Syntactic +structure +Multi-head +Attention +Feed Forward +W +Word +embedding +W +W +W +W +W +Linguistic-style- +aware document +embedding +Recursive +aggregation +Bi-GRU +Avg pooling +Bi-GRU +Avg pooling +Bi-GRU +Avg pooling +W +W +Bi-GRU +Avg pooling +Bi-GRU +Avg pooling +Bi-GRU +Avg pooling +Fig. 2: Framework overview, which contains a top-bottom building process of hierarchical linguistic trees, a bottom-top feature +extraction process using the proposed hierarchical recursive neural network, and a classifier to predict fake news. The neural +network’s architecture adaptively preserves various news documents’ global and hierarchical linguistic tree structures. The +Bi-GRU aggregator catches text’s local sequentiality that is linguistically valuable and often short (explained in Section III-B), +which is more effective than self-attention here (see Section IV-B for details). +have focused on the sequentiality or locality of news text but +not on its linguistic structure. In comparison, the proposed +neural network explicitly catches this structure; it also captures +text’s sequentiality and locality, which will be detailed in +Section III-B. We point out that the proposed neural network +provides a fundamental approach to news text representation +learning and thus can be easily extended for multimodal fake +news prediction. +III. METHODOLOGY +We specify the proposed model in this section, which can +be divided into three steps. For each news document, we first +construct its hierarchical linguistic tree (see Section III-A), +then extract its features via the proposed hierarchical recursive +neural network that preserves the hierarchical linguistic tree +information (see Section III-B), and finally predict it as fake +news or the truth (see Section III-C). Figure 2 presents the +framework overview. +A. Hierarchical Linguistic Tree Construction +Given a news document D, we first generate its hierarchical +linguistic tree. The tree can explicitly present the order of +words used in the document and how these words shape +EDUs (meaningful phrases, sentences, or paragraphs) and +further shape the entire document. An example is shown +in Figure 1. Specifically, our first attempt is to obtain D’s +discourse (rhetorical) structure, which identifies D’s EDUs and +reveals how these EDUs recursively form the document D. To +this end, we first utilize Standford CoreNLP [22] to segment +D into EDUs. Then, we apply a modified transition-based +system [23] to identify span (S) and nuclearity (N), based +on which D’s rhetorical structure can be obtained without +recognizing specific RRs (e.g., elaboration in Figure 1). This +semi-naked tree structure allows us to divide each RR node +into within-sentence, across-sentence, and across-paragraph +levels in terms of its left and right subtrees, extract structural +features for each RR node, and ultimately adopt level-specific +SVM classifiers [23] to predict the node attribute (i.e., the +specific RR). This multi-stage approach outperforms the well- +established one [24] in our experiments, where [24] works +as an integrated system. Finally, we employ a state-of-the-art +discriminative constituency parser for each identified EDU of +the document D to obtain its syntactic structure [25]. The +parser consists of a self-attentive encoder [25] and a chart +decoder [26] (see Figure 2 for the detailed architecture). The +syntactic structure reveals how the EDU’s words recursively +form the entire EDU. +B. Feature Extraction via Hierarchical Recursive Neural Network +We propose the hierarchical recursive neural network to +extract features of news documents, whose architecture adap- +tively maintains the global structure of the hierarchical lin- +guistic trees of news documents. + +Given a news document D, its feature extraction using +the hierarchical recursive neural network is bottom-top. We +first encode D’s words, which are the leaf nodes of D’s +hierarchical linguistic tree. Then, we aggregate the obtained +embeddings of the words that attach to the same parent node, +forming the embedding of their parent node. The hierarchical +recursive neural network will repeat such aggregations from +the lower (syntactic) level to the upper (discourse) level until +the document D as the tree’s root node is embedded. +Hence, the question arises of how the aggregator performs +on a recurring fundamental component (i.e., a depth-one +parent-children structure). Note that for each parent node in +a hierarchical linguistic tree, its children contain local and +sequential information of the corresponding news document. +The information is local because it reveals partial information +about the overall news content that is linguistically valuable. +It is sequential as we keep the order of words of the news +document. Naturally, recurrent neural networks can be adopted +as the aggregator to catch the sequentiality of children sharing +the same parent. The information locality further relieves the +pressure on recurrent neural networks to keep the dependency +of long entities since the number of children for a parent node +is no more than the EDU length and essentially less than +the document length. As seen from Figure 1, the maximum +length that recurrent neural networks require to process is four, +whereas the document has 29 tokens. +With the above considerations, we develop Bi-GRU (bidi- +rectional gated recurrent unit), one of the well-established +recurrent neural networks [27], to aggregate the embeddings +of all the child nodes to represent their parent node. We also +empirically compare Bi-GRU with multi-head self-attention +that has remarkably performed in many tasks; Bi-GRU is more +effective for the proposed model. Formally, the embedding of +a parent node is computed as +xp = +� +c∈Cp[−−−→ +GRU{xc} ⊕ ←−−− +GRU{xc}] +|Cp| +, +(1) +where p denotes the parent node and Cp is the set of child +nodes of p. Vectors xc ∈ Rd and xp ∈ Rd refer to the features +of the child and parent node, respectively. The operator ⊕ +denotes concatenation. The GRU is formulated as follows: +ri = σ(Wrxi + Urhi−1), +zi = σ(Wzxi + Uzhi−1), +ˆhi = tanh(Whxi + Uh(hi−1 ⊙ ri)), +hi = (1 − zi) ⊙ hi−1 + zi ⊙ ˆhi, +(2) +where hi ∈ Rd/2 is the output hidden state of the i-th +child, with h0 += 0. The symbol ⊙ denotes Hadamard +product. Matrices W∗ ∈ R(d/2)×d and U∗ ∈ R(d/2)×(d/2) +(∗ ∈ {r, z, h}) are learnable parameters. ri and zi are the reset +gate and update gate, respectively. σ and tanh are the sigmoid +and hyperbolic tangent activation functions, respectively. In a +nutshell, the above architecture first employs the Bi-GRU to +capture “deep” sequential feature interactions of all the child +features and then uses a mean pooling layer over all the hidden +states to obtain the parent node’s features. +After determining the aggregation within each recurring fun- +damental component, we introduce three specific hierarchical +recursive neural networks (HEROs): +• Unified HERO: The first hierarchical recursive neural +network is the one with unified aggregators. In other +words, all the Bi-GRUs in the neural network share the +same set of W∗ and U∗ (∗ ∈ {r, z, h}, see Equation (2)). +• Level-specific HERO: It is the hierarchical recursive neu- +ral network with level-specific aggregators. As detailed, +the hierarchical linguistic tree presents both syntactic +and rhetorical structures of news content, and the hier- +archical recursive neural network preserves such struc- +tures. Hence, all the Bi-GRUs in a hierarchical recursive +neural network can be grouped by the level (syntax +or discourse) they belong to the corresponding tree. +We define L(v) as the function that maps a certain +vertex in the hierarchical linguistic tree to its linguistic +level (i.e., L(v) ∈ {syntax, discourse}). Then, Equation +(1) can be reformulated as xp = +1 +|Cp| +� +c∈Cp[−−−→ +GRU L(c) +{xc} ⊕ ←−−− +GRU L(c){xc}]. +• Attribute-specific HERO: It stands for the hierarchical +recursive neural network with attribute-specific aggrega- +tors. In other words, we categorize the hierarchical recur- +sive neural network’s recurring fundamental components +according to the attributes of their parent nodes in the +corresponding hierarchical linguistic tree, which can be +various POSs and RRs. We deploy the same Bi-GRU for +the components within each category and the different Bi- +GRU for the components falling into different categories. +Mathematically, we define A(v) as the function that +maps a certain vertex in the hierarchical linguistic tree +to its attributes. Assume there are m different POSs +and n RRs, we have A(v) ∈ {POSi, RRj +: i = +1, 2, · · · , m, j = 1, 2, · · · , n}. The root vertex would +be assigned with a RR in discourse parsing. For EDU +vertices, they would not be assigned with any RRs in +discourse parsing but with some POSs in constituency +parsing. In this way, Equation (1) is rewritten as xp = +1 +|Cp| +� +c∈Cp[−−−→ +GRU A(p){xc} ⊕ ←−−− +GRU A(p){xc}]. +C. Fake News Prediction +We add a softmax classifier on the top of the proposed hier- +archical recursive neural network to predict the document D as +fake news or the truth. Let hD denote D’s features extracted +via the proposed hierarchical recursive neural network. The +softmax function maps hD to the probability of D being a +fake news document by pD = Softmax(WhD + b), where +W and b are learnable parameters. +To learn the parameters Θ = {W∗, U∗, W, b} within the +neural network and classifier, we employ cross-entropy to +calculate the classification loss in the model training process. +Assume we have q verified news documents D = {Di}q +i=1 +with the ground-truth labels Y = {yi : yi ∈ {0, 1}}q +i=1 (yi = 0 +for true news, and yi = 1 for fake news), the loss is computed + +TABLE I: Data statistics. +Recovery +MM-COVID +# news documents +2,029 +3,536 +- true news +1,364 +1,444 +- fake news +665 +2,092 +Avg. # words per EDU +24 +17 +Avg. # EDUs per document +38 +2 +Avg. # words per document +841 +16 +by L = − 1 +q +�q +i=1[yDi log pDi +(1−yDi) log(1−pDi)]. Based +on it, the parameter set Θ is estimated by ˆΘ = arg minΘ L. +IV. EMPIRICAL EVALUATION +We aim to evaluate the proposed method by answering the +following three questions. +1) How effective is the proposed model in fake news +prediction compared to the-state-of-art approaches? +2) Is the hierarchical linguistic structure of news documents +essential in representing their linguistic styles? +3) What characterizes the linguistic style of fake news as +distinguishable from the truth? +To that end, we first detail our experimental setup in Sec- +tion IV-A and then compare the proposed unified, level- +specific, and attribute-specific HEROs in predicting fake news +(see Section IV-B). Subsequently, we compare the proposed +model with the baselines to verify its effectiveness in predict- +ing fake news (to answer RQ1, see Section IV-C) and conduct +the ablation study to assess the importance of our developed +hierarchical linguistic trees (to answer RQ2, see Section IV-D). +Finally, we characterize the linguistic style of fake news +as distinguishable from the truth by doing quantitative and +comparative analyses (to answer RQ3, see Section IV-E). +A. Experimental Setup +We first introduce the datasets used for evaluation (see +Section IV-A1), followed by the baselines for comparison (see +Section IV-A2). Finally, we detail our implementation details +in Section IV-A3. +1) Datasets: We conduct experiments on two benchmark +datasets in fake news prediction: Recovery [9] and MM- +COVID [28]. Both datasets contain labeled news documents. +Differently, news documents collected in Recovery are articles +(long text, often including multiple paragraphs) but in MM- +COVID are statements (short text, often formed by one or two +sentences). We present the detailed statistics of two datasets +in Table I. +2) Baselines: We involve the following well-received and +state-of-the-art methods as baselines in our experiments. +• HCLF [5]: HCLF stands for hand-crafted linguistic fea- +ture. Each news document’s HCLFs include the frequen- +cies of words (i.e., bag-of-word features), POSs, RRs, and +production rules. The extracted features are used to pre- +dict fake news by employing well-established classifiers. +Here we examine a comprehensive list of classifiers– +logistic regression, SVM, k-nearest neighbors, decision +trees, naive Bayes, random forest, and AdaBoost–and +select the one performing best. +• EANN [20]: The event adversarial neural network con- +tains three components: feature extraction by Text-CNN +(for text) and VGG-19 (for images), event discrimination +to learn event-invariant features of news content, and fake +news prediction. We exclude the visual features for a fair +comparison. +• HAN [29]: HAN exploits attention-GRU for news clas- +sification. It captures the hierarchical sequence of docu- +ments; i.e., each document is a sequence of its sentences, +and each of its sentences is a sequence of words. +• DRNN [30]: DRNN is a discourse-structure-aware neu- +ral network, which focuses on the tree with rhetorical +relationships as edge attributes and leverages an atten- +tion mechanism for news classification. In other words, +DRNN differs from HERO in the aggregation rule. Com- +pared to DRNN’s tree, the hierarchical linguistic tree +integrates syntax-level structures and has RRs as nodes +and non-attributed edges at the discourse level. DRNN is +developed to categorize news documents with more than +one elementary discourse unit; otherwise, it is reduced to +Bi-LSTM. +• Text-GCN [31]: The approach develops the graph convo- +lutional neural network for news classification. The graph +investigates the co-occurrence relationship among news +documents and the words within the documents. +• Transformer [32]: It is a deep neural network model +with a self-attention-based encoder-decoder architecture, +which has excellently performed in diverse natural lan- +guage processing tasks. Here, we consider Transformer’s +encoder–applicable for classification tasks–as a baseline +to predict fake news, a non-pretrained version rather than +pretrained models (e.g., BERT) for a fair comparison +as the pretrained Transformers have learned large-scale +external resources. +3) Implementation Details: +We randomly divide each +dataset into 0.7:0.1:0.2 proportions for model training, vali- +dation, and testing. Macro-F1, micro-F1, and AUC are used +to evaluate the performance of methods in news classification. +The discourse parser is pretrained using RST-DT [23], and the +constituency parser is pretrained using the Penn Treebank [25]. +For the neural-network-based models, we uniformly utilize the +pretrained GloVe [33] to obtain semantic-aware embeddings of +words, with 100 as the embedding dimension. The hidden di- +mension within neural networks is set as 100, correspondingly. +We deploy Adam optimizer to learn parameters, with 50 as the +maximum number of epochs. We perform a grid search over +the learning rate ∈ {0.1, 0.01, 0.001, 0.0001} with validation +data. In the end, 0.0001 performs best for our models and +most of the baselines other than Transformer (0.001) and Text- +GCN (0.01). All the experiments of the neural networks are +implemented with PyTorch and are conducted on an NVIDIA +Quadro RTX 6000 GPU (24 GB memory), Intel(R) Xeon(R) +Gold 6248R CPU (3.00 GHz), and with 64 GB of RAM. For + +TABLE +II: +Performance +of +unified, +level-specific, +and +attribute-specific HEROs in fake news prediction. Attribute- +specific HERO performs best, demonstrating that the node +attributes (POSs or RRs) in hierarchical linguistic trees are +essential. MAF1: Macro-F1. MIF1: Micro-F1 +Recovery +MM-COVID +HERO +MAF1 +MIF1 +AUC +MAF1 +MIF1 +AUC +Unified +0.801 +0.822 +0.827 +0.889 +0.891 +0.899 +Level-specific +0.817 +0.838 +0.841 +0.878 +0.878 +0.892 +Attribute-specfic +0.850 +0.869 +0.866 +0.894 +0.896 +0.896 +TABLE III: Performance of the proposed model, HERO, and +baselines in fake news prediction. HERO outperforms the +baselines by 2–17% in AUC on Recovery and by 3–30% in +MM-COVID. MAF1: Macro-F1. MIF1: Micro-F1 +Recovery +MM-COVID +MAF1 +MIF1 +AUC +MAF1 +MIF1 +AUC +HCLF +0.752 +0.801 +0.746 +0.566 +0.624 +0.577 +Transformer +0.774 +0.793 +0.810 +0.804 +0.809 +0.806 +Text-GCN +0.841 +0.869 +0.835 +0.826 +0.836 +0.817 +EANN +0.811 +0.864 +0.795 +0.825 +0.926 +0.833 +HAN +0.847 +0.869 +0.844 +0.840 +0.856 +0.846 +DRNN +0.711 +0.778 +0.698 +0.845 +0.846 +0.848 +HERO +0.850 +0.869 +0.866 +0.894 +0.896 +0.896 +HCLFs, classifiers are used with the default hyperparameters +presented in the scikit-learn library. Z-score normalization is +applied for the feature matrix to enhance the classification +performance. +B. Determining the Best HERO +We compare the performance of the proposed neural net- +works with unified, level-specific, and attribute-specific Bi- +GRUs in predicting fake news. Table II presents the results. +The results indicate that with Recovery data, the performance +ranking is attribute-specific HERO > level-specific HERO > +unified HERO. Specifically, attribute-specific HERO correctly +predicts news as fake or true with 0.85 macro-F1 and 0.87 +micro-F1 and AUC, outperforming unified HERO by ∼4% +and level-specific HERO by ∼3%. With MM-COVID data, +the performance ranking is attribute-specific HERO ≈ unified +HERO > level-specific HERO. Attribute-specific and unified +HEROs achieve ∼0.89–0.90% in macro-F1, micro-F1, and +AUC, outperforming level-specific HERO by ∼1%. In con- +clusion, attribute-specific HERO performs best in classifying +long articles and short statements as fake news or the truth. +This result demonstrates the importance of the node attributes +(POSs or RRs) in developed hierarchical linguistic trees. +Additionally, +we +compare +Bi-GRU +and +self-attention +(#heads=10) as aggregators in the proposed hierarchical re- +cursive neural network for fake news prediction. The results +indicate that Bi-GRU performs better than self-attention by at +least 1% in AUC on both datasets. +Unified +Level-specific +Attribute-specific +0.0 +0.2 +0.4 +0.6 +0.8 +AUC +HERO\(Syn+Dis) +HERO\Syn +HERO\Dis +HERO +(a) Recovery +Unified +Level-specific +Attribute-specific +0.0 +0.2 +0.4 +0.6 +0.8 +AUC +HERO\(Syn+Dis) +HERO\Syn +HERO\Dis +HERO +(b) MM-COVID +Fig. 3: Ablation study. (a) The proposed HERO outper- +forms HERO\Dis by 1% in AUC for unified HERO and by +3% for level- and attribute-specific HEROs. It outperforms +HERO\Syn by 7–9% and HERO\Syn by 30%+ in AUC. +(b) HERO performs similarly to HERO\Dis as MM-COVID +contains short statements having minimal discourse struc- +tures (i.e., syntax-level structures dominate). It outperforms +HERO\Syn and HERO\(Syn+Dis) by 10%+ in AUC. Thus, +syntax- and discourse-level structures are both essential. +C. Comparing HERO with Baselines +We compare the proposed model with the baselines in +predicting fake news. The results presented in Table III re- +veal that the proposed model can generally outperform the +baselines. Specifically, the proposed model has an AUC score +approaching 0.87, outperforming HAN by more than 2%, Text- +GCN by more than 3%, Transformer by more than 5%, EANN +by more than 7%, HCLF by 12%, and DRNN by 17%. With +MM-COVID data, the proposed model has an AUC score +approaching 0.90, outperforming EANN by more than 6%, +DRNN and HAN by ∼5%, Text-GCN and Transformer by +∼8-9%, and HCLF by more than 30%. From the table, we +also observe that the proposed model outperforms EANN by +6–7% in macro-F1 and AUC but underperforms it by ∼3% +in micro-F1 on MM-COVID. This result suggests that EANN +tends to predict given news statements as the major class. +D. Ablation Study +We compare the proposed model, HERO, which contains +hierarchical linguistic (syntax- and discourse-level) structures +with its following variants. +• HERO\Dis: It stands for the variant of HERO with only +syntax-level structures. In this variant, the embedding of a +news document is obtained by averaging its embeddings +of EDUs. +• HERO\Syn: It stands for the variant of HERO with only +discourse-level structures. In this variant, the embedding +of each EDU of a news document is obtained by averag- +ing its words. +• HERO\(Syn+Dis): It stands for the variant of HERO +with no structures, the embedding of a news document is +directly obtained by averaging its word embeddings. + +Fake True +1.500 +1.525 +1.550 +Avg Child Cnt +Articles +Fake True +1.3 +1.4 +1.5 +1.6 +Statements +(a) Children of parent nodes. +Fake True +0.010 +0.015 +0.020 +% EDUs +Articles +Fake True +0.02 +0.04 +0.06 +Statements +Fake True +0.02 +0.04 +0.06 +% POS:NNS +Articles +Fake True +0.00 +0.05 +0.10 +0.15 +Statements +Fake True +0.04 +0.06 +0.08 +% POS:IN +Articles +Fake True +0.00 +0.05 +0.10 +0.15 +Statements +(b) Note attributes. +Fake True +50 +100 +# Nodes (Syn) +Articles +Fake True +0 +50 +100 +Statements +Fake True +10 +20 +30 +40 +# Leafs (Syn) +Articles +Fake True +0 +20 +40 +Statements +Fake True +10 +20 +30 +Max Width (Syn) +Articles +Fake True +5 +10 +15 +Statements +Fake True +10 +20 +30 +Depth (Syn) +Articles +Fake True +5 +10 +15 +20 +Statements +Fake True +0 +2000 +4000 +6000 +# Nodes +Articles +Fake True +0 +100 +# Nodes (Dis) +Articles +Fake True +0 +100 +200 +300 +Max Width +Articles +Fake True +5 +10 +Max Width (Dis) +Articles +(c) Size, width, and depth of trees. +Fig. 4: Hierarchical linguistic trees of fake and true news. Orange solid line: Median. Green dashed line: Mean. +The results are visualized in Figure 3. We observe that with +Recovery data, the proposed HERO outperforms HERO\Dis +by 1% in AUC for unified HERO and by 3% for level- and +attribute-specific HEROs. It outperforms HERO\Syn by 7– +9% and notably outperforms HERO\Syn by above 30% in +AUC. With MM-COVID data, the proposed HERO performs +similarly to HERO\Dis since the statements presented in MM- +COVID are short with two EDUs on average and hence +have minimal discourse structures (i.e., syntax-level structures +dominate hierarchical linguistic structures). Meanwhile, it out- +performs HERO\Syn and HERO\(Syn+Dis) by more than +10% in AUC. Therefore, we conclude that the proposed HERO +is better than its variants, demonstrating the importance of +hierarchical linguistic structures. +E. Characterizing Linguistic Style of Fake News +Fake news has been theoretically identified with a linguistic +style distinguishable from the truth [3]. This experiment aims +to specify this different linguistic style of fake news. We com- +pare the hierarchical linguistic trees generated by fake news +and the truth, which we develop to represent the linguistic +style of news documents systematically. The comparison is +from the (i) children of parent nodes, (ii) attributes of nodes, +and (iii) size, width, and depth of trees. +a) Children of Parent Nodes: We compare fake news +with real news in the average and the maximum number of +children of parent nodes in hierarchical linguistic trees. Results +that are statistically significant with a p-value < 0.001 (using +t-test, unless otherwise specified) are in Figure 4a. We observe +that the hierarchical linguistic trees of fake news have more +child nodes for each parent node than true news on average. +News here indicates long news articles in the Recovery dataset +and short statements in the MM-COVID dataset. +b) Attributes of Nodes: Considering the nodes within +a hierarchical linguistic tree can indicate the document (as +the root), RRs, EDUs, POSs, and words (as the leaf nodes), +we first compare fake news with the truth in the proportion +of RRs, EDUs, POSs, and words, respectively. The reason +for computing their proportions rather than the numbers is +to eliminate the impact of the size of trees (discussed in the +next paragraph). We observe that compared to true news, the +hierarchical linguistic trees of fake news have a significantly +smaller proportion of EDU nodes (p-value < 0.05) and POS +nodes indicating NNS (noun in the plural, p-value < 0.001) +but have a significantly larger proportion of nodes indicat- +ing specific POSs such as IN (preposition or subordinating +conjunction), PP (prepositional phrase), and DT (determiner, +p-value < 0.001). We illustrate the results in Figure 4b. +c) Size, Width, Depth of Trees: We compare fake news +with the truth in the size, maximum width, and depth of +hierarchical linguistic trees. Since hierarchical linguistic trees +contain two-level structures, we also compare fake and true +news in the size, maximum width, and depth of discourse- +and syntactic-level trees. +We observe that the syntactic-level tree of fake news is +generally greater with more nodes, broader, and deeper than +true news. In particular, the syntactic-level tree of fake news +has more leaf nodes than true news, which reveals that fake +news often has longer EDUs with more words than true news. +The above conclusions hold for long news articles (using +Recovery data) and short statements (with MM-COVID data) +with a p-value < 0.01; news files in both datasets are rich +in syntactic information. Figure 4c (the upper ones) presents + +the details. Moreover, we observe that fake news articles +generate smaller and narrower discourse-level trees that lead +to smaller and narrower hierarchical linguistic trees than true +news articles (p-value < 0.01, see the bottom figures in +Figure 4c). We point out that the discourse structures of short +statements are plain with two EDUs on average and hence +have trivial impacts on the shape of the entire hierarchical +linguistic structures. Lastly, we point out that comparing trees’ +maximum and average widths leads to the same conclusions. +Comparing the longest (i.e., depth) and the average distance +between the root and leaves also leads to the same conclusions. +V. CONCLUSION +We propose a psychology-informed neural network to pre- +dict fake news. The proposed neural network learns the +linguistic style of news documents represented by hierarchical +linguistic trees, which explicitly captures the writers’ usage of +words and the linguistically meaningful ways these words are +structured as phrases, sentences, paragraphs, and, ultimately, +documents. We conduct experiments on public real-world +datasets. The results demonstrate the effectiveness of the +proposed neural network, with 0.87–0.90 AUC scores, and the +importance of the developed hierarchical linguistic tree. The +proposed neural network can outperform the previous (recur- +rent, convolutional, graph, and self-attentive) neural networks +and feature-engineering-based approach in predicting news–as +long articles or short statements–as fake news or the truth. +We observe from the data that the hierarchical linguistic trees +of fake news can significantly differ from true news in the +children of parent nodes, the attributes of nodes, and the size, +width, and depth of the trees. In our future work, we aim to +enhance the proposed model’s performance with multimodal +and social-context information. +REFERENCES +[1] X. Zhou and R. Zafarani, “A survey of fake news: Fundamental the- +ories, detection methods, and opportunities,” ACM Computing Surveys +(CSUR), vol. 53, no. 5, pp. 1–40, 2020. +[2] X. Zhou, K. Shu, V. V. Phoha, H. Liu, and R. Zafarani, ““this is fake! +shared it by mistake”: Assessing the intent of fake news spreaders,” in +Proceedings of the ACM Web Conference 2022, 2022, pp. 3685–3694. +[3] U. Undeutsch, “Beurteilung der glaubhaftigkeit von aussagen,” Hand- +buch der psychologie, vol. 11, pp. 26–181, 1967. +[4] V. P´erez-Rosas, B. Kleinberg, A. Lefevre, and R. Mihalcea, “Automatic +detection of fake news,” in Proceedings of the 27th International +Conference on Computational Linguistics, 2018, pp. 3391–3401. +[5] X. Zhou, A. Jain, V. V. Phoha, and R. Zafarani, “Fake news early detec- +tion: A theory-driven model,” Digital Threats: Research and Practice, +vol. 1, no. 2, pp. 1–25, 2020. +[6] M. Potthast, J. Kiesel, K. Reinartz, J. Bevendorff, and B. Stein, “A +stylometric inquiry into hyperpartisan and fake news,” in ACL, 2018, +pp. 231–240. +[7] P. Przybyla, “Capturing the style of fake news,” in AAAI, vol. 34, no. 01, +2020, pp. 490–497. +[8] V. L. Rubin and T. Lukoianova, “Truth and deception at the rhetorical +structure level,” Journal of the Association for Information Science and +Technology, vol. 66, no. 5, pp. 905–917, 2015. +[9] X. Zhou, A. Mulay, E. Ferrara, and R. Zafarani, “ReCOVery: A mul- +timodal repository for COVID-19 news credibility research,” in CIKM, +2020, pp. 3205–3212. +[10] L. Cheng, R. Guo, K. Shu, and H. Liu, “Causal understanding of fake +news dissemination on social media,” in KDD, 2021, pp. 148–157. +[11] E. Min, Y. Rong, Y. Bian, T. Xu, P. Zhao, J. Huang, and S. Ananiadou, +“Divide-and-conquer: Post-user interaction network for fake news de- +tection on social media,” in Proceedings of the ACM Web Conference +2022, 2022, pp. 1148–1158. +[12] K. Shu, L. Cui, S. Wang, D. Lee, and H. Liu, “defend: Explainable fake +news detection,” in KDD, 2019, pp. 395–405. +[13] F. Qian, C. Gong, K. Sharma, and Y. Liu, “Neural user response +generator: fake news detection with collective user intelligence,” in +Proceedings of the 27th International Joint Conference on Artificial +Intelligence, 2018, pp. 3834–3840. +[14] Q. Zhang, A. Lipani, S. Liang, and E. Yilmaz, “Reply-aided detection +of misinformation via bayesian deep learning,” in The World Wide Web +Conference, 2019, pp. 2333–2343. +[15] A. Tommasel, J. M. Rodriguez, and F. Menczer, “Following the trail of +fake news spreaders in social media: A deep learning model,” in Adjunct +Proceedings of the 30th ACM Conference on User Modeling, Adaptation +and Personalization, 2022, pp. 29–34. +[16] C. Naumzik and S. Feuerriegel, “Detecting false rumors from retweet +dynamics on social media,” in Proceedings of the ACM Web Conference +2022, 2022, pp. 2798–2809. +[17] Y. Liu and Y.-F. B. Wu, “Early detection of fake news on social media +through propagation path classification with recurrent and convolutional +networks,” in AAAI, 2018. +[18] Y. Zhu, Q. Sheng, J. Cao, Q. Nan, K. Shu, M. Wu, J. Wang, and +F. Zhuang, “Memory-guided multi-view multi-domain fake news detec- +tion,” IEEE Transactions on Knowledge and Data Engineering, 2022. +[19] H. Karimi and J. Tang, “Learning hierarchical discourse-level structure +for fake news detection,” in NAACL, vol. 1, 2019, pp. 3432–3442. +[20] Y. Wang, F. Ma, Z. Jin, Y. Yuan, G. Xun, K. Jha, L. Su, and J. Gao, +“EANN: Event adversarial neural networks for multi-modal fake news +detection,” in KDD, 2018, pp. 849–857. +[21] L. Cui, S. Wang, and D. Lee, “Same: Sentiment-aware multi-modal +embedding for detecting fake news,” in ASONAM, 2019, pp. 41–48. +[22] C. D. Manning, M. Surdeanu, J. Bauer, J. R. Finkel, S. Bethard, and +D. McClosky, “The Stanford CoreNLP natural language processing +toolkit,” in Proceedings of 52nd Annual Meeting of the Association for +Computational Linguistics: System Demonstrations, 2014, pp. 55–60. +[23] Y. Wang, S. Li, and H. Wang, “A two-stage parsing method for text- +level discourse analysis,” in Proceedings of the 55th Annual Meeting of +the Association for Computational Linguistics (Volume 2: Short Papers), +2017, pp. 184–188. +[24] Y. Ji and J. Eisenstein, “Representation learning for text-level discourse +parsing,” in Proceedings of the 52nd Annual Meeting of the Association +for Computational Linguistics, vol. 1, 2014, pp. 13–24. +[25] N. Kitaev and D. Klein, “Constituency parsing with a self-attentive +encoder,” in ACL, 2018, pp. 2676–2686. +[26] D. Gaddy, M. Stern, and D. Klein, “What’s going on in neural con- +stituency parsers? an analysis,” in Proceedings of the 2018 Conference +of the North American Chapter of the Association for Computational +Linguistics: Human Language Technologies, Volume 1 (Long Papers), +2018, pp. 999–1010. +[27] K. Cho, B. Van Merri¨enboer, C. Gulcehre, D. Bahdanau, F. Bougares, +H. Schwenk, and Y. Bengio, “Learning phrase representations using +RNN encoder–decoder for statistical machine translation,” in Proceed- +ings of the 2014 Conference on Empirical Methods in Natural Language +Processing (EMNLP), 2014, pp. 1724–1734. +[28] Y. Li, B. Jiang, K. Shu, and H. Liu, “MM-COVID: A multilingual and +multimodal data repository for combating COVID-19 disinformation,” +arXiv preprint arXiv:2011.04088, 2020. +[29] Z. Yang, D. Yang, C. Dyer, X. He, A. Smola, and E. Hovy, “Hierarchical +attention networks for document classification,” in NAACL, 2016, pp. +1480–1489. +[30] Y. Ji and N. A. Smith, “Neural discourse structure for text categoriza- +tion,” in ACL, 2017, pp. 996–1005. +[31] L. Yao, C. Mao, and Y. Luo, “Graph convolutional networks for text +classification,” in Proceedings of the AAAI Conference on Artificial +Intelligence, vol. 33, no. 01, 2019, pp. 7370–7377. +[32] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, +Ł. Kaiser, and I. Polosukhin, “Attention is all you need,” in Advances +in Neural Information Processing Systems, 2017, pp. 5998–6008. +[33] J. Pennington, R. Socher, and C. D. Manning, “Glove: Global vectors +for word representation,” in Proceedings of the 2014 Conference on +Empirical Methods in Natural Language Processing, 2014, pp. 1532– +1543. + diff --git a/oNE0T4oBgHgl3EQf8wJq/content/tmp_files/load_file.txt b/oNE0T4oBgHgl3EQf8wJq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f4e0a6a0fd521ef8321c99ff01b2e9590608df23 --- /dev/null +++ b/oNE0T4oBgHgl3EQf8wJq/content/tmp_files/load_file.txt @@ -0,0 +1,836 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf,len=835 +page_content='Linguistic-style-aware Neural Networks for Fake News Detection Xinyi Zhou1*†, Jiayu Li2*, Qinzhou Li3†, Reza Zafarani2 1Paul G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Allen School of Computer Science and Engineering, University of Washington, Seattle, WA 98195 2Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244 3Google, Durham, NC 27701 Abstract—We propose the hierarchical recursive neural net- work (HERO) to predict fake news by learning its linguistic style, which is distinguishable from the truth, as psychological theories reveal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We first generate the hierarchical linguistic tree of news documents;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' by doing so, we translate each news document’s lin- guistic style into its writer’s usage of words and how these words are recursively structured as phrases, sentences, paragraphs, and, ultimately, the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' By integrating the hierarchical linguis- tic tree with the neural network, the proposed method learns and classifies the representation of news documents by capturing their locally sequential and globally recursive structures that are linguistically meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' It is the first work offering the hierarchical linguistic tree and the neural network preserving the tree information to our best knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Experimental results based on public real-world datasets demonstrate the proposed method’s effectiveness, which can outperform state-of-the-art techniques in classifying short and long news documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We also examine the differential linguistic style of fake news and the truth and observe some patterns of fake news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='1 Index Terms—fake news, neural network, linguistic style I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' INTRODUCTION “Fake news,” as deceptive and misleading news articles (or statements at times), has been broadly discussed along with its influence on democracies and economies [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Public health has also been negatively impacted, especially with the “infodemic” that we face along with the pandemic [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Effective fake news detection has thus become an urgent task to mitigate such detrimental impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Psychological theories, such as Undeutsch hypothesis [3], have suggested that the linguistic style of fake news is distinguishable from that of the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Therefore, effective techniques can be designed to identify fake news by analyzing the linguistic style of news articles [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Linguistic style can be captured by looking at the writer’s usage of words (lexically and semantically) and the way these words are further formed into sentences (syntactic level) and the document (discourse level) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Within a machine learning framework, existing studies have captured a news article’s linguistic style by computing the frequencies of each word [4], [5], part of speech (POS, at the syntactic level) [5]–[7], and rhetorical relationship (RR, at the discourse level) [5], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' These frequencies form a news article’s representation, which is further classified by, The authors are equally contributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' †This work was completed while the authors were at Syracuse University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 1The code and data are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='com/Code4Graph/HERO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=', support vector machines (SVM) and random forests to predict the news as fake news or the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' These studies have advanced linguistic-style-aware fake news prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' However, translating a news article’s linguistic style into the appearances of words, POSs, and RRs overlooks the linguistic structure that reveals how the article’s words, POSs, and RRs are assembled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Specifically, we can form a hierarchical linguistic tree for each news article;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' see Section III-A for the details and Figure 1 for an illustrated tree for the news piece “Vitamin D determines severity in COVID-19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' so government advice needs to change,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' experts urge: Researchers point to changes in government advice in Wales,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' England,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' and Scotland.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' This tree explicitly presents the order of words used in the article,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' syntactic structure revealing how these words are recursively structured as the elementary discourse units (EDUs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' which are meaningful phrases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' sentences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' or paragraphs) through POSs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' and discourse structure exhibiting how these EDUs are recursively structured as the entire article through RRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Previous approaches paid full attention to the tree’s node information by looking if this news piece used a specific word (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=', “COVID-19”), POS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=', “NNP”), or RR (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=', “NS-elaboration”) in the corpus or how many times it appears without considering the relational (edge) information among the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Although Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [9] and P´erez-Rosas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [4] computed the frequencies of production rules (at the syntactic level only), each rule can merely show the structure within a fundamental component of the tree (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=', parent-children, such as VP → VBZ NP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Each fundamental component is investigated independently by overlooking how components are connected to form the tree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' the tree’s structure is hence preserved locally rather than globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' In addition, the representation of news articles obtained by the frequencies of these local structures is often high-dimensional and sparse, which can be adverse to the prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' To address the above problems, we propose the hierarchical recursive neural network (HERO) for fake news prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The architecture of the proposed neural network adaptively preserves the global structure of the hierarchical linguistic tree of various news articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' To our best knowledge, 2Source: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='politifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='com/factchecks/2020/may/01/rudy-giuliani/ rudy-giuliani-wrong-about-us-policy-grant-amount-w/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='02792v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='CL] 7 Jan 2023 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='7 Why did the US in 2017 give $3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='7m to the Wuhan Lab in China?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Such grants were prohibited in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Did President Obama grant an exception?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Why did the US in 2017 give $3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='7m to the Wuhan Lab in China?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' NS-Elaboration Such grants were prohibited in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Did President Obama grant an exception?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' WHADVP SP ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Why did VP PP VP the US in 2017 give NP PP $ m to NP NP PP the Wuhan Lab in China NP VP VP Such grants were VP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' prohibited PP in 2014 Did NP VP ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' president Obama grant NP an exception Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 1: The hierarchical linguistic tree for the news piece “Why did the US in 2017 give $3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='7m to the Wuhan Lab in China?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Such grants were prohibited in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Did President Obama grant an exception?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' verified as a false statement by PolitiFact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='2Blue nodes: RRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Green nodes: POSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' this is the first work that develops hierarchical linguistic trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Leveraging the developed trees, the proposed neural network can learn the linguistic-style-aware representations of news articles by explicitly capturing the writers’ usage of words and the linguistically meaningful ways in which these words are structured as phrases, sentences, paragraphs, and, ultimately, the documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We conduct extensive experiments on real-world datasets with well-established and state-of-the- art approaches, which demonstrate the effectiveness of the proposed neural network in predicting fake news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Additionally, we examine the differential linguistic style of fake news and the truth and identify statistically significant and consistent patterns of fake news across datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We review re- lated work in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We introduce the proposed method in Section III and detail the experiments designed and conducted to evaluate the proposed method in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We conclude in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' RELATED WORK Fake news prediction methods can be categorized as content-based or propagation-based depending on whether the method focuses on investigating news content or its propaga- tion on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Propagation-based methods can utilize rich auxiliary social media information, including news spreaders’ intent [2] or profiles [10], relationships between news spreaders and their posts [11], social feedback [12]–[14], social networks [15], and propagation paths [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Nevertheless, they can only be deployed after news articles published on news outlets have been disseminated on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' In comparison, content- based methods have the primary advantage of predicting fake news early when news articles have been published online but have not been spread [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Additionally, an effective content- based method can be easily extended further by incorporating social context information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' With this consideration, we focus on analyzing news content to predict fake news and review re- lated work on content-based fake news prediction approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' As news articles are mainly text, content-based methods start by manually extracting linguistic features and predicting fake news using common classifiers such as SVM [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Such linguistic features have been related to lexicons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=', bag-of- words) [5], POSs [5], [6], context-free grammars (production rules) [4], [5], RRs [5], [8], readability [6], [18], and n- grams that preserve the sequences of words or POSs [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Though news features can be easily interpreted within this machine learning framework, features cannot be automatically extracted, which can significantly impact the prediction per- formance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' hence, the performance heavily relies on experts’ involvement and experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' More importantly, as detailed in Section I, it is difficult to capture the global structure of news text (language) at any of the syntactic and discourse levels with these hand-crafted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Compared to these methods, the proposed neural network can learn the features of news articles, which capture the global and hierarchical structures that news linguistic styles carry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Recently, neural networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=', Bi-LSTM [7], [19] and Text-CNN [20]) have been frequently employed to identify fake news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' These models can learn the features of news text (sometimes, combined with other modalities in news content, such as images [20], [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' These neural networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='News ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='document ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='EDU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='RR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='EDU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='EDU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Discourse parsing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='EDU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='POS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='POS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='EDU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='POS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='POS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='EDU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='POS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='POS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Constituency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='parsing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Softmax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Fake/true news ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Hierarchical linguistic tree construction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Feature extraction via hierarchical recursive neural network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Fake news prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Document ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='EDUs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='CoreNLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Discourse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='w/o RRs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Discourse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Level-specific classifiers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Transition-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='LayerNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='LayerNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='word ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='tag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='position ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Decoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Syntactic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Multi-head ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Feed Forward ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Word ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Linguistic-style- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='aware document ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Recursive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='aggregation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Bi-GRU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Avg pooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Bi-GRU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Avg pooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Bi-GRU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Avg pooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Bi-GRU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Avg pooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Bi-GRU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Avg pooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Bi-GRU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Avg pooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 2: Framework overview, which contains a top-bottom building process of hierarchical linguistic trees, a bottom-top feature extraction process using the proposed hierarchical recursive neural network, and a classifier to predict fake news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The neural network’s architecture adaptively preserves various news documents’ global and hierarchical linguistic tree structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The Bi-GRU aggregator catches text’s local sequentiality that is linguistically valuable and often short (explained in Section III-B), which is more effective than self-attention here (see Section IV-B for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' have focused on the sequentiality or locality of news text but not on its linguistic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' In comparison, the proposed neural network explicitly catches this structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' it also captures text’s sequentiality and locality, which will be detailed in Section III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We point out that the proposed neural network provides a fundamental approach to news text representation learning and thus can be easily extended for multimodal fake news prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' METHODOLOGY We specify the proposed model in this section, which can be divided into three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' For each news document, we first construct its hierarchical linguistic tree (see Section III-A), then extract its features via the proposed hierarchical recursive neural network that preserves the hierarchical linguistic tree information (see Section III-B), and finally predict it as fake news or the truth (see Section III-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Figure 2 presents the framework overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Hierarchical Linguistic Tree Construction Given a news document D, we first generate its hierarchical linguistic tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The tree can explicitly present the order of words used in the document and how these words shape EDUs (meaningful phrases, sentences, or paragraphs) and further shape the entire document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' An example is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Specifically, our first attempt is to obtain D’s discourse (rhetorical) structure, which identifies D’s EDUs and reveals how these EDUs recursively form the document D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' To this end, we first utilize Standford CoreNLP [22] to segment D into EDUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Then, we apply a modified transition-based system [23] to identify span (S) and nuclearity (N), based on which D’s rhetorical structure can be obtained without recognizing specific RRs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=', elaboration in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' This semi-naked tree structure allows us to divide each RR node into within-sentence, across-sentence, and across-paragraph levels in terms of its left and right subtrees, extract structural features for each RR node, and ultimately adopt level-specific SVM classifiers [23] to predict the node attribute (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=', the specific RR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' This multi-stage approach outperforms the well- established one [24] in our experiments, where [24] works as an integrated system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Finally, we employ a state-of-the-art discriminative constituency parser for each identified EDU of the document D to obtain its syntactic structure [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The parser consists of a self-attentive encoder [25] and a chart decoder [26] (see Figure 2 for the detailed architecture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The syntactic structure reveals how the EDU’s words recursively form the entire EDU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Feature Extraction via Hierarchical Recursive Neural Network We propose the hierarchical recursive neural network to extract features of news documents, whose architecture adap- tively maintains the global structure of the hierarchical lin- guistic trees of news documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Given a news document D, its feature extraction using the hierarchical recursive neural network is bottom-top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We first encode D’s words, which are the leaf nodes of D’s hierarchical linguistic tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Then, we aggregate the obtained embeddings of the words that attach to the same parent node, forming the embedding of their parent node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The hierarchical recursive neural network will repeat such aggregations from the lower (syntactic) level to the upper (discourse) level until the document D as the tree’s root node is embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Hence, the question arises of how the aggregator performs on a recurring fundamental component (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=', a depth-one parent-children structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Note that for each parent node in a hierarchical linguistic tree, its children contain local and sequential information of the corresponding news document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The information is local because it reveals partial information about the overall news content that is linguistically valuable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' It is sequential as we keep the order of words of the news document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Naturally, recurrent neural networks can be adopted as the aggregator to catch the sequentiality of children sharing the same parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The information locality further relieves the pressure on recurrent neural networks to keep the dependency of long entities since the number of children for a parent node is no more than the EDU length and essentially less than the document length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' As seen from Figure 1, the maximum length that recurrent neural networks require to process is four, whereas the document has 29 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' With the above considerations, we develop Bi-GRU (bidi- rectional gated recurrent unit), one of the well-established recurrent neural networks [27], to aggregate the embeddings of all the child nodes to represent their parent node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We also empirically compare Bi-GRU with multi-head self-attention that has remarkably performed in many tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Bi-GRU is more effective for the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Formally, the embedding of a parent node is computed as xp = � c∈Cp[−−−→ GRU{xc} ⊕ ←−−− GRU{xc}] |Cp| , (1) where p denotes the parent node and Cp is the set of child nodes of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Vectors xc ∈ Rd and xp ∈ Rd refer to the features of the child and parent node, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The operator ⊕ denotes concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The GRU is formulated as follows: ri = σ(Wrxi + Urhi−1), zi = σ(Wzxi + Uzhi−1), ˆhi = tanh(Whxi + Uh(hi−1 ⊙ ri)), hi = (1 − zi) ⊙ hi−1 + zi ⊙ ˆhi, (2) where hi ∈ Rd/2 is the output hidden state of the i-th child, with h0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The symbol ⊙ denotes Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Matrices W∗ ∈ R(d/2)×d and U∗ ∈ R(d/2)×(d/2) (∗ ∈ {r, z, h}) are learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' ri and zi are the reset gate and update gate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' σ and tanh are the sigmoid and hyperbolic tangent activation functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' In a nutshell, the above architecture first employs the Bi-GRU to capture “deep” sequential feature interactions of all the child features and then uses a mean pooling layer over all the hidden states to obtain the parent node’s features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' After determining the aggregation within each recurring fun- damental component, we introduce three specific hierarchical recursive neural networks (HEROs): Unified HERO: The first hierarchical recursive neural network is the one with unified aggregators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' In other words, all the Bi-GRUs in the neural network share the same set of W∗ and U∗ (∗ ∈ {r, z, h}, see Equation (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Level-specific HERO: It is the hierarchical recursive neu- ral network with level-specific aggregators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' As detailed, the hierarchical linguistic tree presents both syntactic and rhetorical structures of news content, and the hier- archical recursive neural network preserves such struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Hence, all the Bi-GRUs in a hierarchical recursive neural network can be grouped by the level (syntax or discourse) they belong to the corresponding tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We define L(v) as the function that maps a certain vertex in the hierarchical linguistic tree to its linguistic level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=', L(v) ∈ {syntax, discourse}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Then, Equation (1) can be reformulated as xp = 1 |Cp| � c∈Cp[−−−→ GRU L(c) {xc} ⊕ ←−−− GRU L(c){xc}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Attribute-specific HERO: It stands for the hierarchical recursive neural network with attribute-specific aggrega- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' In other words, we categorize the hierarchical recur- sive neural network’s recurring fundamental components according to the attributes of their parent nodes in the corresponding hierarchical linguistic tree, which can be various POSs and RRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We deploy the same Bi-GRU for the components within each category and the different Bi- GRU for the components falling into different categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Mathematically, we define A(v) as the function that maps a certain vertex in the hierarchical linguistic tree to its attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Assume there are m different POSs and n RRs, we have A(v) ∈ {POSi, RRj : i = 1, 2, · · · , m, j = 1, 2, · · · , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The root vertex would be assigned with a RR in discourse parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' For EDU vertices, they would not be assigned with any RRs in discourse parsing but with some POSs in constituency parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' In this way, Equation (1) is rewritten as xp = 1 |Cp| � c∈Cp[−−−→ GRU A(p){xc} ⊕ ←−−− GRU A(p){xc}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Fake News Prediction We add a softmax classifier on the top of the proposed hier- archical recursive neural network to predict the document D as fake news or the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Let hD denote D’s features extracted via the proposed hierarchical recursive neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The softmax function maps hD to the probability of D being a fake news document by pD = Softmax(WhD + b), where W and b are learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' To learn the parameters Θ = {W∗, U∗, W, b} within the neural network and classifier, we employ cross-entropy to calculate the classification loss in the model training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Assume we have q verified news documents D = {Di}q i=1 with the ground-truth labels Y = {yi : yi ∈ {0, 1}}q i=1 (yi = 0 for true news, and yi = 1 for fake news), the loss is computed TABLE I: Data statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Recovery MM-COVID # news documents 2,029 3,536 true news 1,364 1,444 fake news 665 2,092 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' # words per EDU 24 17 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' # EDUs per document 38 2 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' # words per document 841 16 by L = − 1 q �q i=1[yDi log pDi +(1−yDi) log(1−pDi)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Based on it, the parameter set Θ is estimated by ˆΘ = arg minΘ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' EMPIRICAL EVALUATION We aim to evaluate the proposed method by answering the following three questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 1) How effective is the proposed model in fake news prediction compared to the-state-of-art approaches?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 2) Is the hierarchical linguistic structure of news documents essential in representing their linguistic styles?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 3) What characterizes the linguistic style of fake news as distinguishable from the truth?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' To that end, we first detail our experimental setup in Sec- tion IV-A and then compare the proposed unified, level- specific, and attribute-specific HEROs in predicting fake news (see Section IV-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Subsequently, we compare the proposed model with the baselines to verify its effectiveness in predict- ing fake news (to answer RQ1, see Section IV-C) and conduct the ablation study to assess the importance of our developed hierarchical linguistic trees (to answer RQ2, see Section IV-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Finally, we characterize the linguistic style of fake news as distinguishable from the truth by doing quantitative and comparative analyses (to answer RQ3, see Section IV-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Experimental Setup We first introduce the datasets used for evaluation (see Section IV-A1), followed by the baselines for comparison (see Section IV-A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Finally, we detail our implementation details in Section IV-A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 1) Datasets: We conduct experiments on two benchmark datasets in fake news prediction: Recovery [9] and MM- COVID [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Both datasets contain labeled news documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Differently, news documents collected in Recovery are articles (long text, often including multiple paragraphs) but in MM- COVID are statements (short text, often formed by one or two sentences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We present the detailed statistics of two datasets in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 2) Baselines: We involve the following well-received and state-of-the-art methods as baselines in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' HCLF [5]: HCLF stands for hand-crafted linguistic fea- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Each news document’s HCLFs include the frequen- cies of words (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=', bag-of-word features), POSs, RRs, and production rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The extracted features are used to pre- dict fake news by employing well-established classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Here we examine a comprehensive list of classifiers– logistic regression, SVM, k-nearest neighbors, decision trees, naive Bayes, random forest, and AdaBoost–and select the one performing best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' EANN [20]: The event adversarial neural network con- tains three components: feature extraction by Text-CNN (for text) and VGG-19 (for images), event discrimination to learn event-invariant features of news content, and fake news prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We exclude the visual features for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' HAN [29]: HAN exploits attention-GRU for news clas- sification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' It captures the hierarchical sequence of docu- ments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=', each document is a sequence of its sentences, and each of its sentences is a sequence of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' DRNN [30]: DRNN is a discourse-structure-aware neu- ral network, which focuses on the tree with rhetorical relationships as edge attributes and leverages an atten- tion mechanism for news classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' In other words, DRNN differs from HERO in the aggregation rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Com- pared to DRNN’s tree, the hierarchical linguistic tree integrates syntax-level structures and has RRs as nodes and non-attributed edges at the discourse level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' DRNN is developed to categorize news documents with more than one elementary discourse unit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' otherwise, it is reduced to Bi-LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Text-GCN [31]: The approach develops the graph convo- lutional neural network for news classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The graph investigates the co-occurrence relationship among news documents and the words within the documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Transformer [32]: It is a deep neural network model with a self-attention-based encoder-decoder architecture, which has excellently performed in diverse natural lan- guage processing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Here, we consider Transformer’s encoder–applicable for classification tasks–as a baseline to predict fake news, a non-pretrained version rather than pretrained models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=', BERT) for a fair comparison as the pretrained Transformers have learned large-scale external resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 3) Implementation Details: We randomly divide each dataset into 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='7:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='1:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='2 proportions for model training, vali- dation, and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Macro-F1, micro-F1, and AUC are used to evaluate the performance of methods in news classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The discourse parser is pretrained using RST-DT [23], and the constituency parser is pretrained using the Penn Treebank [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' For the neural-network-based models, we uniformly utilize the pretrained GloVe [33] to obtain semantic-aware embeddings of words, with 100 as the embedding dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The hidden di- mension within neural networks is set as 100, correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We deploy Adam optimizer to learn parameters, with 50 as the maximum number of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We perform a grid search over the learning rate ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='0001} with validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' In the end, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='0001 performs best for our models and most of the baselines other than Transformer (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='001) and Text- GCN (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' All the experiments of the neural networks are implemented with PyTorch and are conducted on an NVIDIA Quadro RTX 6000 GPU (24 GB memory), Intel(R) Xeon(R) Gold 6248R CPU (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='00 GHz), and with 64 GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' For TABLE II: Performance of unified, level-specific, and attribute-specific HEROs in fake news prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Attribute- specific HERO performs best, demonstrating that the node attributes (POSs or RRs) in hierarchical linguistic trees are essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' MAF1: Macro-F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' MIF1: Micro-F1 Recovery MM-COVID HERO MAF1 MIF1 AUC MAF1 MIF1 AUC Unified 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='801 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='822 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='827 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='889 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='891 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='899 Level-specific 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='817 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='838 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='841 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='878 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='878 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='892 Attribute-specfic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='850 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='869 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='866 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='894 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='896 TABLE III: Performance of the proposed model, HERO, and baselines in fake news prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' HERO outperforms the baselines by 2–17% in AUC on Recovery and by 3–30% in MM-COVID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' MAF1: Macro-F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' MIF1: Micro-F1 Recovery MM-COVID MAF1 MIF1 AUC MAF1 MIF1 AUC HCLF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='752 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='801 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='746 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='566 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='624 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='577 Transformer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='774 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='793 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='810 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='804 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='809 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='806 Text-GCN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='841 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='869 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='835 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='826 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='836 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='817 EANN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='811 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='864 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='795 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='833 HAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='847 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='869 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='844 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='840 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='856 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='846 DRNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='711 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='778 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='698 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='846 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='848 HERO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='850 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='869 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='866 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='894 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='896 HCLFs, classifiers are used with the default hyperparameters presented in the scikit-learn library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Z-score normalization is applied for the feature matrix to enhance the classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Determining the Best HERO We compare the performance of the proposed neural net- works with unified, level-specific, and attribute-specific Bi- GRUs in predicting fake news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Table II presents the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The results indicate that with Recovery data, the performance ranking is attribute-specific HERO > level-specific HERO > unified HERO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Specifically, attribute-specific HERO correctly predicts news as fake or true with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='85 macro-F1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='87 micro-F1 and AUC, outperforming unified HERO by ∼4% and level-specific HERO by ∼3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' With MM-COVID data, the performance ranking is attribute-specific HERO ≈ unified HERO > level-specific HERO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Attribute-specific and unified HEROs achieve ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='89–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='90% in macro-F1, micro-F1, and AUC, outperforming level-specific HERO by ∼1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' In con- clusion, attribute-specific HERO performs best in classifying long articles and short statements as fake news or the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' This result demonstrates the importance of the node attributes (POSs or RRs) in developed hierarchical linguistic trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Additionally, we compare Bi-GRU and self-attention (#heads=10) as aggregators in the proposed hierarchical re- cursive neural network for fake news prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The results indicate that Bi-GRU performs better than self-attention by at least 1% in AUC on both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Unified Level-specific Attribute-specific 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='8 AUC HERO\\(Syn+Dis) HERO\\Syn HERO\\Dis HERO (a) Recovery Unified Level-specific Attribute-specific 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='8 AUC HERO\\(Syn+Dis) HERO\\Syn HERO\\Dis HERO (b) MM-COVID Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 3: Ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' (a) The proposed HERO outper- forms HERO\\Dis by 1% in AUC for unified HERO and by 3% for level- and attribute-specific HEROs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' It outperforms HERO\\Syn by 7–9% and HERO\\Syn by 30%+ in AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' (b) HERO performs similarly to HERO\\Dis as MM-COVID contains short statements having minimal discourse struc- tures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=', syntax-level structures dominate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' It outperforms HERO\\Syn and HERO\\(Syn+Dis) by 10%+ in AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Thus, syntax- and discourse-level structures are both essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Comparing HERO with Baselines We compare the proposed model with the baselines in predicting fake news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The results presented in Table III re- veal that the proposed model can generally outperform the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Specifically, the proposed model has an AUC score approaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='87, outperforming HAN by more than 2%, Text- GCN by more than 3%, Transformer by more than 5%, EANN by more than 7%, HCLF by 12%, and DRNN by 17%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' With MM-COVID data, the proposed model has an AUC score approaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='90, outperforming EANN by more than 6%, DRNN and HAN by ∼5%, Text-GCN and Transformer by ∼8-9%, and HCLF by more than 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' From the table, we also observe that the proposed model outperforms EANN by 6–7% in macro-F1 and AUC but underperforms it by ∼3% in micro-F1 on MM-COVID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' This result suggests that EANN tends to predict given news statements as the major class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Ablation Study We compare the proposed model, HERO, which contains hierarchical linguistic (syntax- and discourse-level) structures with its following variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' HERO\\Dis: It stands for the variant of HERO with only syntax-level structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' In this variant, the embedding of a news document is obtained by averaging its embeddings of EDUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' HERO\\Syn: It stands for the variant of HERO with only discourse-level structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' In this variant, the embedding of each EDU of a news document is obtained by averag- ing its words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' HERO\\(Syn+Dis): It stands for the variant of HERO with no structures, the embedding of a news document is directly obtained by averaging its word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Fake True 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='525 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='550 Avg Child Cnt Articles Fake True 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='6 Statements (a) Children of parent nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Fake True 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='020 % EDUs Articles Fake True 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='06 Statements Fake True 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='06 % POS:NNS Articles Fake True 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='15 Statements Fake True 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='08 % POS:IN Articles Fake True 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='15 Statements (b) Note attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Fake True ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='# Nodes (Syn) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Articles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Fake True ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Statements ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Fake True ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='# Leafs (Syn) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Articles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Fake True ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Statements ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Fake True ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Max Width (Syn) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Articles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Fake True ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Statements ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Fake True ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Depth (Syn) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Articles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Fake True ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Statements ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Fake True ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='6000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='# Nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Articles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Fake True ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='# Nodes (Dis) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Articles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Fake True ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Max Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Articles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Fake True ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Max Width (Dis) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='Articles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='(c) Size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' width,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' and depth of trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 4: Hierarchical linguistic trees of fake and true news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Orange solid line: Median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Green dashed line: Mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The results are visualized in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We observe that with Recovery data, the proposed HERO outperforms HERO\\Dis by 1% in AUC for unified HERO and by 3% for level- and attribute-specific HEROs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' It outperforms HERO\\Syn by 7– 9% and notably outperforms HERO\\Syn by above 30% in AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' With MM-COVID data, the proposed HERO performs similarly to HERO\\Dis since the statements presented in MM- COVID are short with two EDUs on average and hence have minimal discourse structures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=', syntax-level structures dominate hierarchical linguistic structures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Meanwhile, it out- performs HERO\\Syn and HERO\\(Syn+Dis) by more than 10% in AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Therefore, we conclude that the proposed HERO is better than its variants, demonstrating the importance of hierarchical linguistic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Characterizing Linguistic Style of Fake News Fake news has been theoretically identified with a linguistic style distinguishable from the truth [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' This experiment aims to specify this different linguistic style of fake news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We com- pare the hierarchical linguistic trees generated by fake news and the truth, which we develop to represent the linguistic style of news documents systematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The comparison is from the (i) children of parent nodes, (ii) attributes of nodes, and (iii) size, width, and depth of trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' a) Children of Parent Nodes: We compare fake news with real news in the average and the maximum number of children of parent nodes in hierarchical linguistic trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Results that are statistically significant with a p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='001 (using t-test, unless otherwise specified) are in Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We observe that the hierarchical linguistic trees of fake news have more child nodes for each parent node than true news on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' News here indicates long news articles in the Recovery dataset and short statements in the MM-COVID dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' b) Attributes of Nodes: Considering the nodes within a hierarchical linguistic tree can indicate the document (as the root), RRs, EDUs, POSs, and words (as the leaf nodes), we first compare fake news with the truth in the proportion of RRs, EDUs, POSs, and words, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The reason for computing their proportions rather than the numbers is to eliminate the impact of the size of trees (discussed in the next paragraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We observe that compared to true news, the hierarchical linguistic trees of fake news have a significantly smaller proportion of EDU nodes (p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='05) and POS nodes indicating NNS (noun in the plural, p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='001) but have a significantly larger proportion of nodes indicat- ing specific POSs such as IN (preposition or subordinating conjunction), PP (prepositional phrase), and DT (determiner, p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We illustrate the results in Figure 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' c) Size, Width, Depth of Trees: We compare fake news with the truth in the size, maximum width, and depth of hierarchical linguistic trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Since hierarchical linguistic trees contain two-level structures, we also compare fake and true news in the size, maximum width, and depth of discourse- and syntactic-level trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We observe that the syntactic-level tree of fake news is generally greater with more nodes, broader, and deeper than true news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' In particular, the syntactic-level tree of fake news has more leaf nodes than true news, which reveals that fake news often has longer EDUs with more words than true news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The above conclusions hold for long news articles (using Recovery data) and short statements (with MM-COVID data) with a p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' news files in both datasets are rich in syntactic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Figure 4c (the upper ones) presents the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Moreover, we observe that fake news articles generate smaller and narrower discourse-level trees that lead to smaller and narrower hierarchical linguistic trees than true news articles (p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='01, see the bottom figures in Figure 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We point out that the discourse structures of short statements are plain with two EDUs on average and hence have trivial impacts on the shape of the entire hierarchical linguistic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Lastly, we point out that comparing trees’ maximum and average widths leads to the same conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Comparing the longest (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=', depth) and the average distance between the root and leaves also leads to the same conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' CONCLUSION We propose a psychology-informed neural network to pre- dict fake news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The proposed neural network learns the linguistic style of news documents represented by hierarchical linguistic trees, which explicitly captures the writers’ usage of words and the linguistically meaningful ways these words are structured as phrases, sentences, paragraphs, and, ultimately, documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We conduct experiments on public real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The results demonstrate the effectiveness of the proposed neural network, with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='87–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='90 AUC scores, and the importance of the developed hierarchical linguistic tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' The proposed neural network can outperform the previous (recur- rent, convolutional, graph, and self-attentive) neural networks and feature-engineering-based approach in predicting news–as long articles or short statements–as fake news or the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' We observe from the data that the hierarchical linguistic trees of fake news can significantly differ from true news in the children of parent nodes, the attributes of nodes, and the size, width, and depth of the trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' In our future work, we aim to enhance the proposed model’s performance with multimodal and social-context information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' REFERENCES [1] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Zhou and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Zafarani, “A survey of fake news: Fundamental the- ories, detection methods, and opportunities,” ACM Computing Surveys (CSUR), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 53, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 1–40, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [2] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Zhou, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Shu, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Phoha, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Liu, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Zafarani, ““this is fake!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' shared it by mistake”: Assessing the intent of fake news spreaders,” in Proceedings of the ACM Web Conference 2022, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 3685–3694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [3] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Undeutsch, “Beurteilung der glaubhaftigkeit von aussagen,” Hand- buch der psychologie, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 26–181, 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [4] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' P´erez-Rosas, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Kleinberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Lefevre, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Mihalcea, “Automatic detection of fake news,” in Proceedings of the 27th International Conference on Computational Linguistics, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 3391–3401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [5] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Zhou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Jain, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Phoha, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Zafarani, “Fake news early detec- tion: A theory-driven model,” Digital Threats: Research and Practice, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 1–25, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Potthast, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Kiesel, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Reinartz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Bevendorff, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Stein, “A stylometric inquiry into hyperpartisan and fake news,” in ACL, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 231–240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [7] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Przybyla, “Capturing the style of fake news,” in AAAI, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 01, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 490–497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [8] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Rubin and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Lukoianova, “Truth and deception at the rhetorical structure level,” Journal of the Association for Information Science and Technology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 66, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 905–917, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [9] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Zhou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Mulay, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Ferrara, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Zafarani, “ReCOVery: A mul- timodal repository for COVID-19 news credibility research,” in CIKM, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 3205–3212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [10] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Cheng, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Guo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Shu, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Liu, “Causal understanding of fake news dissemination on social media,” in KDD, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 148–157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [11] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Min, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Rong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Bian, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Xu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Zhao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Huang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Ananiadou, “Divide-and-conquer: Post-user interaction network for fake news de- tection on social media,” in Proceedings of the ACM Web Conference 2022, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 1148–1158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [12] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Shu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Cui, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Lee, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Liu, “defend: Explainable fake news detection,” in KDD, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 395–405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [13] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Qian, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Gong, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Sharma, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Liu, “Neural user response generator: fake news detection with collective user intelligence,” in Proceedings of the 27th International Joint Conference on Artificial Intelligence, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 3834–3840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [14] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Lipani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Liang, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Yilmaz, “Reply-aided detection of misinformation via bayesian deep learning,” in The World Wide Web Conference, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 2333–2343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Tommasel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Rodriguez, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Menczer, “Following the trail of fake news spreaders in social media: A deep learning model,” in Adjunct Proceedings of the 30th ACM Conference on User Modeling, Adaptation and Personalization, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 29–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [16] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Naumzik and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Feuerriegel, “Detecting false rumors from retweet dynamics on social media,” in Proceedings of the ACM Web Conference 2022, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 2798–2809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [17] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Liu and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Wu, “Early detection of fake news on social media through propagation path classification with recurrent and convolutional networks,” in AAAI, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [18] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Zhu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Sheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Cao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Nan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Shu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Wang, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Zhuang, “Memory-guided multi-view multi-domain fake news detec- tion,” IEEE Transactions on Knowledge and Data Engineering, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [19] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Karimi and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Tang, “Learning hierarchical discourse-level structure for fake news detection,” in NAACL, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 1, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 3432–3442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [20] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Ma, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Jin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Yuan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Xun, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Jha, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Su, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Gao, “EANN: Event adversarial neural networks for multi-modal fake news detection,” in KDD, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 849–857.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [21] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Cui, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Wang, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Lee, “Same: Sentiment-aware multi-modal embedding for detecting fake news,” in ASONAM, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 41–48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [22] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Manning, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Surdeanu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Bauer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Finkel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Bethard, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' McClosky, “The Stanford CoreNLP natural language processing toolkit,” in Proceedings of 52nd Annual Meeting of the Association for Computational Linguistics: System Demonstrations, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 55–60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [23] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Li, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Wang, “A two-stage parsing method for text- level discourse analysis,” in Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers), 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 184–188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [24] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Ji and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Eisenstein, “Representation learning for text-level discourse parsing,” in Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 1, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 13–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [25] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Kitaev and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Klein, “Constituency parsing with a self-attentive encoder,” in ACL, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 2676–2686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [26] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Gaddy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Stern, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Klein, “What’s going on in neural con- stituency parsers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' an analysis,” in Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers), 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 999–1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [27] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Cho, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Van Merri¨enboer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Gulcehre, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Bahdanau, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Bougares, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Schwenk, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Bengio, “Learning phrase representations using RNN encoder–decoder for statistical machine translation,” in Proceed- ings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 1724–1734.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [28] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Jiang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Shu, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Liu, “MM-COVID: A multilingual and multimodal data repository for combating COVID-19 disinformation,” arXiv preprint arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content='04088, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [29] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Yang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Dyer, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' He, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Smola, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Hovy, “Hierarchical attention networks for document classification,” in NAACL, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 1480–1489.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [30] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Ji and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Smith, “Neural discourse structure for text categoriza- tion,” in ACL, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 996–1005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [31] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Yao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Mao, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Luo, “Graph convolutional networks for text classification,” in Proceedings of the AAAI Conference on Artificial Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 33, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 01, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 7370–7377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [32] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Vaswani, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Shazeer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Parmar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Uszkoreit, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Jones, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Gomez, Ł.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Kaiser, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Polosukhin, “Attention is all you need,” in Advances in Neural Information Processing Systems, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 5998–6008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' [33] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Pennington, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Socher, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' Manning, “Glove: Global vectors for word representation,” in Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} +page_content=' 1532– 1543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE0T4oBgHgl3EQf8wJq/content/2301.02792v1.pdf'} diff --git a/odFKT4oBgHgl3EQfFy38/content/2301.11723v1.pdf b/odFKT4oBgHgl3EQfFy38/content/2301.11723v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f672d428bd8f0c9340908266fd47124556b072df --- /dev/null +++ b/odFKT4oBgHgl3EQfFy38/content/2301.11723v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3b1999824ff5175ee6531a5e28ee646b70e45bb20452a05d5a892f40493cf786 +size 939811 diff --git a/odFKT4oBgHgl3EQfFy38/vector_store/index.pkl b/odFKT4oBgHgl3EQfFy38/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..3f199717ddf97a16020e043b1538fd5f450bf5da --- /dev/null +++ b/odFKT4oBgHgl3EQfFy38/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d188f643993ab58bab6ff4f8021a817be721585c426c7b2d5cfbdf7f3d15ad87 +size 295377 diff --git a/otFPT4oBgHgl3EQf6zWU/content/2301.13202v1.pdf b/otFPT4oBgHgl3EQf6zWU/content/2301.13202v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7eca4908b1bd273526be0e5b3b8a33a73ddc9523 --- /dev/null +++ b/otFPT4oBgHgl3EQf6zWU/content/2301.13202v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ee3272de73f815b7d7c20085ff69175f33a7ebbca870d14437f95264d460214f +size 9607987 diff --git a/p9E4T4oBgHgl3EQfvg1C/vector_store/index.faiss b/p9E4T4oBgHgl3EQfvg1C/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..4ecc4272742fc6ffabb35c2dea488600202f5a41 --- /dev/null +++ b/p9E4T4oBgHgl3EQfvg1C/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:30db11c142ac20262f2116143fd44eea99631dc98710d1eaf0b89c528a51c902 +size 7012397 diff --git a/p9E4T4oBgHgl3EQfvg1C/vector_store/index.pkl b/p9E4T4oBgHgl3EQfvg1C/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..0473b82376bd7f098e74bb89db2dfbf3c1deba59 --- /dev/null +++ b/p9E4T4oBgHgl3EQfvg1C/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a85fd2a45b2dc756307b8db86bfac75180e280f39e93a72ea953f51cf1d42b51 +size 248038 diff --git a/q9E3T4oBgHgl3EQfMQlY/content/tmp_files/2301.04371v1.pdf.txt b/q9E3T4oBgHgl3EQfMQlY/content/tmp_files/2301.04371v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8618f7cef1bbab785c85e8df91f615e6eb1f3d61 --- /dev/null +++ b/q9E3T4oBgHgl3EQfMQlY/content/tmp_files/2301.04371v1.pdf.txt @@ -0,0 +1,1488 @@ +Recognising geometric primitives in 3D point clouds of mechanical +CAD objects +Chiara Romanengo +∗, Andrea Raffo +∗, Silvia Biasotti +†, and Bianca Falcidieno +Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes”, Consiglio Nazionale delle +Ricerche, Via de Marini 6, 16149 Genova, Italy. +Abstract +The problem faced in this paper concerns the recognition of simple and complex geometric +primitives in point clouds resulting from scans of mechanical CAD objects. A large number of +points, the presence of noise, outliers, missing or redundant parts and uneven distribution are +the main problems to be addressed to meet this need. In this article we propose a solution, +based on the Hough transform, that can recognize simple and complex geometric primitives and +is robust to noise, outliers, and missing parts. Additionally, we can extract a series of geometric +descriptors that uniquely characterize a primitive and, based on them, aggregate the output into +maximal or compound primitives, thus reducing oversegmentation. The results presented in the +paper demonstrate the robustness of the method and its competitiveness with respect to other +solutions proposed in the literature. +Keywords: geometric primitives, point clouds, mechanical CAD objects, Hough transform. +1 +Introduction +The increasing availability of affordable digital scanning devices – such as close-range photogram- +metry and laser scanners – has made point clouds one of the most common ways of representing the +surface of an object. In Computer-Aided Design, this fact results in the need to extract information +from measured data points using higher-level geometric primitives. To give an example, it is highly +convenient to recognise and reconstruct a digital model so that it can be interpreted in terms of +some basic components and easily manipulated by CAD systems. A large number of points, the +presence of noise and outliers, the occurrence of missing or redundant parts and the non-uniform +distribution of the data severely limit the use of tessellations (e.g., meshes) as a means to ease the +analysis and reconstruction of shapes; rather, they make it more convenient to analyze the point +cloud directly [1]. +Many approaches for detecting primitives lack robustness to noise and outliers or deal only with +mesh models. Most of them are able to extract only a few classes of simple geometric primitives, +being planes and cylinders the most commonly recognised, and do not consider more complex basic +shapes, such as surfaces of revolution or generalized cylinders. Moreover, many methods have a +tendency to oversegment, thus producing very fragmented output without aggregating the non- +contiguous subsets of points that lie on the same primitive. The detection of maximal or compound +primitives (e.g. cylinders composed of non-adjacent parts, or patterns) is another open problem, +mostly addressed only for planes or cylinders [2, 3]. For example, if a pipe is interrupted by another +part, as in the block model in Figure 5(b), traditional methods recognise the two parts of the pipe as +two distinct primitives. This is particularly important, for instance, when decomposing patterns that +correspond to scans of assembly CAD models, because it concurs with the recognition of compound +primitives and patterns [4]. When it comes to memory usage and computational complexity, learning +∗These authors have contributed equally to this work. +†Corresponding author +1 +arXiv:2301.04371v1 [cs.GR] 11 Jan 2023 + +approaches require onerous training and large annotated databases; on the other hand, the direct +detection of mathematical primitives in a general space embedding typically faces the computational +limitation of dealing with a number of free parameters that rapidly grows as the complexity of the +primitives increases. +In this paper, we deal with the recognition of simple and complex geometric primitives in point +clouds originating from scans of mechanical CAD objects, where recognition means the detection +of the primitives in a given point cloud and the extraction/computation of the best (geometric) +parameters associated with those primitives. We adopt an approach based on the Hough transform +(HT) to do this. The HT meets the needs of being able to recognize multiple instances of primitive +functions and, through a voting procedure, to be robust to noise and outliers. Originally limited to +straight lines in images [5], it has been generalized and extended in multiple directions to handle +other shapes, also using families of non-parametric templates in images [6] or point clouds from +CAD objects [7]; recently, the extended version proposed in [8] has opened up the HT to a wide +range of algebraic primitives. +Here we generalise the HT to surface primitives – not necessarily algebraic – represented in +parametric form. Our solution is tailored to deal with point clouds and is able to deal with simple +geometric primitives and some complex ones (generalized cylinders and cones, surfaces of revolution, +helical surfaces, etc.), including maximal and compound primitives. To this end, we aggregate the +primitives found on the basis of their size and space embedding, whenever possible. Our strategy also +reduces the oversegmentation of the output and we exploit the strategy proposed in [9] to decrease +the computational complexity of methods based on the HT thanks to an opportune preprocessing +of the point cloud and its subparts. Our experiments confirm the robustness and completeness of +the method, and the comparisons made exhibit its competitiveness with respect to the methods +proposed so far in the literature. To sum up, our main contributions are: +• The introduction of a geometric primitive recognition method that is particularly robust to +noise and outliers and is able to recognize multiple instances of the same primitive. +• The extraction and recognition of complex primitives, in addition to the most common ones +(planes, cylinders, cones, spheres, tori). +• The aggregation of primitives on the basis of their shape, size and position to detect maximal +and compound primitives. +• An extensive experimentation on different datasets and comparison with state-of-the-art meth- +ods. +The rest of the paper is organised as follows. Section 2 overviews previous work in geometric +primitives detection and HT-based recognition methods. Section 3 briefly recalls the basic concepts +of HT. Section 4 introduces and lists the geometric primitives that can be identified by our method, +with emphasis on complex primitives such as general cylinders and cones, surfaces of revolutions, +helical surfaces, and convex combinations of curves. Section 5 describes our approach for the iden- +tification of primitives in point clouds based on the HT, and the search for maximal primitives and +possible relationships between primitives through a clustering technique. Section 6 provides exper- +imental results of our method for the identification of simple and complex primitives and shows a +qualitative and quantitative comparative analysis. Concluding remarks end the paper. +2 +Previous work +Representing an object with a set of geometric components is a long-standing problem in computer +vision, computer graphics and CAD. As we are interested in recognising (simple or complex) geo- +metric primitives in the manufacturing domain, here we focus our attention on those approaches +that share the same goal. +A considerable variety of algorithms have been devised to decompose digitalized point clouds +or meshes representing CAD objects into regions approximated by primitives belonging to some +2 + +given sets. +According to [10], these approaches can be grouped into four families: stochastic, +parameter space, clustering and learning techniques. The first group includes the RANSAC method +[11] and its optimizations. The second family includes Hough-like voting methods and parameter +space clustering, e.g., [12]. The third class gathers all the other clustering techniques, and can be +classified into three main types: primitive-driven region growing, e.g., [13]; automatic clustering +and Lloyd-based algorithms, e.g., [14]; primitive-oblivious segmentation, e.g., [15]. Finally, with +the growing popularity of deep learning techniques, supervised fitting methods have been proposed +even for multi-class primitives [16, 17]. The reader is referred to [10] for a comprehensive historical +taxonomy of methods for simple primitive detection, which is beyond the scope of this paper. +Despite a large number of available solutions, the problem is far from being solved; novel ap- +proaches have been proposed in the very last few years to address the shortcomings of existing +paradigms or to propose novel directions to move in. A recent approach to deal with the recognition +of geometric primitives is described in [18]. It consists of a curvature-based partitioning method +that decomposes an input triangle mesh into maximal surface portions; the decomposition is per- +formed so that each segment corresponds to one of the following seven invariance classes of surfaces: +plane, sphere, cylinder, surface of revolution, prismatic, helix, and complex surface. The method +was adapted to handle point clouds in [19]. In [20], a method for the identification and fitting of +planes and cylinders from unstructured point clouds in manufacturing is presented. It consists of +three subsequent phases: point cloud segmentation; merging of oversegmented regions and estima- +tion of surface parameters; extraction of cylinders and planes. Being the method able to handle +only two primitive types, its applicability is however restricted. To handle the increasing availability +of acquired data, [1] introduces a region-growing-based system for the segmentation of large point +clouds in planar regions. Other approaches, devised to detect only specific types of primitives, are: +[21], which deals with quadric surfaces; [22] which fits surfaces of revolution; and [23], which extracts +cylindrical shapes from non-oriented point clouds. In the field of deep learning, two methods have +shown promising results in the problems of segmentation and recognition: ParSeNet [24] and HPNet +[25]. Beyond their performance, outstanding merit is their capability of handling – together with the +more classical simple geometric primitives – open and closed spline surfaces. Another learning-based +method lately proposed for fitting primitives is PriFit [26], which learns to decompose a point cloud +of various 3D shape categories into a set of geometric primitives, such as ellipsoids and cuboids, or +alternatively deformed planes. However, the natural flexibility of supervised learning approaches in +identifying geometric primitives comes with a considerable cost: the need of having gigantic labelled +training sets, whose difficulty of gathering limits their current application. +As for the methods that use the HT to identify geometric primitives, they were limited, until +recently, to planes [12, 27], combinations of linear subspaces [28], spheres [29] and circular cylinders +[30]; in the case of cylinders, however, the rotational axis needs to be known before the application +of the Hough transform. The recent advances in the use of the algebraic functions [8] are paving the +road to a larger use of the HT for recognising more complex families of geometric primitives. To +the best of our knowledge, the sole approaches that exploit this idea were proposed in [31], for the +recognition of an ellipsoid in a free-form model; and in [9], for the recognition of spheres, (circular) +cylinders, (circular) cones and tori in presegmented point clouds, as a way to post process faulty +segmentations. The work in [9] also addresses the problem of reducing the number of the parameters +of the primitive representation when the point cloud corresponds to a patch of a primitive: in this +case, the point cloud is opportunely rototranslated in a space embedding so that it can be fit by +a primitive in a standard form. The method has been recently compared with other direct and +learning-based approaches, see [32]. However, the point clouds used for this comparison did neither +originate from scans of mechanical CAD objects nor can be assumed to represent presegmented +data, making the benchmark itself not of interest to this paper. +3 +Basic concepts on the Hough transform +We denote by A3 +x(R) and An +A(R), respectively, the 3-dimensional and the n-dimensional affine spaces +over R, where x := (x, y, z) and A := (A1, . . . , An) vectors of indeterminates. Given a surface S +3 + +Figure 1: Simple geometric primitives: plane, cylinder, cone, sphere and torus respectively, along +with their attributes (geometric descriptors). +defined as the zero locus of a function f, a parameter-dependent family of surfaces can be described +by functions fa as F = {Sa : fa(x) = 0 | a ∈ U}, where U is an open set of the parameter space +An +A(R) and a := (a1, ..., an) is the parameter vector. Then, given a point P ∈ A3 +x(R), the HT of +the point P with respect to the family F is ΓP = {fa(P) = 0} ⊂ An +A(R). For sufficiently general +points P ∈ A3 +x(R), it turns out that ΓP are hypersurfaces in An +A(R). If the set of hypersurfaces ΓP +generated by varying P on a given surface meets in one and only one point ¯a ∈ U, the family of +surfaces F is called Hough-regular; the intersection point ¯a of the Hough transforms ΓP uniquely +identifies the surface S¯a from which the points P ∈ A3 +x(R) were sampled in the first place. If the +HT regularity is not guaranteed, the solution to the intersection problem might be non-unique and +a solution must be selected among more potential parameters solutions. +In the discrete scenario, the HT framework deals with the problem of finding a surface – within +a family F of surfaces – that best approximates a particular shape, given in the form of a data +set of N points P, where N >> n. The common strategy to identify the solution (or a solution), +introduced in [8], is based on the so-called accumulator function; it consists in a voting system +whereby each point in P votes a n-uple a = (a1, . . . , an); the most voted n-uple corresponds to the +most representative surface for the profile. +While the HT traditionally deals with one shape at a time, the framework we propose in this +paper can be directly applied to point clouds containing multiple shapes. The family of surfaces we +consider is represented in parametric form and we assume that the system defining Sa with respect +to the Cartesian coordinates x, y, and z can be analytically solved with respect to the parameter +vector a = (a1, . . . , an). A more detailed description of how this works in our case is given in Section +5. +4 +Definition of complex geometric primitives +In addition to the simple geometric primitives of Figure 1 (plane, cylinder, cone, sphere, and torus) +– see [9] for their recognition in presegmented point clouds – in this section, we introduce a set of +complex geometric primitives that to the best of our knowledge have never been used before for +recognition by HT. +In this paper we express surfaces parametrically with respect to the variables u and v. For the +sake of simplicity, some of the surfaces are here presented in their standard form or with respect +to some specific axes; nevertheless, one can easily generalise these equations by applying a generic +transformation of the special orthogonal group SO(3). +• General cylinders. +A cylinder is a surface traced by a straight line of fixed direction, the +generatrix, while moving along a curve, the directrix. +Given a curve (x(u), y(u), z(u)) := +(f1(a, u), f2(a, u), f3(a, u)) and a direction (l, m, n), the parametric representation of the cor- +responding cylinder is given by: +� +� +� +� +� +x = f1(a, u) + lv +y = f2(a, u) + mv +z = f3(a, u) + nv +. +4 + +Imax +CO +工 +rminNote that a general cylinder depends on the parameters defining generatrix and directrix. +The dictionary of curves to be considered as directrix is extremely rich, see [33]. Table 1(a) +considers a 5−convexity curve as directrix, whose equation can be found in [33]. +• General cones. A cone is a surface traced by a straight line, the generatrix, while gliding along a +curve, the directrix, and passing through a fixed point, the vertex. Given a parametrized curve +(x(u), y(u), z(u)) := (f1(a, u), f2(a, u), f3(a, u)) and a point V = (xV , yV , zV ), the parametric +representation of the corresponding cone is given by: +� +� +� +� +� +x = xV + (f1(a, u) − xV )v +y = yV + (f2(a, u) − yV )v +z = zV + (f3(a, u) − zV )v +. +As in the case of cylinders, we can exploit the dictionary of plane curves to create families +of cones. +Then, a general cone depends on the parameters that define the curve and the +components of the vertex. Table 1(b) shows a cone generated by a 5−convexity curve. +• Surfaces of revolution. A family of surfaces of revolution can be created by rotating a fam- +ily of curves around an axis of rotation. +For example, given the family of plane curves +(x(u), y(u), z(u)) := (f1(a, u), 0, f2(a, u)) and the z−axis, we obtain the parametric equations +� +� +� +� +� +x = f1(a, u) cos v +y = f1(a, u) sin v +z = f2(a, u) +. +One of the most known examples is the torus; another example is given by ellipsoids, which +are obtained by rotating an ellipse with respect to one of its principal axes. Table 1(c) shows +the case of a surface obtained by rotating the curve (x(u), y(u), z(u)) := (au, 0, b/u5) around +the z−axis. +• Helical surfaces. Table 1(d) presents a family of equations obtained by modifying the parametri- +sation of a circular cylinder. Precisely, the radius R here varies between [R1, R2], where R1 > 0, +by a cosine function; when radii are fixed, the slope of the output surface is controlled by the +parameters in z(u, v). Note that the radius variation can be adapted to other shapes (e.g., the +triangle wave function). +• Convex combination of curves. It is possible to define surface primitives by considering the con- +vex combination of a pair of parametrised curves (f1(a, u), f2(a, u), f3(a, u)) and (g1(b, u), g2(b, u), g3(b, u)). +This family has the following parametric equations: +� +� +� +� +� +x = vf1(a, u) + (1 − v)g1(b, u) +y = vf2(a, u) + (1 − v)g2(b, u) +z = vf3(a, u) + (1 − v)g3(b, u) +, +where v ∈ [0, 1]. Note that the primitive parametrisation depends on the same parameters +which define the pair of curves, i.e., a and b. A planar example is given by the annulus, i.e., +the region bounded by two concentric circles. A helical strip can be obtained by cutting and +bending an annular strip; this corresponds to considering a convex combination of two helices +of an equal slope but different radii. An example of a helical strip is provided in Table 1(e). +The use of the standard form allows us to diminish the number of unknown parameters in the +recognition process. To give a practical example, the family +� +� +� +� +� +x = a1 cos(u) + b1 sin(u) + c1v + d1 +y = a2 cos(u) + b2 sin(u) + c2v + d2 +z = a3 cos(u) + b3 sin(u) + c3v + d3 +, +5 + +contains (circular) cylinders; it has 12 unknown parameters, which make the representation compu- +tationally and memory-wise unmanageable when used in a voting procedure. By further assuming +that the cylinder can be rototranslated so that it has the rotational axis aligned with the z−axis and +it is centred in the origin, the number of parameters reduces to 1 in the case of a circular cylinder, +i.e., the sole radius: +� +� +� +� +� +x = r cos(u) +y = r sin(u) +z = v +. +A strategy to obtain the standard form in an automatic way starting from a point cloud is provided +in [9], and it can be naturally extended to complex primitives. +Table 1: Parametric representation of some complex geometric primitives. +(a) +(b) +(c) +(d) +(e) +generalized +generalized +surface of +helical +helical +cylinder +cone +revolution +surface +strip +� +� +� +� +� +� +� +x = +a cos u +1+b cos(5u) +y = +a sin u +1+b cos(5u) +z = v +� +� +� +� +� +� +� +x = +av cos u +1+b cos(5u) +y = +av sin u +1+b cos(5u) +z = Av + B +� +� +� +� +� +x = au cos v +y = au sin v +z = +b +u5 +� +� +� +� +� +x = R(u) cos v +y = R(u) sin v +z = k1(u + nv) + k2 +, +where +R(u) := R1 + R2 − R1 +2 +(cos u + 1), +n ∈ Z +� +� +� +� +� +x = R(u) cos v +y = R(u) sin v +z = v +, +where +R(u):=au+(1-u)b +5 +Recognising geometric primitives using Hough transforms +In this section, we describe a method to recognise an input point cloud P with geometric primitives +via the HT technique. It consists of the following main steps: an initial point cloud preprocessing +followed by the iteration of a recognition step and a splitting phase, and a final step which applies +a clustering technique to discover geometric relationships between primitives or parts of them. A +graphical illustration of its pipeline is given in Figure 2. +5.1 +Point cloud preprocessing +First, the input point cloud P is translated to place its barycenter at the origin of the Cartesian +coordinate system. Then, the normals at the input points are estimated. In the literature, there +are several ways to estimate normals; in general, we are interested in using an approach suitable for +handling noisy input. +For our purposes the orientation of the normal vector is not relevant, as we are only interested in +aligning the most voted normal direction to the z-axis. We can therefore use the approach introduced +in [9], which estimates the normal vectors through a voting procedure. For convenience, when P +is clean we can also consider the method presented in [34], which is available in MATLAB with +the pcnormals function; for each input point in P, the method selects the k points of P closest to +that input point and then uses principal component analysis (PCA) to estimate the normal vector. +Note that this does not necessarily generate normal vectors that are coherent in their orientation; +however, this is not a problem for our pipeline since we are not interested in the normal orientation +but rather in the direction of the (estimated) normal vectors. +6 + +The point cloud is then rotated so that the most voted direction among the (just-estimated) +normal vectors coincides with the z−axis. This is an essential step for multiple reasons. Firstly, +our dictionary contains primitives that – in their standard form – are given with the axis coincident +with the z−axis. In addition, this process is necessary to test the existence – as a first check – of +primitives in their standard form (i.e., with centres or vertices in the origin of the Cartesian axes and +with the normal or the principal axis aligned to the z-axis), thus reducing the number of parameters +that will need to be estimated by the HT. +Finally, P is scaled into a unit cube, in order to restrict the range of variation for most parameters. +5.2 +Recognition step +After being preprocessed, P becomes the input of the HT-based recognition step. +Since P can +contain different instances of the same primitive type and primitives of different types, the standard +HT procedure must be adjusted to allow such cases. This step can be summarized as follows: +• Selection of one or more families of primitives. The user can select the families of primitives +to be used for recognition; in its default configuration, the algorithm tests the presence of +simple geometric primitives – one family at a time. By studying the geometric properties of +P and by relating them to the geometric characteristics of the selected family (e.g., bounding +box, radius, diagonal length), it is possible to initialise a region T of the parameter space. +The region T is discretized into cells, which are uniquely identified by the coordinates of their +centre. Then, an accumulator function H, in the discretized form of a matrix, is initialized. +The matrix entries are in a one-to-one correspondence with the cells of the discretization +performed in the previous step. +• Estimation of the accumulator function. An entry of the accumulator function H is increased +by 1 each time the Hough transform ΓP of a point P intersects the corresponding cell. The way +we check if and where a Hough transform intersects some cells depends on whether the family +of surfaces is described in implicit or parametric form. In our case, surfaces are expressed in +parametric form; we adapt to surfaces the method devised for curves in [35]. Specifically, if +the system of equations fa defining the family can be solved analytically with respect to the +unknown parameters a, we automatically calculate a sample of ΓP by exploiting the Moore- +Penrose pseudo-inverse [36] of the matrix that defines the coefficients of a. The evaluation of +the intersection is translated into an inequality between the components of the sample points +of ΓP and the coordinates of the cell endpoints. +• Selection of potential fitting primitives. +The cells corresponding to the peak values of the +accumulator function H have to be identified. When the set of points is assumed to represent +a single surface profile, the traditional HT formulation aims at finding the maximum of the +accumulator function H; when the family of surfaces is not Hough-regular, there might exist +several maxima. On the other hand, if the point cloud is composed of different primitives, +different peaks of H identify potential primitives that might fit different parts. To select these +peaks, we observe that peaks of the accumulator function corresponding to primitive shapes +rise distinctly with respect to their neighbours and are well-characterised as isolated peaks. +Formally said, we proceed by identifying peaks that have high topological persistence1. In our +implementation, peaks that correspond to primitives are automatically recognised by keeping +the local maxima having a persistence higher than 10% of the maximum value of H, by using +the algorithm for persistent maxima proposed in [39]. The coordinates of the cell centres of +the maxima correspond to the parameters of potentially recognised surface primitives. +1The notion of topological persistence was introduced in [37] for encoding and simplifying the points of a filtration +f by classifying them as either a feature or noise depending on its lifetime or persistence within the filtration. In +practice, given a pair of points p and q, their persistence is defined as f(p) − f(q). Pairing is defined in terms of the +topological connection between the points, for details on topological persistence and saliency, we refer to [37, 38]. In +our case, the domain is represented by the grid of T, the role of filtration is played by the accumulator function H, +and we are interested only in the peaks of H. +7 + +• Evaluation of the approximation accuracy. To measure the recognition accuracy of a specific +primitive, we select the set of input points X closer to such a primitive than a given threshold +and study its density via the k-Nearest Neighbor algorithm (see, for example, [40]). If X is +sparse, the recognised primitive is considered a false positive; otherwise, for each dense subset +Xi ⊂ X we define the Mean Fitting Error (MFE) as: +E(Xi, P) := +1 +|Xi| +� +x∈Xi +d(x, P)/li, +(1) +where P is the current primitive, d is the Euclidean distance, and li is the diagonal of the axis- +aligned bounding box containing Xi. What can happen when the selected family of primitives +does not fit any part of the point cloud? There are two possibilities: +– The accumulator function is identically zero, with the result that its persistence is zero; +therefore the selection of potential fitting primitives returns the empty solution. +– The accumulator function H does not present predominant peaks, resulting in false pos- +itives which are identifiable by studying the sparsity of the fitted points. +Given a set of dense points Xi and two candidate primitives Pi,1 and Pi,2, we first calculate the +fitting errors E(Xi, Pi,1) and Ei(Xi, Pi,2) between each primitive and Xi; the primitive having +lowest error is kept. +Each time a primitive is recognized, the points of P close to the recognised primitive less than +a threshold ε are discarded from P. The value of ε typically ranges from 1% to 3% of the +diagonal of the minimum bounding box of the P, according to the type of primitive. +IF ALL +POINTS +ARE +LABELED + primitive type +∀ + IF + SPARSE +𝒫′ +ELSEIF + +∅ ⊊ 𝒫′ ⊊ 𝒫 +SELECTION +OF A FAMILY +• +Plane +• +Cylinder +• +Sphere +• +Cone +• +Torus +• +… +VOTING +PROCEDURE +RECOGNITION +OF POTENTIAL +PRIMITIVES +APPROXIMATION +ACCURACY +PREPROCESSING +UNRECOGNIZED POINTS + +𝒫′ +RANSAC +AGGREGATION +INTO CLUSTERS +(DBSCAN) + +𝒫′ = ⋃ +i +𝒫′ i +SEGMENT +PREPROCESSING + segment +: +SET +:= +∀ +𝒫′ i +𝒫 𝒫′ i +INPUT + +𝒫 +POINTS ARE +UNCLASSIFIED +SET OF +RECOGNIZED +SEGMENTS + 𝒮j +CLUSTERING +RECOGNITION STEP +SPLITTING PHASE +POSTPROCESSING +ELSEIF + +𝒫′ = 𝒫 +Figure 2: Pipeline of the method. +The algorithm returns the parameters of the geometric primitives and the corresponding points +fitted by them, as well as the set of points that were not fitted by any primitive – denoted by P′. +Note that, if more geometric primitives potentially fit the same region of the point cloud, we select +the one with the minimum fitting accuracy (i.e., the lowest MFE). +8 + +X 54.6645 +X100.583 +Y 53.502 +Y98.4437 +Z 3479 +4000 +Z 3385 +3000~ +X131.195 +Y128.405 +2000~ +Z 1424 +1000~ +0 > +50 +140 +120 +100 +100 +80 +60 +40 +20 +150MFE +cylinder 1 +0.0039 +cylinder 2 +0.0032 +cylinder 3 +0.00625.3 +Splitting phase +The algorithm chooses the next step according to the resulting P′: +• If P′ is sparse, then its points are returned as unclassified. +• If ∅ ⊊ P′ ⊊ P, i.e., if P′ is a proper nonempty subset of P, we proceed by aggregating points +in P′ into clusters P′ +j, with j = 1, . . . , Nclust, by adopting the Density-Based Spatial Clustering +of Applications with Noise (DBSCAN) method [41], which groups together nearby points and +marks as outliers isolated points in low-density regions. It requires two parameters: the first is +a threshold used as the radius of the density region, while the second represents the minimum +number of points required to form a dense region. Then, the recognition step from Section 5.2 +is iterated over each cluster P′ +j as long as some geometric primitives are recognised. Before +proceeding with a new recognition round, we preprocess each cluster P′ +j; the preprocessing +adopted here differs from that applied to the entire point cloud P: we exploit the strategy +presented in [9] to estimate its standard form, thus reducing the number of parameters that +have to be estimated in the new iteration of the recognition step. Specifically, it computes the +normals of the points of the cluster P′ +j through a voting procedure and it estimates the position +of vertices or centres and the direction of axes of symmetry, exploiting different geometric rules +specialised for each type of primitive. To reduce the dimension of the parameter space, vertices +and centres are translated to the origin of the coordinate system, while axes are aligned to the +z − axis. Then, it automatically centres and orients a (spherical, cylindrical, conical or toric) +cluster so that it can be fitted with a primitive in standard form. In case the recognition step +involves complex primitives, the strategy proposed in [9] can be naturally extended since they +are characterized by the presence of symmetry axes that can be estimated in a similar way +through the use of the normals of points. +• It is possible that, in some steps, no primitive is recognized. This can happen in two cases: (i) +when applying our algorithm to an input point cloud that has no primitives in their standard +form, or (ii) when the estimation of the standard position of some cluster P′ +j fails – because +P′ +j contains more than one primitive. In both cases, we proceed by oversegmenting the model: +in our experiments, we use the RANSAC algorithm proposed in [11], because of its efficiency +and its tendency to oversegment point clouds (see, for example, [15]); however, we proceed +by estimating primitive types and geometric parameters with a new round of the recognition +step, as voting procedures have shown greater robustness to point cloud artifacts. +This step ends by transforming the parameters found by the Hough transform with respect to the +inverse translations, rotations and scaling applied in the previous steps of the algorithm. The result +of this procedure is the decomposition of an input point cloud P into several subsets, called surface +segments or simply segments, in such a way that points of the same segment are well approximated +by the same primitive. We denote these final segments by Sj. +5.4 +Segment association based on geometric relationships +After decomposing the input point cloud into the segments Sj, a clustering technique is applied +to unveil geometric relationships. The goal of this step is to find maximal primitives that are not +automatically detected in the HT-based recognition process, to detect patterns of primitives or to +relate primitives (or parts of them) even if they do not belong to the same primitive. More specif- +ically, we relate primitives on the basis of their positions, orientation and dimension, as described +in [9] for simple geometric primitives: segments are aggregated only if they are part of the same +primitives; otherwise, we only identify relationships that may be of interest, for example in the +processing phases (e.g., process planning, machining). To give some examples: +• Segments sampled from circular cylinders can be associated with respect to their radii and +rotational axes. For instance, it is possible to check whether such segments: originate from +9 + +the exact same primitive shape, i.e., if they are all sampled from a cylinder of a given radius +and rotational axis (such as for the red segments in Figure 5); are characterized by the same +radius and have parallel rotational axes (such as for the black segments in Figure 8); share the +radius but not the rotational axis, not even in terms of parallelism. +• Segments sampled from helical surfaces can be associated with respect to their parameters R1, +R2 and n, k1, k2 and the direction – or a combination of such geometric descriptors. +• Segments sampled from n-convexity cylinders can be associated with respect to their parame- +ters a and b, the number of convexities n, and the direction of the generatrix – or a combination +of such geometric descriptors. +To perform segment associations, a well-known (hierarchical) clustering strategy is considered – the +complete linkage – as it penalizes chaining effects. +5.5 +An illustrative example +Figure 2 provides a graphical illustration of the pipeline of our method. The main steps of the +algorithm are associated with an example of a point cloud representing a gear. +After preprocessing the point cloud, we start with the first round of the recognition step – which +aims at recognizing the presence of primitives in their canonical position. For the sake of clarity, +however, the graphical illustration only focuses on the recognition of cylinders. Specifically, once +the family of cylinders is selected, the corresponding accumulator function is computed; it exhibits +three peaks, which indicate the presence of three potential solutions: the three cylinders highlighted +in the colours red, green, and blue. The approximation accuracy for this type of primitive is less +than 1%. +Being some segments (the non-axis-aligned planar segments) non in their canonical position, their +points are not recognized in the first round of the recognition step. Instead, these points are collected +in P′ and aggregated into dense clusters in the splitting phase; each of such clusters is individually +studied by a new round of the recognition step and recognized as planar segments. However, being +these segments studied separately, we lose some geometric information. We then apply clustering to +recognise that some planar segments lie on the same parametric plane, by exploiting the geometric +descriptors provided by the HT. +The final result is a segmentation of 17 segments. It is worth mentioning that the method is able +to group some of the non-adjacent parts, which belong to the same primitive in canonical position +– as in the example of the two external cylinders, and of the axis-aligned planes that form four +couples. However, for primitives not in their canonical position, postprocessing based on clustering +is required. +5.6 +Computational complexity +In the preprocessing step, our method includes the estimation of the normal vectors and the indi- +viduation of the most voted normal, which is implemented as explained in [34] – in the case of clean +point clouds – and in [9] – in the presence of point cloud artifacts. For a thorough analysis of the +computational complexities of this estimation, the reader is referred to the corresponding papers. +The formulation of the HT for surface primitives embedded in R3 naturally extends that for plane +curves in R2. In the pipeline presented in this paper, for each point cloud P (both in the case of the +initial point cloud and of single clusters at subsequent iterations) we apply the HT by considering +the different types of geometric primitives one at a time; thus, the dimension of the parameter space +changes accordingly to the type of primitive considered. The quantization of a region of interest +and the dimension of the parameter space determines the size of the accumulator function, and +dominate both the memory usage and the computational complexity of the Hough transform: it +is, therefore, necessary to balance the samples for each parameter and their number. To reduce +the computational cost, primitives are put in their (estimated) standard positions, in this way the +number of parameters does not exceed 3, as explained in [9]; complementarily, adaptive approaches +10 + +can be used to further speed up the search (e.g., [42]). The overall computational complexity of +the HT-based recognition step is O(ML), where M denotes the number of cells of the parameter +space and L represents the number of points (in the initial point cloud or in a cluster obtained at a +subsequent iteration) at which we evaluate the HT accumulator function. More precisely: +• At the first iteration the HT is applied to the whole input point cloud. Supposing that the +number of families of primitives to be used for the recognition is K and the number of points +in P is N, the overall complexity of the first iteration is O(N �K +k=1 Mk), where Mk is the +dimension of the parameter space for the k-th primitive type. +• In the subsequent steps, the HT-based recognition method is applied to each cluster P′ +j and +then the computational cost corresponds to O(�Nclust +j=1 +Nj +�K +k=1 Mk), where Nj denotes the +number of points in cluster P′ +j while Nclust is the number of clusters. +Notice that, being the recognition step performed independently w.r.t. the primitive type, the task +is embarrassingly parallel. The same applies to the clusters of points obtained in the same iteration. +Finally, agglomerative hierarchical clustering approaches require, in their naive implementation, +O(N3 +seg) operations, where Nseg denotes the number of segments in the output segmentation; see, +for example, [43]. When it comes to complete-linkage clustering, one can consider more efficient +implementations, such as the one proposed in [44], which costs O(N2 +seg). Although the dissimilarity- +matrix assembly costs O(N2 +seg), one may note that each entry is computed independently; again, +the task is therefore embarrassingly parallel. +6 +Experimental results +In this section, we show the results obtained with our method. Specifically, Section 6.1.1 presents +some point clouds of mechanical objects containing only simple geometric primitives, while Section +6.1.2 provides some examples with complex primitives. In Section 6.1.3, we exhibit some noisy point +clouds to demonstrate the robustness of our pipeline. In all cases, whenever possible, we show the +maximal primitives of the selected point cloud. Finally, a qualitative and quantitative comparative +analysis is provided in Section 6.2. +6.1 +Overall analysis +6.1.1 +Simple geometric primitives +As a sanity check, we first apply our method to two models from [45], which were selected because +containing all simple geometric primitives – plane, sphere, cylinder, cone and torus – and because +of the availability of ground truth. +A first example is provided in Figure 3. +The point cloud, +corresponding to the set of vertices of the original triangle mesh, is decomposed in 8 surface segments: +5 cylinders, 2 planes and 1 sphere. The high accuracy of our method is proved by comparing the +parameters from the HT with those in the database. In the second example, presented in Figure +4, the corresponding point cloud is subdivided into 9 surface primitives: 4 cylinders, 1 torus, 3 +axis-aligned planes and 1 cone. Again, we are able to recognise all primitives up to a small error in +the parameters with respect to those provided in the dataset. +Figure 5(a-b) shows point clouds that can be segmented into complete geometric primitives +without the need for a further step of aggregation. The first example, shown in Figure 5(a), consists +of 11 segments – 3 cylinders and 8 planes – and it highlights the robustness of our method when +it comes to detecting intersecting cylinders, here arranged similarly to a Steinmetz solid. Complete +geometric primitives, each of which is represented by a specific colour, are automatically detected. +Another point cloud, displayed in Figure 5(b), contains 5 segments: 2 tori, 1 cylinder and 2 axis- +aligned planes. +As for the previous case, the HT-based recognition successfully detects all the +primitives, even those made up of non-adjacent parts. +Figures (6-9) show point clouds wherein the segments produced by the HT approach are post- +processed by hierarchical clustering. In Figure 6(b), the input point cloud is first segmented into 20 +11 + +equation +code +ground truth +HT parameters +x2 + y2 = r2 +C1 +r = 1.50 +r = 1.50 +C2 +r = 8.00 +r = 8.00 +C3 +r = 4.00 +r = 4.00 +C4 +r = 1.50 +r = 1.50 +C5 +r = 5.00 +r = 5.00 +z = k +P1 +k = 20.00 +k = 20.00 +P2 +k = 35.00 +k = 35.00 +x2 + y2 + z2 = r2 +S +r = 5.00 +r = 5.05 +(a) +(b) +(c) +Figure 3: In (a) a mechanical CAD model from the benchmark in [45]; in (b) the vertices of its +triangle mesh decomposed into 8 surface segments; in (c), for each primitive, the HT parameters +are compared with those provided by the database +equation +code +ground truth +HT parameters +x2 + y2 = r2 +C1 +r = 15.01 +r = 15.01 +C2 +r = 3.97 +r = 3.97 +C3 +r = 1.28 +r = 1.26 +C4 +r = 29.25 +r = 29.25 +z = k +P1 +k = 12.00 +k = 12.00 +P2 +k = 9.08 +k = 9.08 +P3 +k = 0.00 +k = 0.00 +(R − +� +x2 + y2)2 + z2 − r2 = 0 +T +R = 27.25 +R = 27.25 +r = 2.00 +r = 2.00 +x2 + y2 + a(z − b)2 = 0 +C +a = −2.77 +a = −2.81 +b = 2.19 +b = 2.22 +(a) +(b) +(c) +Figure 4: In (a) a mechanical CAD model from the benchmark in [45]; in (b) the vertices of its +triangle mesh decomposed into 9 surface segments; in (c), for each primitive, the HT parameters +are compared with those provided by the database +Segments (a) +Segments (b) +Figure 5: Recognition of CAD point clouds containing only simple geometric primitives. +The +identification of maximal segments does not require, in these cases, the application of any clustering +algorithm. +surface primitives via the HT-based recognition algorithm (cylinders and planes), which are then +associated thanks to the comparison of the geometric descriptors that characterize them uniquely. +As expected, no pair of primitives lying on the same surface is found. +Despite the presence of +imperfections in the original model (see Figure 6(a)), we can correctly identify repeating primitives, +here in the form of circular cylinders of equal radii. Figure 6(c) highlights the similarities identified +by associating cylinders: 4 in light blue, 3 in red, 2 in purple and 2 in yellow. +12 + +Cylinder 1 (C1) +Plane 2 (P2) +Plane 1 (P1) +Cylinder 2 (C2) +Plane 3 (P3) +Torus () +Cone (C) +Cylinder4 (C4) +Cylinder 3 (C3)Plane (P1) +Cylinder (C1) +Cylinder (C2) +Plane (P2) +Cylinder (C3) +Cylinder (C4) +Cylinder (C5) +Sphere (S)(a) Model +(b) Segments +(c) Clustering +Figure 6: A linkage arm. In (a), the original model is shown, together with a magnification revealing +some imperfections. The segments identified by the HT are shown in (b) in different colours. (c) +draws attention to cylinders, among which we can recognize segments lying on the same primitive, +up to a translation. +Figure 7 increases the difficulty by considering a point cloud containing a higher number of +segments, some of which are low-quality as the small holes highlighted in Figure 7(a). +In this +example, the geometric primitives identified by the HT algorithm are cylinders, cones, and planes. +The result is a segmentation in 28 surface segments, see Figure 7(b). Note that the central hole +presents some grooves, i.e., a surface detail that was not present as a geometric feature in the +original CAD model; therefore, we recognise it as a cylinder and the HT is able to ignore the +shallow grooves. A final application of clustering makes it possible to identify repeating primitives, +up to translations. Figure 7(c) illustrates the similarity between 6 cylindrical holes (in green) and +between other 6 cylindrical segments (in yellow). +(a) Model +(b) Segments +(c) Clustering +Figure 7: A carter. +In (a), the original model is shown. +The surface segments found by the +HT approach are depicted in (b), while (c) shows the result of primitive association, when one is +interested in identifying the same primitive up to a translational transformation. Different rows +correspond to different points of view. +Another example of gear is shown in Figure 8(a). Here, the total number of extracted geometric +primitives is 68: 19 cylinders; 2 cones; 47 planes, 5 of which are axis-aligned. The result is presented +in Figure 8(b). As highlighted by the original model, in this prototypical version of the NuGear +component, cylinders are roughly approximated by a series of planar primitives; in this specific case, +13 + +0we fit a cylinder instead of many planes, since the former can describe a much larger area without +significantly increasing the error. Clustering identifies here a similarity between the 12 cylindrical +holes – up to translations – and between 2 external cylinders, while the surface segments identified +by non-axis-aligned planes are not grouped in pairs; this is because the tooth inclination prevents +any alignment. Figure 8(c) shows this result, colouring the holes in black and the external cylinders +in yellow. +(a) Model +(b) Segments +(c) Clustering +Figure 8: A prototype of the NuGear component, courtesy of STAM S.r.l. (Genoa, Italy). The +original model is shown in (a). The decomposition in clusters of points produced by the HT approach +is given in (b). The output of the additional clustering association procedure is shown in (c), which +highlights the similarity between 12 cylindrical holes (in black) and between two cylinders (in yellow). +Finally, Figure 9(a) shows another mechanical part. The HT-based decomposition of the input +point cloud consists of 50 surface segments, all of which are tori, cylinders, and planes, see Figure 9 +(b). The clustering method can identify the similarity between 8 red cylindrical holes and between +2 blue cylindrical segments, up to translations – see Figure 9(c). Finally, 2 tori are aggregated, +because they lie on the same surface primitive up to rototranslations. +Table 2 summarises the characteristics of each point cloud processed in this section and the mean +fitting error for all the simple primitives recognised on it. +Table 2: Statistics of the MFES for all models of Section 6.1.1. Being the MFE normalized by +definition, we can conclude that the maximum error for the fitting of the simple primitives is 4.48%, +which corresponds to the noisy holes in the carter of Figure 7. +Model +# points +# segs +min(Ei) +mean(Ei) +max(Ei) +Fig. 3 +15, 216 +8 +0.0006 +0.0021 +0.0046 +Fig. 4 +15, 022 +9 +0.0008 +0.0029 +0.0051 +Fig. 5(a) +25, 000 +11 +0.0009 +0.0024 +0.0056 +Fig. 5(b) +25, 000 +5 +0.0004 +0.0031 +0.0059 +Fig. 6(b) +50, 000 +20 +0.0004 +0.0098 +0.0300 +Fig. 7(b) +50, 000 +28 +0.0013 +0.0107 +0.0448 +Fig. 8(b) +50, 000 +68 +0.0006 +0.0053 +0.0178 +Fig. 9(b) +50, 000 +50 +0.0008 +0.0035 +0.0057 +14 + +0 +0 +0 +0(a) Model +(b) Segments +(c) Clustering +Figure 9: A mechanical part. In (a) the original model is shown, while in (b) the decomposition of +the corresponding point cloud into segments produced by the HT. In (c) the result of the clustering +procedure: 8 cylindrical holes, in red, have a high similarity, up to translations; the same applies for +2 cylindrical segments, in blue; 2 tori, in orange, identify the same primitive, up to a rototranslation. +6.1.2 +Complex geometric primitives +As anticipated, our method can recognize other primitives in addition to the simple ones shown in +Section 6.1.1. Here we show some of the complex primitives identified in our experiments. +The first example – shown in Figure 10(a) – contains an ellipsoid, which is easily recognized by +our method at the price of an additional parameter in the parameter space; additionally, the point +cloud can be partitioned into 4 planes, 1 torus and 1 cylinder. In the point cloud from Figure 10(b), +we are able to identify correctly the yellow segment as a surface of revolution from Table 1(c); the +remaining points are segmented into 2 cylinders and 2 planes. In the point cloud shown in Figure +10(c), we recognize the gold part as a generalized cone, while the blue and the green segments are +fitted with generalized cylinders: all of the three have the same directrix, a 5−convexity curve. This +last point cloud has been segmented into 7 clusters. +Segments (a) +Segments (b) +Segments (c) +Figure 10: Recognition of complex geometric primitives in CAD point clouds. Their identification +does not require, in these cases, the application of any clustering algorithm. +Figure 11(a) displays a mechanical part which can be accurately described by combining a +portion of a helical surface, as introduced in Table 1(d), with a pair of planes and a pair of convex +combinations of helices, see Table 1(e). +The result is a segmentation of the point cloud into 6 +primitives, Figure 11(b). Two of them are then grouped by the clustering technique since the helical +strips have the same equation up to a translation, as shown in Figure 11(c). +Table 3 provides the main characteristics of each point cloud processed in this section and the +mean fitting error for all the simple and complex primitives recognised on it. +15 + +(a) Model +(b) Segments +(c) Clustering +Figure 11: A screw-like part. The original model, (a), is sampled. The surface primitives detected +via HT are shown in (b) using different colours: a helical surface (in purple), two planes (in red and +magenta), and two helical strips (in orange and yellow). Although no pair of them lies on the same +parametrised surface, the 2 helical strips have the same equation up to a translation, as shown in +(c). +Table 3: Statistics of the MFEs for all models of Section 6.1.2. Being the MFE normalized by +definition, we can conclude that the maximum error for the fitting of the simple primitives is 1, 54%, +which corresponds to the helical surface of Figure 11. +Model +# points +# segs +min(Ei) +mean(Ei) +max(Ei) +Fig. 10(a) +25, 524 +7 +0.0009 +0.0042 +0.0063 +Fig. 10(b) +67, 777 +5 +0.0006 +0.0031 +0.0071 +Fig. 10(c) +25, 000 +7 +0.0020 +0.0053 +0.0081 +Fig. 11(b) +25, 000 +6 +0.0033 +0.0086 +0.0154 +6.1.3 +Robustness of the pipeline +The use of the HT naturally leads to a robust method for the recognition of mathematical surfaces, +as suggested in the examples of Figures 6 and 7 – which were characterized by spurious parts. +In the point cloud of Figure 12, the HT recognition correctly identifies the cylinder that fits the +central part, without being negatively influenced by the letters in relief – see the original model in +Figure 12(a). The point cloud is decomposed into 38 segments: 23 cylinders with different axes, +10 planes, 4 cones, and 1 torus that automatically identifies the top and bottom of the cylinder +with the “GRAYLOC” inscription (see Figure 12(b)). The application of the hierarchical clustering +technique allows us to group together: 8 grey cylindrical holes (up to rototranslations); 8 purple +cylindrical segments; 2 aquamarine circular cylinders; 3 violet circular cylinders; 2 orange circular +cones (up to a reflection); 2 black cones (up to a reflection). Moreover, the small imperfections of the +manufacture on the central part of the body (recognised by vertical cones, cylinders and tori at the +top and bottom) and on the lateral holes do not prevent the clustering technique from appropriately +associating the corresponding segments, correctly dealing with rotations and reflections. However +not everything is recognised: the black dots in Figure 12(b) correspond to points that are not fitted +by any of the geometric primitives at our disposal, as they originate from irregular elements that act +as a connection between better-defined segments. We label such points as “unsegmented” because +of the high mean fitting error. +Further proof of the robustness of our method applied to raw data is presented in Figure 13 +and quantitatively analysed in Table 4. In this example, we perturb the point cloud of Figure 5(b) +by adding zero-mean Gaussian noise of standard deviation: 0.01, 0.05, 0.10 and 0.20. The first +row shows the points classified as noise (in black) and the segments found by our method in the +same image, for each level of noise. The second row focuses only on the points that fit the identified +primitives, thus providing a denoised segmentation. The robustness to noise is quantitatively studied +in Table 4: the parameters obtained in the original point cloud are there compared with those found +16 + +(a) Model +(b) Segments +(c) Clustering +Figure 12: A clamp connector. In (a) the original model. In (b) the decomposition of the correspond- +ing point cloud into 38 segments provided by the HT procedure. In (c), the final grouping obtained +by clustering, consisting of 6 clusters of primitives (here, singletons of segments are transparent). +in the perturbed point clouds. +σ = 0.01 +σ = 0.05 +σ = 0.10 +σ = 0.20 +Figure 13: The point cloud in Figure 5(b) is perturbed by adding zero-mean Gaussian noise of +standard deviation: 0.01, 0.05, 0.10 and 0.20. The first row superimposes the points identified as +noise (in black) to the final segments found by our method; the second row depicts the points that +fit the primitives found and provides a denoised segmentation. +6.2 +Comparative analysis +We here present two comparisons with alternative methods, all of which rely on estimating the +normal vectors at the given input points. +The first analysis, proposed in Figure 14, is merely visual. Due to the lack of freely-downloadable +implementations for some methods, it is indeed not possible to present this comparison other than +qualitatively. The analysis consists of a comparison of our method with the RANSAC-based method +introduced in [11], the patch aggregation approach in [15] and two recent deep learning architectures: +ParSeNet [24] and HPNet [25]. In this comparison, we have focused on models that solely present +simple primitives, to show that even on these our approach gives a great performance; on the +other hand, the capability to handle more complex primitives is undoubtedly an added value of our +method. Similarly to [15], we use different colours to represent different primitive typologies: red +for planes, green for cylinders, blue for cones, black for tori, pink for (open and closed) B-splines +and yellow for unsegmented parts. For a simple model like the block of Figure 14(a), all methods +provide decent decompositions, although RANSAC misclassifies some primitives while the deep +learning frameworks run into problems because of an inaccurate estimation of the surface normals: +this assertion is supported by the location of misclassified points, which are largely limited to the +17 + +GRAYLOGTable 4: Parameter comparison between the original point cloud from Figure 5(b) and the perturbed +versions from Figure 13. +Segment +Original +σ = 0.01 +σ = 0.05 +σ = 0.10 +σ = 0.20 +Plane 1 +k = −1.28 +k = −1.28 +k = −1.29 +k = −1.28 +k = −1.31 +Plane 2 +k = 1.28 +k = 1.28 +k = 1.28 +k = 1.28 +k = 1.26 +Cylinder +r = 1.80 +r = 1.79 +r = 1.78 +r = 1.79 +r = 1.78 +Torus 1 +R = 1.49 +R = 1.49 +R = 1.47 +R = 1.48 +R = 1.56 +r = 0.72 +r = 0.73 +r = 0.70 +r = 0.67 +r = 0.74 +Torus 2 +R = 1.05 +R = 1.09 +R = 1.02 +R = 1.17 +R = 1.13 +r = 0.79 +r = 0.78 +r = 0.78 +r = 0.74 +r = 0.80 +sharp edges. In this regard, it is worth noting that – considering that the networks were trained on +clean data – the more accurate the normals, the more precise the final segmentation. In our case, +the use of a voting procedure allows us to withstand possible perturbations. For the remaining more +complex models, our method outperforms the competitors. In Figure 14(b), our approach is the +only capable of correctly identifying portions of tori, misclassified by Le and Duan [15] and partly +unsegmented by RANSAC; ParSeNet and HPNet are able to reveal the presence of tori, but the +resulting segmentation is unreliable along sharp edges – again, mostly because of a deficiency in +the normal estimates. Figures 14(c-d) show a RANSAC tendency to oversegment and misclassify +complex models. While Le and Duan [15] obtain considerably improved results, their algorithm +misses a thin cylinder (see Figure 14(c)) and all the tori in both models. Being these two objects +acquired by low-quality scanners, the corresponding point clouds (and meshes) are affected by point +cloud artifacts which, in turn, lead to an even greater error in the normal estimates. The resulting +segmentations are unreliable and mostly associated with spline segments. As previously mentioned, +note that the two architectures were trained on the ABC dataset, which does not take into account +the presence of noise or other types of perturbation in the data. Due to a lack of a sufficiently rich +benchmark of scanned CAD objects, however, new training is not currently possible. +To offer a quantitative analysis of the robustness of our pipeline, we compare its performance +with that of three state-of-the-art methods whose implementation was made freely available by the +authors: a direct method approach on a primitive-growing (PG) framework [18], adapted to handle +point clouds as described in [19], and the two learning-based methods from the previous comparison: +ParSeNet (PN) and HPNet (HN). We conduct the study over the whole Fit4CAD benchmark [19], +which contains CAD point clouds defined by simple geometric primitives. In addition, Fit4CAD +contains a selection of meaningful and validated models from [45], converted from meshes to point +clouds. Figure 15 summarizes the performance of each method over the test set with respect to the +following classification measures: Sørensen-Dice index (DSC), Positive Predicted Value (PPV), True +Positive Rate (TPR), Negative Predicted Value (NPV), True Negative Rate (TNR) and accuracy +(ACC). Given a point cloud, the benchmark defines these measures by comparing each point cloud +segment in the ground truth to the most overlapping segment returned by a segmentation method, +and then by averaging over all segments contained in that point cloud. Note that, given a point +cloud segment in the ground truth, the points inside that segment are the positives while the points +outside that segment are the negatives. We arrive at the following conclusions: +• Learning-based methods show remarkably high TNRs – said otherwise, the points they predict +as being negative are almost always true negatives. On the other hand, they are more penalized +by NPV: while it is ParSeNet that exhibits the highest variability and the lowest quartiles, both +methods are seriously affected by outliers – with some point clouds having this score below +0.2. A low NPV means that quite a few points predicted as negative are false negatives (e.g., +common in heavily oversegmented models). In this case, these methods have been penalized +by the low density of the point clouds, as well as from the possible presence of point cloud +artifacts (e.g., missing data). +18 + +RANSAC +Le and Duan +ParSeNet +HPNet +Ours +(a) +(b) +(c) +(d) +Figure 14: Primitive type recognition: a comparison between our approach, a RANSAC-based +segmentation [11], and the method in [15]. Different colours correspond to different primitive types: +planes (red), cylinders (green), cones (blue), tori (black), splines (pink), unsegmented (yellow). +• Our approach performs significantly better than the competitors in terms of TPR, i.e., it has +similar proportions of correct (positive) predictions among positives. When it comes to the +PPV, differences are more modest. +• In terms of accuracy, the four methods reach high scores, with the direct methods being +the less prone to outliers: however, this metric is not completely reliable as this is a naturally +unbalanced binary classification problem. A more reliable measure is provided by the Sørensen- +Dice index (DSC). Our method visibly outperforms the competitors in terms of the DSC. +Intuitively, the higher the DSC, the more accurate segments returned by a segmentation +method are – with respect to the most overlapping segment in the ground truth; to put it +differently, DSC penalizes greatly both under- and oversegmentation. +When it comes to execution time, our Hough-based method and the primitive-growing approach +were run on a desktop PC equipped with an Intel Core i9 processor (at 3.6 GHz) and a Windows +10 operating system. The average execution time, per model, are 286.0 seconds for the PG-method +and 50.7 seconds for our pipeline. ParSeNet and HPNet were run on Google Colab pro equipped +with NVIDIA-SMI 460.32.03 and CUDA 11.2. Regarding the average execution times, we have 5 +seconds for ParSeNet and 257.1 seconds for HPNet (when the preprocessing of normals is applied). +19 + +88 +8Figure 15: Comparison of our approach (HT) with other three methods: a primitive growing ap- +proach (PG), ParSeNet (PN) and HPNet (HN). The analysis is performed over the Fit4CAD bench- +mark [19]. +7 +Conclusions +To address the problem of recognition of simple and complex primitives in a point cloud, we have +opportunely extended the family of geometric primitives to which the HT technique can be applied. +The HT is naturally robust to noise and outliers and recognizes multiple instances of the same +primitive. Thanks to an opportune preprocessing of the point cloud and its sub-parts, we are able +to limit the number of parameters that are necessary to represent the primitive, thus reducing the +computational complexity of the HT computation. In addition, the explicit extraction of position- +independent primitive parameters and the use of a hierarchical clustering strategy permits us to +identify maximal and compound primitives, thus reducing the output oversegmentation. Indeed, +differently from spline-based primitives, our strategy is suited for primitive similarity reasoning and +permits us to find maximal primitives. +Although learning methods have performed very well in recent years, when models present +complex combinations of primitives (such as the examples in Figure 14(c,d)), direct methods perform +even better, and our method is very competitive. In addition, our method has been validated on a +whole benchmark and, compared with the others, it turns out to be the best. +The use of geometric primitives whose mathematical representations require a large number +of parameters – even in their standard forms – is currently the main limitation of our pipeline; +while theoretically possible, this would indeed increase the execution time exponentially, making +the algorithm inapplicable in practice. An instructive example, shown in Figure 16, is given by +point clouds that contain segments generated by spline surfaces: while our method cannot deal with +spline surfaces, it does manage to recognize that parts of a point cloud do not originate from any +primitive at our disposal. The first row of Figure 16 shows the ground truth segmentation, while +the second row shows the segmentation we achieve: the black dots correspond to points that are not +recognized as simple geometric primitives, and that are labelled as “unsegmented” because of the +high mean fitting error; note that spline segments are undersampled for the sake of visibility, but +20 + +1.1 +1 +0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.11.1 +1 +0.9 ++ +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1this processing is realized only after segmenting the point clouds. On the contrary, methods that +recognise splines tend to give any segment a label, as shown in Figure 14. +As future work, we are thinking of other strategies to reduce the computational cost in the +HT-based recognition step. +Indeed, the parameter space can be further optimised by reducing +its dimension through the use of a refinement strategy that discretizes the parameter space only +in the relevant regions of interest. Another possible direction of investigation could concern the +applicability of our method to non-orientable surfaces. Indeed we have only applied our method to +objects with orientable surfaces since the datasets we have used exhibit only this type of surface. +However, in theory, our approach could also apply to objects with non-orientable surfaces, as long +as their equation exists in the literature and the system of equations is solvable with respect to the +parameter (as in the case of M¨obius strip). The challenge here would be to study a preprocessing +technique to translate and/or rotate the point cloud into the standard form of that surface type. +For instance, in the case of the M¨obius strip, how to estimate the symmetry axis in order to place +the point cloud so that we can fit it with a primitive in a standard form. +Regarding the automation of the process, we could take advantage of a pre-classification of the +points in the input point cloud using a curvature-based characterization, for instance, the local +surface variation proposed in [46] or the shape index and curvedness as used in [18] for ISO GPS +segmentation. Once points are aggregated by their type, we could further split or aggregate these +regions using our fitting method and possibly considering only a suitable subset of primitives, i.e., the +primitives that are compatible with the recognized type of points, e.g., planar, cylindrical, spherical, +etc. +Segments (a) +Segments (b) +Segments (c) +Segments (d) +GT +US +Figure 16: Four models from the ABC dataset [45] containing spline segments: comparison between +the ground truth (GT) segmentation and our recognition of simple geometric primitives (US). To +enhance the visibility of the primitives correctly recognized, the second row displays the segments +classified as splines with a lower point density. +Acknowledgements +This work has been developed in the CNR research activities DIT.AD004.100, DIT.AD021.080.001 +and DIT.AD021.125. +21 + +The authors thank the VVS shape repository by the AIM@SHAPE consortium (http://visionair. +ge.imati.cnr.it) and Dr. Alexander Leutgeb from RISC Software GmbH (Linz, Austria) for the +models used in this paper. +References +[1] F. Poux, C. Mattes, Z. Selman, L. Kobbelt, Automatic region-growing system for the seg- +mentation of large point clouds, Automation in Construction 138 (2022) 104250. +doi: +10.1016/j.autcon.2022.104250. +[2] Y. Li, X. Wu, Y. Chrysathou, A. Sharf, D. Cohen-Or, N. J. Mitra, GlobFit: Consistently +Fitting Primitives by Discovering Global Relations, ACM Transactions on Graphics 30 (4). +doi:10.1145/2010324.1964947. +[3] A. Monszpart, N. Mellado, G. J. Brostow, N. J. Mitra, Rapter: Rebuilding man-made scenes +with regular arrangements of planes, ACM Trans. Graph. 34 (4). doi:10.1145/2766995. +[4] K. Lupinetti, J.-P. Pernot, M. Monti, F. Giannini, Content-based CAD assembly model re- +trieval: Survey and future challenges, Computer-Aided Design 113 (2019) 62 – 81. +doi: +10.1016/j.cad.2019.03.005. +[5] P. V. C. Hough, Method and means for recognizing complex patterns, US Patent 3,069,654 +(1962). +[6] D. H. Ballard, Generalizing the Hough transform to detect arbitrary shapes, Pattern Recogni- +tion 13 (2) (1981) 111–122. doi:10.1016/0031-3203(81)90009-1. +[7] O. J. Woodford, M. Pham, A. Maki, F. Perbet, B. Stenger, Demisting the Hough Transform +for 3D Shape Recognition and Registration, International Journal of Computer Vision 106 (3) +(2014) 332–341. doi:10.1007/s11263-013-0623-2. +[8] M. C. Beltrametti, L. Robbiano, An algebraic approach to Hough transforms, Journal of Algebra +37 (2012) 669–681. doi:10.1016/j.jalgebra.2012.09.012. +[9] A. Raffo, C. Romanengo, B. Falcidieno, S. Biasotti, Fitting and recognition of geometric prim- +itives in segmented 3d point clouds using a localized voting procedure, Computer Aided Geo- +metric Design 97. doi:10.1016/j.cagd.2022.102123. +[10] A. Kaiser, J. A. Ybanez Zepeda, T. Boubekeur, A Survey of Simple Geometric Primitives +Detection Methods for Captured 3D Data, Computer Graphics Forum 38 (1) (2019) 167–196. +doi:10.1111/cgf.13451. +[11] R. Schnabel, R. Wahl, R. Klein, Efficient RANSAC for Point-Cloud Shape Detection, Computer +Graphics Forum 26 (2) (2007) 214–226. doi:10.1111/j.1467-8659.2007.01016.x. +[12] F. A. Limberger, M. M. Oliveira, Real-time detection of planar regions in unorganized point +clouds, Pattern Recognition 48 (6) (2015) 2043 – 2053. doi:10.1016/j.patcog.2014.12.020. +[13] M. Attene, G. Patan`e, Hierarchical structure recovery of point-sampled surfaces, Computer +Graphics Forum 29 (6) (2010) 1905–1920. doi:10.1111/j.1467-8659.2010.01658.x. +[14] D.-M. Yan, W. Wang, Y. Liu, Z. Yang, Variational mesh segmentation via quadric surface +fitting, Computer-Aided Design 44 (11) (2012) 1072 – 1082. doi:10.1016/j.cad.2012.04.005. +[15] T. Le, Y. Duan, A primitive-based 3D segmentation algorithm for mechanical CAD models, +Computer Aided Geometric Design 52-53 (2017) 231–246, Geometric Modeling and Processing +2017. doi:10.1016/j.cagd.2017.02.009. +22 + +[16] L. Li, M. Sung, A. Dubrovina, L. Yi, L. J. Guibas, Supervised Fitting of Geometric Primitives +to 3D Point Clouds, in: Proceedings – IEEE Computer Society Conference on Computer Vision +and Pattern Recognition, 2019, pp. 2652–2660. doi:10.1109/CVPR.2019.00276. +[17] M. A. Uy, Y.-Y. Chang, M. Sung, P. Goel, J. G. Lambourne, T. Birdal, L. J. Guibas, Point2cyl: +Reverse engineering 3d objects from point clouds to extrusion cylinders, in: Proceedings of the +IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022, pp. 11850–11860. +[18] Y. Qie, L. Qiao, N. Anwer, Enhanced invariance class partitioning using discrete curvatures +and conformal geometry, Computer-Aided Design 133. doi:10.1016/j.cad.2020.102985. +[19] C. Romanengo, A. Raffo, Y. Qie, N. Anwer, B. Falcidieno, Fit4CAD: A point cloud benchmark +for fitting simple geometric primitives in CAD objects, Computers & Graphics 102 (2022) +133–143. doi:10.1016/j.cag.2021.09.013. +[20] V. Markovic, Z. Jakovljevic, I. Budak, Automatic recognition of cylinders and planes from +unstructured point clouds, The Visual Computer (2021) 1–24. +[21] T. Birdal, B. Busam, N. Navab, S. Ilic, P. Sturm, Generic primitive detection in point clouds us- +ing novel minimal quadric fits, IEEE Transactions on Pattern Analysis and Machine Intelligence +42 (6) (2020) 1333–1347. doi:10.1109/TPAMI.2019.2900309. +[22] C. Liu, W. Hu, Real-time geometric fitting and pose estimation for surface of revolution, Pattern +Recognition 85 (2019) 90–108. doi:10.1016/j.patcog.2018.08.002. +[23] F. Bergamasco, M. Pistellato, A. Albarelli, A. Torsello, Cylinders extraction in non-oriented +point clouds as a clustering problem, Pattern Recognition 107 (2020) 107443. +doi:https: +//doi.org/10.1016/j.patcog.2020.107443. +URL https://www.sciencedirect.com/science/article/pii/S0031320320302466 +[24] G. Sharma, D. Liu, S. Maji, E. Kalogerakis, S. Chaudhuri, R. Mˇech, ParSeNet: A Parametric +Surface Fitting Network for 3D Point Clouds, in: Computer Vision – ECCV 2020: 16th Euro- +pean Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part VII, Springer-Verlag, +Berlin, Heidelberg, 2020, p. 261–276. doi:10.1007/978-3-030-58571-6_16. +[25] S. Yan, Z. Yang, C. Ma, H. Huang, E. Vouga, Q. Huang, HPNet: Deep Primitive Segmentation +Using Hybrid Representations, in: Proceedings of the IEEE/CVF International Conference on +Computer Vision (ICCV), 2021, pp. 2753–2762. doi:10.1109/ICCV48922.2021.00275. +[26] G. Sharma, B. Dash, A. RoyChowdhury, M. Gadelha, M. Loizou, L. Cao, R. Wang, E. G. +Learned-Miller, S. Maji, E. Kalogerakis, PriFit: Learning to Fit Primitives Improves Few Shot +Point Cloud Segmentation, Computer Graphics Forumdoi:10.1111/cgf.14601. +[27] D. Borrmann, J. Elseberg, K. Lingemann, A. N¨uchter, The 3D Hough transform for plane +detection in point clouds: A review and a new accumulator design, 3D Research 2 (2). doi: +10.1007/3DRes.02(2011)3. +[28] L. A. Fernandes, M. M. Oliveira, A general framework for subspace detection in unordered mul- +tidimensional data, Pattern Recognition 45 (9) (2012) 3566 – 3579. doi:10.1016/j.patcog. +2012.02.033. +[29] M. Camurri, R. Vezzani, R. Cucchiara, 3D Hough Transform for Sphere Recognition on +Point Clouds, Machine Vision and Applications 25 (7) (2014) 1877–1891. +doi:10.1007/ +s00138-014-0640-3. +[30] T. Rabbani Shah, F. van den Heuvel, Efficient Hough transform for automatic detection of +cylinders in point clouds, in: G. Vosselman, C. Brenner (Eds.), ISPRS Laser Scanning 2005, +ISPRS Working Groups, 2005, pp. 60–65. +23 + +[31] M. C. Beltrametti, J. R. Sendra, J. Sendra, M. Torrente, Moore–Penrose approach in the +Hough transform framework, Applied Mathematics and Computation 375 (2020) 125083. doi: +10.1016/j.amc.2020.125083. +[32] C. Romanengo, A. Raffo, S. Biasotti, B. Falcidieno, V. Fotis, I. Romanelis, E. Psatha, K. Mous- +takas, I. Sipiran, Q.-T. Nguyen, C.-B. Chu, K.-N. Nguyen-Ngoc, D.-K. Vo, T.-A. To, N.-T. +Nguyen, N.-Q. Le-Pham, H.-D. Nguyen, M.-T. Tran, Y. Qie, N. Anwer, SHREC 2022: Fitting +and recognition of simple geometric primitives on point clouds, Computers & Graphics 107 +(2022) 32–49. doi:https://doi.org/10.1016/j.cag.2022.07.004. +URL https://www.sciencedirect.com/science/article/pii/S0097849322001224 +[33] E. V. Shikin, Handbook and atlas of curves, CRC Press, 1995. +[34] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, W. Stuetzle, Surface reconstruction from +unorganized points, in: Proceedings of the 19th Annual Conference on Computer Graphics and +Interactive Techniques, SIGGRAPH ’92, Association for Computing Machinery, New York, NY, +USA, 1992, p. 71–78. doi:10.1145/133994.134011. +URL https://doi.org/10.1145/133994.134011 +[35] C. Romanengo, E. Brunetto, S. Biasotti, C. E. Catalano, B. Falcidieno, Recognition, modelling +and interactive manipulation of motifs or symbols represented by a composition of curves, +in: S. Biasotti, R. Pintus, S. Berretti (Eds.), Italian Chapter Conference 2020 - Smart Tools +and Apps in computer Graphics, STAG 2020, Virtual Event, Italy, November 12-13, 2020, +Eurographics Association, 2020, pp. 27–35. doi:10.2312/stag.20201237. +[36] R. Penrose, On best approximate solutions of linear matrix equations, Mathematical Proceed- +ings of the Cambridge Philosophical Society 52 (1) (1956) 17–19. +[37] Edelsbrunner, Letscher, Zomorodian, Topological persistence and simplification, Discrete Com- +putational Geometry 28 (4) (2002) 511–533. doi:10.1007/s00454-002-2885-2. +[38] H. Doraiswamy, N. Shivashankar, V. Natarajan, Y. Wang, Topological saliency, Computers & +Graphics 37 (7) (2013) 787 – 799. doi:10.1016/j.cag.2013.04.009. +[39] S. Biasotti, A. Cerri, S. Pittaluga, D. Sobrero, M. Spagnuolo, Tracking the evolution of rainfall +precipitation fields using persistent maxima, in: Proceedings of the Conference on Smart Tools +and Applications in Computer Graphics, 2016, pp. 29–37. doi:10.2312/stag.20161361. +[40] J. H. Friedman, J. L. Bentley, R. A. Finkel, An algorithm for finding best matches in logarithmic +expected time, ACM Transactions on Mathematical Software 3 (3) (1977) 209–226. +doi: +10.1145/355744.355745. +[41] M. Ester, H. P. Kriegel, J. Sander, X. Xu, A density-based algorithm for discovering clusters in +large spatial databases with noise, in: 2nd Int. Conf. Knowledge Discovery and Data Mining, +AAAI Press, 1996, pp. 226–231. +[42] H. Li, M. A. Lavin, R. J. Le Master, Fast Hough transform: A hierarchical approach, Computer +Vision, Graphics, and Image Processing 36 (2) (1986) 139–161. doi:10.1016/0734-189X(86) +90073-3. +[43] W. Day, H. Edelsbrunner, Efficient algorithms for agglomerative hierarchical clustering meth- +ods, Journal of Classification 1 (1) (1984) 7–24. doi:10.1007/BF01890115. +[44] D. Defays, An efficient algorithm for a complete link method, The Computer Journal 20 (4) +(1977) 364–366. doi:10.1093/comjnl/20.4.364. +24 + +[45] S. Koch, A. Matveev, Z. Jiang, F. Williams, A. Artemov, E. Burnaev, M. Alexa, D. Zorin, +D. Panozzo, ABC: A Big CAD Model Dataset For Geometric Deep Learning, in: The IEEE +Conference on Computer Vision and Pattern Recognition (CVPR), 2019. doi:10.1109/CVPR. +2019.00983. +[46] M. Pauly, M. Gross, L. P. Kobbelt, Efficient simplification of point-sampled surfaces, in: IEEE +Visualization, 2002, 2002, pp. 163–170. doi:10.1109/VISUAL.2002.1183771. +25 + diff --git a/q9E3T4oBgHgl3EQfMQlY/content/tmp_files/load_file.txt b/q9E3T4oBgHgl3EQfMQlY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..28dbb1f5a143e1bb635434fb6ebf0bd79f029039 --- /dev/null +++ b/q9E3T4oBgHgl3EQfMQlY/content/tmp_files/load_file.txt @@ -0,0 +1,1249 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf,len=1248 +page_content='Recognising geometric primitives in 3D point clouds of mechanical CAD objects Chiara Romanengo ∗, Andrea Raffo ∗, Silvia Biasotti †, and Bianca Falcidieno Istituto di Matematica Applicata e Tecnologie Informatiche “E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Magenes”, Consiglio Nazionale delle Ricerche, Via de Marini 6, 16149 Genova, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Abstract The problem faced in this paper concerns the recognition of simple and complex geometric primitives in point clouds resulting from scans of mechanical CAD objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A large number of points, the presence of noise, outliers, missing or redundant parts and uneven distribution are the main problems to be addressed to meet this need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In this article we propose a solution, based on the Hough transform, that can recognize simple and complex geometric primitives and is robust to noise, outliers, and missing parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Additionally, we can extract a series of geometric descriptors that uniquely characterize a primitive and, based on them, aggregate the output into maximal or compound primitives, thus reducing oversegmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The results presented in the paper demonstrate the robustness of the method and its competitiveness with respect to other solutions proposed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Keywords: geometric primitives, point clouds, mechanical CAD objects, Hough transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 1 Introduction The increasing availability of affordable digital scanning devices – such as close-range photogram- metry and laser scanners – has made point clouds one of the most common ways of representing the surface of an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In Computer-Aided Design, this fact results in the need to extract information from measured data points using higher-level geometric primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' To give an example, it is highly convenient to recognise and reconstruct a digital model so that it can be interpreted in terms of some basic components and easily manipulated by CAD systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A large number of points, the presence of noise and outliers, the occurrence of missing or redundant parts and the non-uniform distribution of the data severely limit the use of tessellations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', meshes) as a means to ease the analysis and reconstruction of shapes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' rather, they make it more convenient to analyze the point cloud directly [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Many approaches for detecting primitives lack robustness to noise and outliers or deal only with mesh models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Most of them are able to extract only a few classes of simple geometric primitives, being planes and cylinders the most commonly recognised, and do not consider more complex basic shapes, such as surfaces of revolution or generalized cylinders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Moreover, many methods have a tendency to oversegment, thus producing very fragmented output without aggregating the non- contiguous subsets of points that lie on the same primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The detection of maximal or compound primitives (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' cylinders composed of non-adjacent parts, or patterns) is another open problem, mostly addressed only for planes or cylinders [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' For example, if a pipe is interrupted by another part, as in the block model in Figure 5(b), traditional methods recognise the two parts of the pipe as two distinct primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' This is particularly important, for instance, when decomposing patterns that correspond to scans of assembly CAD models, because it concurs with the recognition of compound primitives and patterns [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' When it comes to memory usage and computational complexity, learning ∗These authors have contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' †Corresponding author 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='04371v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='GR] 11 Jan 2023 approaches require onerous training and large annotated databases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' on the other hand, the direct detection of mathematical primitives in a general space embedding typically faces the computational limitation of dealing with a number of free parameters that rapidly grows as the complexity of the primitives increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In this paper, we deal with the recognition of simple and complex geometric primitives in point clouds originating from scans of mechanical CAD objects, where recognition means the detection of the primitives in a given point cloud and the extraction/computation of the best (geometric) parameters associated with those primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' We adopt an approach based on the Hough transform (HT) to do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The HT meets the needs of being able to recognize multiple instances of primitive functions and, through a voting procedure, to be robust to noise and outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Originally limited to straight lines in images [5], it has been generalized and extended in multiple directions to handle other shapes, also using families of non-parametric templates in images [6] or point clouds from CAD objects [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' recently, the extended version proposed in [8] has opened up the HT to a wide range of algebraic primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Here we generalise the HT to surface primitives – not necessarily algebraic – represented in parametric form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Our solution is tailored to deal with point clouds and is able to deal with simple geometric primitives and some complex ones (generalized cylinders and cones, surfaces of revolution, helical surfaces, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' ), including maximal and compound primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' To this end, we aggregate the primitives found on the basis of their size and space embedding, whenever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Our strategy also reduces the oversegmentation of the output and we exploit the strategy proposed in [9] to decrease the computational complexity of methods based on the HT thanks to an opportune preprocessing of the point cloud and its subparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Our experiments confirm the robustness and completeness of the method, and the comparisons made exhibit its competitiveness with respect to the methods proposed so far in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' To sum up, our main contributions are: The introduction of a geometric primitive recognition method that is particularly robust to noise and outliers and is able to recognize multiple instances of the same primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The extraction and recognition of complex primitives, in addition to the most common ones (planes, cylinders, cones, spheres, tori).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The aggregation of primitives on the basis of their shape, size and position to detect maximal and compound primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' An extensive experimentation on different datasets and comparison with state-of-the-art meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The rest of the paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Section 2 overviews previous work in geometric primitives detection and HT-based recognition methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Section 3 briefly recalls the basic concepts of HT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Section 4 introduces and lists the geometric primitives that can be identified by our method, with emphasis on complex primitives such as general cylinders and cones, surfaces of revolutions, helical surfaces, and convex combinations of curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Section 5 describes our approach for the iden- tification of primitives in point clouds based on the HT, and the search for maximal primitives and possible relationships between primitives through a clustering technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Section 6 provides exper- imental results of our method for the identification of simple and complex primitives and shows a qualitative and quantitative comparative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Concluding remarks end the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 2 Previous work Representing an object with a set of geometric components is a long-standing problem in computer vision, computer graphics and CAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' As we are interested in recognising (simple or complex) geo- metric primitives in the manufacturing domain, here we focus our attention on those approaches that share the same goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A considerable variety of algorithms have been devised to decompose digitalized point clouds or meshes representing CAD objects into regions approximated by primitives belonging to some 2 given sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' According to [10], these approaches can be grouped into four families: stochastic, parameter space, clustering and learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The first group includes the RANSAC method [11] and its optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The second family includes Hough-like voting methods and parameter space clustering, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The third class gathers all the other clustering techniques, and can be classified into three main types: primitive-driven region growing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', [13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' automatic clustering and Lloyd-based algorithms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', [14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' primitive-oblivious segmentation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Finally, with the growing popularity of deep learning techniques, supervised fitting methods have been proposed even for multi-class primitives [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The reader is referred to [10] for a comprehensive historical taxonomy of methods for simple primitive detection, which is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Despite a large number of available solutions, the problem is far from being solved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' novel ap- proaches have been proposed in the very last few years to address the shortcomings of existing paradigms or to propose novel directions to move in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A recent approach to deal with the recognition of geometric primitives is described in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' It consists of a curvature-based partitioning method that decomposes an input triangle mesh into maximal surface portions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' the decomposition is per- formed so that each segment corresponds to one of the following seven invariance classes of surfaces: plane, sphere, cylinder, surface of revolution, prismatic, helix, and complex surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The method was adapted to handle point clouds in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In [20], a method for the identification and fitting of planes and cylinders from unstructured point clouds in manufacturing is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' It consists of three subsequent phases: point cloud segmentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' merging of oversegmented regions and estima- tion of surface parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' extraction of cylinders and planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Being the method able to handle only two primitive types, its applicability is however restricted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' To handle the increasing availability of acquired data, [1] introduces a region-growing-based system for the segmentation of large point clouds in planar regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Other approaches, devised to detect only specific types of primitives, are: [21], which deals with quadric surfaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [22] which fits surfaces of revolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' and [23], which extracts cylindrical shapes from non-oriented point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In the field of deep learning, two methods have shown promising results in the problems of segmentation and recognition: ParSeNet [24] and HPNet [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Beyond their performance, outstanding merit is their capability of handling – together with the more classical simple geometric primitives – open and closed spline surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Another learning-based method lately proposed for fitting primitives is PriFit [26], which learns to decompose a point cloud of various 3D shape categories into a set of geometric primitives, such as ellipsoids and cuboids, or alternatively deformed planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' However, the natural flexibility of supervised learning approaches in identifying geometric primitives comes with a considerable cost: the need of having gigantic labelled training sets, whose difficulty of gathering limits their current application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' As for the methods that use the HT to identify geometric primitives, they were limited, until recently, to planes [12, 27], combinations of linear subspaces [28], spheres [29] and circular cylinders [30];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' in the case of cylinders, however, the rotational axis needs to be known before the application of the Hough transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The recent advances in the use of the algebraic functions [8] are paving the road to a larger use of the HT for recognising more complex families of geometric primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' To the best of our knowledge, the sole approaches that exploit this idea were proposed in [31], for the recognition of an ellipsoid in a free-form model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' and in [9], for the recognition of spheres, (circular) cylinders, (circular) cones and tori in presegmented point clouds, as a way to post process faulty segmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The work in [9] also addresses the problem of reducing the number of the parameters of the primitive representation when the point cloud corresponds to a patch of a primitive: in this case, the point cloud is opportunely rototranslated in a space embedding so that it can be fit by a primitive in a standard form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The method has been recently compared with other direct and learning-based approaches, see [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' However, the point clouds used for this comparison did neither originate from scans of mechanical CAD objects nor can be assumed to represent presegmented data, making the benchmark itself not of interest to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 3 Basic concepts on the Hough transform We denote by A3 x(R) and An A(R), respectively, the 3-dimensional and the n-dimensional affine spaces over R, where x := (x, y, z) and A := (A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' , An) vectors of indeterminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Given a surface S 3 Figure 1: Simple geometric primitives: plane, cylinder, cone, sphere and torus respectively, along with their attributes (geometric descriptors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' defined as the zero locus of a function f, a parameter-dependent family of surfaces can be described by functions fa as F = {Sa : fa(x) = 0 | a ∈ U}, where U is an open set of the parameter space An A(R) and a := (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', an) is the parameter vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Then, given a point P ∈ A3 x(R), the HT of the point P with respect to the family F is ΓP = {fa(P) = 0} ⊂ An A(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' For sufficiently general points P ∈ A3 x(R), it turns out that ΓP are hypersurfaces in An A(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' If the set of hypersurfaces ΓP generated by varying P on a given surface meets in one and only one point ¯a ∈ U, the family of surfaces F is called Hough-regular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' the intersection point ¯a of the Hough transforms ΓP uniquely identifies the surface S¯a from which the points P ∈ A3 x(R) were sampled in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' If the HT regularity is not guaranteed, the solution to the intersection problem might be non-unique and a solution must be selected among more potential parameters solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In the discrete scenario, the HT framework deals with the problem of finding a surface – within a family F of surfaces – that best approximates a particular shape, given in the form of a data set of N points P, where N >> n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The common strategy to identify the solution (or a solution), introduced in [8], is based on the so-called accumulator function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' it consists in a voting system whereby each point in P votes a n-uple a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' , an);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' the most voted n-uple corresponds to the most representative surface for the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' While the HT traditionally deals with one shape at a time, the framework we propose in this paper can be directly applied to point clouds containing multiple shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The family of surfaces we consider is represented in parametric form and we assume that the system defining Sa with respect to the Cartesian coordinates x, y, and z can be analytically solved with respect to the parameter vector a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' , an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A more detailed description of how this works in our case is given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 4 Definition of complex geometric primitives In addition to the simple geometric primitives of Figure 1 (plane, cylinder, cone, sphere, and torus) – see [9] for their recognition in presegmented point clouds – in this section, we introduce a set of complex geometric primitives that to the best of our knowledge have never been used before for recognition by HT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In this paper we express surfaces parametrically with respect to the variables u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' For the sake of simplicity, some of the surfaces are here presented in their standard form or with respect to some specific axes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' nevertheless, one can easily generalise these equations by applying a generic transformation of the special orthogonal group SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' General cylinders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A cylinder is a surface traced by a straight line of fixed direction, the generatrix, while moving along a curve, the directrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Given a curve (x(u), y(u), z(u)) := (f1(a, u), f2(a, u), f3(a, u)) and a direction (l, m, n), the parametric representation of the cor- responding cylinder is given by: � � � � � x = f1(a, u) + lv y = f2(a, u) + mv z = f3(a, u) + nv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 4 Imax CO 工 rminNote that a general cylinder depends on the parameters defining generatrix and directrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The dictionary of curves to be considered as directrix is extremely rich, see [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Table 1(a) considers a 5−convexity curve as directrix, whose equation can be found in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' General cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A cone is a surface traced by a straight line, the generatrix, while gliding along a curve, the directrix, and passing through a fixed point, the vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Given a parametrized curve (x(u), y(u), z(u)) := (f1(a, u), f2(a, u), f3(a, u)) and a point V = (xV , yV , zV ), the parametric representation of the corresponding cone is given by: � � � � � x = xV + (f1(a, u) − xV )v y = yV + (f2(a, u) − yV )v z = zV + (f3(a, u) − zV )v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' As in the case of cylinders, we can exploit the dictionary of plane curves to create families of cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Then, a general cone depends on the parameters that define the curve and the components of the vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Table 1(b) shows a cone generated by a 5−convexity curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Surfaces of revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A family of surfaces of revolution can be created by rotating a fam- ily of curves around an axis of rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' For example, given the family of plane curves (x(u), y(u), z(u)) := (f1(a, u), 0, f2(a, u)) and the z−axis, we obtain the parametric equations � � � � � x = f1(a, u) cos v y = f1(a, u) sin v z = f2(a, u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' One of the most known examples is the torus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' another example is given by ellipsoids, which are obtained by rotating an ellipse with respect to one of its principal axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Table 1(c) shows the case of a surface obtained by rotating the curve (x(u), y(u), z(u)) := (au, 0, b/u5) around the z−axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Helical surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Table 1(d) presents a family of equations obtained by modifying the parametri- sation of a circular cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Precisely, the radius R here varies between [R1, R2], where R1 > 0, by a cosine function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' when radii are fixed, the slope of the output surface is controlled by the parameters in z(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Note that the radius variation can be adapted to other shapes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', the triangle wave function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Convex combination of curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' It is possible to define surface primitives by considering the con- vex combination of a pair of parametrised curves (f1(a, u), f2(a, u), f3(a, u)) and (g1(b, u), g2(b, u), g3(b, u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' This family has the following parametric equations: � � � � � x = vf1(a, u) + (1 − v)g1(b, u) y = vf2(a, u) + (1 − v)g2(b, u) z = vf3(a, u) + (1 − v)g3(b, u) , where v ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Note that the primitive parametrisation depends on the same parameters which define the pair of curves, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A planar example is given by the annulus, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', the region bounded by two concentric circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A helical strip can be obtained by cutting and bending an annular strip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' this corresponds to considering a convex combination of two helices of an equal slope but different radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' An example of a helical strip is provided in Table 1(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The use of the standard form allows us to diminish the number of unknown parameters in the recognition process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' To give a practical example, the family � � � � � x = a1 cos(u) + b1 sin(u) + c1v + d1 y = a2 cos(u) + b2 sin(u) + c2v + d2 z = a3 cos(u) + b3 sin(u) + c3v + d3 , 5 contains (circular) cylinders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' it has 12 unknown parameters, which make the representation compu- tationally and memory-wise unmanageable when used in a voting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' By further assuming that the cylinder can be rototranslated so that it has the rotational axis aligned with the z−axis and it is centred in the origin, the number of parameters reduces to 1 in the case of a circular cylinder, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', the sole radius: � � � � � x = r cos(u) y = r sin(u) z = v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A strategy to obtain the standard form in an automatic way starting from a point cloud is provided in [9], and it can be naturally extended to complex primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Table 1: Parametric representation of some complex geometric primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' (a) (b) (c) (d) (e) generalized generalized surface of helical helical cylinder cone revolution surface strip � � � � � � � x = a cos u 1+b cos(5u) y = a sin u 1+b cos(5u) z = v � � � � � � � x = av cos u 1+b cos(5u) y = av sin u 1+b cos(5u) z = Av + B � � � � � x = au cos v y = au sin v z = b u5 � � � � � x = R(u) cos v y = R(u) sin v z = k1(u + nv) + k2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' where R(u) := R1 + R2 − R1 2 (cos u + 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' n ∈ Z � � � � � x = R(u) cos v y = R(u) sin v z = v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' where R(u):=au+(1-u)b 5 Recognising geometric primitives using Hough transforms In this section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' we describe a method to recognise an input point cloud P with geometric primitives via the HT technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' It consists of the following main steps: an initial point cloud preprocessing followed by the iteration of a recognition step and a splitting phase, and a final step which applies a clustering technique to discover geometric relationships between primitives or parts of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A graphical illustration of its pipeline is given in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1 Point cloud preprocessing First, the input point cloud P is translated to place its barycenter at the origin of the Cartesian coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Then, the normals at the input points are estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In the literature, there are several ways to estimate normals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' in general, we are interested in using an approach suitable for handling noisy input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' For our purposes the orientation of the normal vector is not relevant, as we are only interested in aligning the most voted normal direction to the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' We can therefore use the approach introduced in [9], which estimates the normal vectors through a voting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' For convenience, when P is clean we can also consider the method presented in [34], which is available in MATLAB with the pcnormals function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' for each input point in P, the method selects the k points of P closest to that input point and then uses principal component analysis (PCA) to estimate the normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Note that this does not necessarily generate normal vectors that are coherent in their orientation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' however, this is not a problem for our pipeline since we are not interested in the normal orientation but rather in the direction of the (estimated) normal vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 6 The point cloud is then rotated so that the most voted direction among the (just-estimated) normal vectors coincides with the z−axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' This is an essential step for multiple reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Firstly, our dictionary contains primitives that – in their standard form – are given with the axis coincident with the z−axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In addition, this process is necessary to test the existence – as a first check – of primitives in their standard form (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', with centres or vertices in the origin of the Cartesian axes and with the normal or the principal axis aligned to the z-axis), thus reducing the number of parameters that will need to be estimated by the HT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Finally, P is scaled into a unit cube, in order to restrict the range of variation for most parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2 Recognition step After being preprocessed, P becomes the input of the HT-based recognition step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Since P can contain different instances of the same primitive type and primitives of different types, the standard HT procedure must be adjusted to allow such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' This step can be summarized as follows: Selection of one or more families of primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The user can select the families of primitives to be used for recognition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' in its default configuration, the algorithm tests the presence of simple geometric primitives – one family at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' By studying the geometric properties of P and by relating them to the geometric characteristics of the selected family (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', bounding box, radius, diagonal length), it is possible to initialise a region T of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The region T is discretized into cells, which are uniquely identified by the coordinates of their centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Then, an accumulator function H, in the discretized form of a matrix, is initialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The matrix entries are in a one-to-one correspondence with the cells of the discretization performed in the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Estimation of the accumulator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' An entry of the accumulator function H is increased by 1 each time the Hough transform ΓP of a point P intersects the corresponding cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The way we check if and where a Hough transform intersects some cells depends on whether the family of surfaces is described in implicit or parametric form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In our case, surfaces are expressed in parametric form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' we adapt to surfaces the method devised for curves in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Specifically, if the system of equations fa defining the family can be solved analytically with respect to the unknown parameters a, we automatically calculate a sample of ΓP by exploiting the Moore- Penrose pseudo-inverse [36] of the matrix that defines the coefficients of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The evaluation of the intersection is translated into an inequality between the components of the sample points of ΓP and the coordinates of the cell endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Selection of potential fitting primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The cells corresponding to the peak values of the accumulator function H have to be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' When the set of points is assumed to represent a single surface profile, the traditional HT formulation aims at finding the maximum of the accumulator function H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' when the family of surfaces is not Hough-regular, there might exist several maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' On the other hand, if the point cloud is composed of different primitives, different peaks of H identify potential primitives that might fit different parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' To select these peaks, we observe that peaks of the accumulator function corresponding to primitive shapes rise distinctly with respect to their neighbours and are well-characterised as isolated peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Formally said, we proceed by identifying peaks that have high topological persistence1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In our implementation, peaks that correspond to primitives are automatically recognised by keeping the local maxima having a persistence higher than 10% of the maximum value of H, by using the algorithm for persistent maxima proposed in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The coordinates of the cell centres of the maxima correspond to the parameters of potentially recognised surface primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 1The notion of topological persistence was introduced in [37] for encoding and simplifying the points of a filtration f by classifying them as either a feature or noise depending on its lifetime or persistence within the filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In practice, given a pair of points p and q, their persistence is defined as f(p) − f(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Pairing is defined in terms of the topological connection between the points, for details on topological persistence and saliency, we refer to [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In our case, the domain is represented by the grid of T, the role of filtration is played by the accumulator function H, and we are interested only in the peaks of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 7 Evaluation of the approximation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' To measure the recognition accuracy of a specific primitive, we select the set of input points X closer to such a primitive than a given threshold and study its density via the k-Nearest Neighbor algorithm (see, for example, [40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' If X is sparse, the recognised primitive is considered a false positive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' otherwise, for each dense subset Xi ⊂ X we define the Mean Fitting Error (MFE) as: E(Xi, P) := 1 |Xi| � x∈Xi d(x, P)/li, (1) where P is the current primitive, d is the Euclidean distance, and li is the diagonal of the axis- aligned bounding box containing Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' What can happen when the selected family of primitives does not fit any part of the point cloud?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' There are two possibilities: – The accumulator function is identically zero, with the result that its persistence is zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' therefore the selection of potential fitting primitives returns the empty solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' – The accumulator function H does not present predominant peaks, resulting in false pos- itives which are identifiable by studying the sparsity of the fitted points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Given a set of dense points Xi and two candidate primitives Pi,1 and Pi,2, we first calculate the fitting errors E(Xi, Pi,1) and Ei(Xi, Pi,2) between each primitive and Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' the primitive having lowest error is kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Each time a primitive is recognized, the points of P close to the recognised primitive less than a threshold ε are discarded from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The value of ε typically ranges from 1% to 3% of the diagonal of the minimum bounding box of the P, according to the type of primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='IF ALL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='POINTS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='ARE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='LABELED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='primitive type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='∀ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='IF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='SPARSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='𝒫′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='ELSEIF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='∅ ⊊ 𝒫′ ⊊ 𝒫 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='SELECTION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='OF A FAMILY Plane Cylinder Sphere Cone Torus … ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='VOTING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='PROCEDURE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='RECOGNITION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='OF POTENTIAL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='PRIMITIVES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='APPROXIMATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='ACCURACY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='PREPROCESSING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='UNRECOGNIZED POINTS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='𝒫′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='RANSAC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='AGGREGATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='INTO CLUSTERS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='(DBSCAN) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='𝒫′ = ⋃ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='𝒫′ i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='SEGMENT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='PREPROCESSING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='segment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=': ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='SET ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=':= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='∀ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='𝒫′ i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='𝒫 𝒫′ i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='INPUT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='𝒫 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='POINTS ARE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='UNCLASSIFIED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='SET OF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='RECOGNIZED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='SEGMENTS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='𝒮j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='CLUSTERING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='RECOGNITION STEP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='SPLITTING PHASE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='POSTPROCESSING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='ELSEIF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='𝒫′ = 𝒫 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='Figure 2: Pipeline of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The algorithm returns the parameters of the geometric primitives and the corresponding points fitted by them, as well as the set of points that were not fitted by any primitive – denoted by P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Note that, if more geometric primitives potentially fit the same region of the point cloud, we select the one with the minimum fitting accuracy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', the lowest MFE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 8 X 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='6645 X100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='583 Y 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='502 Y98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='4437 Z 3479 4000 Z 3385 3000~ X131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='195 Y128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='405 2000~ Z 1424 1000~ 0 > 50 140 120 100 100 80 60 40 20 150MFE cylinder 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0039 cylinder 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0032 cylinder 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='3 Splitting phase The algorithm chooses the next step according to the resulting P′: If P′ is sparse, then its points are returned as unclassified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' If ∅ ⊊ P′ ⊊ P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', if P′ is a proper nonempty subset of P, we proceed by aggregating points in P′ into clusters P′ j, with j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' , Nclust, by adopting the Density-Based Spatial Clustering of Applications with Noise (DBSCAN) method [41], which groups together nearby points and marks as outliers isolated points in low-density regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' It requires two parameters: the first is a threshold used as the radius of the density region, while the second represents the minimum number of points required to form a dense region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Then, the recognition step from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2 is iterated over each cluster P′ j as long as some geometric primitives are recognised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Before proceeding with a new recognition round, we preprocess each cluster P′ j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' the preprocessing adopted here differs from that applied to the entire point cloud P: we exploit the strategy presented in [9] to estimate its standard form, thus reducing the number of parameters that have to be estimated in the new iteration of the recognition step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Specifically, it computes the normals of the points of the cluster P′ j through a voting procedure and it estimates the position of vertices or centres and the direction of axes of symmetry, exploiting different geometric rules specialised for each type of primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' To reduce the dimension of the parameter space, vertices and centres are translated to the origin of the coordinate system, while axes are aligned to the z − axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Then, it automatically centres and orients a (spherical, cylindrical, conical or toric) cluster so that it can be fitted with a primitive in standard form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In case the recognition step involves complex primitives, the strategy proposed in [9] can be naturally extended since they are characterized by the presence of symmetry axes that can be estimated in a similar way through the use of the normals of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' It is possible that, in some steps, no primitive is recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' This can happen in two cases: (i) when applying our algorithm to an input point cloud that has no primitives in their standard form, or (ii) when the estimation of the standard position of some cluster P′ j fails – because P′ j contains more than one primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In both cases, we proceed by oversegmenting the model: in our experiments, we use the RANSAC algorithm proposed in [11], because of its efficiency and its tendency to oversegment point clouds (see, for example, [15]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' however, we proceed by estimating primitive types and geometric parameters with a new round of the recognition step, as voting procedures have shown greater robustness to point cloud artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' This step ends by transforming the parameters found by the Hough transform with respect to the inverse translations, rotations and scaling applied in the previous steps of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The result of this procedure is the decomposition of an input point cloud P into several subsets, called surface segments or simply segments, in such a way that points of the same segment are well approximated by the same primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' We denote these final segments by Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='4 Segment association based on geometric relationships After decomposing the input point cloud into the segments Sj, a clustering technique is applied to unveil geometric relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The goal of this step is to find maximal primitives that are not automatically detected in the HT-based recognition process, to detect patterns of primitives or to relate primitives (or parts of them) even if they do not belong to the same primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' More specif- ically, we relate primitives on the basis of their positions, orientation and dimension, as described in [9] for simple geometric primitives: segments are aggregated only if they are part of the same primitives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' otherwise, we only identify relationships that may be of interest, for example in the processing phases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', process planning, machining).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' To give some examples: Segments sampled from circular cylinders can be associated with respect to their radii and rotational axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' For instance, it is possible to check whether such segments: originate from 9 the exact same primitive shape, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', if they are all sampled from a cylinder of a given radius and rotational axis (such as for the red segments in Figure 5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' are characterized by the same radius and have parallel rotational axes (such as for the black segments in Figure 8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' share the radius but not the rotational axis, not even in terms of parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Segments sampled from helical surfaces can be associated with respect to their parameters R1, R2 and n, k1, k2 and the direction – or a combination of such geometric descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Segments sampled from n-convexity cylinders can be associated with respect to their parame- ters a and b, the number of convexities n, and the direction of the generatrix – or a combination of such geometric descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' To perform segment associations, a well-known (hierarchical) clustering strategy is considered – the complete linkage – as it penalizes chaining effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='5 An illustrative example Figure 2 provides a graphical illustration of the pipeline of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The main steps of the algorithm are associated with an example of a point cloud representing a gear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' After preprocessing the point cloud, we start with the first round of the recognition step – which aims at recognizing the presence of primitives in their canonical position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' For the sake of clarity, however, the graphical illustration only focuses on the recognition of cylinders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Specifically, once the family of cylinders is selected, the corresponding accumulator function is computed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' it exhibits three peaks, which indicate the presence of three potential solutions: the three cylinders highlighted in the colours red, green, and blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The approximation accuracy for this type of primitive is less than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Being some segments (the non-axis-aligned planar segments) non in their canonical position, their points are not recognized in the first round of the recognition step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Instead, these points are collected in P′ and aggregated into dense clusters in the splitting phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' each of such clusters is individually studied by a new round of the recognition step and recognized as planar segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' However, being these segments studied separately, we lose some geometric information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' We then apply clustering to recognise that some planar segments lie on the same parametric plane, by exploiting the geometric descriptors provided by the HT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The final result is a segmentation of 17 segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' It is worth mentioning that the method is able to group some of the non-adjacent parts, which belong to the same primitive in canonical position – as in the example of the two external cylinders, and of the axis-aligned planes that form four couples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' However, for primitives not in their canonical position, postprocessing based on clustering is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='6 Computational complexity In the preprocessing step, our method includes the estimation of the normal vectors and the indi- viduation of the most voted normal, which is implemented as explained in [34] – in the case of clean point clouds – and in [9] – in the presence of point cloud artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' For a thorough analysis of the computational complexities of this estimation, the reader is referred to the corresponding papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The formulation of the HT for surface primitives embedded in R3 naturally extends that for plane curves in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In the pipeline presented in this paper, for each point cloud P (both in the case of the initial point cloud and of single clusters at subsequent iterations) we apply the HT by considering the different types of geometric primitives one at a time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' thus, the dimension of the parameter space changes accordingly to the type of primitive considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The quantization of a region of interest and the dimension of the parameter space determines the size of the accumulator function, and dominate both the memory usage and the computational complexity of the Hough transform: it is, therefore, necessary to balance the samples for each parameter and their number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' To reduce the computational cost, primitives are put in their (estimated) standard positions, in this way the number of parameters does not exceed 3, as explained in [9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' complementarily, adaptive approaches 10 can be used to further speed up the search (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', [42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The overall computational complexity of the HT-based recognition step is O(ML), where M denotes the number of cells of the parameter space and L represents the number of points (in the initial point cloud or in a cluster obtained at a subsequent iteration) at which we evaluate the HT accumulator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' More precisely: At the first iteration the HT is applied to the whole input point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Supposing that the number of families of primitives to be used for the recognition is K and the number of points in P is N, the overall complexity of the first iteration is O(N �K k=1 Mk), where Mk is the dimension of the parameter space for the k-th primitive type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In the subsequent steps, the HT-based recognition method is applied to each cluster P′ j and then the computational cost corresponds to O(�Nclust j=1 Nj �K k=1 Mk), where Nj denotes the number of points in cluster P′ j while Nclust is the number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Notice that, being the recognition step performed independently w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' the primitive type, the task is embarrassingly parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The same applies to the clusters of points obtained in the same iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Finally, agglomerative hierarchical clustering approaches require, in their naive implementation, O(N3 seg) operations, where Nseg denotes the number of segments in the output segmentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' see, for example, [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' When it comes to complete-linkage clustering, one can consider more efficient implementations, such as the one proposed in [44], which costs O(N2 seg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Although the dissimilarity- matrix assembly costs O(N2 seg), one may note that each entry is computed independently;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' again, the task is therefore embarrassingly parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 6 Experimental results In this section, we show the results obtained with our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Specifically, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1 presents some point clouds of mechanical objects containing only simple geometric primitives, while Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2 provides some examples with complex primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='3, we exhibit some noisy point clouds to demonstrate the robustness of our pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In all cases, whenever possible, we show the maximal primitives of the selected point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Finally, a qualitative and quantitative comparative analysis is provided in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1 Overall analysis 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1 Simple geometric primitives As a sanity check, we first apply our method to two models from [45], which were selected because containing all simple geometric primitives – plane, sphere, cylinder, cone and torus – and because of the availability of ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A first example is provided in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The point cloud, corresponding to the set of vertices of the original triangle mesh, is decomposed in 8 surface segments: 5 cylinders, 2 planes and 1 sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The high accuracy of our method is proved by comparing the parameters from the HT with those in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In the second example, presented in Figure 4, the corresponding point cloud is subdivided into 9 surface primitives: 4 cylinders, 1 torus, 3 axis-aligned planes and 1 cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Again, we are able to recognise all primitives up to a small error in the parameters with respect to those provided in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Figure 5(a-b) shows point clouds that can be segmented into complete geometric primitives without the need for a further step of aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The first example, shown in Figure 5(a), consists of 11 segments – 3 cylinders and 8 planes – and it highlights the robustness of our method when it comes to detecting intersecting cylinders, here arranged similarly to a Steinmetz solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Complete geometric primitives, each of which is represented by a specific colour, are automatically detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Another point cloud, displayed in Figure 5(b), contains 5 segments: 2 tori, 1 cylinder and 2 axis- aligned planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' As for the previous case, the HT-based recognition successfully detects all the primitives, even those made up of non-adjacent parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Figures (6-9) show point clouds wherein the segments produced by the HT approach are post- processed by hierarchical clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In Figure 6(b), the input point cloud is first segmented into 20 11 equation code ground truth HT parameters x2 + y2 = r2 C1 r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='50 r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='50 C2 r = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00 r = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00 C3 r = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00 r = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00 C4 r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='50 r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='50 C5 r = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00 r = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00 z = k P1 k = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00 k = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00 P2 k = 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00 k = 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00 x2 + y2 + z2 = r2 S r = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00 r = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='05 (a) (b) (c) Figure 3: In (a) a mechanical CAD model from the benchmark in [45];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' in (b) the vertices of its triangle mesh decomposed into 8 surface segments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' in (c), for each primitive, the HT parameters are compared with those provided by the database equation code ground truth HT parameters x2 + y2 = r2 C1 r = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='01 r = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='01 C2 r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='97 r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='97 C3 r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='28 r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='26 C4 r = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='25 r = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='25 z = k P1 k = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00 k = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00 P2 k = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='08 k = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='08 P3 k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00 k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00 (R − � x2 + y2)2 + z2 − r2 = 0 T R = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='25 R = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='25 r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00 r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00 x2 + y2 + a(z − b)2 = 0 C a = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='77 a = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='81 b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='19 b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='22 (a) (b) (c) Figure 4: In (a) a mechanical CAD model from the benchmark in [45];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' in (b) the vertices of its triangle mesh decomposed into 9 surface segments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' in (c), for each primitive, the HT parameters are compared with those provided by the database Segments (a) Segments (b) Figure 5: Recognition of CAD point clouds containing only simple geometric primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The identification of maximal segments does not require, in these cases, the application of any clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' surface primitives via the HT-based recognition algorithm (cylinders and planes), which are then associated thanks to the comparison of the geometric descriptors that characterize them uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' As expected, no pair of primitives lying on the same surface is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Despite the presence of imperfections in the original model (see Figure 6(a)), we can correctly identify repeating primitives, here in the form of circular cylinders of equal radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Figure 6(c) highlights the similarities identified by associating cylinders: 4 in light blue, 3 in red, 2 in purple and 2 in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 12 Cylinder 1 (C1) Plane 2 (P2) Plane 1 (P1) Cylinder 2 (C2) Plane 3 (P3) Torus () Cone (C) Cylinder4 (C4) Cylinder 3 (C3)Plane (P1) Cylinder (C1) Cylinder (C2) Plane (P2) Cylinder (C3) Cylinder (C4) Cylinder (C5) Sphere (S)(a) Model (b) Segments (c) Clustering Figure 6: A linkage arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In (a), the original model is shown, together with a magnification revealing some imperfections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The segments identified by the HT are shown in (b) in different colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' (c) draws attention to cylinders, among which we can recognize segments lying on the same primitive, up to a translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Figure 7 increases the difficulty by considering a point cloud containing a higher number of segments, some of which are low-quality as the small holes highlighted in Figure 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In this example, the geometric primitives identified by the HT algorithm are cylinders, cones, and planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The result is a segmentation in 28 surface segments, see Figure 7(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Note that the central hole presents some grooves, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', a surface detail that was not present as a geometric feature in the original CAD model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' therefore, we recognise it as a cylinder and the HT is able to ignore the shallow grooves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A final application of clustering makes it possible to identify repeating primitives, up to translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Figure 7(c) illustrates the similarity between 6 cylindrical holes (in green) and between other 6 cylindrical segments (in yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' (a) Model (b) Segments (c) Clustering Figure 7: A carter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In (a), the original model is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The surface segments found by the HT approach are depicted in (b), while (c) shows the result of primitive association, when one is interested in identifying the same primitive up to a translational transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Different rows correspond to different points of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Another example of gear is shown in Figure 8(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Here, the total number of extracted geometric primitives is 68: 19 cylinders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 2 cones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 47 planes, 5 of which are axis-aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The result is presented in Figure 8(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' As highlighted by the original model, in this prototypical version of the NuGear component, cylinders are roughly approximated by a series of planar primitives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' in this specific case, 13 0we fit a cylinder instead of many planes, since the former can describe a much larger area without significantly increasing the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Clustering identifies here a similarity between the 12 cylindrical holes – up to translations – and between 2 external cylinders, while the surface segments identified by non-axis-aligned planes are not grouped in pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' this is because the tooth inclination prevents any alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Figure 8(c) shows this result, colouring the holes in black and the external cylinders in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' (a) Model (b) Segments (c) Clustering Figure 8: A prototype of the NuGear component, courtesy of STAM S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' (Genoa, Italy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The original model is shown in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The decomposition in clusters of points produced by the HT approach is given in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The output of the additional clustering association procedure is shown in (c), which highlights the similarity between 12 cylindrical holes (in black) and between two cylinders (in yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Finally, Figure 9(a) shows another mechanical part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The HT-based decomposition of the input point cloud consists of 50 surface segments, all of which are tori, cylinders, and planes, see Figure 9 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The clustering method can identify the similarity between 8 red cylindrical holes and between 2 blue cylindrical segments, up to translations – see Figure 9(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Finally, 2 tori are aggregated, because they lie on the same surface primitive up to rototranslations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Table 2 summarises the characteristics of each point cloud processed in this section and the mean fitting error for all the simple primitives recognised on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Table 2: Statistics of the MFES for all models of Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Being the MFE normalized by definition, we can conclude that the maximum error for the fitting of the simple primitives is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='48%, which corresponds to the noisy holes in the carter of Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Model # points # segs min(Ei) mean(Ei) max(Ei) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 3 15, 216 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0046 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 4 15, 022 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0051 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 5(a) 25, 000 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0056 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 5(b) 25, 000 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0059 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 6(b) 50, 000 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0300 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 7(b) 50, 000 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0448 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 8(b) 50, 000 68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0178 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 9(b) 50, 000 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0057 14 0 0 0 0(a) Model (b) Segments (c) Clustering Figure 9: A mechanical part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In (a) the original model is shown, while in (b) the decomposition of the corresponding point cloud into segments produced by the HT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In (c) the result of the clustering procedure: 8 cylindrical holes, in red, have a high similarity, up to translations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' the same applies for 2 cylindrical segments, in blue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 2 tori, in orange, identify the same primitive, up to a rototranslation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2 Complex geometric primitives As anticipated, our method can recognize other primitives in addition to the simple ones shown in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Here we show some of the complex primitives identified in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The first example – shown in Figure 10(a) – contains an ellipsoid, which is easily recognized by our method at the price of an additional parameter in the parameter space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' additionally, the point cloud can be partitioned into 4 planes, 1 torus and 1 cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In the point cloud from Figure 10(b), we are able to identify correctly the yellow segment as a surface of revolution from Table 1(c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' the remaining points are segmented into 2 cylinders and 2 planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In the point cloud shown in Figure 10(c), we recognize the gold part as a generalized cone, while the blue and the green segments are fitted with generalized cylinders: all of the three have the same directrix, a 5−convexity curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' This last point cloud has been segmented into 7 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Segments (a) Segments (b) Segments (c) Figure 10: Recognition of complex geometric primitives in CAD point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Their identification does not require, in these cases, the application of any clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Figure 11(a) displays a mechanical part which can be accurately described by combining a portion of a helical surface, as introduced in Table 1(d), with a pair of planes and a pair of convex combinations of helices, see Table 1(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The result is a segmentation of the point cloud into 6 primitives, Figure 11(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Two of them are then grouped by the clustering technique since the helical strips have the same equation up to a translation, as shown in Figure 11(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Table 3 provides the main characteristics of each point cloud processed in this section and the mean fitting error for all the simple and complex primitives recognised on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 15 (a) Model (b) Segments (c) Clustering Figure 11: A screw-like part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The original model, (a), is sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The surface primitives detected via HT are shown in (b) using different colours: a helical surface (in purple), two planes (in red and magenta), and two helical strips (in orange and yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Although no pair of them lies on the same parametrised surface, the 2 helical strips have the same equation up to a translation, as shown in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Table 3: Statistics of the MFEs for all models of Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Being the MFE normalized by definition, we can conclude that the maximum error for the fitting of the simple primitives is 1, 54%, which corresponds to the helical surface of Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Model # points # segs min(Ei) mean(Ei) max(Ei) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 10(a) 25, 524 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0063 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 10(b) 67, 777 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0071 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 10(c) 25, 000 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0081 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 11(b) 25, 000 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0154 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='3 Robustness of the pipeline The use of the HT naturally leads to a robust method for the recognition of mathematical surfaces, as suggested in the examples of Figures 6 and 7 – which were characterized by spurious parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In the point cloud of Figure 12, the HT recognition correctly identifies the cylinder that fits the central part, without being negatively influenced by the letters in relief – see the original model in Figure 12(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The point cloud is decomposed into 38 segments: 23 cylinders with different axes, 10 planes, 4 cones, and 1 torus that automatically identifies the top and bottom of the cylinder with the “GRAYLOC” inscription (see Figure 12(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The application of the hierarchical clustering technique allows us to group together: 8 grey cylindrical holes (up to rototranslations);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 8 purple cylindrical segments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 2 aquamarine circular cylinders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 3 violet circular cylinders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 2 orange circular cones (up to a reflection);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 2 black cones (up to a reflection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Moreover, the small imperfections of the manufacture on the central part of the body (recognised by vertical cones, cylinders and tori at the top and bottom) and on the lateral holes do not prevent the clustering technique from appropriately associating the corresponding segments, correctly dealing with rotations and reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' However not everything is recognised: the black dots in Figure 12(b) correspond to points that are not fitted by any of the geometric primitives at our disposal, as they originate from irregular elements that act as a connection between better-defined segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' We label such points as “unsegmented” because of the high mean fitting error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Further proof of the robustness of our method applied to raw data is presented in Figure 13 and quantitatively analysed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In this example, we perturb the point cloud of Figure 5(b) by adding zero-mean Gaussian noise of standard deviation: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='10 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The first row shows the points classified as noise (in black) and the segments found by our method in the same image, for each level of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The second row focuses only on the points that fit the identified primitives, thus providing a denoised segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The robustness to noise is quantitatively studied in Table 4: the parameters obtained in the original point cloud are there compared with those found 16 (a) Model (b) Segments (c) Clustering Figure 12: A clamp connector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In (a) the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In (b) the decomposition of the correspond- ing point cloud into 38 segments provided by the HT procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In (c), the final grouping obtained by clustering, consisting of 6 clusters of primitives (here, singletons of segments are transparent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' in the perturbed point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='01 σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='05 σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='10 σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='20 Figure 13: The point cloud in Figure 5(b) is perturbed by adding zero-mean Gaussian noise of standard deviation: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='10 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The first row superimposes the points identified as noise (in black) to the final segments found by our method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' the second row depicts the points that fit the primitives found and provides a denoised segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2 Comparative analysis We here present two comparisons with alternative methods, all of which rely on estimating the normal vectors at the given input points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The first analysis, proposed in Figure 14, is merely visual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Due to the lack of freely-downloadable implementations for some methods, it is indeed not possible to present this comparison other than qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The analysis consists of a comparison of our method with the RANSAC-based method introduced in [11], the patch aggregation approach in [15] and two recent deep learning architectures: ParSeNet [24] and HPNet [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In this comparison, we have focused on models that solely present simple primitives, to show that even on these our approach gives a great performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' on the other hand, the capability to handle more complex primitives is undoubtedly an added value of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Similarly to [15], we use different colours to represent different primitive typologies: red for planes, green for cylinders, blue for cones, black for tori, pink for (open and closed) B-splines and yellow for unsegmented parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' For a simple model like the block of Figure 14(a), all methods provide decent decompositions, although RANSAC misclassifies some primitives while the deep learning frameworks run into problems because of an inaccurate estimation of the surface normals: this assertion is supported by the location of misclassified points, which are largely limited to the 17 GRAYLOGTable 4: Parameter comparison between the original point cloud from Figure 5(b) and the perturbed versions from Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Segment Original σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='01 σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='05 σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='10 σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='20 Plane 1 k = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='28 k = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='28 k = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='29 k = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='28 k = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='31 Plane 2 k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='28 k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='28 k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='28 k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='28 k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='26 Cylinder r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='80 r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='79 r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='78 r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='79 r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='78 Torus 1 R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='49 R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='49 R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='47 R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='48 R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='56 r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='72 r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='73 r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='70 r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='67 r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='74 Torus 2 R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='05 R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='09 R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='02 R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='17 R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='13 r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='79 r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='78 r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='78 r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='74 r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='80 sharp edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In this regard, it is worth noting that – considering that the networks were trained on clean data – the more accurate the normals, the more precise the final segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In our case, the use of a voting procedure allows us to withstand possible perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' For the remaining more complex models, our method outperforms the competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In Figure 14(b), our approach is the only capable of correctly identifying portions of tori, misclassified by Le and Duan [15] and partly unsegmented by RANSAC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' ParSeNet and HPNet are able to reveal the presence of tori, but the resulting segmentation is unreliable along sharp edges – again, mostly because of a deficiency in the normal estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Figures 14(c-d) show a RANSAC tendency to oversegment and misclassify complex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' While Le and Duan [15] obtain considerably improved results, their algorithm misses a thin cylinder (see Figure 14(c)) and all the tori in both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Being these two objects acquired by low-quality scanners, the corresponding point clouds (and meshes) are affected by point cloud artifacts which, in turn, lead to an even greater error in the normal estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The resulting segmentations are unreliable and mostly associated with spline segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' As previously mentioned, note that the two architectures were trained on the ABC dataset, which does not take into account the presence of noise or other types of perturbation in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Due to a lack of a sufficiently rich benchmark of scanned CAD objects, however, new training is not currently possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' To offer a quantitative analysis of the robustness of our pipeline, we compare its performance with that of three state-of-the-art methods whose implementation was made freely available by the authors: a direct method approach on a primitive-growing (PG) framework [18], adapted to handle point clouds as described in [19], and the two learning-based methods from the previous comparison: ParSeNet (PN) and HPNet (HN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' We conduct the study over the whole Fit4CAD benchmark [19], which contains CAD point clouds defined by simple geometric primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In addition, Fit4CAD contains a selection of meaningful and validated models from [45], converted from meshes to point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Figure 15 summarizes the performance of each method over the test set with respect to the following classification measures: Sørensen-Dice index (DSC), Positive Predicted Value (PPV), True Positive Rate (TPR), Negative Predicted Value (NPV), True Negative Rate (TNR) and accuracy (ACC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Given a point cloud, the benchmark defines these measures by comparing each point cloud segment in the ground truth to the most overlapping segment returned by a segmentation method, and then by averaging over all segments contained in that point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Note that, given a point cloud segment in the ground truth, the points inside that segment are the positives while the points outside that segment are the negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' We arrive at the following conclusions: Learning-based methods show remarkably high TNRs – said otherwise, the points they predict as being negative are almost always true negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' On the other hand, they are more penalized by NPV: while it is ParSeNet that exhibits the highest variability and the lowest quartiles, both methods are seriously affected by outliers – with some point clouds having this score below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A low NPV means that quite a few points predicted as negative are false negatives (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', common in heavily oversegmented models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In this case, these methods have been penalized by the low density of the point clouds, as well as from the possible presence of point cloud artifacts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', missing data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 18 RANSAC Le and Duan ParSeNet HPNet Ours (a) (b) (c) (d) Figure 14: Primitive type recognition: a comparison between our approach, a RANSAC-based segmentation [11], and the method in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Different colours correspond to different primitive types: planes (red), cylinders (green), cones (blue), tori (black), splines (pink), unsegmented (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Our approach performs significantly better than the competitors in terms of TPR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', it has similar proportions of correct (positive) predictions among positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' When it comes to the PPV, differences are more modest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In terms of accuracy, the four methods reach high scores, with the direct methods being the less prone to outliers: however, this metric is not completely reliable as this is a naturally unbalanced binary classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A more reliable measure is provided by the Sørensen- Dice index (DSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Our method visibly outperforms the competitors in terms of the DSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Intuitively, the higher the DSC, the more accurate segments returned by a segmentation method are – with respect to the most overlapping segment in the ground truth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' to put it differently, DSC penalizes greatly both under- and oversegmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' When it comes to execution time, our Hough-based method and the primitive-growing approach were run on a desktop PC equipped with an Intel Core i9 processor (at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='6 GHz) and a Windows 10 operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The average execution time, per model, are 286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='0 seconds for the PG-method and 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='7 seconds for our pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' ParSeNet and HPNet were run on Google Colab pro equipped with NVIDIA-SMI 460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='03 and CUDA 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Regarding the average execution times, we have 5 seconds for ParSeNet and 257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1 seconds for HPNet (when the preprocessing of normals is applied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 19 88 8Figure 15: Comparison of our approach (HT) with other three methods: a primitive growing ap- proach (PG), ParSeNet (PN) and HPNet (HN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The analysis is performed over the Fit4CAD bench- mark [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 7 Conclusions To address the problem of recognition of simple and complex primitives in a point cloud, we have opportunely extended the family of geometric primitives to which the HT technique can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The HT is naturally robust to noise and outliers and recognizes multiple instances of the same primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Thanks to an opportune preprocessing of the point cloud and its sub-parts, we are able to limit the number of parameters that are necessary to represent the primitive, thus reducing the computational complexity of the HT computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In addition, the explicit extraction of position- independent primitive parameters and the use of a hierarchical clustering strategy permits us to identify maximal and compound primitives, thus reducing the output oversegmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Indeed, differently from spline-based primitives, our strategy is suited for primitive similarity reasoning and permits us to find maximal primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Although learning methods have performed very well in recent years, when models present complex combinations of primitives (such as the examples in Figure 14(c,d)), direct methods perform even better, and our method is very competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' In addition, our method has been validated on a whole benchmark and, compared with the others, it turns out to be the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The use of geometric primitives whose mathematical representations require a large number of parameters – even in their standard forms – is currently the main limitation of our pipeline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' while theoretically possible, this would indeed increase the execution time exponentially, making the algorithm inapplicable in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' An instructive example, shown in Figure 16, is given by point clouds that contain segments generated by spline surfaces: while our method cannot deal with spline surfaces, it does manage to recognize that parts of a point cloud do not originate from any primitive at our disposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The first row of Figure 16 shows the ground truth segmentation, while the second row shows the segmentation we achieve: the black dots correspond to points that are not recognized as simple geometric primitives, and that are labelled as “unsegmented” because of the high mean fitting error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' note that spline segments are undersampled for the sake of visibility, but 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='9 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1this processing is realized only after segmenting the point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' On the contrary, methods that recognise splines tend to give any segment a label, as shown in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' As future work, we are thinking of other strategies to reduce the computational cost in the HT-based recognition step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Indeed, the parameter space can be further optimised by reducing its dimension through the use of a refinement strategy that discretizes the parameter space only in the relevant regions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Another possible direction of investigation could concern the applicability of our method to non-orientable surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Indeed we have only applied our method to objects with orientable surfaces since the datasets we have used exhibit only this type of surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' However, in theory, our approach could also apply to objects with non-orientable surfaces, as long as their equation exists in the literature and the system of equations is solvable with respect to the parameter (as in the case of M¨obius strip).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' The challenge here would be to study a preprocessing technique to translate and/or rotate the point cloud into the standard form of that surface type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' For instance, in the case of the M¨obius strip, how to estimate the symmetry axis in order to place the point cloud so that we can fit it with a primitive in a standard form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Regarding the automation of the process, we could take advantage of a pre-classification of the points in the input point cloud using a curvature-based characterization, for instance, the local surface variation proposed in [46] or the shape index and curvedness as used in [18] for ISO GPS segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Once points are aggregated by their type, we could further split or aggregate these regions using our fitting method and possibly considering only a suitable subset of primitives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', the primitives that are compatible with the recognized type of points, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=', planar, cylindrical, spherical, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Segments (a) Segments (b) Segments (c) Segments (d) GT US Figure 16: Four models from the ABC dataset [45] containing spline segments: comparison between the ground truth (GT) segmentation and our recognition of simple geometric primitives (US).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' To enhance the visibility of the primitives correctly recognized, the second row displays the segments classified as splines with a lower point density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Acknowledgements This work has been developed in the CNR research activities DIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='AD004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='100, DIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='AD021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='001 and DIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='AD021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 21 The authors thank the VVS shape repository by the AIM@SHAPE consortium (http://visionair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' ge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='imati.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='cnr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='it) and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Alexander Leutgeb from RISC Software GmbH (Linz, Austria) for the models used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Poux, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Mattes, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Selman, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Kobbelt, Automatic region-growing system for the seg- mentation of large point clouds, Automation in Construction 138 (2022) 104250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='autcon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='104250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [2] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Chrysathou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Sharf, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Cohen-Or, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Mitra, GlobFit: Consistently Fitting Primitives by Discovering Global Relations, ACM Transactions on Graphics 30 (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1145/2010324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1964947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Monszpart, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Mellado, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Brostow, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Mitra, Rapter: Rebuilding man-made scenes with regular arrangements of planes, ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 34 (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1145/2766995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [4] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Lupinetti, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Pernot, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Monti, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Giannini, Content-based CAD assembly model re- trieval: Survey and future challenges, Computer-Aided Design 113 (2019) 62 – 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='cad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [5] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Hough, Method and means for recognizing complex patterns, US Patent 3,069,654 (1962).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Ballard, Generalizing the Hough transform to detect arbitrary shapes, Pattern Recogni- tion 13 (2) (1981) 111–122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1016/0031-3203(81)90009-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [7] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Woodford, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Pham, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Maki, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Perbet, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Stenger, Demisting the Hough Transform for 3D Shape Recognition and Registration, International Journal of Computer Vision 106 (3) (2014) 332–341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1007/s11263-013-0623-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Beltrametti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Robbiano, An algebraic approach to Hough transforms, Journal of Algebra 37 (2012) 669–681.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='jalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Raffo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Romanengo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Falcidieno, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Biasotti, Fitting and recognition of geometric prim- itives in segmented 3d point clouds using a localized voting procedure, Computer Aided Geo- metric Design 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='cagd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='102123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Kaiser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Ybanez Zepeda, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Boubekeur, A Survey of Simple Geometric Primitives Detection Methods for Captured 3D Data, Computer Graphics Forum 38 (1) (2019) 167–196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1111/cgf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='13451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [11] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Schnabel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Wahl, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Klein, Efficient RANSAC for Point-Cloud Shape Detection, Computer Graphics Forum 26 (2) (2007) 214–226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1467-8659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='01016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [12] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Limberger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Oliveira, Real-time detection of planar regions in unorganized point clouds, Pattern Recognition 48 (6) (2015) 2043 – 2053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='patcog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Attene, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Patan`e, Hierarchical structure recovery of point-sampled surfaces, Computer Graphics Forum 29 (6) (2010) 1905–1920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1467-8659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='01658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [14] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Yan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Yang, Variational mesh segmentation via quadric surface fitting, Computer-Aided Design 44 (11) (2012) 1072 – 1082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='cad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [15] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Le, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Duan, A primitive-based 3D segmentation algorithm for mechanical CAD models, Computer Aided Geometric Design 52-53 (2017) 231–246, Geometric Modeling and Processing 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='cagd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 22 [16] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Sung, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Dubrovina, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Yi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Guibas, Supervised Fitting of Geometric Primitives to 3D Point Clouds, in: Proceedings – IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 2652–2660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1109/CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Uy, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Chang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Sung, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Goel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Lambourne, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Birdal, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Guibas, Point2cyl: Reverse engineering 3d objects from point clouds to extrusion cylinders, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 11850–11860.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [18] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Qie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Qiao, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Anwer, Enhanced invariance class partitioning using discrete curvatures and conformal geometry, Computer-Aided Design 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='cad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='102985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [19] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Romanengo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Raffo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Qie, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Anwer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Falcidieno, Fit4CAD: A point cloud benchmark for fitting simple geometric primitives in CAD objects, Computers & Graphics 102 (2022) 133–143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='cag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [20] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Markovic, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Jakovljevic, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Budak, Automatic recognition of cylinders and planes from unstructured point clouds, The Visual Computer (2021) 1–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [21] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Birdal, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Busam, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Navab, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Ilic, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Sturm, Generic primitive detection in point clouds us- ing novel minimal quadric fits, IEEE Transactions on Pattern Analysis and Machine Intelligence 42 (6) (2020) 1333–1347.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1109/TPAMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2900309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [22] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Liu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Hu, Real-time geometric fitting and pose estimation for surface of revolution, Pattern Recognition 85 (2019) 90–108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='patcog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [23] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Bergamasco, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Pistellato, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Albarelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Torsello, Cylinders extraction in non-oriented point clouds as a clustering problem, Pattern Recognition 107 (2020) 107443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='patcog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='107443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='com/science/article/pii/S0031320320302466 [24] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Sharma, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Maji, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Kalogerakis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Chaudhuri, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Mˇech, ParSeNet: A Parametric Surface Fitting Network for 3D Point Clouds, in: Computer Vision – ECCV 2020: 16th Euro- pean Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part VII, Springer-Verlag, Berlin, Heidelberg, 2020, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 261–276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1007/978-3-030-58571-6_16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [25] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Yan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Ma, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Huang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Vouga, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Huang, HPNet: Deep Primitive Segmentation Using Hybrid Representations, in: Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 2753–2762.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1109/ICCV48922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [26] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Sharma, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Dash, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' RoyChowdhury, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Gadelha, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Loizou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Cao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Wang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Learned-Miller, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Maji, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Kalogerakis, PriFit: Learning to Fit Primitives Improves Few Shot Point Cloud Segmentation, Computer Graphics Forumdoi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1111/cgf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='14601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [27] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Borrmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Elseberg, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Lingemann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' N¨uchter, The 3D Hough transform for plane detection in point clouds: A review and a new accumulator design, 3D Research 2 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1007/3DRes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='02(2011)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [28] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Fernandes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Oliveira, A general framework for subspace detection in unordered mul- tidimensional data, Pattern Recognition 45 (9) (2012) 3566 – 3579.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='patcog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Camurri, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Vezzani, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Cucchiara, 3D Hough Transform for Sphere Recognition on Point Clouds, Machine Vision and Applications 25 (7) (2014) 1877–1891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1007/ s00138-014-0640-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [30] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Rabbani Shah, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' van den Heuvel, Efficient Hough transform for automatic detection of cylinders in point clouds, in: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Vosselman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Brenner (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' ), ISPRS Laser Scanning 2005, ISPRS Working Groups, 2005, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 60–65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 23 [31] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Beltrametti, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Sendra, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Sendra, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Torrente, Moore–Penrose approach in the Hough transform framework, Applied Mathematics and Computation 375 (2020) 125083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='amc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='125083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [32] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Romanengo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Raffo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Biasotti, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Falcidieno, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Fotis, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Romanelis, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Psatha, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Mous- takas, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Sipiran, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Nguyen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Chu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Nguyen-Ngoc, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Vo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' To, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Nguyen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Le-Pham, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Nguyen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Tran, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Qie, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Anwer, SHREC 2022: Fitting and recognition of simple geometric primitives on point clouds, Computers & Graphics 107 (2022) 32–49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='cag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='com/science/article/pii/S0097849322001224 [33] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Shikin, Handbook and atlas of curves, CRC Press, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [34] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Hoppe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' DeRose, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Duchamp, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' McDonald, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Stuetzle, Surface reconstruction from unorganized points, in: Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’92, Association for Computing Machinery, New York, NY, USA, 1992, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 71–78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1145/133994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='134011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1145/133994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='134011 [35] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Romanengo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Brunetto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Biasotti, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Catalano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Falcidieno, Recognition, modelling and interactive manipulation of motifs or symbols represented by a composition of curves, in: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Biasotti, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Pintus, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Berretti (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' ), Italian Chapter Conference 2020 - Smart Tools and Apps in computer Graphics, STAG 2020, Virtual Event, Italy, November 12-13, 2020, Eurographics Association, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 27–35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2312/stag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='20201237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [36] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Penrose, On best approximate solutions of linear matrix equations, Mathematical Proceed- ings of the Cambridge Philosophical Society 52 (1) (1956) 17–19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [37] Edelsbrunner, Letscher, Zomorodian, Topological persistence and simplification, Discrete Com- putational Geometry 28 (4) (2002) 511–533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1007/s00454-002-2885-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [38] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Doraiswamy, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Shivashankar, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Natarajan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Wang, Topological saliency, Computers & Graphics 37 (7) (2013) 787 – 799.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='cag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [39] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Biasotti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Cerri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Pittaluga, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Sobrero, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Spagnuolo, Tracking the evolution of rainfall precipitation fields using persistent maxima, in: Proceedings of the Conference on Smart Tools and Applications in Computer Graphics, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 29–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2312/stag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='20161361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [40] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Friedman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Bentley, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Finkel, An algorithm for finding best matches in logarithmic expected time, ACM Transactions on Mathematical Software 3 (3) (1977) 209–226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1145/355744.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='355745.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [41] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Ester, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Kriegel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Sander, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Xu, A density-based algorithm for discovering clusters in large spatial databases with noise, in: 2nd Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Knowledge Discovery and Data Mining, AAAI Press, 1996, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 226–231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [42] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Lavin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Le Master, Fast Hough transform: A hierarchical approach, Computer Vision, Graphics, and Image Processing 36 (2) (1986) 139–161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1016/0734-189X(86) 90073-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [43] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Day, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Edelsbrunner, Efficient algorithms for agglomerative hierarchical clustering meth- ods, Journal of Classification 1 (1) (1984) 7–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1007/BF01890115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [44] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Defays, An efficient algorithm for a complete link method, The Computer Journal 20 (4) (1977) 364–366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1093/comjnl/20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 24 [45] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Koch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Matveev, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Jiang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Williams, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Artemov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Burnaev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Alexa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Zorin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Panozzo, ABC: A Big CAD Model Dataset For Geometric Deep Learning, in: The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1109/CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='00983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' [46] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Pauly, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Gross, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' Kobbelt, Efficient simplification of point-sampled surfaces, in: IEEE Visualization, 2002, 2002, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 163–170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1109/VISUAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content='1183771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} +page_content=' 25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E3T4oBgHgl3EQfMQlY/content/2301.04371v1.pdf'} diff --git a/qtFRT4oBgHgl3EQfezdF/vector_store/index.faiss b/qtFRT4oBgHgl3EQfezdF/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..b04df7052a7f1c2382551ea4fb4f35cc4ed24098 --- /dev/null +++ b/qtFRT4oBgHgl3EQfezdF/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:86cb8dda6e543c96cde7f3edc6ecff1293328d3c96aa9560aed6c4bad2671aa2 +size 6225965 diff --git a/stFKT4oBgHgl3EQfKC06/content/tmp_files/2301.11740v1.pdf.txt b/stFKT4oBgHgl3EQfKC06/content/tmp_files/2301.11740v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..65a7813976d0458c972b9a7b5c4f150bc7d14101 --- /dev/null +++ b/stFKT4oBgHgl3EQfKC06/content/tmp_files/2301.11740v1.pdf.txt @@ -0,0 +1,625 @@ +arXiv:2301.11740v1 [math.LO] 27 Jan 2023 +Implicative models of intuitionistic set theory +Samuele Maschio +Abstract +In this paper we will show that using implicative algebras one can +produce models of intuitionistic set theory generalizing both realizability +and Heyting-valued models. This has as consequence that if one assumes +the inaccessible cardinal axiom, then every topos which is obtained from +a Set-based tripos as the result of the tripos-to-topos construction hosts +a model of intuitionistic set theory. +1 +Introduction +Implicative algebras were introduced by A.Miquel in [Miq20a] in order to provide +a common foundation for realizability and forcing. +Given a complete Heyting algebra, one can define a tripos (see e.g. [HJP80]) +of Heyting-valued predicates over Set. Given a PCA one can obtain a real- +izability tripos, as shown e.g. in [vO08]. If one applies a construction called +the tripos-to-topos construction ([HJP80]) to these triposes one obtains forc- +ing toposes and realizability toposes, respectively. +In [Miq20a] it is shown +that both triposes are particular cases of a more general notion of tripos in- +duced by an implicative algebra (which we call here an implicative tripos). In +[Miq20b] Miquel proved much more, namely that every set-based tripos is in +fact (isomorphic to) an implicative tripos. +Forcing toposes and realizability +toposes host models of intuitionistic set theory IZF (and the same holds for +a larger class of toposes as shows in [MS15]), provided there exists an enough +big strongly inaccessible cardinal. In the first case, the models of IZF are the +so-called Heyting-valued models of set theory (see [Bel11]) while in the sec- +ond case they are Friedman/Rosolini/McCarty realizability models of IZF (see +[Fri73, Ros82, McC84]). If one takes a look to how these models are defined +then it can notice some similarities. In this paper we show that these similarities +comes from the fact that we can generalize the construction of Heyting-valued +and realizability models of IZF to toposes coming from implicative triposes. +We work in ZFC as metatheory. +2 +Intuitionistic set theory +In the language of intuitionistic set theory IZF the only terms are variables +and there are two binary predicate symbols: equality = and membership ∈. As +1 + +usual in the language of set theory ∀x ∈ y ϕ is a shorhand for ∀x(x ∈ y → ϕ) +and ∃x ∈ y ϕ is a shorhand for ∃x(x ∈ y ∧ ϕ), while x ⊆ y is a shorthand for +∀z ∈ x (z ∈ y). +We consider the following presentation of the axioms of IZF. +Emp) ∃x∀y ∈ x ⊥ +Ext) ∀x∀y (x ⊆ y ∧ y ⊆ x → x = y) +Pair) ∀x∀y∃z (x ∈ z ∧ y ∈ z) +Union) ∀x∃u∀y ∈ x∀z ∈ y (z ∈ u) +Pow) ∀x∃z∀y (y ⊆ x → y ∈ z) +Inf) ∃uInf(u) where Inf(u) is the conjunction of Inf 1(u) :≡ ∃x ∈ u∀y ∈ x ⊥ +and Inf 2(u) :≡ ∀x ∈ u∃y ∈ u(x ⊆ y ∧ x ∈ y ∧ ∀z ∈ y(z ∈ x ∨ z = x)) +Sepϕ) ∀w1....∀wn∀x∃y∀z (∀z ∈ y (z ∈ x ∧ ϕ) ∧ ∀z ∈ x (ϕ → z ∈ y)) for all formu- +las in context ϕ[w1, ...wn, x, z]. +Col) ∀w1....∀wn (∀x ∈ y∃z ϕ → ∃u∀x ∈ y∃z ∈ u ϕ) for all formulas in context +ϕ[w1, ...wn, x, y, z]. +∈ -Indϕ) ∀w1...∀wn(∀x(∀y ∈ x ϕ[y/x] → ϕ) → ∀x ϕ) for all formulas in context +ϕ[w1, ..., wn, x] formula in context. +3 +Implicative algebras and implicative triposes +An implicative algebra is a 4-tuple A = (A, ≤, →, Σ) where +1. (A, ≤) is a complete lattice; +2. →: A × A → A is a function which is monotone in the second component +and anti-monotone in the first component, and which satisfies the following +condition +a → +� +i∈I +bi = +� +i∈I +(a → bi) +for every indexed family (bi)i∈I of elements of A and every a ∈ A; +3. Σ ⊆ A is upward closed, it contains also b as soon as it contains a → b +and a, and it contains K := � +a,b∈A(a → (b → a)) and S := � +a,b∈A((a → +(b → c)) → ((a → b) → (a → c))). +Every complete Heyting algebra (H, ≤) with Heyting implication → gives rise +to an implicative algebra (H, ≤, →, {⊤}). Moreover, every total combinatory al- +gebra (R, ·) gives rise to an implicative algebra (P(R), ⊆, ⇒, P(R) \ {∅}), where +A ⇒ B := {r ∈ R| r · a ∈ B for every a ∈ A} for every A, B ⊆ R. Many +other examples can be found in [Miq20a]. In the case of a partial combinatory +2 + +algebra (R, ·), the 4-uple (P(R), ⊆, ⇒, P(R) \ {∅}) is not in general an implica- +tive algebra, but a quasi-implicative algebra (see [Miq20a]). However, there is a +standard way to transform it into an implicative algebra in such a way that the +tripos one obtains is equivalent to the realizability tripos built from (R, ·). +Closed λ-terms with constant parameters in A can be encoded in an im- +plicative algebra as follows: aA := a for every a ∈ A, (ts)A := tA · sA and +(λx.t)A := � +a∈A +� +a → (t[a/x])A� +where application · is defined as follows for +every a, b ∈ A: +a · b := +� +{x ∈ A| a ≤ b → x} +Under this encoding, if we define k as λx.λy.x and s as λx.λy.λz.xz(yz), one +can show that K = kA and S = sA. +Useful properties of the encoding of λ-terms in A are the following: +1. if t β-reduces to s, then tA ≤ sA; +2. if t is a pure-λ term with free variables x1, ..., xn and a1, ..., an ∈ Σ, then +(t[x1 : a1, ..., xn : an])A ∈ Σ1; in particular the encodings of closed pure +λ-terms are elements of Σ. +In what follows we will remove the superscript A from the encoding of λ-terms +in order to lighten the notation. +Moreover for every a, b ∈ A following [Miq20a] we define: +a × b := +� +x∈A +((a → (b → x)) → x) +a + b := +� +x∈A +((a → x) → ((b → x) → x)) +and for every set indexed family (ai)i∈I we define +∀i∈Iai := +� +i∈I +ai +∃i∈Iai := +� +x∈A +�� +i∈I +(ai → x) → x +� +We introduce the following shorthands for some λ-terms: k := λx.λy.x, +k := λx.λy.y, p := λx.λy.λz.zxy, p1 := λx.xk, p2 := λx.xk, j1 := λx.λz.λw.zx, +j2 := λx.λz.λw.wx, e := λx.λz.zx. Notice that (the encoding of) all of them +belong to Σ. +If Γ is a finite list of variable assignments x1 : a1, ..., xn : an with a1, ..., an ∈ +A and t is a λ-term with parameters in A and free variables among x1, .., xn, +we write Γ ⊢ t : a as a shorthand for t[Γ]A ≤ a (where t[Γ] is the result of the +substitution corresponding to Γ applied to t) and the following rules are sound +(this is a little variation of the system of rules presented in [Miq20a]). +1We denote with t[x1 : a1, ..., xn : an] the λ-term obtained by substituting the variables +x1, ..., xn with a1, ..., an. +3 + +x : a ∈ Γ +Γ ⊢ x : A +Γ ⊢ a : a +Γ ⊢ t : a +a ≤ b +Γ ⊢ t : b +Γ ⊢ t : ⊥ +Γ ⊢ t : a +Γ ⊢ t : a +Γ ⊢ t : ⊤ +Γ ⊢ t : a → b +Γ ⊢ s : a +Γ ⊢ ts : b +Γ, x : a ⊢ t : b +Γ ⊢ λx.t : a → b +Γ ⊢ t : a +Γ ⊢ s : b +Γ ⊢ pts : a × b +Γ ⊢ t : a × b +Γ ⊢ p1t : a +Γ ⊢ t : a × b +Γ ⊢ p2t : b +Γ ⊢ t : a +Γ ⊢ j1t : a + b +Γ ⊢ t : b +Γ ⊢ j2t : a + b +Γ ⊢ t : a + b +Γ, x : a ⊢ u : c +Γ, y : b ⊢ v : c +Γ ⊢ t(λx.u)(λy.v) : c +Γ ⊢ t : ai (for all i ∈ I) +Γ ⊢ t : ∀i∈Iai +Γ ⊢ t : ∀i∈Iai +Γ ⊢ t : ai +i ∈ I +Γ ⊢ t : ai +Γ ⊢ et : ∃i∈Iai +Γ ⊢ t : ∃i∈Iai +Γ, x : ai ⊢ u : b ( for all i ∈ I) +Γ ⊢ t(λx.u) : b +As shown in [Miq20a], to every implicative algebra A can be associated a +tripos (see [HJP80] or [vO08]) +pA : Setop → Heyt +by sending every set I to the posetal reflection of the preordered set (AI, ⊢Σ[I]) +where ϕ ⊢Σ[I] ψ if and only if � +i∈I(ϕ(i) → ψ(i)) ∈ Σ and every function f : I → +J to the function induced by the pre-composition function (−) ◦ f : AJ → AI. +Component-wise use of →, × and + defines a Heyting prealgebra structure +on every preorder (AI, ⊢Σ[I]), which is preserved by precomposition. ∃ and ∀ +are used to produce left and right adjoints to reindexing maps satisfying Beck- +Chevalley condition, while a generic predicate is given by (the equivalence class +of) the identity function on A. +A remarkable result in [Miq20b] is the following +Theorem 3.1. Let p : Setop → Heyt be a tripos. +Then, there exists an +implicative algebra A such that p is isomorphic to pA. +Recall also (see e.g. [vO08]) that to every tripos p over Set is associated an +elementary topos Set[p] obtained by means of the so-called “tripos-to-topos” +construction. +4 +Implicative-valued models of IZF-Col +Let us assume a strongly inaccessible cardinal κ exists and let A be a +fixed implicative algebra such that |A| < κ. +We define the following hierarchy of sets indexed by ordinals: +W A +α := + + + + + +∅ if α = 0 +Part(W A +β , A) if α = β + 1 +� +β<α W A +β if α is a limit ordinal +4 + +where Part(X, Y ) denotes the set of partial functions from X to Y . We take +W to be W A +κ . Since W A +α ⊆ W A +β if α < β, one can assign a rank in the hier- +archy to every element of W in the obvious way. In particular, we can define +simultaneously, by recursion on rank, two functions ∈W, =W: W × W → A: +1. α ∈W β := ∃t∈∂0(β) (β(t) × t =W α) +2. α =W β := α ⊆W β×β ⊆W α where α ⊆W β := ∀t∈∂0(α) (α(t) → t ∈W β). +We interpret the language of set theory in such a way that to every formula in +context ϕ[x1, ..., xn] we associate a function +∥ϕ[x1, ..., xn]∥ : Wn → A +by recursion on complexity of formulas as follows: +1. ∥xi ∈ xj [x1, ..., xn]∥ (α1, ..., αn) :≡ αi ∈W αj +2. ∥xi = xj [x1, ..., xn]∥ (α1, ..., αn) :≡ αi =W αj +3. ∥ϕ ∧ ψ[x]∥ (α) :≡ ∥ϕ[x]∥ (α) × ∥ψ[x]∥ (α) +4. ∥ϕ ∨ ψ[x]∥ (α) :≡ ∥ϕ[x]∥ (α) + ∥ψ[x]∥ (α) +5. ∥ϕ → ψ[x]∥ (α) :≡ ∥ϕ[x]∥ (α) → ∥ψ[x]∥ (α) +6. ∥∃y ϕ [x]∥ (α) :≡ ∃β∈W (∥ϕ [x, y]∥ (α, β)) +7. ∥∀y ϕ [x]∥ (α) :≡ ∀β∈W (∥ϕ [x, y]∥ (α, β))2 +We write W ⊨ ϕ [x] if � +α∈Wℓ(x) (∥ϕ [x]∥ (α)) ∈ Σ when [x] is non-empty and +for every statement ϕ we write W ⊨ ϕ if ∥ϕ [ ]∥ ∈ Σ, that is W ⊨ ϕ [x] means +∥ϕ[x]∥ is in the maximum class of pA(Wn) where n is the lenght of [x]. +We will often write ∥ϕ∥ instead of ∥ϕ[ ]∥. +Notice that the definition of the interpretation we proposed above coincides +with that of Heyting-valued models in the case in which A = (A, ≤, →, {⊤}) +where (A, ≤) is a complete Heyting algebra with Heyting implication →, while +it coincides with the realizability interpretation mentioned in the introduction +when A = (P(R), ⊆, ⇒, P(R) \ {∅}) with (R, ·) a total combinatory algebra +and ⇒ is the usual implication between sets of realizers. The case of a partial +combinatory algebra can be recovered by using the techniques illustrated in +[Miq20a]. +2In these two clauses, we can assume, without loss of generality, that y is not a variable +appearing in the context [x]. +5 + +4.1 +Useful lemmas +Lemma 4.1. There exist ρ, j, σ, s1, s2, s3 ∈ Σ such that +1. ρ ≤ � +α∈W (α =W α) +2. j ≤ � +α∈W +� +u∈∂0(α) (α(u) → u ∈W α) +3. σ ∈ � +α,β∈W (α =W β → β =W α) +4. s1 ≤ � +α,β,γ∈W (α =W β × γ ∈W α → γ ∈W β) +5. s2 ≤ � +α,β,γ∈W (α =W β × α ∈W γ → β ∈W γ) +6. s3 ≤ � +α,β,γ∈W (α =W β × γ =W α → γ =W β) +Proof. +1. Let ρ be yf ∈ Σ where f := λr.p(λx.e(pxr))(λx.e(pxr)) and y +is a pure closed λ-term which is a fixed point operator (see e.g. [vO08]) +for which yf β-reduces to f(yf). We claim that ρ ≤ α =W α for every +α ∈ W. Let α be an arbitrary element of W and let us assume that +ρ ≤ β =W β for every β ∈ W with rank in the hierarchy less than that of +α. Then we can consider the following derivation tree in which we used +only rules from the previous section. +x : α(u) ⊢ x : α(u) (for all u ∈ ∂0(α)) +x : α(u) ⊢ ρ : u =W u (for all u ∈ ∂0(α)) +x : α(u) ⊢ pxρ : α(u) × u =W u (for all u ∈ ∂0(α)) +x : α(u) ⊢ e(pxρ) : u ∈W α (for all u ∈ ∂0(α)) +⊢ λx.e(pxρ) : α(u) → u ∈W α (for all u ∈ ∂0(α)) +⊢ λx.e(pxρ) : α ⊆W α +⊢ p(λx.e(pxρ))(λx.e(pxρ)) : α =W α +The last λ-term in the deduction tree is a β reduction of ρ. Thus we can +conclude that ρ ≤ α =W α. +2. Let j be defined as λx.e(pxρ) ∈ Σ. Assume α ∈ W and u ∈ ∂0(α). Then +x : α(u) ⊢ pxρ : α(u) × u =W u . Hence, x : α(u) ⊢ e(pxρ) : u ∈W α. +Thus +j ≤ +� +α∈W +� +u∈∂0(α) +(α(u) → u ∈W α) +3. σ can be just defined as λx.p(p1x)(p2x) ∈ Σ +4,5,6. Assume s3 to exist. +Let α, β, γ ∈ W and let Γ(u) be a shorthand for +x : α =W β × α ∈W γ, y : γ(u) × u =W α +where u an arbitrary element of the domain of γ. Since +⊢ s3 : α =W β × u =W α → u =W β +6 + +we obtain Γ(u) ⊢ p(p1y)(s3(p(p1x)(p2y))) : γ(u) × u =W β from which +it follows that +Γ(u) ⊢ e(p(p1y)(s3(p(p1x)(p2y)))) : β ∈W γ +Since x : α =W β × α ∈W γ ⊢ p2x : γ ∈W α, we get +x : α =W β × α ∈W γ ⊢ (p2x)(λy.e(p(p1y)(s3(p(p1x)(p2y))))) : β ∈W γ +from which it follows that +⊢ λx.(p2x)(λy.(e(p(p1y)(s3(p(p1x)(p2y)))))) : α =W β×α ∈W γ → β ∈W γ +From this it follows that s2 can be defined as λx.(p2x)(λy.(e(p(p1y)(s3(p(p1x)(p2y)))))). +Assume now s2 to exist and consider Γ′(u) a shorthand for +x : α =W β × γ ∈W α, y : α(u) × u =W γ +where u is an arbitrary element of the domain of α. We easily see that +Γ′(u) ⊢ p(p2y)((p1(p1x))(p1y)) : u =W γ × u ∈W β. Thus +Γ′(u) ⊢ s2(p(p2y)((p1(p1x))(p1y))) : γ ∈W β +Since x : α =W β × γ ∈W α ⊢ p2x : γ ∈W α, we have that +x : α =W β × γ ∈W α ⊢ (p2x)(λy.s2(p(p2y)((p1(p1x))(p1y)))) : γ ∈W β +from which it follows that +⊢ λx.(p2x)(λy.s2(p(p2y)((p1(p1x))(p1y)))) : α =W β×γ ∈W α → γ ∈W β +Thus s1 can be defined as λx.(p2x)(λy.s2(p(p2y)((p1(p1x))(p1y)))). +Similarly, one can prove that if s1 is assumed to exist, then one can define +s3 as a λ-term containing s1 as the unique parameter. +Thus it is sufficient to compose this mutual dependence to define by a fix +point yg one among s1, s2 and s3 and then define the other two using +that one exploiting the interdefinability. So if we define s1 as a fixpoint, +we can then define s3 using s1 and then s2 using s3. +Lemma 4.2. For every formula in context ϕ [x] where x has length n, there +exists rϕ[x] ∈ Σ such that +rϕ[x] ≤ +� +α∈Wn +� +β∈Wn +� +α =W β × ∥ϕ [x]∥ (α) → ∥ϕ [x]∥ (β) +� +Proof. By induction on complexity of formulas by using the previous lemma for +the atomic cases. +7 + +Also the following lemma can easily been proved as a consequence of the +previous result and of the rules in the previous section. +Lemma 4.3. Let ϕ[x] and ψ[x] be formulas in context in the language of set +theory and let n be the lenght of [x], If ϕ ⊢x +IL= ψ, then ∥ϕ [x]∥ ⊢Σ[Wn] ∥ψ [x]∥. +Lemma 4.4. If [x] has lenght n, then +∥∃z ∈ y ϕ [x, y]∥ ≡Σ[Wn+1] Λα.Λβ.∃u∈∂0(β) (β(u) × ∥ϕ [x, y, z]∥ (α, β, u)) 3 +∥∀z ∈ y ϕ [x, y]∥ (α, β) ≡Σ[Wn+1] Λα.Λβ.∀u∈∂0(β) (β(u) → ∥ϕ [x, y, z]∥ (α, β, u)) +Proof. We consider the case of the existential quantifier and we leave the anal- +ogous proof of the universal case to the reader. We also restrict to the case in +which x is empty. The general case is analogous, but only heavier in notation. +By definition of the interpretation we have that ∥∃z ∈ y ϕ [y]∥ (β) is +η := ∃γ∈W +�∃u∈∂0(β)(β(u) × u =W γ) × ∥ϕ [y, z]∥ (β, γ) +� +We denote with η1(γ) the scope of the quantifier ∃γ∈W, while we denote with +η2(u, γ) the scope of the quantifier ∃u∈∂0(β). It is immediate to check that the +following sequent holds for every β, γ ∈ W and every u ∈ ∂0(β): +x : η, y : η1(γ), z : η2(u, γ) ⊢ l : (β =W β × γ =W u) × ∥ϕ [y, z]∥ (β, γ)) +where l := p(pρ(σ(p2z)))(p2y). With the notation from Lemma 4.2, we can +conclude that x : η, y : η1(γ), z : η2(u, γ) ⊢ rϕ[y,z]l : ∥ϕ [y, z]∥ (β, u). +Thus +x : η, y : η1(γ), z : η2(u, γ) ⊢ p(p1z)(rϕ[y,z]l) : β(u) × ∥ϕ [y, z]∥ (β, u). Hence +x : η, y : η1(γ), z : η2(u, γ) ⊢ e(p(p1z)(rϕ[y,z]l)) : ∃w∈∂0(β(w) × ∥ϕ [y, z]∥ (β, w)) +Using the rules of elimination of existential quantification, one can conclude +that +η → ∃w∈∂0(β(w) × ∥ϕ [y, z]∥ (β, w)) ≥ l′ ∈ Σ +where l′ := λx.xλy.(p1y)(λz.e(p(p1z)(rϕ[y,z]l))). +It is easier to show that +∃w∈∂0(β(w) × ∥ϕ [y, z]∥ (β, w)) → η ≥ λx.xλy.p(p(p1y)ρ)(p2y) ∈ Σ +One can in fact write a deduction tree in which the existential quantifiers of the +consequent are both witnessed by a w ∈ ∂0(β) for which β(w) × ∥ϕ [y, z]∥ (β, w) +is assumed to hold. +Corollary 4.5. ∥x ⊆ y [x, y]∥ ≡Σ[W2] Λα.Λβ. (α ⊆W β) for every α, β ∈ W. +3We use the notation Λα.f(α) to denote the function sending each α in the domain to +f(α). +8 + +4.2 +Validity of axioms +4.2.1 +Empty-set +Thanks to Lemma 4.4 we know that +∥Emp∥ ≡Σ ∃α∈W∀u∈∂0(α) (α(u) → ⊥) +Consider ∅ ∈ W. Then ∀u∈∂0(∅) (∅(u) → ⊥) = � ∅ = ⊤. This entails that e⊤ ≤ +∃α∈W∀u∈∂0(α) (α(u) → ⊥). Since e⊤ ∈ Σ, we can conclude that W ⊨ Emp. +4.2.2 +Extensionality +Thanks to Corollary 4.5 we know that +∥Ext∥ ≡Σ ∀α∈W∀β∈W (α ⊆W β × β ⊆W α → α =W β) ≥ λx.x ∈ Σ +Thus, W ⊨ Ext. +4.2.3 +Pair +∥Pair∥ ≡Σ ∀α∈W∀β∈W∃γ∈W (α ∈W γ × β ∈W γ) +Let us consider arbitrary α, β ∈ W and the partial function ηα,β ∈ W defined +as follows: ∂0(ηα,β) = {α, β} and ηα,β(u) = ⊤ for every u in the domain. By +Lemma 4.1, ⊢ ρ : α =W α, from which it follows that +⊢ q := e(p⊤ρ) : ∃t∈{α,β} (⊤ × t =W α) = α ∈W ηα,β +In the same way, we can show that ⊢ q : β ∈W ηα,β. As a consequence +⊢ q′ := pqq : α ∈W ηα,β × β ∈W ηα,β +Thus, ⊢ eq′ : ∃γ∈W (α ∈W γ × β ∈W γ). Since eq′ does not depend on α and +β and belongs to Σ, +⊢ eq′ : ∀α∈W∀β∈W∃γ∈W (α ∈W γ × β ∈W γ) +Hence W ⊨ Pair. +4.2.4 +Union +Thanks to Lemma 4.4 we know that +∥Union∥ ≡Σ ∀α∈W∃β∈W∀u∈∂0(α) +� +α(u) → ∀w∈∂0(u) (u(w) → w ∈W β) +� +Let us fix now an arbitrary α ∈ W and let us define ζα ∈ W as follows. The +domain of ζα is � +u∈∂0(α) ∂0(u) and for every element of such domain ζα(u) = ⊤. +9 + +Let w ∈ � +u∈∂0(α) ∂0(u), then ⊢ p⊤ρ : ⊤ × w =W w (we are using Lemma +4.1), from which it follows that ⊢ e(p⊤r)) : w ∈W ζα. From this it follows that +⊢ λv′.λv.e(p⊤r)) : ∀u∈∂0(α) +� +α(u) → ∀w∈∂0(u) (u(w) → w ∈W ζα) +� +and thus that +⊢ e(λv′.λv.e(p⊤r))) : ∃β∈W∀u∈∂0(α) +� +α(u) → ∀w∈∂0(u) (u(w) → w ∈W β) +� +Since e(λv′.λv.e(p⊤r))) does not depend on α and it is an element of Σ, we +have +⊢ e(λv′.λv.e(p⊤r))) : ∀α∈W∃β∈W∀u∈∂0(α) +� +α(u) → ∀w∈∂0(u) (u(w) → w ∈W β) +� +and W ⊨ Union. +4.2.5 +Powerset +Using Lemma 4.5 +∥Pow∥ ≡Σ ∀α∈W∃β∈W∀γ∈W (γ ⊆W α → γ ∈W β) +Let us consider an arbitrary α ∈ W and define πα ∈ W as that partial function +having domain A∂0(α) and for which πα(u) = ⊤ for every u in the domain. πα is +in W since κ is strongly inaccessible. For every γ ∈ W we also define γα ∈ W +as follows. The domain of γα is ∂0(α) ∪ ∂0(γ) and γα(u) := u ∈W α × u ∈W γ +for every u in the domain. +We now use Lemma 4.1. Let u ∈ ∂0(α). Clearly: +1. x : γ ⊆W α, y : γ(u) ⊢ xy : u ∈W α, +2. x : γ ⊆W α, y : γ(u) ⊢ jy : u ∈ γ, +3. x : γ ⊆W α, y : γ(u) ⊢ ρ : u =W u. +From this it follows that +x : γ ⊆W α, y : γ(u) ⊢ r := p(p(xy)(jy))ρ : (u ∈W α × u ∈W γ) × u =W u +and thus that x : γ ⊆W α, y : γ(u) ⊢ er : u ∈ γα from which it follows that +x : γ ⊆W α ⊢ λy.er : γ(u) → u ∈ γα +and since λy.er does not depend on u ∈ ∂0(α) +x : γ ⊆W α ⊢ λy.er : γ ⊆W γα +One can also easily show that ⊢ λz.(p2z) : γα ⊆W γ. Thus +x : γ ⊆W α ⊢ r := p⊤(p(λz.(p2z))(λy.er)) : ⊤ × γα =W γ +10 + +Since γα is in the domain of πα we hence have that +x : γ ⊆W α ⊢ er : γ ∈ πα +We can thus conclude that ⊢ λx.er : γ ⊆W α → γ ∈W πα. Since λx.er and +e(λx.er) do not depend on γ and α we get, +⊢ e(λx.er) : ∀α∈W∃β∈W∀γ∈W (γ ⊆W α → γ ∈W β) +Since e(λx.er) ∈ Σ, we can conclude that W ⊨ Pow. +4.2.6 +Infinity +For every n ∈ ω, we define �n ∈ W as follows: ∂0(�n) = { �m| m < n} and +�n( �m) := m where m ∈ Σ is Krivine’s encoding of the natural number m. We +define �ω as the element of W with domain {�n| n ∈ ω} and defined by �ω(�n) := n. +First, if we consider �0 = ∅, we can easily see that ⊢ e(p0⊤) : ∥Inf 1(u)[u]∥ (�ω). +Consider now a closed λ-term f ∈ Σ, which exists by a recursion theorem +(and which is in Σ, because ρ ∈ Σ), such that +� +fnm ։β j1(pmρ) if m ̸= n +fnm ։β j2ρ if m = n +for every n, m ∈ ω. +Then, for every n ∈ ω +⊢ λu.fnu : ∀i=0,...,n( � +n + 1(�i) → (�i ∈W �n +�i =W �n)) +Moreover +⊢ λx.e(pxρ) : �n ⊆W � +n + 1 +⊢ e(pnρ) : �n ∈W � +n + 1 +Thus t ≤ ∥Inf 2(u)[u]∥ (�ω) for some t ∈ Σ, and we can conclude that W ⊨ Inf. +4.2.7 +Separation +Assume ϕ [w, x, z] be a formula in context with w a list of variable of length n. +��Sepϕ +�� ≡Σ ∀ω∈Wn∀α∈W∃β∈W(∀u∈∂0(β)(β(u) → u ∈W α× ∥ϕ [w, x, z]∥ (ω, α, u))× +∀u′∈∂0(α)(α(u′) → (∥ϕ [w, x, z]∥ (ω, α, u′) → u′ ∈W β))) +For an arbitrary α ∈ W and ω ∈ Wn we define αω +ϕ ∈ W as follows: its domain +is equal to the domain of α, while αω +ϕ(u) := α(u) × ∥ϕ [w, x, z]∥ (ω, α, u). In +order to show that W ⊩ Sepϕ, it is sufficient to find a t ∈ Σ not depending on +ω and α such that +⊢ t : ∀u∈∂0(α)(αω +ϕ(u) → u ∈W α × ∥ϕ [w, x, z]∥ (ω, α, u))× +11 + +∀u′∈∂0(α)(α(u′) → (∥ϕ [w, x, z]∥ (ω, α, u′) → u′ ∈W αω +ϕ)) +But this is immediate to prove, since using Lemma 4.1 +⊢ λx.p(j(p1x))(p2x) : ∀u∈∂0(α)(αω +ϕ(u) → u ∈W α × ∥ϕ [w, x, z]∥ (ω, α, u)) +⊢ λx.λy.e(p(pxy)ρ) : ∀u′∈∂0(α)(α(u′) → (∥ϕ [w, x, z]∥ (ω, α, u′) → u′ ∈W αω +ϕ)) +4.2.8 +∈-Induction +Let y be the fix-point operator such that yf β-reduces to f(yf) for every f and +consider +h := y(λh.λx.x(λy.hx)) ∈ Σ +in such a way that h ≤ (λh.λx.x(λy.hx))h ≤ λx.x(λy.hx). +We want to prove that h ≤ ∥∈ -Indϕ∥ to conclude that W ⊨∈ -Indϕ. +We prove this by induction on the rank in W. Fix an arbitrary α and assume +that +h ≤ ∀α∈W +�∀u∈∂0(α)(α(u) → ∥ϕ[x]∥ (u)) → ∥ϕ[x]∥ (α) +� +→ ∥ϕ[x]∥ (β) +for every β with rank strictly less than that of α. Let us use εα as a shorthand for +∀u∈∂0(α)(α(u) → ∥ϕ[x]∥ (u)) → ∥ϕ[x]∥ (α) and ε as a shorthand for ∀α∈Wεα. +If we consider the following derivation tree: +x : ε ⊢ x : ε +x : ε ⊢ x : εα +⊢ h : ε → ∥ϕ [x]∥ (u) ( for every u ∈ ∂0(α)) +x : ε ⊢ h : ε → ∥ϕ [x]∥ (u)( for every u ∈ ∂0(α)) +x : ε ⊢ x : ε +x : ε ⊢ hx : ∥ϕ [x]∥ (u)( for every u ∈ ∂0(α)) +x : ε, y : α(u) ⊢ hx : ∥ϕ [x]∥ (u)( for every u ∈ ∂0(α)) +x : ε ⊢ λy.hx : α(u) → ∥ϕ [x]∥ (u)( for every u ∈ ∂0(α)) +x : ε ⊢ λy.hx : ∀u∈∂0(α)(α(u) → ∥ϕ [x]∥ (u)) +x : ε ⊢ x(λy.hx) : ∥ϕ[x]∥ (α) +⊢ λx.x(λy.hx) : ε → ∥ϕ[x]∥ (α) +we can conclude that h ≤ ε → ∥ϕ[x]∥ (α). +4.2.9 +Collection +In order to lighten the notation we will consider Colϕ for a formula ϕ in context +[x, y] (so without any additional parameter). Moreover we will write ϕ(a, b) +instad of ∥ϕ [x, y]∥ (a, b). +Assume α ∈ W and u ∈ ∂0(α). +Since κ in inaccessible, |A| < κ and +{ϕ(u, γ)| u ∈ W} ⊆ A, there exists η < κ such that ∃γ∈W(⊤ × ϕ(u, γ)) = +∃γ∈W A +η (⊤ × ϕ(u, γ)). We define ηu to be the minimum such an η and we define +ηα := �{ηu| u ∈ ∂0(α)} which is strictly less than κ, since the cardinality of A +12 + +is strictly less than κ. We define βα ∈ W as the constant function with value +⊤ and domain W A +ηα. Using the calculus we can show that there is an element +r ∈ Σ not depending on α such that +⊢ r : ∀u∈∂0(α) +� +α(u) → ∃γ∈Wϕ(u, γ) +� +→ +∀u∈∂0(α) +� +α(u) → ∃w∈∂0(βα)(βα(w) × ϕ(u, w)) +� +and using this fact one can easily show that Colϕ is validated in the model. +5 +Models of IZF in a class of toposes +Theorem 5.1. Every topos E obtained from an implicative tripos by means of +the tripos-to-topos construction from an implicative algebra A = (A, ≤, →, Σ) +with |A| < κ for a strongly inaccassible cardinal κ hosts a model of IZF. If Σ +is classical (see [Miq20a]), then E hosts a model of ZF. +Proof. In such a topos E an internal model of IZF is given by the object +(W, [=W]) together with the mono embedding the object +(W × W, [((x, y), (x′, y′)) �→ x ∈W y × x =W x′ × y =W y′]) +into (W, [=W]) × (W, [=W]) which inteprets the membership relation. +Using Theorem 3.1, we obtain the following: +Corollary 5.2. If for every cardinal κ′ there exists a strongly inaccessible car- +dinal κ such that κ′ < κ (i.e. if the inaccessible cardinal axiom holds), then +every topos obtained from a set-based tripos by means of the tripos-to-topos con- +struction hosts a model of IZF. +Acknowledgements +The author would like to acknowledge T. Streicher and F.Ciraulo for useful +discussions on the subject of this paper. +References +[Bel11] +John L. Bell. Set theory, volume 47 of Oxford Logic Guides. Oxford +University Press, Oxford, 2011. Boolean-valued models and indepen- +dence proofs, With a foreword by Dana Scott, Paperback of the third +(2005) edition. +[Fri73] +Harvey Friedman. Some applications of Kleene’s methods for intu- +itionistic systems. +In Cambridge Summer School in Mathematical +Logic (Cambridge, 1971), Lecture Notes in Math., Vol. 337, pages +113–170. Springer, Berlin, 1973. +13 + +[HJP80] +J. M. E. Hyland, P. T. Johnstone, and A. M. Pitts. Tripos theory. +Math. Proc. Cambridge Philos. Soc., 88(2):205–231, 1980. +[McC84] D.C. McCarty. Realizability and recursive mathematics. 1984. +[Miq20a] Alexandre Miquel. Implicative algebras: a new foundation for real- +izability and forcing. Math. Structures Comput. Sci., 30(5):458–510, +2020. +[Miq20b] Alexandre Miquel. Implicative algebras ii: completeness w.r.t. set- +based triposes, 2020. +[MS15] +Samuele Maschio and Thomas Streicher. Models of intuitionistic set +theory in subtoposes of nested realizability toposes. Ann. Pure Appl. +Logic, 166(6):729–739, 2015. +[Ros82] +G Rosolini. Un modello per la teoria intuizionista degli insiemi. In +Atti degli Incontri di Logica Matematica. 1982. +[vO08] +J. van Oosten. Realizability: an introduction to its categorical side, +volume 152 of Studies in Logic and the Foundations of Mathematics. +Elsevier B. V., Amsterdam, 2008. +14 + diff --git a/stFKT4oBgHgl3EQfKC06/content/tmp_files/load_file.txt b/stFKT4oBgHgl3EQfKC06/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..63656cc8966aa81abad690c76171859f45c2926e --- /dev/null +++ b/stFKT4oBgHgl3EQfKC06/content/tmp_files/load_file.txt @@ -0,0 +1,422 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf,len=421 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='11740v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='LO] 27 Jan 2023 Implicative models of intuitionistic set theory Samuele Maschio Abstract In this paper we will show that using implicative algebras one can produce models of intuitionistic set theory generalizing both realizability and Heyting-valued models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' This has as consequence that if one assumes the inaccessible cardinal axiom, then every topos which is obtained from a Set-based tripos as the result of the tripos-to-topos construction hosts a model of intuitionistic set theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 1 Introduction Implicative algebras were introduced by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='Miquel in [Miq20a] in order to provide a common foundation for realizability and forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Given a complete Heyting algebra, one can define a tripos (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' [HJP80]) of Heyting-valued predicates over Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Given a PCA one can obtain a real- izability tripos, as shown e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' in [vO08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' If one applies a construction called the tripos-to-topos construction ([HJP80]) to these triposes one obtains forc- ing toposes and realizability toposes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' In [Miq20a] it is shown that both triposes are particular cases of a more general notion of tripos in- duced by an implicative algebra (which we call here an implicative tripos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' In [Miq20b] Miquel proved much more, namely that every set-based tripos is in fact (isomorphic to) an implicative tripos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Forcing toposes and realizability toposes host models of intuitionistic set theory IZF (and the same holds for a larger class of toposes as shows in [MS15]), provided there exists an enough big strongly inaccessible cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' In the first case, the models of IZF are the so-called Heyting-valued models of set theory (see [Bel11]) while in the sec- ond case they are Friedman/Rosolini/McCarty realizability models of IZF (see [Fri73, Ros82, McC84]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' If one takes a look to how these models are defined then it can notice some similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' In this paper we show that these similarities comes from the fact that we can generalize the construction of Heyting-valued and realizability models of IZF to toposes coming from implicative triposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' We work in ZFC as metatheory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 2 Intuitionistic set theory In the language of intuitionistic set theory IZF the only terms are variables and there are two binary predicate symbols: equality = and membership ∈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' As 1 usual in the language of set theory ∀x ∈ y ϕ is a shorhand for ∀x(x ∈ y → ϕ) and ∃x ∈ y ϕ is a shorhand for ∃x(x ∈ y ∧ ϕ), while x ⊆ y is a shorthand for ∀z ∈ x (z ∈ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' We consider the following presentation of the axioms of IZF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Emp) ∃x∀y ∈ x ⊥ Ext) ∀x∀y (x ⊆ y ∧ y ⊆ x → x = y) Pair) ∀x∀y∃z (x ∈ z ∧ y ∈ z) Union) ∀x∃u∀y ∈ x∀z ∈ y (z ∈ u) Pow) ∀x∃z∀y (y ⊆ x → y ∈ z) Inf) ∃uInf(u) where Inf(u) is the conjunction of Inf 1(u) :≡ ∃x ∈ u∀y ∈ x ⊥ and Inf 2(u) :≡ ∀x ∈ u∃y ∈ u(x ⊆ y ∧ x ∈ y ∧ ∀z ∈ y(z ∈ x ∨ z = x)) Sepϕ) ∀w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='.∀wn∀x∃y∀z (∀z ∈ y (z ∈ x ∧ ϕ) ∧ ∀z ∈ x (ϕ → z ∈ y)) for all formu- las in context ϕ[w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='wn, x, z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Col) ∀w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='.∀wn (∀x ∈ y∃z ϕ → ∃u∀x ∈ y∃z ∈ u ϕ) for all formulas in context ϕ[w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='wn, x, y, z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' ∈ -Indϕ) ∀w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='∀wn(∀x(∀y ∈ x ϕ[y/x] → ϕ) → ∀x ϕ) for all formulas in context ϕ[w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=', wn, x] formula in context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 3 Implicative algebras and implicative triposes An implicative algebra is a 4-tuple A = (A, ≤, →, Σ) where 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' (A, ≤) is a complete lattice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' →: A × A → A is a function which is monotone in the second component and anti-monotone in the first component, and which satisfies the following condition a → � i∈I bi = � i∈I (a → bi) for every indexed family (bi)i∈I of elements of A and every a ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Σ ⊆ A is upward closed, it contains also b as soon as it contains a → b and a, and it contains K := � a,b∈A(a → (b → a)) and S := � a,b∈A((a → (b → c)) → ((a → b) → (a → c))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Every complete Heyting algebra (H, ≤) with Heyting implication → gives rise to an implicative algebra (H, ≤, →, {⊤}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Moreover, every total combinatory al- gebra (R, ·) gives rise to an implicative algebra (P(R), ⊆, ⇒, P(R) \\ {∅}), where A ⇒ B := {r ∈ R| r · a ∈ B for every a ∈ A} for every A, B ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Many other examples can be found in [Miq20a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' In the case of a partial combinatory 2 algebra (R, ·), the 4-uple (P(R), ⊆, ⇒, P(R) \\ {∅}) is not in general an implica- tive algebra, but a quasi-implicative algebra (see [Miq20a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' However, there is a standard way to transform it into an implicative algebra in such a way that the tripos one obtains is equivalent to the realizability tripos built from (R, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Closed λ-terms with constant parameters in A can be encoded in an im- plicative algebra as follows: aA := a for every a ∈ A, (ts)A := tA · sA and (λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='t)A := � a∈A � a → (t[a/x])A� where application · is defined as follows for every a, b ∈ A: a · b := � {x ∈ A| a ≤ b → x} Under this encoding, if we define k as λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='x and s as λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='xz(yz), one can show that K = kA and S = sA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Useful properties of the encoding of λ-terms in A are the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' if t β-reduces to s, then tA ≤ sA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' if t is a pure-λ term with free variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=', xn and a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=', an ∈ Σ, then (t[x1 : a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=', xn : an])A ∈ Σ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' in particular the encodings of closed pure λ-terms are elements of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' In what follows we will remove the superscript A from the encoding of λ-terms in order to lighten the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Moreover for every a, b ∈ A following [Miq20a] we define: a × b := � x∈A ((a → (b → x)) → x) a + b := � x∈A ((a → x) → ((b → x) → x)) and for every set indexed family (ai)i∈I we define ∀i∈Iai := � i∈I ai ∃i∈Iai := � x∈A �� i∈I (ai → x) → x � We introduce the following shorthands for some λ-terms: k := λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='x, k := λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='y, p := λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='zxy, p1 := λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='xk, p2 := λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='xk, j1 := λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='λw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='zx, j2 := λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='λw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='wx, e := λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='zx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Notice that (the encoding of) all of them belong to Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' If Γ is a finite list of variable assignments x1 : a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=', xn : an with a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=', an ∈ A and t is a λ-term with parameters in A and free variables among x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='., xn, we write Γ ⊢ t : a as a shorthand for t[Γ]A ≤ a (where t[Γ] is the result of the substitution corresponding to Γ applied to t) and the following rules are sound (this is a little variation of the system of rules presented in [Miq20a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 1We denote with t[x1 : a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=', xn : an] the λ-term obtained by substituting the variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=', xn with a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=', an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 3 x : a ∈ Γ Γ ⊢ x : A Γ ⊢ a : a Γ ⊢ t : a a ≤ b Γ ⊢ t : b Γ ⊢ t : ⊥ Γ ⊢ t : a Γ ⊢ t : a Γ ⊢ t : ⊤ Γ ⊢ t : a → b Γ ⊢ s : a Γ ⊢ ts : b Γ, x : a ⊢ t : b Γ ⊢ λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='t : a → b Γ ⊢ t : a Γ ⊢ s : b Γ ⊢ pts : a × b Γ ⊢ t : a × b Γ ⊢ p1t : a Γ ⊢ t : a × b Γ ⊢ p2t : b Γ ⊢ t : a Γ ⊢ j1t : a + b Γ ⊢ t : b Γ ⊢ j2t : a + b Γ ⊢ t : a + b Γ, x : a ⊢ u : c Γ, y : b ⊢ v : c Γ ⊢ t(λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='u)(λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='v) : c Γ ⊢ t : ai (for all i ∈ I) Γ ⊢ t : ∀i∈Iai Γ ⊢ t : ∀i∈Iai Γ ⊢ t : ai i ∈ I Γ ⊢ t : ai Γ ⊢ et : ∃i∈Iai Γ ⊢ t : ∃i∈Iai Γ, x : ai ⊢ u : b ( for all i ∈ I) Γ ⊢ t(λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='u) : b As shown in [Miq20a], to every implicative algebra A can be associated a tripos (see [HJP80] or [vO08]) pA : Setop → Heyt by sending every set I to the posetal reflection of the preordered set (AI, ⊢Σ[I]) where ϕ ⊢Σ[I] ψ if and only if � i∈I(ϕ(i) → ψ(i)) ∈ Σ and every function f : I → J to the function induced by the pre-composition function (−) ◦ f : AJ → AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Component-wise use of →, × and + defines a Heyting prealgebra structure on every preorder (AI, ⊢Σ[I]), which is preserved by precomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' ∃ and ∀ are used to produce left and right adjoints to reindexing maps satisfying Beck- Chevalley condition, while a generic predicate is given by (the equivalence class of) the identity function on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' A remarkable result in [Miq20b] is the following Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Let p : Setop → Heyt be a tripos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Then, there exists an implicative algebra A such that p is isomorphic to pA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Recall also (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' [vO08]) that to every tripos p over Set is associated an elementary topos Set[p] obtained by means of the so-called “tripos-to-topos” construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 4 Implicative-valued models of IZF-Col Let us assume a strongly inaccessible cardinal κ exists and let A be a fixed implicative algebra such that |A| < κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' We define the following hierarchy of sets indexed by ordinals: W A α := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ∅ if α = 0 Part(W A β , A) if α = β + 1 � β<α W A β if α is a limit ordinal 4 where Part(X, Y ) denotes the set of partial functions from X to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' We take W to be W A κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Since W A α ⊆ W A β if α < β, one can assign a rank in the hier- archy to every element of W in the obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' In particular, we can define simultaneously, by recursion on rank, two functions ∈W, =W: W × W → A: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' α ∈W β := ∃t∈∂0(β) (β(t) × t =W α) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' α =W β := α ⊆W β×β ⊆W α where α ⊆W β := ∀t∈∂0(α) (α(t) → t ∈W β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' We interpret the language of set theory in such a way that to every formula in context ϕ[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=', xn] we associate a function ∥ϕ[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=', xn]∥ : Wn → A by recursion on complexity of formulas as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' ∥xi ∈ xj [x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=', xn]∥ (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=', αn) :≡ αi ∈W αj 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' ∥xi = xj [x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=', xn]∥ (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=', αn) :≡ αi =W αj 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' ∥ϕ ∧ ψ[x]∥ (α) :≡ ∥ϕ[x]∥ (α) × ∥ψ[x]∥ (α) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' ∥ϕ ∨ ψ[x]∥ (α) :≡ ∥ϕ[x]∥ (α) + ∥ψ[x]∥ (α) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' ∥ϕ → ψ[x]∥ (α) :≡ ∥ϕ[x]∥ (α) → ∥ψ[x]∥ (α) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' ∥∃y ϕ [x]∥ (α) :≡ ∃β∈W (∥ϕ [x, y]∥ (α, β)) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' ∥∀y ϕ [x]∥ (α) :≡ ∀β∈W (∥ϕ [x, y]∥ (α, β))2 We write W ⊨ ϕ [x] if � α∈Wℓ(x) (∥ϕ [x]∥ (α)) ∈ Σ when [x] is non-empty and for every statement ϕ we write W ⊨ ϕ if ∥ϕ [ ]∥ ∈ Σ, that is W ⊨ ϕ [x] means ∥ϕ[x]∥ is in the maximum class of pA(Wn) where n is the lenght of [x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' We will often write ∥ϕ∥ instead of ∥ϕ[ ]∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Notice that the definition of the interpretation we proposed above coincides with that of Heyting-valued models in the case in which A = (A, ≤, →, {⊤}) where (A, ≤) is a complete Heyting algebra with Heyting implication →, while it coincides with the realizability interpretation mentioned in the introduction when A = (P(R), ⊆, ⇒, P(R) \\ {∅}) with (R, ·) a total combinatory algebra and ⇒ is the usual implication between sets of realizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' The case of a partial combinatory algebra can be recovered by using the techniques illustrated in [Miq20a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 2In these two clauses, we can assume, without loss of generality, that y is not a variable appearing in the context [x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='1 Useful lemmas Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' There exist ρ, j, σ, s1, s2, s3 ∈ Σ such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' ρ ≤ � α∈W (α =W α) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' j ≤ � α∈W � u∈∂0(α) (α(u) → u ∈W α) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' σ ∈ � α,β∈W (α =W β → β =W α) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' s1 ≤ � α,β,γ∈W (α =W β × γ ∈W α → γ ∈W β) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' s2 ≤ � α,β,γ∈W (α =W β × α ∈W γ → β ∈W γ) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' s3 ≤ � α,β,γ∈W (α =W β × γ =W α → γ =W β) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Let ρ be yf ∈ Σ where f := λr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='p(λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='e(pxr))(λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='e(pxr)) and y is a pure closed λ-term which is a fixed point operator (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' [vO08]) for which yf β-reduces to f(yf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' We claim that ρ ≤ α =W α for every α ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Let α be an arbitrary element of W and let us assume that ρ ≤ β =W β for every β ∈ W with rank in the hierarchy less than that of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Then we can consider the following derivation tree in which we used only rules from the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' x : α(u) ⊢ x : α(u) (for all u ∈ ∂0(α)) x : α(u) ⊢ ρ : u =W u (for all u ∈ ∂0(α)) x : α(u) ⊢ pxρ : α(u) × u =W u (for all u ∈ ∂0(α)) x : α(u) ⊢ e(pxρ) : u ∈W α (for all u ∈ ∂0(α)) ⊢ λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='e(pxρ) : α(u) → u ∈W α (for all u ∈ ∂0(α)) ⊢ λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='e(pxρ) : α ⊆W α ⊢ p(λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='e(pxρ))(λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='e(pxρ)) : α =W α The last λ-term in the deduction tree is a β reduction of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Thus we can conclude that ρ ≤ α =W α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Let j be defined as λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='e(pxρ) ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Assume α ∈ W and u ∈ ∂0(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Then x : α(u) ⊢ pxρ : α(u) × u =W u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Hence, x : α(u) ⊢ e(pxρ) : u ∈W α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Thus j ≤ � α∈W � u∈∂0(α) (α(u) → u ∈W α) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' σ can be just defined as λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='p(p1x)(p2x) ∈ Σ 4,5,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Assume s3 to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Let α, β, γ ∈ W and let Γ(u) be a shorthand for x : α =W β × α ∈W γ, y : γ(u) × u =W α where u an arbitrary element of the domain of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Since ⊢ s3 : α =W β × u =W α → u =W β 6 we obtain Γ(u) ⊢ p(p1y)(s3(p(p1x)(p2y))) : γ(u) × u =W β from which it follows that Γ(u) ⊢ e(p(p1y)(s3(p(p1x)(p2y)))) : β ∈W γ Since x : α =W β × α ∈W γ ⊢ p2x : γ ∈W α, we get x : α =W β × α ∈W γ ⊢ (p2x)(λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='e(p(p1y)(s3(p(p1x)(p2y))))) : β ∈W γ from which it follows that ⊢ λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='(p2x)(λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' (e(p(p1y)(s3(p(p1x)(p2y)))))) : α =W β×α ∈W γ → β ∈W γ From this it follows that s2 can be defined as λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='(p2x)(λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' (e(p(p1y)(s3(p(p1x)(p2y)))))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Assume now s2 to exist and consider Γ′(u) a shorthand for x : α =W β × γ ∈W α, y : α(u) × u =W γ where u is an arbitrary element of the domain of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' We easily see that Γ′(u) ⊢ p(p2y)((p1(p1x))(p1y)) : u =W γ × u ∈W β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Thus Γ′(u) ⊢ s2(p(p2y)((p1(p1x))(p1y))) : γ ∈W β Since x : α =W β × γ ∈W α ⊢ p2x : γ ∈W α, we have that x : α =W β × γ ∈W α ⊢ (p2x)(λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='s2(p(p2y)((p1(p1x))(p1y)))) : γ ∈W β from which it follows that ⊢ λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' (p2x)(λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='s2(p(p2y)((p1(p1x))(p1y)))) : α =W β×γ ∈W α → γ ∈W β Thus s1 can be defined as λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' (p2x)(λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='s2(p(p2y)((p1(p1x))(p1y)))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Similarly, one can prove that if s1 is assumed to exist, then one can define s3 as a λ-term containing s1 as the unique parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Thus it is sufficient to compose this mutual dependence to define by a fix point yg one among s1, s2 and s3 and then define the other two using that one exploiting the interdefinability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' So if we define s1 as a fixpoint, we can then define s3 using s1 and then s2 using s3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' For every formula in context ϕ [x] where x has length n, there exists rϕ[x] ∈ Σ such that rϕ[x] ≤ � α∈Wn � β∈Wn � α =W β × ∥ϕ [x]∥ (α) → ∥ϕ [x]∥ (β) � Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' By induction on complexity of formulas by using the previous lemma for the atomic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 7 Also the following lemma can easily been proved as a consequence of the previous result and of the rules in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Let ϕ[x] and ψ[x] be formulas in context in the language of set theory and let n be the lenght of [x], If ϕ ⊢x IL= ψ, then ∥ϕ [x]∥ ⊢Σ[Wn] ∥ψ [x]∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' If [x] has lenght n, then ∥∃z ∈ y ϕ [x, y]∥ ≡Σ[Wn+1] Λα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='Λβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='∃u∈∂0(β) (β(u) × ∥ϕ [x, y, z]∥ (α, β, u)) 3 ∥∀z ∈ y ϕ [x, y]∥ (α, β) ≡Σ[Wn+1] Λα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='Λβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='∀u∈∂0(β) (β(u) → ∥ϕ [x, y, z]∥ (α, β, u)) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' We consider the case of the existential quantifier and we leave the anal- ogous proof of the universal case to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' We also restrict to the case in which x is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' The general case is analogous, but only heavier in notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' By definition of the interpretation we have that ∥∃z ∈ y ϕ [y]∥ (β) is η := ∃γ∈W �∃u∈∂0(β)(β(u) × u =W γ) × ∥ϕ [y, z]∥ (β, γ) � We denote with η1(γ) the scope of the quantifier ∃γ∈W, while we denote with η2(u, γ) the scope of the quantifier ∃u∈∂0(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' It is immediate to check that the following sequent holds for every β, γ ∈ W and every u ∈ ∂0(β): x : η, y : η1(γ), z : η2(u, γ) ⊢ l : (β =W β × γ =W u) × ∥ϕ [y, z]∥ (β, γ)) where l := p(pρ(σ(p2z)))(p2y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' With the notation from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='2, we can conclude that x : η, y : η1(γ), z : η2(u, γ) ⊢ rϕ[y,z]l : ∥ϕ [y, z]∥ (β, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Thus x : η, y : η1(γ), z : η2(u, γ) ⊢ p(p1z)(rϕ[y,z]l) : β(u) × ∥ϕ [y, z]∥ (β, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Hence x : η, y : η1(γ), z : η2(u, γ) ⊢ e(p(p1z)(rϕ[y,z]l)) : ∃w∈∂0(β(w) × ∥ϕ [y, z]∥ (β, w)) Using the rules of elimination of existential quantification, one can conclude that η → ∃w∈∂0(β(w) × ∥ϕ [y, z]∥ (β, w)) ≥ l′ ∈ Σ where l′ := λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='xλy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' (p1y)(λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='e(p(p1z)(rϕ[y,z]l))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' It is easier to show that ∃w∈∂0(β(w) × ∥ϕ [y, z]∥ (β, w)) → η ≥ λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='xλy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='p(p(p1y)ρ)(p2y) ∈ Σ One can in fact write a deduction tree in which the existential quantifiers of the consequent are both witnessed by a w ∈ ∂0(β) for which β(w) × ∥ϕ [y, z]∥ (β, w) is assumed to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' ∥x ⊆ y [x, y]∥ ≡Σ[W2] Λα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='Λβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' (α ⊆W β) for every α, β ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 3We use the notation Λα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='f(α) to denote the function sending each α in the domain to f(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='2 Validity of axioms 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='1 Empty-set Thanks to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='4 we know that ∥Emp∥ ≡Σ ∃α∈W∀u∈∂0(α) (α(u) → ⊥) Consider ∅ ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Then ∀u∈∂0(∅) (∅(u) → ⊥) = � ∅ = ⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' This entails that e⊤ ≤ ∃α∈W∀u∈∂0(α) (α(u) → ⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Since e⊤ ∈ Σ, we can conclude that W ⊨ Emp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='2 Extensionality Thanks to Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='5 we know that ∥Ext∥ ≡Σ ∀α∈W∀β∈W (α ⊆W β × β ⊆W α → α =W β) ≥ λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='x ∈ Σ Thus, W ⊨ Ext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='3 Pair ∥Pair∥ ≡Σ ∀α∈W∀β∈W∃γ∈W (α ∈W γ × β ∈W γ) Let us consider arbitrary α, β ∈ W and the partial function ηα,β ∈ W defined as follows: ∂0(ηα,β) = {α, β} and ηα,β(u) = ⊤ for every u in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='1, ⊢ ρ : α =W α, from which it follows that ⊢ q := e(p⊤ρ) : ∃t∈{α,β} (⊤ × t =W α) = α ∈W ηα,β In the same way, we can show that ⊢ q : β ∈W ηα,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' As a consequence ⊢ q′ := pqq : α ∈W ηα,β × β ∈W ηα,β Thus, ⊢ eq′ : ∃γ∈W (α ∈W γ × β ∈W γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Since eq′ does not depend on α and β and belongs to Σ, ⊢ eq′ : ∀α∈W∀β∈W∃γ∈W (α ∈W γ × β ∈W γ) Hence W ⊨ Pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='4 Union Thanks to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='4 we know that ∥Union∥ ≡Σ ∀α∈W∃β∈W∀u∈∂0(α) � α(u) → ∀w∈∂0(u) (u(w) → w ∈W β) � Let us fix now an arbitrary α ∈ W and let us define ζα ∈ W as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' The domain of ζα is � u∈∂0(α) ∂0(u) and for every element of such domain ζα(u) = ⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 9 Let w ∈ � u∈∂0(α) ∂0(u), then ⊢ p⊤ρ : ⊤ × w =W w (we are using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='1), from which it follows that ⊢ e(p⊤r)) : w ∈W ζα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' From this it follows that ⊢ λv′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='λv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='e(p⊤r)) : ∀u∈∂0(α) � α(u) → ∀w∈∂0(u) (u(w) → w ∈W ζα) � and thus that ⊢ e(λv′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='λv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='e(p⊤r))) : ∃β∈W∀u∈∂0(α) � α(u) → ∀w∈∂0(u) (u(w) → w ∈W β) � Since e(λv′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='λv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='e(p⊤r))) does not depend on α and it is an element of Σ, we have ⊢ e(λv′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='λv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='e(p⊤r))) : ∀α∈W∃β∈W∀u∈∂0(α) � α(u) → ∀w∈∂0(u) (u(w) → w ∈W β) � and W ⊨ Union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='5 Powerset Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='5 ∥Pow∥ ≡Σ ∀α∈W∃β∈W∀γ∈W (γ ⊆W α → γ ∈W β) Let us consider an arbitrary α ∈ W and define πα ∈ W as that partial function having domain A∂0(α) and for which πα(u) = ⊤ for every u in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' πα is in W since κ is strongly inaccessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' For every γ ∈ W we also define γα ∈ W as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' The domain of γα is ∂0(α) ∪ ∂0(γ) and γα(u) := u ∈W α × u ∈W γ for every u in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' We now use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Let u ∈ ∂0(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Clearly: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' x : γ ⊆W α, y : γ(u) ⊢ xy : u ∈W α, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' x : γ ⊆W α, y : γ(u) ⊢ jy : u ∈ γ, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' x : γ ⊆W α, y : γ(u) ⊢ ρ : u =W u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' From this it follows that x : γ ⊆W α, y : γ(u) ⊢ r := p(p(xy)(jy))ρ : (u ∈W α × u ∈W γ) × u =W u and thus that x : γ ⊆W α, y : γ(u) ⊢ er : u ∈ γα from which it follows that x : γ ⊆W α ⊢ λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='er : γ(u) → u ∈ γα and since λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='er does not depend on u ∈ ∂0(α) x : γ ⊆W α ⊢ λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='er : γ ⊆W γα One can also easily show that ⊢ λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' (p2z) : γα ⊆W γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Thus x : γ ⊆W α ⊢ r := p⊤(p(λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' (p2z))(λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='er)) : ⊤ × γα =W γ 10 Since γα is in the domain of πα we hence have that x : γ ⊆W α ⊢ er : γ ∈ πα We can thus conclude that ⊢ λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='er : γ ⊆W α → γ ∈W πα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Since λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='er and e(λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='er) do not depend on γ and α we get, ⊢ e(λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='er) : ∀α∈W∃β∈W∀γ∈W (γ ⊆W α → γ ∈W β) Since e(λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='er) ∈ Σ, we can conclude that W ⊨ Pow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='6 Infinity For every n ∈ ω, we define �n ∈ W as follows: ∂0(�n) = { �m| m < n} and �n( �m) := m where m ∈ Σ is Krivine’s encoding of the natural number m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' We define �ω as the element of W with domain {�n| n ∈ ω} and defined by �ω(�n) := n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' First, if we consider �0 = ∅, we can easily see that ⊢ e(p0⊤) : ∥Inf 1(u)[u]∥ (�ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Consider now a closed λ-term f ∈ Σ, which exists by a recursion theorem (and which is in Σ, because ρ ∈ Σ), such that � fnm ։β j1(pmρ) if m ̸= n fnm ։β j2ρ if m = n for every n, m ∈ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Then, for every n ∈ ω ⊢ λu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='fnu : ∀i=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=',n( � n + 1(�i) → (�i ∈W �n +�i =W �n)) Moreover ⊢ λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='e(pxρ) : �n ⊆W � n + 1 ⊢ e(pnρ) : �n ∈W � n + 1 Thus t ≤ ∥Inf 2(u)[u]∥ (�ω) for some t ∈ Σ, and we can conclude that W ⊨ Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='7 Separation Assume ϕ [w, x, z] be a formula in context with w a list of variable of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' ��Sepϕ �� ≡Σ ∀ω∈Wn∀α∈W∃β∈W(∀u∈∂0(β)(β(u) → u ∈W α× ∥ϕ [w, x, z]∥ (ω, α, u))× ∀u′∈∂0(α)(α(u′) → (∥ϕ [w, x, z]∥ (ω, α, u′) → u′ ∈W β))) For an arbitrary α ∈ W and ω ∈ Wn we define αω ϕ ∈ W as follows: its domain is equal to the domain of α, while αω ϕ(u) := α(u) × ∥ϕ [w, x, z]∥ (ω, α, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' In order to show that W ⊩ Sepϕ, it is sufficient to find a t ∈ Σ not depending on ω and α such that ⊢ t : ∀u∈∂0(α)(αω ϕ(u) → u ∈W α × ∥ϕ [w, x, z]∥ (ω, α, u))× 11 ∀u′∈∂0(α)(α(u′) → (∥ϕ [w, x, z]∥ (ω, α, u′) → u′ ∈W αω ϕ)) But this is immediate to prove, since using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='1 ⊢ λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='p(j(p1x))(p2x) : ∀u∈∂0(α)(αω ϕ(u) → u ∈W α × ∥ϕ [w, x, z]∥ (ω, α, u)) ⊢ λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='e(p(pxy)ρ) : ∀u′∈∂0(α)(α(u′) → (∥ϕ [w, x, z]∥ (ω, α, u′) → u′ ∈W αω ϕ)) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='8 ∈-Induction Let y be the fix-point operator such that yf β-reduces to f(yf) for every f and consider h := y(λh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='x(λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='hx)) ∈ Σ in such a way that h ≤ (λh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='x(λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='hx))h ≤ λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='x(λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='hx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' We want to prove that h ≤ ∥∈ -Indϕ∥ to conclude that W ⊨∈ -Indϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' We prove this by induction on the rank in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Fix an arbitrary α and assume that h ≤ ∀α∈W �∀u∈∂0(α)(α(u) → ∥ϕ[x]∥ (u)) → ∥ϕ[x]∥ (α) � → ∥ϕ[x]∥ (β) for every β with rank strictly less than that of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Let us use εα as a shorthand for ∀u∈∂0(α)(α(u) → ∥ϕ[x]∥ (u)) → ∥ϕ[x]∥ (α) and ε as a shorthand for ∀α∈Wεα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' If we consider the following derivation tree: x : ε ⊢ x : ε x : ε ⊢ x : εα ⊢ h : ε → ∥ϕ [x]∥ (u) ( for every u ∈ ∂0(α)) x : ε ⊢ h : ε → ∥ϕ [x]∥ (u)( for every u ∈ ∂0(α)) x : ε ⊢ x : ε x : ε ⊢ hx : ∥ϕ [x]∥ (u)( for every u ∈ ∂0(α)) x : ε, y : α(u) ⊢ hx : ∥ϕ [x]∥ (u)( for every u ∈ ∂0(α)) x : ε ⊢ λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='hx : α(u) → ∥ϕ [x]∥ (u)( for every u ∈ ∂0(α)) x : ε ⊢ λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='hx : ∀u∈∂0(α)(α(u) → ∥ϕ [x]∥ (u)) x : ε ⊢ x(λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='hx) : ∥ϕ[x]∥ (α) ⊢ λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='x(λy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='hx) : ε → ∥ϕ[x]∥ (α) we can conclude that h ≤ ε → ∥ϕ[x]∥ (α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='9 Collection In order to lighten the notation we will consider Colϕ for a formula ϕ in context [x, y] (so without any additional parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Moreover we will write ϕ(a, b) instad of ∥ϕ [x, y]∥ (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Assume α ∈ W and u ∈ ∂0(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Since κ in inaccessible, |A| < κ and {ϕ(u, γ)| u ∈ W} ⊆ A, there exists η < κ such that ∃γ∈W(⊤ × ϕ(u, γ)) = ∃γ∈W A η (⊤ × ϕ(u, γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' We define ηu to be the minimum such an η and we define ηα := �{ηu| u ∈ ∂0(α)} which is strictly less than κ, since the cardinality of A 12 is strictly less than κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' We define βα ∈ W as the constant function with value ⊤ and domain W A ηα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Using the calculus we can show that there is an element r ∈ Σ not depending on α such that ⊢ r : ∀u∈∂0(α) � α(u) → ∃γ∈Wϕ(u, γ) � → ∀u∈∂0(α) � α(u) → ∃w∈∂0(βα)(βα(w) × ϕ(u, w)) � and using this fact one can easily show that Colϕ is validated in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 5 Models of IZF in a class of toposes Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Every topos E obtained from an implicative tripos by means of the tripos-to-topos construction from an implicative algebra A = (A, ≤, →, Σ) with |A| < κ for a strongly inaccassible cardinal κ hosts a model of IZF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' If Σ is classical (see [Miq20a]), then E hosts a model of ZF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' In such a topos E an internal model of IZF is given by the object (W, [=W]) together with the mono embedding the object (W × W, [((x, y), (x′, y′)) �→ x ∈W y × x =W x′ × y =W y′]) into (W, [=W]) × (W, [=W]) which inteprets the membership relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='1, we obtain the following: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' If for every cardinal κ′ there exists a strongly inaccessible car- dinal κ such that κ′ < κ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' if the inaccessible cardinal axiom holds), then every topos obtained from a set-based tripos by means of the tripos-to-topos con- struction hosts a model of IZF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Acknowledgements The author would like to acknowledge T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Streicher and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='Ciraulo for useful discussions on the subject of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' References [Bel11] John L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Bell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Set theory, volume 47 of Oxford Logic Guides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Oxford University Press, Oxford, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Boolean-valued models and indepen- dence proofs, With a foreword by Dana Scott, Paperback of the third (2005) edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' [Fri73] Harvey Friedman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Some applications of Kleene’s methods for intu- itionistic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' In Cambridge Summer School in Mathematical Logic (Cambridge, 1971), Lecture Notes in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 337, pages 113–170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Springer, Berlin, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 13 [HJP80] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Hyland, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Johnstone, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Pitts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Tripos theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Cambridge Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=', 88(2):205–231, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' [McC84] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' McCarty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Realizability and recursive mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' [Miq20a] Alexandre Miquel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Implicative algebras: a new foundation for real- izability and forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Structures Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=', 30(5):458–510, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' [Miq20b] Alexandre Miquel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Implicative algebras ii: completeness w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' set- based triposes, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' [MS15] Samuele Maschio and Thomas Streicher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Models of intuitionistic set theory in subtoposes of nested realizability toposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Logic, 166(6):729–739, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' [Ros82] G Rosolini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Un modello per la teoria intuizionista degli insiemi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' In Atti degli Incontri di Logica Matematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' [vO08] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' van Oosten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Realizability: an introduction to its categorical side, volume 152 of Studies in Logic and the Foundations of Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' Elsevier B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=', Amsterdam, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} +page_content=' 14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQfKC06/content/2301.11740v1.pdf'} diff --git a/tdAyT4oBgHgl3EQfZ_fS/content/tmp_files/2301.00235v1.pdf.txt b/tdAyT4oBgHgl3EQfZ_fS/content/tmp_files/2301.00235v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7c06dcac71219086b1a8370d2dfdc239c1c68e9d --- /dev/null +++ b/tdAyT4oBgHgl3EQfZ_fS/content/tmp_files/2301.00235v1.pdf.txt @@ -0,0 +1,2078 @@ +Prepared for submission to JHEP +Quasi-topological Gravities on General Spherically +Symmetric Metric +Feiyu Chen,a,b,1 +aInstitute of High Energy Physics, Chinese Academy of Sciences, +Beijing 100049, P.R. China +bSchool of Physics, University of Chinese Academy of Sciences, +Beijing 100049, P.R. China +E-mail: chenfy@ihep.ac.cn +Abstract: In this work we study a more restricted class of quasi-topological gravity the- +ories where the higher curvature terms have no contribution to the equation of motion on +general static spherically symmetric metric where gttgrr ̸= constant. We construct such +theories up to quintic order in Riemann tensor and observe an important property of these +theories: the higher order term in the Lagrangian vanishes identically when evaluated on +the most general non-stationary spherically symmetric metric ansatz. This not only signals +the higher terms could only have non-trivial effects when considering perturbations, but +also makes the theories quasi-topological on a much wider range of metrics. As an example +of the holographic effects of such theories, we consider a general Einstein-scalar theory and +calculate it’s holographic shear viscosity. +Keywords: Higher Curvature Gravity, Black Holes, AdS-CFT Correspondence +1Corresponding author. +arXiv:2301.00235v1 [hep-th] 31 Dec 2022 + +Contents +1 +Introduction +1 +2 +Construction of the theory +3 +2.1 +Cubic order +4 +2.2 +Higher orders +6 +3 +Properties and discussions +6 +4 +Holographic shear viscosity +8 +5 +Conclusions +10 +A Quartic and quintic quasi-topological gravities +11 +B Results of holographic shear viscosity +20 +1 +Introduction +Einstein gravity theory extended with higher order curvature terms plays a relevant role +among modified gravity theories. It’s predicted by string theory that Einstein gravity should +be corrected by an infinite series of higher curvature terms [1, 2]. Higher curvature terms +have also attracted attentions in holography, they may introduce various new phenomena +on the boundary theory. For example, it’s shown that including higher curvature terms can +lead to the violation of the Kovtun–Son–Starinets (KSS) shear-viscosity-to-entropy bound +η/s ⩾ 1/4π [3–5]. Holography has also been used to determine the physical bounds of +higher curvature couplings by demanding the consistency of the dual CFT [5–7]. +However, gravity theories with higher curvature terms are generally hard to study. +One common way to study a gravity theory is through its black hole solution, however +for higher curvature gravities the equations of motion are usually fourth-order differential +equations, making analytical solutions hard to come by. It’s thus of interest to construct +higher curvature theories that admit analytical black hole solutions. On the other hand, the +linearized equations of motion of these theories around maximally symmetric spacetimes +typically contain fourth derivatives too, so besides the usual massless spin-2 graviton mode, +two extra massive modes might appear, the scalar mode and the ghost-like spin-2 mode [8]. +The existence of ghost-like mode signals instability of the AdS vacua and causes unitarity +breaking of the dual CFT, it is thus mandatory to decouple (set the mass to infinity) the +ghost-like mode when studying holography. The well-known example where these extra +modes are absent is Lovelock gravity [9, 10]. Lovelock term of order k vanishes identically +when D ⩽ 2k − 1 and becomes a total derivative that does not contribute to the equations +– 1 – + +of motion for D = 2k. A total derivative further reduces to a surface term in the action +and only contribute topological characteristics, so higher curvature term of this kind is +also called topological term, no physical effects could emerge when introducing such higher +curvature terms. +Quasi-topological gravity (QTG), on the other hand, is a more intriguing theory in +that the equations of motion are drastically simplified when evaluated on some special +metric ansatz, and also gives non-trivial contribution at perturbation level. In the broader +literature, such theory is defined by that it admits Schwarzschild-like solutions, i.e., the +special spherically symmetric (SSS) metric 1 +ds2 = −f(r)dt2 + +1 +f(r)dr2 + r2dΣ2 +D−2,k +(1.1) +and the equation of f(r) is algebraic. Cubic quasi-topological gravity was first constructed +in [11], it’s holographic properties was later studied in [12]. Higher order ones also exist +and they have been studied extensively [13–16]. Besides quasi-topological gravities, there’s +another closely related class of theory worth mentioning, known as the generalized quasi- +topological gravity (GQTG), satisfying [17, 18] +δSf +δf = 0, +or +Et +t = Er +r +(1.2) +where Sf denotes the action evaluated on the SSS metric, and Eab = 1/ +� +|g|δS/δgab is the +equation of motion. It can be shown that quasi-topological gravity satisfies this condition +and thus is a subclass of GQTG. Features of GQTG have been studied comprehensively +at present [18–27]. In particular, (1.2) implies [26] the equation of f(r) is at most second +order, the existence of Schwarzschild-like solutions, and most importantly, the decoupling +of the extra massive modes. +We are more interested in quasi-topological gravities whose higher curvature terms do +not contribute to the equation of motion. These theories obviously satisfy (1.2) and thus +ghost-free. They are first considered in [28] for Ricci polynomials, where quasi-topological +terms were constructed up to tenth order in Ricci tensor both on the SSS metric and general +spherically symmetric (GSS) metric +ds2 = −h(r)dt2 + +1 +f(r)dr2 + r2dΣ2 +D−2,k +(1.3) +The advantage of this definition of quasi-topological gravity is that black hole solutions in +the original gravity theory in the form of (1.1) or (1.3) simply continue to be solutions when +the corresponding quasi-topological terms are introduced. This could be relevant when mat- +ter are included, where even with the equation of f(r) being algebraic, the inclusion of mat- +ter terms could make the system unintegratable. In this work we are specifically interested +in quasi-topological gravities on GSS metric (1.3), but not just limited to Ricci gravities. +Such metric is the most general ansatz for spacetimes with spherical/planar/hyperbolic +1Unless otherwise noted, when we say spherically symmetry, we actually mean spherically, planar, or +hyperbolic symmetry, corresponding to the curvature of dΣ2 being positive, zero, or negative, respectively. +– 2 – + +symmetry, thus could include a wider range of solutions. An important class of GSS metric +is black hole with scalar hair. Hairy solutions typically have a rich phase structure and in +holography they may be used to describe superconductors [29, 30]. It was shown that even +Einstein gravity with a minimally coupled self-interacting scalar field could result in a hairy +solution [31]. It’s thus interesting to investigate the effects of including higher curvature +terms in these solutions. There has been works on the hairy solutions with conformally +coupled scalar in higher curvature gravities [32, 33]. +In this work we focus on quasi-topological gravities on GSS metric and construct such +quasi-topological terms up to quintic order in Riemann tensor, including dimension-generic +ones and dimension-specific ones. For the latter only D ⩾ 3 is considered since at D = 2 +the Riemann tensor has only one non-zero component. We also find it possible to construct +dimension-independent combinations at quartic and quintic order. We then notice that +these theories satisfy a much stronger condition: the quasi-topological terms vanish identi- +cally when evaluated on the non-stationary spherically symmetric metric! That is, metric +of the following form +ds2 = −h(t, r)dt2 + 2b(t, r)dtdr + +1 +f(t, r)dr2 + r2dΣ2 +D−2,k +(1.4) +On the one hand, this further restricts the effects the quasi-topological term could possibly +have, such as no thermodynamics contribution. This possibly simplifies the problem since +introducing the quasi-topological terms won’t lead to any new phase transitions. +Thus +one may only seek for non-trivial effects of quasi-topological terms by considering per- +turbations. On the other hand, this indicates that these quasi-topological terms are also +quasi-topological on a much wider range kinds of metrics, e.g., Friedmann-Roberson-Walker +metric, it’s thus possible to study the effects of quasi-topological terms on cosmic pertur- +bations. +This paper is organized as follows. +In section 2 we construct explicitly the quasi- +topological gravity theories, up to quintic order. In section 3 we discuss the basic properties +of the obtained theories, mainly the implications of the vanishing of the quasi-topological +terms evaluated on the metric (1.4). As an example to study the physical effects of quasi- +topological terms, in section 4 we consider a general Einstein-scalar theory extended with +quasi-topological terms and calculate its holographic shear viscosity. +2 +Construction of the theory +For a general Lagrangian constructed from metric and Riemann tensor L(gab, Rabcd) the +equation of motion can be written as [34] +Eab[L] = +1 +� +|g| +δS +δgab += P a +cde Rbcde − 1 +2gabL + ∇c∇dP acdb, +P abcd = +∂L +∂Rabcd +(2.1) +where S = +� +dDx +� +|g|L is the action. In our definition of quasi-topological gravities, a +quasi-topological term Q in the Lagrangian satisfies that it does not contribute to the +equation of motion when evaluated on (1.3), namely +Ett[Q]h,f = Err[Q]h,f = 0 +(2.2) +– 3 – + +which is equivalent to +δ +δh +� +dDx +� +|g|Q +��� +h,f = δ +δf +� +dDx +� +|g|Q +��� +h,f = 0 +(2.3) +where . . . |h,f denotes . . . evaluated on the metric (1.3). At a given order, we first write +down the most general Riemann polynomial of that order with undetermined coefficients +and substitute the ansatz (1.3) into it. The non-zero Riemann tensor components of (1.3) +are +Rˆtˆrˆtˆr = f(r)h′(r)2 − h(r) [f′(r)h′(r) + 2f(r)h′′(r)] +4h(r)2 +Rˆtiˆtj = −δi +j +f(r)h′(r) +2rh(r) +Rˆriˆrj = −δi +j +f′(r) +2r +Rijkl = (δi +kδj +l − δi +lδj +k)k − f(r) +r2 +where ijkl are indices of the (D − 2) dimension subspace, and equivalent components are +not shown. By varying and integrating by parts with respect to h(r) and f(r) we get two +algebraic equations containing the undetermined coefficients, we then further convert them +into a linear system about the undetermined coefficients by regarding them as polynomials +in r, h(r), f(r) and their derivatives and requiring all coefficients vanish. The solution space +is given by the null space of the resulting linear system. For dimension-generic solutions, we +take null space directly, and for dimension-specific ones we substitute the dimension first +and then take null space, since there may be more linear independent solutions at lower +dimensions. +2.1 +Cubic order +There are 8 Riemann scalars at cubic order and the most general cubic Riemann polynomial +is given by their linear combination +Q(3) = e1R b d +a c R e f +b d R a b +e f + e2R +cd +ab R +ef +cd +R +ab +ef ++ e3RabcdRabc +eRde + e4RabcdRabcdR ++ e5RabcdRacRbd + e6Rb +aRc +bRa +c + e7Rb +aRa +bR + e8R3 +(2.4) +We found only one dimension-generic solution in this case, the coefficients ei are given by +e1 = 22 − 26D + 9D2 − D3, e2 = 3D2 +4 +− 15D +4 ++ 4, e3 = −3(D − 3)(D − 1) +e4 = 3(D − 3) +2 +, e5 = 3 +� +D2 − 5D + 8 +� +, e6 = 6D − 14, e7 = 3 − 3D, e8 = 1 +(2.5) +As mentioned earlier, quasi-topological gravity is a subclass of generalized quasi-topological +gravity, so the solution (2.5) must be a special case of cubic generalized quasi-topological +gravity. In fact, by setting +c1 = 22 − 26D + 9D2 − D3, c2 = 3D2 +4 +− 15D +4 ++ 4, c3 = −3(D − 3)(D − 1) +– 4 – + +D +{ei} +3 +(−8, 5, −12, 0, 0, 0, 0, 1) +(−2, 3 +2, −4, 0, 0, 0, 1, 0) +(0, 1 +2, − 3 +2, 0, 0, 1, 0, 0) +(−1, 1 +4, −1, 0, 1, 0, 0, 0) +(0, 1, −4, 1, 0, 0, 0, 0) +4 +(16, −8, 36, −3, −24, −8, 0, 1) +(2, −1, 5, − 1 +2, −4, −2, 1, 0) +5 +(−8, 4, −24, 3, 24, 16, −12, 1) +6 +(64, −14, 60, −3, −48, −8, 0, 1) +(6, − 3 +2, 7, − 1 +2, −6, −2, 1, 0) +Table 1. Dimension-specific solutions of cubic quasi-topological gravity +in (2.6) of [17] we get our solution (2.5). As there’s only one solution, it’s not possible to +construct dimension independent solution at cubic order. +Now we turn to dimension-specific solutions. First we found that for D > 6 the number +of independent solutions is always one, meaning they are covered by the dimension-generic +solution, so we only need to consider 3 ⩽ D ⩽ 6. We get two linear independent solutions +at D = 4 and D = 6 respectively, five solutions at D = 3 and one solution again at D = 5. +The solutions are given in table 1. However, not all solutions are non-trivial. It could +happen that some solutions vanish identically on any metric, just like Lovelock terms in +D < 2n, this is possible for dimension-specific cases. Firstly we note that all the solutions +in table 1 have included the cubic Lovelock term, especially the five dimensional solution +which simply coincides with it. So we are left with 4 solution for D = 3, one solution for +D = 4 and D = 6 respectively. +Besides Lovelock terms themselves, another kind of combinations that vanish in lower +dimensions may be constructed from their equation of motion. For example, in D ⩽ 4, the +4D Lovelock, or Gauss-Bonnet term +X (4) = R2 − 4RabRab + RabcdRabcd +(2.6) +is topological, so its equation of motion contribution should vanish +Eab[X (4)] = +1 +� +|g| +δ +δgab +� +dDx +� +|g|X (4) += −4RacRb +c + 2RabR − 4RcdRa b +c d + 2RacdeRb +cde + 1 +2gabX (4) = 0 +(2.7) +we can thus construct another vanishing Riemann polynomial +Eab[X (4)]Rab = −4Ra +bRb +cRc +a + 4RabRabR − 1 +2R3 − 4RabRcdRacbd +− 1 +2RabcdRabcdR + 2RabR cde +a +Rbcde = 0 +(2.8) +– 5 – + +It can be shown that the space spanned by (2.8) and 4D Lovelock term is isomorphic to +the D = 4 solution in table 1, thus both solutions are trivial. In three dimensions, the +Gauss-Bonnet term vanishes, there are three more vanishing Riemann polynomials 2 +RX (4), +∂X (4) +∂Rabcd RacRbd, +∂X (4) +∂Rabcd RcdefR +ab +ef +(2.9) +Again, the space spanned by these three terms, (2.8) and 4D Lovelock, is isomorphic to the +D = 3 solution in table 1, thus only one (linear combination) of the D = 6 solution in table +1 is non-trivial. This 6D solution is also covered by the dimension-generic solution (2.5). +We finally conclude that cubic quasi-topological term is completely given by (2.5) and it’s +only non-trivial for D ⩾ 6. +2.2 +Higher orders +The method of solving higher order quasi-topological terms is exactly the same as we used +in the cubic case. The only difficulty is enumerating all possible Riemann scalars, as the +number of independent Riemann scalars grows rapidly in higher orders. We get 26 scalars +at quartic order [35] and 85 scalars at quintic order. +For quartic case we get 12 linear independent dimension-generic solutions, and at D = +3, D = 4 and D = 8 we get 22, 15 and 13 solutions respectively, all other dimensions have +the same number of solutions as dimension-generic case. Again, the extra solution at D = 8 +is the 8D Lovelock term. The explicit list of solutions is lengthy and given in appendix. +Remarkably, we also found 3 dimension-independent solutions +Q(4),∗,1 = Rabcd +� +R e f +a c R g h +b e Rdgfh − R +ef +ab +R g h +c e Rdgfh − 1 +4R +e +abc R fgh +d +Refgh +� +(2.10) +Q(4),∗,2 = R +ef +ab +Rabcd +� +R +gh +ce +Rdfgh − 1 +2R +gh +cd +Refgh +� +(2.11) +Q(4),∗,3 = RabRcd +� +R e f +a c Rbedf − 1 +2R +ef +ac +Rbdef +� +− 1 +2RabRc +aR def +b +Rcdef +(2.12) +The situation is similar for quintic case, we get 61 dimension-generic solutions and 80, 67, +62 dimension-specific solutions at D = 3, D = 4, D = 10 respectively, for dimension- +independent case we get 29 solutions. However as the solutions of the quintic case are too +lengthy, we only present some representative solutions in the appendix and the full solution +set can be found in the supplementary material. +3 +Properties and discussions +Having constructed the desired theories we now move on to their physical effects. The first +property we noticed is the quasi-topological term vanishes when evaluated on the metric +(1.3) +Q(n) +h,f = 0 +(3.1) +2L = 0 implies ∂L/∂Rabcd = 0 if the identity ∂L/∂gmn = (∂L/∂Rabcd)(∂Rabcd/∂gmn) gives no less +equations than the independent components of ∂L/∂Rabcd, which is true for D ⩽ 3. +– 6 – + +The free energy can be obtained by evaluating the Euclidean action with compactified time +direction. Since our metric is static, the Euclidean action only differs from the Minkowski +action by a minus sign, (3.1) implies the vanishing of the free energy contribution from +quasi-topological term, which then further implies the entropy and thermodynamic energy +contribution should also vanish, thus quasi-topological term completely has no thermody- +namics effects. +To verify the consistency of the above conclusion we need to evaluate the Wald entropy +and thermodynamic energy. The Wald entropy is given by [36] +SWald = −2π +� +P abcdϵabϵcd dΣ, +P abcd = +∂L +∂Rabcd +(3.2) +where the integration is taken at the horizon, ϵab is the binormal to the horizon, dΣ is the +volume form of the horizon surface. Using the method similar to [18] one can show that +Q(n) +h,f = 0 implies +∂Q(n) +∂Rabcd +����� +h,f += 0 +(3.3) +thus the Q(n) contribution to (3.2) must also vanish. It remains to calculate the energy, +which can be done holographically by calculating the tt component of the boundary stress +tensor T tt = (2/ +� +|h|)δS/δhtt where hab is the boundary metric. The surface term and +counter term also need to be taken into account, but since Q(n) vanishes, no new diverges +appear, the counter term contribution is zero. The surface term can be constructed by +introducing an auxiliary field Φab = Pacbdncnd [37] +S∂ = 1 +8π +� +∂M +dD−1x +� +|h|ΦabKab +(3.4) +where na is the normal vector of the boundary, Kab = ∇anb is the exterior curvature +and hab = gab − nanb. Note that when varying this term, Φab should be kept fixed. So +we immediately see from (3.3) that the surface term contribution to T tt should vanish. +Furthermore, because Q(n) vanishes on (1.3), it’s invariant under the variation h(r) → +h(r) + δh(r), so we have δ +� +|g|Q(n)/δhtt = 0. We thus conclude the energy obtained via +holography also vanishes. +As mentioned earlier, we actually found a much stronger conclusion than (3.1), that +is Q(n) also vanishes when evaluated on the general non-stationary spherically symmetric +metric (1.4). It’s straightforward to evaluate a given quasi-topological terms on (1.4) and +check that it vanishes. In practice, the check was done using an equivalent metric +ds2 = −h(t, r)dt2 + 2b(t, r)dtdr + +� +1 +f(t, r) − b2(t, r) +h(t, r) +� +dr2 + r2dΣ2 +D−2,k +(3.5) +The advantage of it is the components of the inverse metric contain no fraction, reducing +the computation cost. The check was done for all cubic and quartic quasi-topological terms, +but at quintic order we encountered extreme computation difficulties so we ended up only +checked the solutions listed in (A.3). It’s then strongly suggested that the condition for +quasi-topological terms (2.3) implies that they vanish when evaluated on (1.4). +– 7 – + +The vanishing of Q(n) on (1.4) makes it quasi-topological on a much wider range of +metrics, e.g. the FRW metric +ds2 = −dt2 + a2(t) +� +dr2 +1 − kr2 + r2dΩ2 +� +(3.6) +which can be put into the form of (1.4) by redefining r → r/a(t). It also implies Q(n) +has no a-charge contribution. In 2n dimensional CFTs, the central a, c charges appears as +coefficients in the trace of the stress tensor [38] +� +T µ +µ +� +∼ −aX (2n) + +� +i +ciI(2n) +i +(3.7) +where X (2n) is the 2n dimensional Lovelock term, I(2n) +i +are conformal invariants in 2n +dimensional space. +Generally the central charges can be calculated holographically by +evaluating the action on the the FG expansion metric and identifying the ρ−1 term as the +trace anomaly [39], but to solely extract the a-charge one may use a specific metric with +conformally flat boundary, e.g., an S2n [40] +ds2 = L2 +4ρ2 dρ2 + f(ρ) +ρ +� dr2 +1 − r2 + r2dΩ2 +D−2 +� +(3.8) +again, by redefining r → r +� +ρ/f(ρ) this metric can be put into the form of (1.4), thus Q(n) +also vanish on (3.8), it doesn’t contribute to the a-charge. +The vanishing of Q(n) on the more general metric (1.4) largely reduces the possible +effects it could have when introduced to some gravity theory. To seek for non-trivial effects +one may only consider the perturbations of it around the metric (1.3), which in holography +includes shear viscosity and heat current, corresponding to perturbations hx1x2 and htx1 +respectively. In the next section we’ll consider the holographic shear viscosity as an example +to study. +4 +Holographic shear viscosity +We consider a general Einstein-scalar theory with the Lagrangian +LES = +1 +16π +� +R − 1 +2∇aφ∇aφ − V (φ) +� +(4.1) +we are interested in the non-extremal 3 asymptotic AdS black hole solutions of (4.1), so we +consider the following planar black hole ansatz +ds2 = −f(r)e−η(r)dt2 + +1 +f(r)dr2 + U(r)d⃗x2 +D−2 +(4.2) +3The extremal limit T → 0 and hydrodynamic limit ω → 0 generally don’t commute, which will compli- +cate the discussion. +– 8 – + +Note that there’s one gauge freedom in the three functions f(r), η(r), U(r). +Near the +boundary we have f(r → ∞) = U(r → ∞) = r2/L2, η(r → ∞) = 0. The horizon is at +r = rh, satisfies f(rh) = 0. The temperature and entropy density are respectively given by +T = 1 +4πf′(rh)e−η(rh)/2, +s = 1 +4U D/2−1(rh) +(4.3) +Assuming φ only depends on r, the equation of motion gives +(D − 2)fU′φ′ + U +� +2f′φ′ − fη′φ′ + 2fφ′′ − 2V ′� += 0 +� +D2 − 7D + 10 +� +fU′2 + 2(D − 2)U +� +f′U ′ + 2fU′′� ++ 2U 2 � +fφ′2 + 2V +� += 0 +2U +� +(D − 2)U ′′(r) + U(r)φ′2� ++ (D − 2)UU ′η′ − (D − 2)U ′2 = 0 +U +� +−(D − 4)f′U ′ + f +� +(D − 3)η′U ′ + 2U ′′�� ++ (D − 4)fU′2 ++U 2 � +−2f′′ + 3f′η′ − f +� +η′2 − 2η′′�� += 0 +(4.4) +The last equation can be integrated to give a radially conserved quantity, as in [41] +Q = e−η/2U D/2−2 �� +f′ − fη′� +U − fU′� +(4.5) +Evaluating it at the horizon gives +Q = e−η(rh)/2U D/2−1(rh)f′(rh) = 16πTs +(4.6) +To calculate the holographic shear viscosity we employ the pole method as proposed in [42]. +Define a new radial coordinate z by r = rh/(1 − z), (4.2) becomes +ds2 = +r2 +h +(1 − z)4 +1 +f( rh +1−z)dz2 − f +� rh +1 − z +� +exp +� +−η +� rh +1 − z +�� +dt2 + U +� rh +1 − z +� +d⃗x2 +D−2 (4.7) +Now add perturbation to (4.7) by shifting the basis dx1 → dx1 + εe−iωtdx2, substitute the +resulting metric into the Lagrangian and expand it to quadratic order in ε. Note that the +perturbation should be kept second order in the metric, and since the perturbation only +involves spatial components, the matter sector of the Lagrangian (4.1) has no contribution. +The shear viscosity can be calculated from the residue of the Lagrangian at z = 0 +η = −8πT lim +ω,ε→0 +Resz=0 +� +|g|L +ε2ω2 +(4.8) +Note that the above expression for the shear viscosity is linear in L so we can compute the +contribution to the shear viscosity of different terms in the Lagrangian separately, but keep +in mind only the summed result has physical meaning. For the Einstein-scalar theory in +(4.1), the contribution is given by +η(0) = +1 +16πU D/2−1(rh) +(4.9) +which results in the standard shear-viscosity-to-entropy ratio η(0)/s = 1/4π, i.e., the exis- +tence of the scalar hair have no effect on the shear viscosity. +– 9 – + +Next we introduce quasi-topological terms to (4.1) by defining the new Lagrangian as +L′ = LES + λ/16πQ(n), the equations of motion aren’t altered by Q(n) and it’s straight- +forward to evaluate (4.7) on them and then apply (4.8) to obtain the shear viscosity, the +results are expressed in f(r), η(r), U(r) and their derivatives at r = rh. Interestingly, +by making use of the radially conserved quantity (4.5) we are only left with η(rh), η′(rh), +U(rh) and its derivatives. The contribution from Q(3) is given by +η(3) = +λ +16π +3 +16(D − 5)(D − 4)(D − 2)2Q2eη(rh)U −D/2−1(rh) +� +(D + 2)U ′2(rh) − 2U(rh)U ′′(rh) − U(rh)η′(rh)U ′(rh) +� +(4.10) +Notice that this result is only non-zero for D ⩾ 6, otherwise the quasi-topological term is +trivial, as discussed earlier. The shear-viscosity-to-entropy ratio in this case is given by +η +s = 1 +4π +� +1 + 3λ +16 (D − 5)(D − 4)(D − 2)2Q2eη +� +(D + 2)U ′2 +U 2 − 2U ′′ +U − η′ U ′ +U +�� +(4.11) +where all functions are evaluated at the horizon. This indicates a possible violation of the +KSS bound. This is expected, since it’s now understood that higher curvature terms would +introduce massive graviton modes, which is the source of KSS bound violation. Indeed, +to confirm that (2.5) violates the bound requires determining the physical bound of the +coupling constant λ, we will not discuss it here. +For quartic and quintic case, we found that the D = 4 solutions are all analytical +at z = 0 when evaluated on the metric (4.7) and thus does not contribute to the shear +viscosity. For the dimension-generic solutions, we found their shear viscosity contribution +can written in the form +ηQ(n) = +λ +16πe +n−1 +2 ηQn−1U −1− n−2 +2 DU ′n−3 � +aU′2 + bUU ′η′ + 2bUU ′′� +(4.12) +For a specific quasi-topological term, the coefficients a, b only depends on the dimension +D, their explicit values are given in the appendix. +5 +Conclusions +Quasi-topological gravities can be thought as a class of higher curvature gravity theories +whose higher curvature terms give no contribution to the equations of motion when evalu- +ated on the metric (1.3), but could have non-trivial perturbations around it. In this case +black hole solutions of the corresponding Einstein gravity continues to be solution when +the higher curvature terms are included, making it much easier to study its higher curva- +ture effects. In this work we constructed such theory up to quintic order in the Riemann +tensor. Most remarkably, we found that all quasi-topological terms we constructed actually +vanish when evaluated on the most general non-stationary spherically symmetric metric +(1.4). On the one hand, this makes these terms have no contribution on the thermody- +namics and holographic a-charge. More importantly, on the other hand, this makes them +quasi-topological on a much wider kinds of metrics, e.g., the FRW metric and the Vaidya +– 10 – + +metric. This opens a large gate of possible applications of such quasi-topological gravity +theories, such as one could study the effects of these terms on the cosmic perturbations. +As an example to study the non-trivial effects of the quasi-topological terms we calcu- +lated the holographic shear viscosity of a general Einstein-scalar theory. The results can be +put into a simple form (4.12). As expected, the KSS bound could possibly be violated due +to the nature of higher curvature gravities. +A +Quartic and quintic quasi-topological gravities +In this section we list all solutions of quartic and quintic quasi-topological terms. The full +set of solutions is also available in the supplementary material, in the form of Mathematica +.wl file, with further instructions included in the usage messages. +For quartic order, the most general Riemann polynomial is +Q(4) = e1R4 + e2R2RabRab + e3RRa +bRb +cRc +a + e4(RabRab)2 + e5Ra +bRb +cRc +dRd +a ++ e6RRacRbdRabcd + e7RacRb +eRedRabcd + e8R2RabcdRabcd + e9RRdeRabc +dRabce ++ e10RabRabRcdefRcdef + e11RabRc +aRdef +bRdefc + e12RabRcdRef +acRefbd ++ e13RabRcdRe f +a bRecfe + e14RabRcdRe f +a cRebfd + e15RRabcdR +ef +ab +Refcd ++ e16RRabcdR e f +a c Rbedf + e17RabR c d +a b Refg +cRefgd + e18RabRcdefR +g +cd aRefgb ++ e19RabRcdefR g +c eaRdgfb + e20(RabcdRabcd)2 + e21RabcdR +e +abc Rfgh +dRfghe ++ e22RabcdR +ef +ab +R +gh +ef +Rcdgh + e23RabcdR +ef +ab +R +gh +ce +Rdfgh + e24RabcdR +ef +ab +R g h +c e Rdgfh ++ e25RabcdR e f +a c R g h +e f Rbgdh + e26RabcdR e f +a c R g h +e b Rfgdh +(A.1) +There are totally 12 dimension-generic solutions, their coefficients ei,j are listed in table 2 +below, where i labels different solutions and j labels the 26 coefficients of one solution. +e1,1 +2D9−61D8+773D7−5451D6+23821D5−67174D4+121930D3−135736D2+81920D−19456 +2(D−4)(D−3)(D−2)5(D−1)D(D3−9D2+26D−22) +e1,2 +−3D10+94D9−1237D8+9168D7−42780D6+131846D5−271580D4+367060D3−308704D2+145024D−29184 +(D−4)(D−3)(D−2)5(D−1)D(D3−9D2+26D−22) +e1,3 +4(2D9−59D8+733D7−5103D6+22103D5−61918D4+111738D3−123512D2+73632D−17024) +(D−4)(D−3)(D−2)4(D−1)D(D3−9D2+26D−22) +e1,4 +2D8−56D7+633D6−3948D5+15253D4−37812D3+58752D2−51840D+19456 +(D−4)(D−3)(D−2)5(D−1)D +e1,5 +−2D7+45D6−398D5+1895D4−5476D3+9776D2−10112D+4864 +(D−4)(D−3)(D−2)4(D−1)D +e1,6 +4(2D8−59D7+730D6−5041D5+21496D4−58348D3+98636D2−94560D+38912) +(D−4)(D−3)(D−2)3(D−1)D(D3−9D2+26D−22) +e1,7 +− +4(D6−19D5+131D4−409D3+520D2+88D−608) +(D−4)(D−3)(D−2)3(D−1)D +e1,8 +D10−36D9+557D8−4979D7+28834D6−113919D5+312276D4−587102D3+723424D2−525760D+170240 +2(D−4)(D−3)(D−2)4(D−1)D(D3−9D2+26D−22) +e1,9 +− +2(D8−30D7+376D6−2636D5+11493D4−32254D3+57146D2−58160D+25536) +(D−4)(D−3)(D−2)2(D−1)D(D3−9D2+26D−22) +e1,10 +− D8−30D7+353D6−2250D5+8748D4−21514D3+32672D2−27712D+9728 +2(D−4)(D−3)(D−2)4(D−1)D +e1,11 +D6−23D5+199D4−861D3+1996D2−2216D+608 +(D−4)(D−3)(D−2)2(D−1)D +e1,15 +D3−12D2+41D−38 +(D−1)D(D3−9D2+26D−22) +e1,17 +76−20D +D3−5D2+6D +– 11 – + +e1,20 +− D5−10D4+28D3+18D2−173D+160 +8(D−3)(D−2)3(D−1) +e1,26 +1 +e2,1 +12D8−244D7+2138D6−10521D5+31695D4−59494D3+67138D2−40696D+9728 +(D−4)(D−3)(D−2)5(D−1)D(D3−9D2+26D−22) +e2,2 +− +4(9D9−191D8+1766D7−9316D6+30809D5−65961D4+90838D3−76978D2+36256D−7296) +(D−4)(D−3)(D−2)5(D−1)D(D3−9D2+26D−22) +e2,3 +8(11D8−224D7+1964D6−9662D5+29067D4−54398D3+61026D2−36552D+8512) +(D−4)(D−3)(D−2)4(D−1)D(D3−9D2+26D−22) +e2,4 +2(13D7−228D6+1730D5−7392D4+19201D3−30196D2+26400D−9728) +(D−4)(D−3)(D−2)5(D−1)D +e2,5 +− +2(14D6−185D5+1018D4−3059D3+5380D2−5344D+2432) +(D−4)(D−3)(D−2)4(D−1)D +e2,6 +− +4(D8−45D7+671D6−5095D5+22712D4−62348D3+104224D2−97464D+38912) +(D−4)(D−3)(D−2)3(D−1)D(D3−9D2+26D−22) +e2,7 +− +8(4D5−41D4+144D3−167D2−116D+304) +(D−4)(D−3)(D−2)3(D−1)D +e2,8 +7D9−178D8+2013D7−13299D6+56610D5−161107D4+306544D3−375726D2+268688D−85120 +(D−4)(D−3)(D−2)4(D−1)D(D3−9D2+26D−22) +e2,9 +− +8(3D7−62D6+551D5−2733D4+8181D3−14787D2+14903D−6384) +(D−4)(D−3)(D−2)2(D−1)D(D3−9D2+26D−22) +e2,10 +− +2(3D7−56D6+445D5−1959D4+5160D3−8093D2+6928D−2432) +(D−4)(D−3)(D−2)4(D−1)D +e2,11 +2(7D5−89D4+437D3−1031D2+1108D−304) +(D−4)(D−3)(D−2)2(D−1)D +e2,15 +−D3+12D2−41D+38 +D5−10D4+35D3−48D2+22D +e2,17 +− +4(D2−8D+19) +(D−3)(D−2)D +e2,20 +− D5−12D4+61D3−167D2+242D−137 +4(D−3)(D−2)3(D−1) +e2,25 +1 +e3,1 +D9−30D8+380D7−2695D6+11858D5−33610D4+61132D3−68000D2+40960D−9728 +(D−4)(D−3)(D−2)5(D−1)D(D3−9D2+26D−22) +e3,2 +−3D10+92D9−1207D8+8988D7−42238D6+131008D5−270984D4+366928D3−308704D2+145024D−29184 +(D−4)(D−3)(D−2)5(D−1)D(D3−9D2+26D−22) +e3,3 +8(D9−29D8+360D7−2521D6+10999D5−30982D4+56036D3−61888D2+36816D−8512) +(D−4)(D−3)(D−2)4(D−1)D(D3−9D2+26D−22) +e3,4 +2(D8−27D7+307D6−1947D5+7638D4−19104D3+29728D2−26112D+9728) +(D−4)(D−3)(D−2)5(D−1)D +e3,5 +− +2(D7−22D6+199D5−970D4+2832D3−5040D2+5152D−2432) +(D−4)(D−3)(D−2)4(D−1)D +e3,6 +4(2D8−59D7+736D6−5129D5+22006D4−59796D3+100632D2−95616D+38912) +(D−4)(D−3)(D−2)3(D−1)D(D3−9D2+26D−22) +e3,7 +− +4(D6−18D5+123D4−382D3+468D2+136D−608) +(D−4)(D−3)(D−2)3(D−1)D +e3,8 +D10−35D9+541D8−4887D7+28686D6−114666D5+316648D4−596688D3+733520D2−529984D+170240 +2(D−4)(D−3)(D−2)4(D−1)D(D3−9D2+26D−22) +e3,9 +− +2(D2−7D+14)(D6−22D5+195D4−906D3+2364D2−3280D+1824) +(D−4)(D−3)(D−2)2(D−1)D(D3−9D2+26D−22) +e3,10 +− D8−28D7+329D6−2138D5+8496D4−21248D3+32576D2−27712D+9728 +2(D−4)(D−3)(D−2)4(D−1)D +e3,11 +D6−22D5+195D4−866D3+2020D2−2216D+608 +(D−4)(D−3)(D−2)2(D−1)D +e3,15 +D3−12D2+41D−38 +(D−1)D(D3−9D2+26D−22) +e3,17 +2(D2−13D+38) +D(D2−5D+6) +e3,20 +− +(D−4)(D3−10D2+31D−26) +4(D−3)(D−2)3(D−1) +e3,24 +1 +e4,1 +2(11D5−140D4+668D3−1502D2+1544D−512) +(D−4)(D−3)(D−2)5(D−1)D +e4,2 +− +16(4D6−55D5+293D4−780D3+1083D2−728D+192) +(D−4)(D−3)(D−2)5(D−1)D +– 12 – + +e4,3 +32(5D5−64D4+306D3−687D2+700D−224) +(D−4)(D−3)(D−2)4(D−1)D +e4,4 +8(5D7−100D6+838D5−3856D4+10553D3−17148D2+15232D−5632) +(D−4)(D−3)(D−2)5(D−1)D +e4,5 +− +8(6D6−89D5+530D4−1675D3+3036D2−3072D+1408) +(D−4)(D−3)(D−2)4(D−1)D +e4,6 +64(3D4−37D3+167D2−335D+256) +(D−4)(D−3)(D−2)3(D−1)D +e4,7 +− +32(2D5−23D4+88D3−111D2−60D+176) +(D−4)(D−3)(D−2)3(D−1)D +e4,8 +2(5D6−99D5+778D4−3182D3+7250D2−8800D+4480) +(D−4)(D−3)(D−2)4(D−1)D +e4,9 +− +16(2D−7)(D3−12D2+41D−48) +(D−4)(D−3)(D−2)2(D−1)D +e4,10 +− +8(D6−22D5+188D4−827D3+2013D2−2600D+1408) +(D−4)(D−3)(D−2)4D +e4,11 +8(3D5−45D4+245D3−607D2+652D−176) +(D−4)(D−3)(D−2)2(D−1)D +e4,15 +4 +D−D2 +e4,17 +− +16(D2−6D+11) +(D−3)(D−2)D +e4,20 +− D5−14D4+79D3−224D2+316D−170 +2(D−3)(D−2)3(D−1) +e4,23 +1 +e5,1 +4(11D5−140D4+668D3−1502D2+1544D−512) +(D−4)(D−3)(D−2)5(D−1)D +e5,2 +− +32(4D6−55D5+293D4−780D3+1083D2−728D+192) +(D−4)(D−3)(D−2)5(D−1)D +e5,3 +64(5D5−64D4+306D3−687D2+700D−224) +(D−4)(D−3)(D−2)4(D−1)D +e5,4 +16(5D7−100D6+838D5−3856D4+10553D3−17148D2+15232D−5632) +(D−4)(D−3)(D−2)5(D−1)D +e5,5 +− +16(6D6−89D5+530D4−1675D3+3036D2−3072D+1408) +(D−4)(D−3)(D−2)4(D−1)D +e5,6 +128(3D4−37D3+167D2−335D+256) +(D−4)(D−3)(D−2)3(D−1)D +e5,7 +− +64(2D5−23D4+88D3−111D2−60D+176) +(D−4)(D−3)(D−2)3(D−1)D +e5,8 +4(5D6−99D5+778D4−3182D3+7250D2−8800D+4480) +(D−4)(D−3)(D−2)4(D−1)D +e5,9 +− +32(2D−7)(D3−12D2+41D−48) +(D−4)(D−3)(D−2)2(D−1)D +e5,10 +− +16(D6−22D5+188D4−827D3+2013D2−2600D+1408) +(D−4)(D−3)(D−2)4D +e5,11 +16(3D5−45D4+245D3−607D2+652D−176) +(D−4)(D−3)(D−2)2(D−1)D +e5,15 +8 +D−D2 +e5,17 +− +32(D2−6D+11) +(D−3)(D−2)D +e5,20 +−D5+14D4−79D3+224D2−316D+170 +(D−3)(D−2)3(D−1) +e5,22 +1 +e6,1 +− +2D(D2−D−4) +(D−4)(D−2)5(D−1) +e6,2 +8D2(D2−3D+1) +(D−4)(D−2)5(D−1) +e6,3 +− +16D(D2−D−4) +(D−4)(D−2)4(D−1) +e6,4 +− +4(2D5−13D4+15D3+68D2−192D+128) +(D−4)(D−2)5(D−1) +– 13 – + +e6,5 +4(D4+3D3−36D2+80D−64) +(D−4)(D−2)4(D−1) +e6,6 +− +32(3D−8) +(D−4)(D−2)3(D−1) +e6,7 +16(D3−5D2+12D−16) +(D−4)(D−2)3(D−1) +e6,8 +− +2(D4−4D3−9D2+56D−64) +(D−4)(D−2)4(D−1) +e6,9 +8(D2−D−8) +(D−4)(D−2)2(D−1) +e6,10 +4D(D4−9D3+29D2−39D+16) +(D−4)(D−2)4(D−1) +e6,11 +− +4D(D2−D−8) +(D−4)(D−2)2(D−1) +e6,17 +− +8 +D−2 +e6,20 +− D3−8D2+21D−16 +2(D−2)3 +e6,21 +1 +e7,1 +− +D6−16D5+103D4−336D3+570D2−458D+128 +(D−4)(D−3)(D−2)2(D−1)D(D3−9D2+26D−22) +e7,2 +3D7−51D6+356D5−1305D4+2665D3−2984D2+1692D−384 +(D−4)(D−3)(D−2)2(D−1)D(D3−9D2+26D−22) +e7,3 +−7D6+114D5−743D4+2440D3−4140D2+3296D−896 +(D−4)(D−3)(D−2)(D−1)D(D3−9D2+26D−22) +e7,4 +− +2(D2−5D+8)(D3−9D2+23D−16) +(D−4)(D−3)(D−2)2(D−1)D +e7,5 +2(D4−10D3+33D2−48D+32) +D(D4−10D3+35D2−50D+24) +e7,6 +−9D5+139D4−851D3+2561D2−3728D+2048 +(D−4)(D−3)(D−1)D(D3−9D2+26D−22) +e7,7 +2(D3−7D2+6D+16) +(D−4)(D−3)(D−1)D +e7,8 +− D7−22D6+206D5−1059D4+3213D3−5733D2+5554D−2240 +2(D−4)(D−3)(D−2)(D−1)D(D3−9D2+26D−22) +e7,9 +(D−2)(4D5−65D4+428D3−1409D2+2258D−1344) +2(D−4)(D−3)(D−1)D(D3−9D2+26D−22) +e7,10 +D5−15D4+85D3−229D2+290D−128 +2(D−4)(D−3)(D−2)(D−1)D +e7,11 +− +(D−2)(D3−10D2+25D−8) +(D−4)(D−3)(D−1)D +e7,15 +3D2−15D+16 +−4D4+36D3−104D2+88D +e7,17 +4−D +(D−3)D +e7,19 +1 +e8,1 +2(3D2−9D+4) +(D−3)(D−2)2(D−1)D +e8,2 +−18D3+70D2−72D+24 +(D−3)(D−2)2(D−1)D +e8,3 +4(11D2−33D+14) +(D−3)(D−2)(D−1)D +e8,4 +4(3D4−28D3+101D2−160D+88) +(D−3)(D−2)2(D−1)D +e8,5 +− +4(3D3−14D2+25D−22) +(D−3)(D−2)(D−1)D +e8,6 +16(3D−8) +(D−3)(D−1)D +e8,7 +−20D2+40D+44 +D3−4D2+3D +e8,8 +3D3−26D2+73D−70 +D(D3−6D2+11D−6) +e8,9 +− 2(D−2)(5D−21) +(D−3)(D−1)D +e8,10 +−3D4+32D3−119D2+182D−88 +D(D3−6D2+11D−6) +e8,11 +6D3−42D2+74D−22 +D(D2−4D+3) +– 14 – + +e8,15 +− 1 +D +e8,17 +− +2(D2−6D+11) +(D−3)D +e8,18 +1 +e9,1 +1 +−D3+9D2−26D+22 +e9,2 +3(D−1) +D3−9D2+26D−22 +e9,3 +14−6D +D3−9D2+26D−22 +e9,6 +− +3(D2−5D+8) +D3−9D2+26D−22 +e9,8 +− +3(D−3) +2(D3−9D2+26D−22) +e9,9 +3(D−3)(D−1) +D3−9D2+26D−22 +e9,15 +3D2−15D+16 +−4D3+36D2−104D+88 +e9,16 +1 +e10,1 +D2−4D+2 +2(D−4)(D−2)2(D−1) +e10,2 +− +(3D−2)(D2−4D+2) +2(D−4)(D−2)2(D−1) +e10,3 +4(D2−4D+2) +(D−4)(D−2)(D−1) +e10,4 +D4−10D3+39D2−68D+40 +(D−4)(D−2)2(D−1) +e10,5 +−D3+5D2−8D+8 +(D−4)(D−2)(D−1) +e10,6 +2(2D−7) +D2−5D+4 +e10,7 +−2D2+6D+4 +D2−5D+4 +e10,8 +(D−3)(D2−6D+10) +4(D−4)(D−2)(D−1) +e10,9 +− +(D−3)2 +(D−4)(D−1) +e10,10 +− +(D−3)(D3−8D2+20D−12) +4(D−4)(D−2)(D−1) +e10,11 +D3−8D2+19D−8 +2(D−4)(D−1) +e10,14 +1 +e11,1 +D +2(D−4)(D−2)2(D−1) +e11,2 +− +D(3D−2) +2(D−4)(D−2)2(D−1) +e11,3 +4D +(D−4)(D−2)(D−1) +e11,4 +2(D3−7D2+17D−12) +(D−4)(D−2)2(D−1) +e11,5 +− +2(D2−3D+4) +D3−7D2+14D−8 +e11,6 +− +2(D−6) +D2−5D+4 +e11,7 +− +8 +D2−5D+4 +e11,8 +(D−3)(3D−8) +4(D−4)(D−2)(D−1) +e11,9 +6−2D +D2−5D+4 +e11,10 +− +(D−3)(3D2−12D+8) +4(D−4)(D−2)(D−1) +e11,11 +(D−3)D +(D−4)(D−1) +e11,13 +1 +e12,1 +D2−4D+2 +(D−4)(D−2)2(D−1) +e12,2 +− +(3D−2)(D2−4D+2) +(D−4)(D−2)2(D−1) +– 15 – + +e12,3 +8(D2−4D+2) +(D−4)(D−2)(D−1) +e12,4 +2(D4−10D3+39D2−68D+40) +(D−4)(D−2)2(D−1) +e12,5 +− +2(D3−5D2+8D−8) +(D−4)(D−2)(D−1) +e12,6 +4(2D−7) +D2−5D+4 +e12,7 +−4D2+12D+8 +D2−5D+4 +e12,8 +(D−3)(D2−6D+10) +2(D−4)(D−2)(D−1) +e12,9 +− +2(D−3)2 +(D−4)(D−1) +e12,10 +− +(D−3)(D3−8D2+20D−12) +2(D−4)(D−2)(D−1) +e12,11 +D3−7D2+14D−4 +(D−4)(D−1) +e12,12 +1 +Table 2: +Coefficients of dimension-generic quartic quasi- +topological gravity solutions. Zero coefficients are omitted. +There are also dimension-specific solutions for D = 3 and D = 4. +{ei} +( 1 +6, − 1 +2, 4 +3, − 3 +2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1) +( 1 +6, 0, 4 +3, −3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0) +( 1 +12, − 1 +2, 2 +3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0) +( 7 +6, −4, 16 +3 , −4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0) +( 7 +3, −8, 32 +3 , −8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0) +( 1 +3, 0, 8 +3, −6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0) +(−1, 8, 0, −16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0) +( 1 +12, − 1 +2, 2 +3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0) +( 4 +3, −5, 14 +3 , −2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0) +( 1 +3, −1, 5 +3, −2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0) +( 1 +4, − 3 +2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) +(3, −12, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) +( 7 +12, − 9 +4, 5 +3, − 1 +2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) +( 1 +12, − 1 +4, 2 +3, −1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) +( 5 +6, − 7 +2, 8 +3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) +( 1 +3, −1, 2 +3, −1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) +(0, 1, 0, −4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) +(1, −4, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) +(1, −4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) +( 1 +3, − 3 +2, 7 +6, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) +( 1 +2, − 5 +2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) +(− 1 +6, 1, − 4 +3, − 1 +2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) +Table 3: Coefficients of quartic quasi-topological gravity for +D = 3 +– 16 – + +{ei} +( 35 +48, − 23 +4 , 20 +3 , 5 +2, −4, 3, 0, 1 +4, 0, − 1 +4, 0, −2, 0, 0, − 1 +12, 0, 0, 0, 0, − 1 +16, 0, 0, 0, 0, 0, 1) +( 19 +32, −5, 20 +3 , 13 +4 , − 11 +2 , 2, 0, 3 +16, 0, − 1 +4, 0, − 5 +2, 0, 0, 1 +12, 0, 0, 0, 0, − 5 +32, 0, 0, 0, 0, 1, 0) +( 11 +24, − 7 +2, 4, 1, −2, 2, 0, 1 +8, 0, 0, 0, −1, 0, 0, − 1 +12, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0) +( 41 +24, −14, 16, 7, −10, 8, 0, 3 +4, 0, −1, 0, −6, 0, 0, − 1 +3, 0, 0, 0, 0, − 1 +8, 0, 0, 1, 0, 0, 0) +( 41 +12, −28, 32, 14, −20, 16, 0, 3 +2, 0, −2, 0, −12, 0, 0, − 2 +3, 0, 0, 0, 0, − 1 +4, 0, 1, 0, 0, 0, 0) +( 13 +12, −9, 32 +3 , 6, −8, 4, 0, 1 +2, 0, −1, 0, −4, 0, 0, 0, 0, 0, 0, 0, − 1 +4, 1, 0, 0, 0, 0, 0) +( 1 +3, − 5 +2, 8 +3, 1 +2, −1, 3 +2, 0, 1 +8, 0, 0, 0, − 1 +2, 0, 0, − 1 +8, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0) +( 5 +8, − 21 +4 , 6, 3, −4, 3, 0, 3 +8, 0, − 3 +4, 0, −3, 0, 0, − 1 +4, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0) +( 7 +12, − 19 +4 , 17 +3 , 3, −4, 2, 0, 1 +4, 0, − 3 +4, 0, −2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0) +( 5 +8, − 9 +2, 4, 0, 0, 3, 0, 3 +8, 0, 0, 0, 0, 0, 0, − 1 +2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) +(− 1 +12, 5 +8, − 2 +3, 0, 0, − 1 +2, 0, 0, 0, − 1 +8, 0, − 1 +2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) +( 3 +8, −3, 4, 3 +2, −3, 1, 0, 1 +8, 0, − 1 +4, 0, −1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) +(− 1 +6, 5 +4, − 4 +3, 0, 0, −1, 0, 0, 0, − 1 +4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) +(− 1 +4, 2, −2, 0, 0, −2, 0, − 1 +4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) +(− 1 +12, 3 +4, − 7 +6, − 1 +2, 1, − 1 +2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) +Table 4: Coefficients of quartic quasi-topological gravity for +D = 4 +For the quintic order, we first need to enumerate all the possible Riemann scalars since +the explicit list is never mentioned in the literature. We employ the following inefficient but +straightforward way: first enumerate all possible scalars that can be formed by the contrac- +tion of their indices, taking into the account the symmetries, but not the cyclic identity. +Then for each resulting scalar we replace each of its Riemann tensor factor respectively +with the cyclic identity +Rabcd + Racdb + Radbc = 0 +(A.2) +with each replacement two new terms are obtained. If both terms are contained in the +Riemann scalar list then this term is equivalent to the already known scalars by the cyclic +identity and should be removed. +As mentioned in the text, we get too many solutions at quintic order, so we are not +going to present the full set of them here, which is included in the supplementary material. +Instead, we only present a small portion of the dimension-generic solutions below. +Q(5),1 = −2RR +ef +ab +RabcdR +gh +ce +Rdfgh + RR +ef +ab +RabcdR +gh +cd +Refgh +(A.3a) +Q(5),2 = 2RabRcdR +ef +ac +R +gh +be +Rdfgh − 2RabRcdR e f +a c R +gh +be +Rdfgh ++ R c +a RabR def +b +R +gh +cd +Refgh +(A.3b) +Q(5),3 = +8(3D − 7)R2 +3D2 − 15D + 16R c +a RabRbc − +12(D − 1)R3 +3D2 − 15D + 16RabRab + +4R5 +3D2 − 15D + 16 ++ 12 +� +D2 − 5D + 8 +� +R2 +3D2 − 15D + 16 +RabRcdRacbd + +6(D − 3)R3 +3D2 − 15D + 16RabcdRabcd +− 12(D − 3)(D − 1)R2 +3D2 − 15D + 16 +RabR cde +a +Rbcde +– 17 – + +− 4 +� +D3 − 9D2 + 26D − 22 +� +R2 +3D2 − 15D + 16 +R e f +a c RabcdRbedf ++ R2R +ef +ab +RabcdRcdef +(A.3c) +Q(5),4 = +4 +� +D3 − 5D2 + 8D − 8 +� +D4 − 11D3 + 44D2 − 72D + 36R c +a RabR d +b RcdR +− +4 +� +D4 − 10D3 + 39D2 − 68D + 40 +� +(D − 2) (D4 − 11D3 + 44D2 − 72D + 36)RabRabRcdRcdR +− +16 +� +D2 − 4D + 2 +� +D4 − 11D3 + 44D2 − 72D + 36R c +a RabRbcR2 ++ +2 +� +3D3 − 14D2 + 14D − 4 +� +(D − 2) (D4 − 11D3 + 44D2 − 72D + 36)RabRabR3 +− +2 +� +D2 − 4D + 2 +� +(D − 2) (D4 − 11D3 + 44D2 − 72D + 36)R5 +− +8 +� +2D2 − 11D + 14 +� +D4 − 11D3 + 44D2 − 72D + 36RabRcdR2Racbd +− +D2 − 6D + 10 +D3 − 8D2 + 20D − 12R3RabcdRabcd + +4 +� +D2 − 5D + 6 +� +D3 − 8D2 + 20D − 12RabR2R cde +a +Rbcde ++ +8 +� +D3 − 5D2 + 4D + 4 +� +D4 − 11D3 + 44D2 − 72D + 36R c +a RabRdeRRbdce ++ 2 +� +D4 − 10D3 + 35D2 − 46D + 16 +� +D4 − 11D3 + 44D2 − 72D + 36 +RabRcdRR +ef +ac +Rbdef +− 4 +� +D4 − 9D3 + 28D2 − 32D + 8 +� +D4 − 11D3 + 44D2 − 72D + 36 RabRcdRR e f +a c Rbedf + RabRabRRcdefRcdef +(A.3d) +Q(5),5 = +2 +� +D5 − 14D4 + 67D3 − 142D2 + 132D − 32 +� +(D − 4)(D − 3)(D − 2) (D3 − 8D2 + 20D − 12)R c +a RabR d +b RcdR ++ +2 +� +5D4 − 45D3 + 154D2 − 236D + 128 +� +(D − 4)(D − 3)(D − 2)2 (D3 − 8D2 + 20D − 12)RabRabRcdRcdR +− +D5 − 13D4 + 50D3 − 80D2 + 68D − 8 +(D − 4)(D − 3)(D − 2) (D3 − 8D2 + 20D − 12)R c +a RabRbcR2 +− +−D5 + 19D4 − 100D3 + 222D2 − 232D + 88 +(D − 4)(D − 3)(D − 2)2 (D3 − 8D2 + 20D − 12)RabRabR3 ++ +2 +� +D3 − 5D2 + 7D − 4 +� +(D − 4)(D − 3)(D − 2)2 (D3 − 8D2 + 20D − 12)R5 +− +2 +� +D4 − 13D3 + 56D2 − 106D + 88 +� +(D − 4)(D − 3) (D3 − 8D2 + 20D − 12)RabRcdR2Racbd +− +D2 − 10D + 22 +2(D − 4) (D3 − 8D2 + 20D − 12)R3RabcdRabcd +− +−3D3 + 28D2 − 64D + 12 +2(D − 4) (D3 − 8D2 + 20D − 12)RabR2R cde +a +Rbcde +– 18 – + ++ 2 +� +D5 − 13D4 + 60D3 − 126D2 + 116D − 16 +� +(D − 4)(D − 3) (D3 − 8D2 + 20D − 12) +R c +a RabRdeRRbdce +− −3D4 + 34D3 − 141D2 + 238D − 112 +(D − 4)(D − 3) (D3 − 8D2 + 20D − 12)RabRcdRR +ef +ac +Rbdef +− 2 +� +D4 − 8D3 + 19D2 − 12D + 4 +� +(D − 3) (D3 − 8D2 + 20D − 12) RabRcdRR e f +a c Rbedf +− 3 − D +D − 4R2R e f +a c RabcdRbedf +− D2 − 3D +D − 4 RabRR cde +a +R f g +b d Rcfeg ++ RabRR c d +a b R efg +c +Rdefg +(A.3e) +Q(5),6 = −16(2D9 − 57D8 + 639D7 − 3863D6 + 14059D5 − 31812D4 + 43724D3 +− 32916D2 + 9424D + 1088)/[(D − 4)(D − 3)(D − 2) +� +D3 − 8D2 + 20D − 12 +� +� +D5 − 14D4 + 79D3 − 224D2 + 316D − 170 +� +]R c +a RabR d +b RcdR +− 16(D10 − 23D9 + 262D8 − 1914D7 + 9661D6 − 34319D5 + 85300D4 − 144708D3 ++ 159028D2 − 101552D + 28480)/[(D − 4)(D − 3)(D − 2)2 � +D3 − 8D2 + 20D − 12 +� +� +D5 − 14D4 + 79D3 − 224D2 + 316D − 170 +� +]RabRabRcdRcdR ++ 32(3D11 − 99D10 + 1368D9 − 10752D8 + 54161D7 − 184897D6 + 437712D5 +− 717804D4 + 794228D3 − 556336D2 + 214928D − 32032)/[(D − 4)(D − 3)(D − 2) +� +3D2 − 15D + 16 +� � +D3 − 8D2 + 20D − 12 +� +(D5 − 14D4 + 79D3 − 224D2 ++ 316D − 170)]R c +a RabRbcR2 ++ 32(3D11 − 45D10 + 183D9 + 869D8 − 12856D7 + 66448D6 − 201530D5 ++ 395034D4 − 507818D3 + 413656D2 − 193024D + 39008)/[(D − 4)(D − 3) +(D − 2)2 � +3D2 − 15D + 16 +� � +D3 − 8D2 + 20D − 12 +� +� +D5 − 14D4 + 79D3 − 224D2 + 316D − 170 +� +]RabRabR3 +− 4(17D10 − 356D9 + 3319D8 − 18114D7 + 63896D6 − 151338D5 + 240888D4 +− 247784D3 + 147192D2 − 35360D − 2816)/[(D − 4)(D − 3)(D − 2)2 +� +3D2 − 15D + 16 +� � +D3 − 8D2 + 20D − 12 +� +(D5 − 14D4 + 79D3 − 224D2 ++ 316D − 170)]R5 ++ 32(3D10 − 99D9 + 1361D8 − 10615D7 + 52980D6 − 178926D5 + 417676D4 +− 669720D3 + 708764D2 − 447392D + 127552)/[(D − 4)(D − 3) +� +3D2 − 15D + 16 +� +� +D3 − 8D2 + 20D − 12 +� � +D5 − 14D4 + 79D3 − 224D2 + 316D − 170 +� +]RabRcdR2Racbd +− 4(3D9 − 78D8 + 943D7 − 6838D6 + 32308D5 − 102114D4 + 214432D3 +− 286992D2 + 221088D − 74336)/[(D − 4) +� +3D2 − 15D + 16 +� +� +D3 − 8D2 + 20D − 12 +� � +D5 − 14D4 + 79D3 − 224D2 + 316D − 170 +� +]R3RabcdRabcd +− 16(3D9 − 84D8 + 948D7 − 5760D6 + 20693D5 − 44340D4 + 52004D3 +− 21480D2 − 14272D + 13008)/[(D − 4) +� +3D2 − 15D + 16 +� � +D3 − 8D2 + 20D − 12 +� +– 19 – + +� +D5 − 14D4 + 79D3 − 224D2 + 316D − 170 +� +]RabR2R cde +a +Rbcde +− 64(D9 − 24D8 + 245D7 − 1405D6 + 4982D5 − 11217D4 + 15640D3 +− 12198D2 + 3768D + 352)/[D − 4)(D − 3) +� +D3 − 8D2 + 20D − 12 +� +(D5 +− 14D4 + 79D3 − 224D2 + 316D − 170)]R c +a RabRdeRRbdce ++ 16(D9 − 28D8 + 336D7 − 2286D6 + 9737D5 − 26860D4 + 47638D3 +− 51642D2 + 30272D − 7024)/[(D − 4)(D − 3) +� +D3 − 8D2 + 20D − 12 +� +� +D5 − 14D4 + 79D3 − 224D2 + 316D − 170 +� +]RabRcdRR +ef +ac +Rbdef ++ 32 +� +D8 − 18D7 + 134D6 − 532D5 + 1203D4 − 1534D3 + 1082D2 − 544D + 256 +� +(D − 3) (D3 − 8D2 + 20D − 12) (D5 − 14D4 + 79D3 − 224D2 + 316D − 170) +RabRcdRR e f +a c Rbedf +− 32 +� +2D7 − 35D6 + 263D5 − 1098D4 + 2743D3 − 4087D2 + 3352D − 1164 +� +(D − 4) (3D2 − 15D + 16) (D5 − 14D4 + 79D3 − 224D2 + 316D − 170) +R2R e f +a c RabcdRbedf ++ 32 +� +D6 − 14D5 + 82D4 − 254D3 + 433D2 − 380D + 132 +� +(D − 4) (D5 − 14D4 + 79D3 − 224D2 + 316D − 170) +RabRR cde +a +R f g +b d Rcfeg +− 2 +� +D5 − 10D4 + 39D3 − 74D2 + 68D − 24 +� +D5 − 14D4 + 79D3 − 224D2 + 316D − 170 RR +ef +ab +RabcdR +gh +ce +Rdfgh ++ RRabcdRabcdRefghRefgh +(A.3f) +B +Results of holographic shear viscosity +In this section we present the values of the coefficients a, b in (4.12) for dimension-generic +quartic and quintic quasi-topological terms. The coefficients for quartic case are given in +table 5 below. +a1 +− +(D−4)(2D8−29D7+145D6−245D5−101D4+226D3+1362D2−1984D+304) +16(D−2)(D−1)D(D3−9D2+26D−22) +b1 +(D−4)(2D7−33D6+188D5−375D4−224D3+1722D2−1744D+304) +16(D−2)(D−1)D(D3−9D2+26D−22) +a2 +− +(D−4)(D8−15D7+98D6−344D5+573D4+9D3−1376D2+1366D−152) +8(D−2)(D−1)D(D3−9D2+26D−22) +b2 +(D−4)(D7−15D6+108D5−481D4+1303D3−1930D2+1246D−152) +8(D−2)(D−1)D(D3−9D2+26D−22) +a3 +− +(D−4)(D8−13D7+45D6+51D5−466D4+178D3+1828D2−2248D+304) +16(D−2)(D−1)D(D3−9D2+26D−22) +b3 +(D−4)(D7−16D6+71D5+38D4−1002D3+2452D2−2008D+304) +16(D−2)(D−1)D(D3−9D2+26D−22) +a4 +− +(D−4)(D5−8D4+17D3+3D2−37D+4) +2(D−2)(D−1)D +b4 +(D−4)(D4−10D3+32D2−37D+4) +2(D−2)(D−1)D +a5 +− +(D−4)(D5−8D4+17D3+3D2−37D+4) +(D−2)(D−1)D +b5 +(D−4)(D4−10D3+32D2−37D+4) +(D−2)(D−1)D +a6 +− (D−4)2(D−3)(D+1) +4(D−2) +b6 +(D−4)2(D−3) +4(D−2) +– 20 – + +a7 +(D−4)(D−2)2(4D3−9D2−55D+8) +32D(D3−9D2+26D−22) +b7 +− +(D−4)(D−2)2(D3+3D2−40D+8) +32D(D3−9D2+26D−22) +a8 +(D−4)(D−2)2 +16D +b8 +− (D−4)(D−2)2 +16D +a9 +3(D−5)(D−4)(D−2)2(D2+5D−2) +32(D3−9D2+26D−22) +b9 +− 3(D−5)(D−4)(D−2)2(2D−1) +16(D3−9D2+26D−22) +Table 5: Coefficients a, b of the shear viscosity contribution +from quartic quasi-topological terms. Each entry corresponds +to the solution with the same index in table 2. Entries with +zero coefficients are omitted. +For quintic quasi-topological terms, we only present the coefficients of the terms pre- +sented in the previous section in (A.3). +a3 +3(D−5)(D−4)(D−2)2D(D2+8D−4) +16(3D2−15D+16) +b3 +− 3(D−5)(D−4)(D−2)2D(7D−4) +16(3D2−15D+16) +a5 +1 +64(D − 3)(D − 2)2D +b5 +− 1 +64(D − 3)(D − 2)2D +a6 +− +(D−3)(D−2)2(6D8−81D7+380D6−508D5−1660D4+7593D3−12210D2+8512D−1792) +4(3D2−15D+16)(D5−14D4+79D3−224D2+316D−170) +b6 +(D−3)(D−2)2(21D7−336D6+2194D5−7582D4+15001D3−16834D2+9472D−1792) +4(3D2−15D+16)(D5−14D4+79D3−224D2+316D−170) +Table 6: Coefficients a, b of the shear viscosity contribution +from quintic quasi-topological terms. Each entry corresponds +to the solution with the same indices in (A.3). Entries with +vanishing coefficients are omitted. +Acknowledgments +I thank Yi Ling and Hong Lü for helpful discussions, I am also grateful to Ying Chen for +supporting my own study projects. +References +[1] D. J. Gross and J. H. Sloan, The Quartic Effective Action for the Heterotic String, Nucl. +Phys. B 291 (1987) 41–89. +[2] M. T. Grisaru and D. Zanon, σ Model Superstring Corrections to the Einstein-hilbert Action, +Phys. Lett. B 177 (1986) 347–351. +[3] P. Kovtun, D. T. Son, and A. O. Starinets, Viscosity in strongly interacting quantum field +theories from black hole physics, Phys. Rev. Lett. 94 (2005) 111601, [hep-th/0405231]. +– 21 – + +[4] A. Buchel, J. Escobedo, R. C. Myers, M. F. Paulos, A. Sinha, and M. Smolkin, Holographic +GB gravity in arbitrary dimensions, JHEP 03 (2010) 111, [arXiv:0911.4257]. +[5] R. C. Myers, M. F. Paulos, and A. Sinha, Holographic studies of quasi-topological gravity, +JHEP 08 (2010) 035, [arXiv:1004.2055]. +[6] X. O. Camanho and J. D. Edelstein, Causality constraints in AdS/CFT from conformal +collider physics and Gauss-Bonnet gravity, JHEP 04 (2010) 007, [arXiv:0911.3160]. +[7] X. O. Camanho and J. D. Edelstein, Causality in AdS/CFT and Lovelock theory, JHEP 06 +(2010) 099, [arXiv:0912.1944]. +[8] P. Bueno, P. A. Cano, V. S. Min, and M. R. Visser, Aspects of general higher-order gravities, +Phys. Rev. D 95 (2017), no. 4 044010, [arXiv:1610.08519]. +[9] D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498–501. +[10] D. Lovelock, Divergence-free tensorial concomitants, aequationes mathematicae 4 (Feb., +1970) 127–138. +[11] R. C. Myers and B. Robinson, Black Holes in Quasi-topological Gravity, JHEP 08 (2010) +067, [arXiv:1003.5357]. +[12] R. C. Myers, M. F. Paulos, and A. Sinha, Holographic studies of quasi-topological gravity, +JHEP 08 (2010) 035, [arXiv:1004.2055]. +[13] J. Oliva and S. Ray, A new cubic theory of gravity in five dimensions: Black hole, Birkhoff’s +theorem and C-function, Class. Quant. Grav. 27 (2010) 225002, [arXiv:1003.4773]. +[14] M. H. Dehghani, A. Bazrafshan, R. B. Mann, M. R. Mehdizadeh, M. Ghanaatian, and M. H. +Vahidinia, Black Holes in Quartic Quasitopological Gravity, Phys. Rev. D 85 (2012) 104009, +[arXiv:1109.4708]. +[15] J. Ahmed, R. A. Hennigar, R. B. Mann, and M. Mir, Quintessential Quartic +Quasi-topological Quartet, JHEP 05 (2017) 134, [arXiv:1703.11007]. +[16] A. Cisterna, L. Guajardo, M. Hassaine, and J. Oliva, Quintic quasi-topological gravity, JHEP +04 (2017) 066, [arXiv:1702.04676]. +[17] R. A. Hennigar, D. Kubizňák, and R. B. Mann, Generalized quasitopological gravity, Phys. +Rev. D 95 (2017), no. 10 104042, [arXiv:1703.01631]. +[18] P. Bueno, P. A. Cano, and R. A. Hennigar, (Generalized) quasi-topological gravities at all +orders, Class. Quant. Grav. 37 (2020), no. 1 015002, [arXiv:1909.07983]. +[19] P. Bueno and P. A. Cano, Einsteinian cubic gravity, Phys. Rev. D 94 (2016), no. 10 104005, +[arXiv:1607.06463]. +[20] R. A. Hennigar and R. B. Mann, Black holes in Einsteinian cubic gravity, Phys. Rev. D 95 +(2017), no. 6 064055, [arXiv:1610.06675]. +[21] P. Bueno and P. A. Cano, Four-dimensional black holes in Einsteinian cubic gravity, Phys. +Rev. D 94 (2016), no. 12 124051, [arXiv:1610.08019]. +[22] P. Bueno, P. A. Cano, and A. Ruipérez, Holographic studies of Einsteinian cubic gravity, +JHEP 03 (2018) 150, [arXiv:1802.00018]. +[23] M. Mir, R. A. Hennigar, J. Ahmed, and R. B. Mann, Black hole chemistry and holography in +generalized quasi-topological gravity, JHEP 08 (2019) 068, [arXiv:1902.02005]. +– 22 – + +[24] M. Mir and R. B. Mann, On generalized quasi-topological cubic-quartic gravity: +thermodynamics and holography, JHEP 07 (2019) 012, [arXiv:1902.10906]. +[25] P. A. Cano, A. J. Murcia, A. Rivadulla Sánchez, and X. Zhang, Higher-derivative holography +with a chemical potential, JHEP 07 (2022) 010, [arXiv:2202.10473]. +[26] P. Bueno, P. A. Cano, J. Moreno, and A. Murcia, All higher-curvature gravities as +Generalized quasi-topological gravities, JHEP 11 (2019) 062, [arXiv:1906.00987]. +[27] P. Bueno, P. A. Cano, R. A. Hennigar, M. Lu, and J. Moreno, Generalized quasi-topological +gravities: the whole shebang, Class. Quant. Grav. 40 (2023), no. 1 015004, +[arXiv:2203.05589]. +[28] Y.-Z. Li, H.-S. Liu, and H. Lu, Quasi-Topological Ricci Polynomial Gravities, JHEP 02 +(2018) 166, [arXiv:1708.07198]. +[29] S. A. Hartnoll, C. P. Herzog, and G. T. Horowitz, Building a Holographic Superconductor, +Phys. Rev. Lett. 101 (2008) 031601, [arXiv:0803.3295]. +[30] S. A. Hartnoll, C. P. Herzog, and G. T. Horowitz, Holographic Superconductors, JHEP 12 +(2008) 015, [arXiv:0810.1563]. +[31] C. Martinez, R. Troncoso, and J. Zanelli, Exact black hole solution with a minimally coupled +scalar field, Phys. Rev. D 70 (2004) 084035, [hep-th/0406111]. +[32] H. Dykaar, R. A. Hennigar, and R. B. Mann, Hairy black holes in cubic quasi-topological +gravity, JHEP 05 (2017) 045, [arXiv:1703.01633]. +[33] N. Caceres, J. Figueroa, J. Oliva, M. Oyarzo, and R. Stuardo, Quadratic gravity and +conformally coupled scalar fields, JHEP 04 (2020) 157, [arXiv:2001.01478]. +[34] T. Padmanabhan and D. Kothawala, Lanczos-Lovelock models of gravity, Phys. Rept. 531 +(2013) 115–171, [arXiv:1302.2151]. +[35] S. A. Fulling, R. C. King, B. G. Wybourne, and C. J. Cummins, Normal forms for tensor +polynomials. i. the riemann tensor, Classical and Quantum Gravity 9 (may, 1992) 1151. +[36] R. M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993), no. 8 +R3427–R3431, [gr-qc/9307038]. +[37] N. Deruelle, M. Sasaki, Y. Sendouda, and D. Yamauchi, Hamiltonian formulation of +f(Riemann) theories of gravity, Prog. Theor. Phys. 123 (2010) 169–185, [arXiv:0908.0679]. +[38] M. J. Duff, Observations on Conformal Anomalies, Nucl. Phys. B 125 (1977) 334–348. +[39] C. Imbimbo, A. Schwimmer, S. Theisen, and S. Yankielowicz, Diffeomorphisms and +holographic anomalies, Class. Quant. Grav. 17 (2000) 1129–1138, [hep-th/9910267]. +[40] Y.-Z. Li, H. Lu, and J.-B. Wu, Causality and a-theorem Constraints on Ricci Polynomial and +Riemann Cubic Gravities, Phys. Rev. D 97 (2018), no. 2 024023, [arXiv:1711.03650]. +[41] L. Li, On Thermodynamics of AdS Black Holes with Scalar Hair, Phys. Lett. B 815 (2021) +136123, [arXiv:2008.05597]. +[42] M. F. Paulos, Transport coefficients, membrane couplings and universality at extremality, +JHEP 02 (2010) 067, [arXiv:0910.4602]. +– 23 – + diff --git a/tdAyT4oBgHgl3EQfZ_fS/content/tmp_files/load_file.txt b/tdAyT4oBgHgl3EQfZ_fS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..54f16e201932bd6669f24ed2f217143352f7f135 --- /dev/null +++ b/tdAyT4oBgHgl3EQfZ_fS/content/tmp_files/load_file.txt @@ -0,0 +1,2098 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf,len=2097 +page_content='Prepared for submission to JHEP Quasi-topological Gravities on General Spherically Symmetric Metric Feiyu Chen,a,b,1 aInstitute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' China bSchool of Physics, University of Chinese Academy of Sciences, Beijing 100049, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' China E-mail: chenfy@ihep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='cn Abstract: In this work we study a more restricted class of quasi-topological gravity the- ories where the higher curvature terms have no contribution to the equation of motion on general static spherically symmetric metric where gttgrr ̸= constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' We construct such theories up to quintic order in Riemann tensor and observe an important property of these theories: the higher order term in the Lagrangian vanishes identically when evaluated on the most general non-stationary spherically symmetric metric ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' This not only signals the higher terms could only have non-trivial effects when considering perturbations, but also makes the theories quasi-topological on a much wider range of metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' As an example of the holographic effects of such theories, we consider a general Einstein-scalar theory and calculate it’s holographic shear viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Keywords: Higher Curvature Gravity, Black Holes, AdS-CFT Correspondence 1Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='00235v1 [hep-th] 31 Dec 2022 Contents 1 Introduction 1 2 Construction of the theory 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1 Cubic order 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 Higher orders 6 3 Properties and discussions 6 4 Holographic shear viscosity 8 5 Conclusions 10 A Quartic and quintic quasi-topological gravities 11 B Results of holographic shear viscosity 20 1 Introduction Einstein gravity theory extended with higher order curvature terms plays a relevant role among modified gravity theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' It’s predicted by string theory that Einstein gravity should be corrected by an infinite series of higher curvature terms [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Higher curvature terms have also attracted attentions in holography, they may introduce various new phenomena on the boundary theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' For example, it’s shown that including higher curvature terms can lead to the violation of the Kovtun–Son–Starinets (KSS) shear-viscosity-to-entropy bound η/s ⩾ 1/4π [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Holography has also been used to determine the physical bounds of higher curvature couplings by demanding the consistency of the dual CFT [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' However, gravity theories with higher curvature terms are generally hard to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' One common way to study a gravity theory is through its black hole solution, however for higher curvature gravities the equations of motion are usually fourth-order differential equations, making analytical solutions hard to come by.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' It’s thus of interest to construct higher curvature theories that admit analytical black hole solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' On the other hand, the linearized equations of motion of these theories around maximally symmetric spacetimes typically contain fourth derivatives too, so besides the usual massless spin-2 graviton mode, two extra massive modes might appear, the scalar mode and the ghost-like spin-2 mode [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The existence of ghost-like mode signals instability of the AdS vacua and causes unitarity breaking of the dual CFT, it is thus mandatory to decouple (set the mass to infinity) the ghost-like mode when studying holography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The well-known example where these extra modes are absent is Lovelock gravity [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Lovelock term of order k vanishes identically when D ⩽ 2k − 1 and becomes a total derivative that does not contribute to the equations – 1 – of motion for D = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A total derivative further reduces to a surface term in the action and only contribute topological characteristics, so higher curvature term of this kind is also called topological term, no physical effects could emerge when introducing such higher curvature terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Quasi-topological gravity (QTG), on the other hand, is a more intriguing theory in that the equations of motion are drastically simplified when evaluated on some special metric ansatz, and also gives non-trivial contribution at perturbation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' In the broader literature, such theory is defined by that it admits Schwarzschild-like solutions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=', the special spherically symmetric (SSS) metric 1 ds2 = −f(r)dt2 + 1 f(r)dr2 + r2dΣ2 D−2,k (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1) and the equation of f(r) is algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Cubic quasi-topological gravity was first constructed in [11], it’s holographic properties was later studied in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Higher order ones also exist and they have been studied extensively [13–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Besides quasi-topological gravities, there’s another closely related class of theory worth mentioning, known as the generalized quasi- topological gravity (GQTG), satisfying [17, 18] δSf δf = 0, or Et t = Er r (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2) where Sf denotes the action evaluated on the SSS metric, and Eab = 1/ � |g|δS/δgab is the equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' It can be shown that quasi-topological gravity satisfies this condition and thus is a subclass of GQTG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Features of GQTG have been studied comprehensively at present [18–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' In particular, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2) implies [26] the equation of f(r) is at most second order, the existence of Schwarzschild-like solutions, and most importantly, the decoupling of the extra massive modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' We are more interested in quasi-topological gravities whose higher curvature terms do not contribute to the equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' These theories obviously satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2) and thus ghost-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' They are first considered in [28] for Ricci polynomials, where quasi-topological terms were constructed up to tenth order in Ricci tensor both on the SSS metric and general spherically symmetric (GSS) metric ds2 = −h(r)dt2 + 1 f(r)dr2 + r2dΣ2 D−2,k (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3) The advantage of this definition of quasi-topological gravity is that black hole solutions in the original gravity theory in the form of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1) or (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3) simply continue to be solutions when the corresponding quasi-topological terms are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' This could be relevant when mat- ter are included, where even with the equation of f(r) being algebraic, the inclusion of mat- ter terms could make the system unintegratable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' In this work we are specifically interested in quasi-topological gravities on GSS metric (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3), but not just limited to Ricci gravities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Such metric is the most general ansatz for spacetimes with spherical/planar/hyperbolic 1Unless otherwise noted, when we say spherically symmetry, we actually mean spherically, planar, or hyperbolic symmetry, corresponding to the curvature of dΣ2 being positive, zero, or negative, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' – 2 – symmetry, thus could include a wider range of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' An important class of GSS metric is black hole with scalar hair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Hairy solutions typically have a rich phase structure and in holography they may be used to describe superconductors [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' It was shown that even Einstein gravity with a minimally coupled self-interacting scalar field could result in a hairy solution [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' It’s thus interesting to investigate the effects of including higher curvature terms in these solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' There has been works on the hairy solutions with conformally coupled scalar in higher curvature gravities [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' In this work we focus on quasi-topological gravities on GSS metric and construct such quasi-topological terms up to quintic order in Riemann tensor, including dimension-generic ones and dimension-specific ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' For the latter only D ⩾ 3 is considered since at D = 2 the Riemann tensor has only one non-zero component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' We also find it possible to construct dimension-independent combinations at quartic and quintic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' We then notice that these theories satisfy a much stronger condition: the quasi-topological terms vanish identi- cally when evaluated on the non-stationary spherically symmetric metric!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' That is, metric of the following form ds2 = −h(t, r)dt2 + 2b(t, r)dtdr + 1 f(t, r)dr2 + r2dΣ2 D−2,k (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4) On the one hand, this further restricts the effects the quasi-topological term could possibly have, such as no thermodynamics contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' This possibly simplifies the problem since introducing the quasi-topological terms won’t lead to any new phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Thus one may only seek for non-trivial effects of quasi-topological terms by considering per- turbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' On the other hand, this indicates that these quasi-topological terms are also quasi-topological on a much wider range kinds of metrics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=', Friedmann-Roberson-Walker metric, it’s thus possible to study the effects of quasi-topological terms on cosmic pertur- bations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' In section 2 we construct explicitly the quasi- topological gravity theories, up to quintic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' In section 3 we discuss the basic properties of the obtained theories, mainly the implications of the vanishing of the quasi-topological terms evaluated on the metric (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' As an example to study the physical effects of quasi- topological terms, in section 4 we consider a general Einstein-scalar theory extended with quasi-topological terms and calculate its holographic shear viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 2 Construction of the theory For a general Lagrangian constructed from metric and Riemann tensor L(gab, Rabcd) the equation of motion can be written as [34] Eab[L] = 1 � |g| δS δgab = P a cde Rbcde − 1 2gabL + ∇c∇dP acdb, P abcd = ∂L ∂Rabcd (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1) where S = � dDx � |g|L is the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' In our definition of quasi-topological gravities, a quasi-topological term Q in the Lagrangian satisfies that it does not contribute to the equation of motion when evaluated on (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3), namely Ett[Q]h,f = Err[Q]h,f = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2) – 3 – which is equivalent to δ δh � dDx � |g|Q ��� h,f = δ δf � dDx � |g|Q ��� h,f = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3) where .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' |h,f denotes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' evaluated on the metric (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' At a given order, we first write down the most general Riemann polynomial of that order with undetermined coefficients and substitute the ansatz (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3) into it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The non-zero Riemann tensor components of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3) are Rˆtˆrˆtˆr = f(r)h′(r)2 − h(r) [f′(r)h′(r) + 2f(r)h′′(r)] 4h(r)2 Rˆtiˆtj = −δi j f(r)h′(r) 2rh(r) Rˆriˆrj = −δi j f′(r) 2r Rijkl = (δi kδj l − δi lδj k)k − f(r) r2 where ijkl are indices of the (D − 2) dimension subspace, and equivalent components are not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' By varying and integrating by parts with respect to h(r) and f(r) we get two algebraic equations containing the undetermined coefficients, we then further convert them into a linear system about the undetermined coefficients by regarding them as polynomials in r, h(r), f(r) and their derivatives and requiring all coefficients vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The solution space is given by the null space of the resulting linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' For dimension-generic solutions, we take null space directly, and for dimension-specific ones we substitute the dimension first and then take null space, since there may be more linear independent solutions at lower dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1 Cubic order There are 8 Riemann scalars at cubic order and the most general cubic Riemann polynomial is given by their linear combination Q(3) = e1R b d a c R e f b d R a b e f + e2R cd ab R ef cd R ab ef + e3RabcdRabc eRde + e4RabcdRabcdR + e5RabcdRacRbd + e6Rb aRc bRa c + e7Rb aRa bR + e8R3 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4) We found only one dimension-generic solution in this case, the coefficients ei are given by e1 = 22 − 26D + 9D2 − D3, e2 = 3D2 4 − 15D 4 + 4, e3 = −3(D − 3)(D − 1) e4 = 3(D − 3) 2 , e5 = 3 � D2 − 5D + 8 � , e6 = 6D − 14, e7 = 3 − 3D, e8 = 1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5) As mentioned earlier, quasi-topological gravity is a subclass of generalized quasi-topological gravity, so the solution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5) must be a special case of cubic generalized quasi-topological gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' In fact, by setting c1 = 22 − 26D + 9D2 − D3, c2 = 3D2 4 − 15D 4 + 4, c3 = −3(D − 3)(D − 1) – 4 – D {ei} 3 (−8, 5, −12, 0, 0, 0, 0, 1) (−2, 3 2, −4, 0, 0, 0, 1, 0) (0, 1 2, − 3 2, 0, 0, 1, 0, 0) (−1, 1 4, −1, 0, 1, 0, 0, 0) (0, 1, −4, 1, 0, 0, 0, 0) 4 (16, −8, 36, −3, −24, −8, 0, 1) (2, −1, 5, − 1 2, −4, −2, 1, 0) 5 (−8, 4, −24, 3, 24, 16, −12, 1) 6 (64, −14, 60, −3, −48, −8, 0, 1) (6, − 3 2, 7, − 1 2, −6, −2, 1, 0) Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Dimension-specific solutions of cubic quasi-topological gravity in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='6) of [17] we get our solution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' As there’s only one solution, it’s not possible to construct dimension independent solution at cubic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Now we turn to dimension-specific solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' First we found that for D > 6 the number of independent solutions is always one, meaning they are covered by the dimension-generic solution, so we only need to consider 3 ⩽ D ⩽ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' We get two linear independent solutions at D = 4 and D = 6 respectively, five solutions at D = 3 and one solution again at D = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The solutions are given in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' However, not all solutions are non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' It could happen that some solutions vanish identically on any metric, just like Lovelock terms in D < 2n, this is possible for dimension-specific cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Firstly we note that all the solutions in table 1 have included the cubic Lovelock term, especially the five dimensional solution which simply coincides with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' So we are left with 4 solution for D = 3, one solution for D = 4 and D = 6 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Besides Lovelock terms themselves, another kind of combinations that vanish in lower dimensions may be constructed from their equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' For example, in D ⩽ 4, the 4D Lovelock, or Gauss-Bonnet term X (4) = R2 − 4RabRab + RabcdRabcd (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='6) is topological, so its equation of motion contribution should vanish Eab[X (4)] = 1 � |g| δ δgab � dDx � |g|X (4) = −4RacRb c + 2RabR − 4RcdRa b c d + 2RacdeRb cde + 1 2gabX (4) = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='7) we can thus construct another vanishing Riemann polynomial Eab[X (4)]Rab = −4Ra bRb cRc a + 4RabRabR − 1 2R3 − 4RabRcdRacbd − 1 2RabcdRabcdR + 2RabR cde a Rbcde = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8) – 5 – It can be shown that the space spanned by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8) and 4D Lovelock term is isomorphic to the D = 4 solution in table 1, thus both solutions are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' In three dimensions, the Gauss-Bonnet term vanishes, there are three more vanishing Riemann polynomials 2 RX (4), ∂X (4) ∂Rabcd RacRbd, ∂X (4) ∂Rabcd RcdefR ab ef (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='9) Again, the space spanned by these three terms, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8) and 4D Lovelock, is isomorphic to the D = 3 solution in table 1, thus only one (linear combination) of the D = 6 solution in table 1 is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' This 6D solution is also covered by the dimension-generic solution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' We finally conclude that cubic quasi-topological term is completely given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5) and it’s only non-trivial for D ⩾ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 Higher orders The method of solving higher order quasi-topological terms is exactly the same as we used in the cubic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The only difficulty is enumerating all possible Riemann scalars, as the number of independent Riemann scalars grows rapidly in higher orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' We get 26 scalars at quartic order [35] and 85 scalars at quintic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' For quartic case we get 12 linear independent dimension-generic solutions, and at D = 3, D = 4 and D = 8 we get 22, 15 and 13 solutions respectively, all other dimensions have the same number of solutions as dimension-generic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Again, the extra solution at D = 8 is the 8D Lovelock term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The explicit list of solutions is lengthy and given in appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Remarkably, we also found 3 dimension-independent solutions Q(4),∗,1 = Rabcd � R e f a c R g h b e Rdgfh − R ef ab R g h c e Rdgfh − 1 4R e abc R fgh d Refgh � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='10) Q(4),∗,2 = R ef ab Rabcd � R gh ce Rdfgh − 1 2R gh cd Refgh � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='11) Q(4),∗,3 = RabRcd � R e f a c Rbedf − 1 2R ef ac Rbdef � − 1 2RabRc aR def b Rcdef (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='12) The situation is similar for quintic case, we get 61 dimension-generic solutions and 80, 67, 62 dimension-specific solutions at D = 3, D = 4, D = 10 respectively, for dimension- independent case we get 29 solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' However as the solutions of the quintic case are too lengthy, we only present some representative solutions in the appendix and the full solution set can be found in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 3 Properties and discussions Having constructed the desired theories we now move on to their physical effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The first property we noticed is the quasi-topological term vanishes when evaluated on the metric (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3) Q(n) h,f = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1) 2L = 0 implies ∂L/∂Rabcd = 0 if the identity ∂L/∂gmn = (∂L/∂Rabcd)(∂Rabcd/∂gmn) gives no less equations than the independent components of ∂L/∂Rabcd, which is true for D ⩽ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' – 6 – The free energy can be obtained by evaluating the Euclidean action with compactified time direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Since our metric is static, the Euclidean action only differs from the Minkowski action by a minus sign, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1) implies the vanishing of the free energy contribution from quasi-topological term, which then further implies the entropy and thermodynamic energy contribution should also vanish, thus quasi-topological term completely has no thermody- namics effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' To verify the consistency of the above conclusion we need to evaluate the Wald entropy and thermodynamic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The Wald entropy is given by [36] SWald = −2π � P abcdϵabϵcd dΣ, P abcd = ∂L ∂Rabcd (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2) where the integration is taken at the horizon, ϵab is the binormal to the horizon, dΣ is the volume form of the horizon surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Using the method similar to [18] one can show that Q(n) h,f = 0 implies ∂Q(n) ∂Rabcd ����� h,f = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3) thus the Q(n) contribution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2) must also vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' It remains to calculate the energy, which can be done holographically by calculating the tt component of the boundary stress tensor T tt = (2/ � |h|)δS/δhtt where hab is the boundary metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The surface term and counter term also need to be taken into account, but since Q(n) vanishes, no new diverges appear, the counter term contribution is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The surface term can be constructed by introducing an auxiliary field Φab = Pacbdncnd [37] S∂ = 1 8π � ∂M dD−1x � |h|ΦabKab (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4) where na is the normal vector of the boundary, Kab = ∇anb is the exterior curvature and hab = gab − nanb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Note that when varying this term, Φab should be kept fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' So we immediately see from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3) that the surface term contribution to T tt should vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Furthermore, because Q(n) vanishes on (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3), it’s invariant under the variation h(r) → h(r) + δh(r), so we have δ � |g|Q(n)/δhtt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' We thus conclude the energy obtained via holography also vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' As mentioned earlier, we actually found a much stronger conclusion than (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1), that is Q(n) also vanishes when evaluated on the general non-stationary spherically symmetric metric (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' It’s straightforward to evaluate a given quasi-topological terms on (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4) and check that it vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' In practice, the check was done using an equivalent metric ds2 = −h(t, r)dt2 + 2b(t, r)dtdr + � 1 f(t, r) − b2(t, r) h(t, r) � dr2 + r2dΣ2 D−2,k (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5) The advantage of it is the components of the inverse metric contain no fraction, reducing the computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The check was done for all cubic and quartic quasi-topological terms, but at quintic order we encountered extreme computation difficulties so we ended up only checked the solutions listed in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' It’s then strongly suggested that the condition for quasi-topological terms (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3) implies that they vanish when evaluated on (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' – 7 – The vanishing of Q(n) on (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4) makes it quasi-topological on a much wider range of metrics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' the FRW metric ds2 = −dt2 + a2(t) � dr2 1 − kr2 + r2dΩ2 � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='6) which can be put into the form of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4) by redefining r → r/a(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' It also implies Q(n) has no a-charge contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' In 2n dimensional CFTs, the central a, c charges appears as coefficients in the trace of the stress tensor [38] � T µ µ � ∼ −aX (2n) + � i ciI(2n) i (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='7) where X (2n) is the 2n dimensional Lovelock term, I(2n) i are conformal invariants in 2n dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Generally the central charges can be calculated holographically by evaluating the action on the the FG expansion metric and identifying the ρ−1 term as the trace anomaly [39], but to solely extract the a-charge one may use a specific metric with conformally flat boundary, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=', an S2n [40] ds2 = L2 4ρ2 dρ2 + f(ρ) ρ � dr2 1 − r2 + r2dΩ2 D−2 � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8) again, by redefining r → r � ρ/f(ρ) this metric can be put into the form of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4), thus Q(n) also vanish on (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8), it doesn’t contribute to the a-charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The vanishing of Q(n) on the more general metric (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4) largely reduces the possible effects it could have when introduced to some gravity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' To seek for non-trivial effects one may only consider the perturbations of it around the metric (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3), which in holography includes shear viscosity and heat current, corresponding to perturbations hx1x2 and htx1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' In the next section we’ll consider the holographic shear viscosity as an example to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 4 Holographic shear viscosity We consider a general Einstein-scalar theory with the Lagrangian LES = 1 16π � R − 1 2∇aφ∇aφ − V (φ) � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1) we are interested in the non-extremal 3 asymptotic AdS black hole solutions of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1), so we consider the following planar black hole ansatz ds2 = −f(r)e−η(r)dt2 + 1 f(r)dr2 + U(r)d⃗x2 D−2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2) 3The extremal limit T → 0 and hydrodynamic limit ω → 0 generally don’t commute, which will compli- cate the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' – 8 – Note that there’s one gauge freedom in the three functions f(r), η(r), U(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Near the boundary we have f(r → ∞) = U(r → ∞) = r2/L2, η(r → ∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The horizon is at r = rh, satisfies f(rh) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The temperature and entropy density are respectively given by T = 1 4πf′(rh)e−η(rh)/2, s = 1 4U D/2−1(rh) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3) Assuming φ only depends on r, the equation of motion gives (D − 2)fU′φ′ + U � 2f′φ′ − fη′φ′ + 2fφ′′ − 2V ′� = 0 � D2 − 7D + 10 � fU′2 + 2(D − 2)U � f′U ′ + 2fU′′� + 2U 2 � fφ′2 + 2V � = 0 2U � (D − 2)U ′′(r) + U(r)φ′2� + (D − 2)UU ′η′ − (D − 2)U ′2 = 0 U � −(D − 4)f′U ′ + f � (D − 3)η′U ′ + 2U ′′�� + (D − 4)fU′2 +U 2 � −2f′′ + 3f′η′ − f � η′2 − 2η′′�� = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4) The last equation can be integrated to give a radially conserved quantity, as in [41] Q = e−η/2U D/2−2 �� f′ − fη′� U − fU′� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5) Evaluating it at the horizon gives Q = e−η(rh)/2U D/2−1(rh)f′(rh) = 16πTs (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='6) To calculate the holographic shear viscosity we employ the pole method as proposed in [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Define a new radial coordinate z by r = rh/(1 − z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2) becomes ds2 = r2 h (1 − z)4 1 f( rh 1−z)dz2 − f � rh 1 − z � exp � −η � rh 1 − z �� dt2 + U � rh 1 − z � d⃗x2 D−2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='7) Now add perturbation to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='7) by shifting the basis dx1 → dx1 + εe−iωtdx2, substitute the resulting metric into the Lagrangian and expand it to quadratic order in ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Note that the perturbation should be kept second order in the metric, and since the perturbation only involves spatial components, the matter sector of the Lagrangian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1) has no contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The shear viscosity can be calculated from the residue of the Lagrangian at z = 0 η = −8πT lim ω,ε→0 Resz=0 � |g|L ε2ω2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8) Note that the above expression for the shear viscosity is linear in L so we can compute the contribution to the shear viscosity of different terms in the Lagrangian separately, but keep in mind only the summed result has physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' For the Einstein-scalar theory in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1), the contribution is given by η(0) = 1 16πU D/2−1(rh) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='9) which results in the standard shear-viscosity-to-entropy ratio η(0)/s = 1/4π, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=', the exis- tence of the scalar hair have no effect on the shear viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' – 9 – Next we introduce quasi-topological terms to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1) by defining the new Lagrangian as L′ = LES + λ/16πQ(n), the equations of motion aren’t altered by Q(n) and it’s straight- forward to evaluate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='7) on them and then apply (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8) to obtain the shear viscosity, the results are expressed in f(r), η(r), U(r) and their derivatives at r = rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Interestingly, by making use of the radially conserved quantity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5) we are only left with η(rh), η′(rh), U(rh) and its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The contribution from Q(3) is given by η(3) = λ 16π 3 16(D − 5)(D − 4)(D − 2)2Q2eη(rh)U −D/2−1(rh) � (D + 2)U ′2(rh) − 2U(rh)U ′′(rh) − U(rh)η′(rh)U ′(rh) � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='10) Notice that this result is only non-zero for D ⩾ 6, otherwise the quasi-topological term is trivial, as discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The shear-viscosity-to-entropy ratio in this case is given by η s = 1 4π � 1 + 3λ 16 (D − 5)(D − 4)(D − 2)2Q2eη � (D + 2)U ′2 U 2 − 2U ′′ U − η′ U ′ U �� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='11) where all functions are evaluated at the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' This indicates a possible violation of the KSS bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' This is expected, since it’s now understood that higher curvature terms would introduce massive graviton modes, which is the source of KSS bound violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Indeed, to confirm that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5) violates the bound requires determining the physical bound of the coupling constant λ, we will not discuss it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' For quartic and quintic case, we found that the D = 4 solutions are all analytical at z = 0 when evaluated on the metric (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='7) and thus does not contribute to the shear viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' For the dimension-generic solutions, we found their shear viscosity contribution can written in the form ηQ(n) = λ 16πe n−1 2 ηQn−1U −1− n−2 2 DU ′n−3 � aU′2 + bUU ′η′ + 2bUU ′′� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='12) For a specific quasi-topological term, the coefficients a, b only depends on the dimension D, their explicit values are given in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 5 Conclusions Quasi-topological gravities can be thought as a class of higher curvature gravity theories whose higher curvature terms give no contribution to the equations of motion when evalu- ated on the metric (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3), but could have non-trivial perturbations around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' In this case black hole solutions of the corresponding Einstein gravity continues to be solution when the higher curvature terms are included, making it much easier to study its higher curva- ture effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' In this work we constructed such theory up to quintic order in the Riemann tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Most remarkably, we found that all quasi-topological terms we constructed actually vanish when evaluated on the most general non-stationary spherically symmetric metric (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' On the one hand, this makes these terms have no contribution on the thermody- namics and holographic a-charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' More importantly, on the other hand, this makes them quasi-topological on a much wider kinds of metrics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=', the FRW metric and the Vaidya – 10 – metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' This opens a large gate of possible applications of such quasi-topological gravity theories, such as one could study the effects of these terms on the cosmic perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' As an example to study the non-trivial effects of the quasi-topological terms we calcu- lated the holographic shear viscosity of a general Einstein-scalar theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The results can be put into a simple form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' As expected, the KSS bound could possibly be violated due to the nature of higher curvature gravities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A Quartic and quintic quasi-topological gravities In this section we list all solutions of quartic and quintic quasi-topological terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The full set of solutions is also available in the supplementary material, in the form of Mathematica .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='wl file, with further instructions included in the usage messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' For quartic order,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' the most general Riemann polynomial is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='Q(4) = e1R4 + e2R2RabRab + e3RRa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='bRb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='cRc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a + e4(RabRab)2 + e5Ra ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='bRb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='cRc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='dRd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ e6RRacRbdRabcd + e7RacRb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='eRedRabcd + e8R2RabcdRabcd + e9RRdeRabc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='dRabce ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ e10RabRabRcdefRcdef + e11RabRc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='aRdef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='bRdefc + e12RabRcdRef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='acRefbd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ e13RabRcdRe f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a bRecfe + e14RabRcdRe f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a cRebfd + e15RRabcdR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ab ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='Refcd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ e16RRabcdR e f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a c Rbedf + e17RabR c d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a b Refg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='cRefgd + e18RabRcdefR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='cd aRefgb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ e19RabRcdefR g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='c eaRdgfb + e20(RabcdRabcd)2 + e21RabcdR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='abc Rfgh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='dRfghe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ e22RabcdR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ab ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='gh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='Rcdgh + e23RabcdR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ab ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='gh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ce ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='Rdfgh + e24RabcdR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ab ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='R g h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='c e Rdgfh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ e25RabcdR e f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a c R g h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='e f Rbgdh + e26RabcdR e f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a c R g h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='e b Rfgdh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1) There are totally 12 dimension-generic solutions, their coefficients ei,j are listed in table 2 below, where i labels different solutions and j labels the 26 coefficients of one solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' e1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1 2D9−61D8+773D7−5451D6+23821D5−67174D4+121930D3−135736D2+81920D−19456 2(D−4)(D−3)(D−2)5(D−1)D(D3−9D2+26D−22) e1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 −3D10+94D9−1237D8+9168D7−42780D6+131846D5−271580D4+367060D3−308704D2+145024D−29184 (D−4)(D−3)(D−2)5(D−1)D(D3−9D2+26D−22) e1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3 4(2D9−59D8+733D7−5103D6+22103D5−61918D4+111738D3−123512D2+73632D−17024) (D−4)(D−3)(D−2)4(D−1)D(D3−9D2+26D−22) e1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4 2D8−56D7+633D6−3948D5+15253D4−37812D3+58752D2−51840D+19456 (D−4)(D−3)(D−2)5(D−1)D e1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5 −2D7+45D6−398D5+1895D4−5476D3+9776D2−10112D+4864 (D−4)(D−3)(D−2)4(D−1)D e1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='6 4(2D8−59D7+730D6−5041D5+21496D4−58348D3+98636D2−94560D+38912) (D−4)(D−3)(D−2)3(D−1)D(D3−9D2+26D−22) e1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='7 − 4(D6−19D5+131D4−409D3+520D2+88D−608) (D−4)(D−3)(D−2)3(D−1)D e1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8 D10−36D9+557D8−4979D7+28834D6−113919D5+312276D4−587102D3+723424D2−525760D+170240 2(D−4)(D−3)(D−2)4(D−1)D(D3−9D2+26D−22) e1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='9 − 2(D8−30D7+376D6−2636D5+11493D4−32254D3+57146D2−58160D+25536) (D−4)(D−3)(D−2)2(D−1)D(D3−9D2+26D−22) e1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='10 − D8−30D7+353D6−2250D5+8748D4−21514D3+32672D2−27712D+9728 2(D−4)(D−3)(D−2)4(D−1)D e1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='11 D6−23D5+199D4−861D3+1996D2−2216D+608 (D−4)(D−3)(D−2)2(D−1)D e1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='15 D3−12D2+41D−38 (D−1)D(D3−9D2+26D−22) e1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='17 76−20D D3−5D2+6D – 11 – e1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='20 − D5−10D4+28D3+18D2−173D+160 8(D−3)(D−2)3(D−1) e1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='26 1 e2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1 12D8−244D7+2138D6−10521D5+31695D4−59494D3+67138D2−40696D+9728 (D−4)(D−3)(D−2)5(D−1)D(D3−9D2+26D−22) e2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 − 4(9D9−191D8+1766D7−9316D6+30809D5−65961D4+90838D3−76978D2+36256D−7296) (D−4)(D−3)(D−2)5(D−1)D(D3−9D2+26D−22) e2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3 8(11D8−224D7+1964D6−9662D5+29067D4−54398D3+61026D2−36552D+8512) (D−4)(D−3)(D−2)4(D−1)D(D3−9D2+26D−22) e2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4 2(13D7−228D6+1730D5−7392D4+19201D3−30196D2+26400D−9728) (D−4)(D−3)(D−2)5(D−1)D e2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5 − 2(14D6−185D5+1018D4−3059D3+5380D2−5344D+2432) (D−4)(D−3)(D−2)4(D−1)D e2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='6 − 4(D8−45D7+671D6−5095D5+22712D4−62348D3+104224D2−97464D+38912) (D−4)(D−3)(D−2)3(D−1)D(D3−9D2+26D−22) e2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='7 − 8(4D5−41D4+144D3−167D2−116D+304) (D−4)(D−3)(D−2)3(D−1)D e2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8 7D9−178D8+2013D7−13299D6+56610D5−161107D4+306544D3−375726D2+268688D−85120 (D−4)(D−3)(D−2)4(D−1)D(D3−9D2+26D−22) e2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='9 − 8(3D7−62D6+551D5−2733D4+8181D3−14787D2+14903D−6384) (D−4)(D−3)(D−2)2(D−1)D(D3−9D2+26D−22) e2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='10 − 2(3D7−56D6+445D5−1959D4+5160D3−8093D2+6928D−2432) (D−4)(D−3)(D−2)4(D−1)D e2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='11 2(7D5−89D4+437D3−1031D2+1108D−304) (D−4)(D−3)(D−2)2(D−1)D e2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='15 −D3+12D2−41D+38 D5−10D4+35D3−48D2+22D e2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='17 − 4(D2−8D+19) (D−3)(D−2)D e2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='20 − D5−12D4+61D3−167D2+242D−137 4(D−3)(D−2)3(D−1) e2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='25 1 e3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1 D9−30D8+380D7−2695D6+11858D5−33610D4+61132D3−68000D2+40960D−9728 (D−4)(D−3)(D−2)5(D−1)D(D3−9D2+26D−22) e3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 −3D10+92D9−1207D8+8988D7−42238D6+131008D5−270984D4+366928D3−308704D2+145024D−29184 (D−4)(D−3)(D−2)5(D−1)D(D3−9D2+26D−22) e3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3 8(D9−29D8+360D7−2521D6+10999D5−30982D4+56036D3−61888D2+36816D−8512) (D−4)(D−3)(D−2)4(D−1)D(D3−9D2+26D−22) e3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4 2(D8−27D7+307D6−1947D5+7638D4−19104D3+29728D2−26112D+9728) (D−4)(D−3)(D−2)5(D−1)D e3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5 − 2(D7−22D6+199D5−970D4+2832D3−5040D2+5152D−2432) (D−4)(D−3)(D−2)4(D−1)D e3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='6 4(2D8−59D7+736D6−5129D5+22006D4−59796D3+100632D2−95616D+38912) (D−4)(D−3)(D−2)3(D−1)D(D3−9D2+26D−22) e3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='7 − 4(D6−18D5+123D4−382D3+468D2+136D−608) (D−4)(D−3)(D−2)3(D−1)D e3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8 D10−35D9+541D8−4887D7+28686D6−114666D5+316648D4−596688D3+733520D2−529984D+170240 2(D−4)(D−3)(D−2)4(D−1)D(D3−9D2+26D−22) e3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='9 − 2(D2−7D+14)(D6−22D5+195D4−906D3+2364D2−3280D+1824) (D−4)(D−3)(D−2)2(D−1)D(D3−9D2+26D−22) e3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='10 − D8−28D7+329D6−2138D5+8496D4−21248D3+32576D2−27712D+9728 2(D−4)(D−3)(D−2)4(D−1)D e3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='11 D6−22D5+195D4−866D3+2020D2−2216D+608 (D−4)(D−3)(D−2)2(D−1)D e3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='15 D3−12D2+41D−38 (D−1)D(D3−9D2+26D−22) e3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='17 2(D2−13D+38) D(D2−5D+6) e3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='20 − (D−4)(D3−10D2+31D−26) 4(D−3)(D−2)3(D−1) e3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='24 1 e4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1 2(11D5−140D4+668D3−1502D2+1544D−512) (D−4)(D−3)(D−2)5(D−1)D e4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 − 16(4D6−55D5+293D4−780D3+1083D2−728D+192) (D−4)(D−3)(D−2)5(D−1)D – 12 – e4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3 32(5D5−64D4+306D3−687D2+700D−224) (D−4)(D−3)(D−2)4(D−1)D e4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4 8(5D7−100D6+838D5−3856D4+10553D3−17148D2+15232D−5632) (D−4)(D−3)(D−2)5(D−1)D e4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5 − 8(6D6−89D5+530D4−1675D3+3036D2−3072D+1408) (D−4)(D−3)(D−2)4(D−1)D e4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='6 64(3D4−37D3+167D2−335D+256) (D−4)(D−3)(D−2)3(D−1)D e4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='7 − 32(2D5−23D4+88D3−111D2−60D+176) (D−4)(D−3)(D−2)3(D−1)D e4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8 2(5D6−99D5+778D4−3182D3+7250D2−8800D+4480) (D−4)(D−3)(D−2)4(D−1)D e4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='9 − 16(2D−7)(D3−12D2+41D−48) (D−4)(D−3)(D−2)2(D−1)D e4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='10 − 8(D6−22D5+188D4−827D3+2013D2−2600D+1408) (D−4)(D−3)(D−2)4D e4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='11 8(3D5−45D4+245D3−607D2+652D−176) (D−4)(D−3)(D−2)2(D−1)D e4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='15 4 D−D2 e4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='17 − 16(D2−6D+11) (D−3)(D−2)D e4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='20 − D5−14D4+79D3−224D2+316D−170 2(D−3)(D−2)3(D−1) e4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='23 1 e5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1 4(11D5−140D4+668D3−1502D2+1544D−512) (D−4)(D−3)(D−2)5(D−1)D e5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 − 32(4D6−55D5+293D4−780D3+1083D2−728D+192) (D−4)(D−3)(D−2)5(D−1)D e5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3 64(5D5−64D4+306D3−687D2+700D−224) (D−4)(D−3)(D−2)4(D−1)D e5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4 16(5D7−100D6+838D5−3856D4+10553D3−17148D2+15232D−5632) (D−4)(D−3)(D−2)5(D−1)D e5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5 − 16(6D6−89D5+530D4−1675D3+3036D2−3072D+1408) (D−4)(D−3)(D−2)4(D−1)D e5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='6 128(3D4−37D3+167D2−335D+256) (D−4)(D−3)(D−2)3(D−1)D e5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='7 − 64(2D5−23D4+88D3−111D2−60D+176) (D−4)(D−3)(D−2)3(D−1)D e5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8 4(5D6−99D5+778D4−3182D3+7250D2−8800D+4480) (D−4)(D−3)(D−2)4(D−1)D e5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='9 − 32(2D−7)(D3−12D2+41D−48) (D−4)(D−3)(D−2)2(D−1)D e5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='10 − 16(D6−22D5+188D4−827D3+2013D2−2600D+1408) (D−4)(D−3)(D−2)4D e5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='11 16(3D5−45D4+245D3−607D2+652D−176) (D−4)(D−3)(D−2)2(D−1)D e5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='15 8 D−D2 e5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='17 − 32(D2−6D+11) (D−3)(D−2)D e5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='20 −D5+14D4−79D3+224D2−316D+170 (D−3)(D−2)3(D−1) e5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='22 1 e6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1 − 2D(D2−D−4) (D−4)(D−2)5(D−1) e6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 8D2(D2−3D+1) (D−4)(D−2)5(D−1) e6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3 − 16D(D2−D−4) (D−4)(D−2)4(D−1) e6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4 − 4(2D5−13D4+15D3+68D2−192D+128) (D−4)(D−2)5(D−1) – 13 – e6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5 4(D4+3D3−36D2+80D−64) (D−4)(D−2)4(D−1) e6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='6 − 32(3D−8) (D−4)(D−2)3(D−1) e6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='7 16(D3−5D2+12D−16) (D−4)(D−2)3(D−1) e6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8 − 2(D4−4D3−9D2+56D−64) (D−4)(D−2)4(D−1) e6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='9 8(D2−D−8) (D−4)(D−2)2(D−1) e6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='10 4D(D4−9D3+29D2−39D+16) (D−4)(D−2)4(D−1) e6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='11 − 4D(D2−D−8) (D−4)(D−2)2(D−1) e6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='17 − 8 D−2 e6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='20 − D3−8D2+21D−16 2(D−2)3 e6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='21 1 e7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1 − D6−16D5+103D4−336D3+570D2−458D+128 (D−4)(D−3)(D−2)2(D−1)D(D3−9D2+26D−22) e7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 3D7−51D6+356D5−1305D4+2665D3−2984D2+1692D−384 (D−4)(D−3)(D−2)2(D−1)D(D3−9D2+26D−22) e7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3 −7D6+114D5−743D4+2440D3−4140D2+3296D−896 (D−4)(D−3)(D−2)(D−1)D(D3−9D2+26D−22) e7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4 − 2(D2−5D+8)(D3−9D2+23D−16) (D−4)(D−3)(D−2)2(D−1)D e7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5 2(D4−10D3+33D2−48D+32) D(D4−10D3+35D2−50D+24) e7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='6 −9D5+139D4−851D3+2561D2−3728D+2048 (D−4)(D−3)(D−1)D(D3−9D2+26D−22) e7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='7 2(D3−7D2+6D+16) (D−4)(D−3)(D−1)D e7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8 − D7−22D6+206D5−1059D4+3213D3−5733D2+5554D−2240 2(D−4)(D−3)(D−2)(D−1)D(D3−9D2+26D−22) e7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='9 (D−2)(4D5−65D4+428D3−1409D2+2258D−1344) 2(D−4)(D−3)(D−1)D(D3−9D2+26D−22) e7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='10 D5−15D4+85D3−229D2+290D−128 2(D−4)(D−3)(D−2)(D−1)D e7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='11 − (D−2)(D3−10D2+25D−8) (D−4)(D−3)(D−1)D e7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='15 3D2−15D+16 −4D4+36D3−104D2+88D e7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='17 4−D (D−3)D e7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='19 1 e8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1 2(3D2−9D+4) (D−3)(D−2)2(D−1)D e8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 −18D3+70D2−72D+24 (D−3)(D−2)2(D−1)D e8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3 4(11D2−33D+14) (D−3)(D−2)(D−1)D e8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4 4(3D4−28D3+101D2−160D+88) (D−3)(D−2)2(D−1)D e8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5 − 4(3D3−14D2+25D−22) (D−3)(D−2)(D−1)D e8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='6 16(3D−8) (D−3)(D−1)D e8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='7 −20D2+40D+44 D3−4D2+3D e8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8 3D3−26D2+73D−70 D(D3−6D2+11D−6) e8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='9 − 2(D−2)(5D−21) (D−3)(D−1)D e8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='10 −3D4+32D3−119D2+182D−88 D(D3−6D2+11D−6) e8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='11 6D3−42D2+74D−22 D(D2−4D+3) – 14 – e8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='15 − 1 D e8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='17 − 2(D2−6D+11) (D−3)D e8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='18 1 e9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1 1 −D3+9D2−26D+22 e9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 3(D−1) D3−9D2+26D−22 e9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3 14−6D D3−9D2+26D−22 e9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='6 − 3(D2−5D+8) D3−9D2+26D−22 e9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8 − 3(D−3) 2(D3−9D2+26D−22) e9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='9 3(D−3)(D−1) D3−9D2+26D−22 e9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='15 3D2−15D+16 −4D3+36D2−104D+88 e9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='16 1 e10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1 D2−4D+2 2(D−4)(D−2)2(D−1) e10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 − (3D−2)(D2−4D+2) 2(D−4)(D−2)2(D−1) e10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3 4(D2−4D+2) (D−4)(D−2)(D−1) e10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4 D4−10D3+39D2−68D+40 (D−4)(D−2)2(D−1) e10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5 −D3+5D2−8D+8 (D−4)(D−2)(D−1) e10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='6 2(2D−7) D2−5D+4 e10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='7 −2D2+6D+4 D2−5D+4 e10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8 (D−3)(D2−6D+10) 4(D−4)(D−2)(D−1) e10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='9 − (D−3)2 (D−4)(D−1) e10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='10 − (D−3)(D3−8D2+20D−12) 4(D−4)(D−2)(D−1) e10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='11 D3−8D2+19D−8 2(D−4)(D−1) e10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='14 1 e11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1 D 2(D−4)(D−2)2(D−1) e11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 − D(3D−2) 2(D−4)(D−2)2(D−1) e11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3 4D (D−4)(D−2)(D−1) e11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4 2(D3−7D2+17D−12) (D−4)(D−2)2(D−1) e11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5 − 2(D2−3D+4) D3−7D2+14D−8 e11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='6 − 2(D−6) D2−5D+4 e11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='7 − 8 D2−5D+4 e11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8 (D−3)(3D−8) 4(D−4)(D−2)(D−1) e11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='9 6−2D D2−5D+4 e11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='10 − (D−3)(3D2−12D+8) 4(D−4)(D−2)(D−1) e11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='11 (D−3)D (D−4)(D−1) e11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='13 1 e12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1 D2−4D+2 (D−4)(D−2)2(D−1) e12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 − (3D−2)(D2−4D+2) (D−4)(D−2)2(D−1) – 15 – e12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3 8(D2−4D+2) (D−4)(D−2)(D−1) e12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4 2(D4−10D3+39D2−68D+40) (D−4)(D−2)2(D−1) e12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5 − 2(D3−5D2+8D−8) (D−4)(D−2)(D−1) e12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='6 4(2D−7) D2−5D+4 e12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='7 −4D2+12D+8 D2−5D+4 e12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8 (D−3)(D2−6D+10) 2(D−4)(D−2)(D−1) e12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='9 − 2(D−3)2 (D−4)(D−1) e12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='10 − (D−3)(D3−8D2+20D−12) 2(D−4)(D−2)(D−1) e12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='11 D3−7D2+14D−4 (D−4)(D−1) e12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='12 1 Table 2: Coefficients of dimension-generic quartic quasi- topological gravity solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Zero coefficients are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' There are also dimension-specific solutions for D = 3 and D = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' {ei} ( 1 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 4 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1) ( 1 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 4 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 1 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 2 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 7 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 16 3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 7 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 32 3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 1 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 8 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) (−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 1 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 2 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 4 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 14 3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 1 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 5 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 1 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 7 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 9 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 5 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 1 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 2 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 5 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 7 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 8 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 1 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 2 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 1 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 7 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 5 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) (− 1 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 4 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) Table 3: Coefficients of quartic quasi-topological gravity for D = 3 – 16 – {ei} ( 35 48,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 23 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 20 3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 5 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1) ( 19 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 20 3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 13 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 11 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 3 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 5 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 5 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 11 24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 7 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 41 24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 3 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 41 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 2 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 13 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 32 3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 1 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 5 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 8 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 5 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 21 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 3 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 3 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 7 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 19 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 17 3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 3 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 5 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 9 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 3 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) (− 1 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 5 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 2 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) ( 3 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) (− 1 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 5 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 4 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) (− 1 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' −2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) (− 1 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 3 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 7 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' − 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 0) Table 4: Coefficients of quartic quasi-topological gravity for D = 4 For the quintic order,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' we first need to enumerate all the possible Riemann scalars since the explicit list is never mentioned in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' We employ the following inefficient but straightforward way: first enumerate all possible scalars that can be formed by the contrac- tion of their indices, taking into the account the symmetries, but not the cyclic identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Then for each resulting scalar we replace each of its Riemann tensor factor respectively with the cyclic identity Rabcd + Racdb + Radbc = 0 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2) with each replacement two new terms are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' If both terms are contained in the Riemann scalar list then this term is equivalent to the already known scalars by the cyclic identity and should be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' As mentioned in the text, we get too many solutions at quintic order, so we are not going to present the full set of them here, which is included in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Instead, we only present a small portion of the dimension-generic solutions below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Q(5),1 = −2RR ef ab RabcdR gh ce Rdfgh + RR ef ab RabcdR gh cd Refgh (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3a) Q(5),2 = 2RabRcdR ef ac R gh be Rdfgh − 2RabRcdR e f a c R gh be Rdfgh + R c a RabR def b R gh cd Refgh (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3b) Q(5),3 = 8(3D − 7)R2 3D2 − 15D + 16R c a RabRbc − 12(D − 1)R3 3D2 − 15D + 16RabRab + 4R5 3D2 − 15D + 16 + 12 � D2 − 5D + 8 � R2 3D2 − 15D + 16 RabRcdRacbd + 6(D − 3)R3 3D2 − 15D + 16RabcdRabcd − 12(D − 3)(D − 1)R2 3D2 − 15D + 16 RabR cde a Rbcde – 17 – − 4 � D3 − 9D2 + 26D − 22 � R2 3D2 − 15D + 16 R e f a c RabcdRbedf + R2R ef ab RabcdRcdef (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3c) Q(5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D3 − 5D2 + 8D − 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D4 − 11D3 + 44D2 − 72D + 36R c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a RabR d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='b RcdR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D4 − 10D3 + 39D2 − 68D + 40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D − 2) (D4 − 11D3 + 44D2 − 72D + 36)RabRabRcdRcdR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D2 − 4D + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D4 − 11D3 + 44D2 − 72D + 36R c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a RabRbcR2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3D3 − 14D2 + 14D − 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D − 2) (D4 − 11D3 + 44D2 − 72D + 36)RabRabR3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D2 − 4D + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D − 2) (D4 − 11D3 + 44D2 − 72D + 36)R5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2D2 − 11D + 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D4 − 11D3 + 44D2 − 72D + 36RabRcdR2Racbd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D2 − 6D + 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D3 − 8D2 + 20D − 12R3RabcdRabcd + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D2 − 5D + 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D3 − 8D2 + 20D − 12RabR2R cde ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='Rbcde ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D3 − 5D2 + 4D + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D4 − 11D3 + 44D2 − 72D + 36R c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a RabRdeRRbdce ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D4 − 10D3 + 35D2 − 46D + 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D4 − 11D3 + 44D2 − 72D + 36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='RabRcdRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ac ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='Rbdef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D4 − 9D3 + 28D2 − 32D + 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D4 − 11D3 + 44D2 − 72D + 36 RabRcdRR e f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a c Rbedf + RabRabRRcdefRcdef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3d) Q(5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D5 − 14D4 + 67D3 − 142D2 + 132D − 32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D − 4)(D − 3)(D − 2) (D3 − 8D2 + 20D − 12)R c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a RabR d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='b RcdR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5D4 − 45D3 + 154D2 − 236D + 128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D − 4)(D − 3)(D − 2)2 (D3 − 8D2 + 20D − 12)RabRabRcdRcdR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D5 − 13D4 + 50D3 − 80D2 + 68D − 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D − 4)(D − 3)(D − 2) (D3 − 8D2 + 20D − 12)R c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a RabRbcR2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='−D5 + 19D4 − 100D3 + 222D2 − 232D + 88 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D − 4)(D − 3)(D − 2)2 (D3 − 8D2 + 20D − 12)RabRabR3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D3 − 5D2 + 7D − 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D − 4)(D − 3)(D − 2)2 (D3 − 8D2 + 20D − 12)R5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D4 − 13D3 + 56D2 − 106D + 88 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D − 4)(D − 3) (D3 − 8D2 + 20D − 12)RabRcdR2Racbd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D2 − 10D + 22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2(D − 4) (D3 − 8D2 + 20D − 12)R3RabcdRabcd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='−3D3 + 28D2 − 64D + 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2(D − 4) (D3 − 8D2 + 20D − 12)RabR2R cde ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='Rbcde ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='– 18 – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D5 − 13D4 + 60D3 − 126D2 + 116D − 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D − 4)(D − 3) (D3 − 8D2 + 20D − 12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='R c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a RabRdeRRbdce ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− −3D4 + 34D3 − 141D2 + 238D − 112 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D − 4)(D − 3) (D3 − 8D2 + 20D − 12)RabRcdRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ac ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='Rbdef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D4 − 8D3 + 19D2 − 12D + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D − 3) (D3 − 8D2 + 20D − 12) RabRcdRR e f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a c Rbedf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 3 − D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D − 4R2R e f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a c RabcdRbedf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− D2 − 3D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D − 4 RabRR cde ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='R f g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='b d Rcfeg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ RabRR c d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a b R efg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='Rdefg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3e) Q(5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='6 = −16(2D9 − 57D8 + 639D7 − 3863D6 + 14059D5 − 31812D4 + 43724D3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 32916D2 + 9424D + 1088)/[(D − 4)(D − 3)(D − 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D3 − 8D2 + 20D − 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D5 − 14D4 + 79D3 − 224D2 + 316D − 170 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=']R c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a RabR d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='b RcdR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 16(D10 − 23D9 + 262D8 − 1914D7 + 9661D6 − 34319D5 + 85300D4 − 144708D3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ 159028D2 − 101552D + 28480)/[(D − 4)(D − 3)(D − 2)2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D3 − 8D2 + 20D − 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D5 − 14D4 + 79D3 − 224D2 + 316D − 170 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=']RabRabRcdRcdR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ 32(3D11 − 99D10 + 1368D9 − 10752D8 + 54161D7 − 184897D6 + 437712D5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 717804D4 + 794228D3 − 556336D2 + 214928D − 32032)/[(D − 4)(D − 3)(D − 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3D2 − 15D + 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D3 − 8D2 + 20D − 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D5 − 14D4 + 79D3 − 224D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ 316D − 170)]R c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a RabRbcR2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ 32(3D11 − 45D10 + 183D9 + 869D8 − 12856D7 + 66448D6 − 201530D5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ 395034D4 − 507818D3 + 413656D2 − 193024D + 39008)/[(D − 4)(D − 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D − 2)2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3D2 − 15D + 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D3 − 8D2 + 20D − 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D5 − 14D4 + 79D3 − 224D2 + 316D − 170 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=']RabRabR3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 4(17D10 − 356D9 + 3319D8 − 18114D7 + 63896D6 − 151338D5 + 240888D4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 247784D3 + 147192D2 − 35360D − 2816)/[(D − 4)(D − 3)(D − 2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3D2 − 15D + 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D3 − 8D2 + 20D − 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D5 − 14D4 + 79D3 − 224D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ 316D − 170)]R5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ 32(3D10 − 99D9 + 1361D8 − 10615D7 + 52980D6 − 178926D5 + 417676D4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 669720D3 + 708764D2 − 447392D + 127552)/[(D − 4)(D − 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3D2 − 15D + 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D3 − 8D2 + 20D − 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D5 − 14D4 + 79D3 − 224D2 + 316D − 170 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=']RabRcdR2Racbd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 4(3D9 − 78D8 + 943D7 − 6838D6 + 32308D5 − 102114D4 + 214432D3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 286992D2 + 221088D − 74336)/[(D − 4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3D2 − 15D + 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D3 − 8D2 + 20D − 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D5 − 14D4 + 79D3 − 224D2 + 316D − 170 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=']R3RabcdRabcd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 16(3D9 − 84D8 + 948D7 − 5760D6 + 20693D5 − 44340D4 + 52004D3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 21480D2 − 14272D + 13008)/[(D − 4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3D2 − 15D + 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D3 − 8D2 + 20D − 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='– 19 – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D5 − 14D4 + 79D3 − 224D2 + 316D − 170 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=']RabR2R cde ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='Rbcde ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 64(D9 − 24D8 + 245D7 − 1405D6 + 4982D5 − 11217D4 + 15640D3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 12198D2 + 3768D + 352)/[D − 4)(D − 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D3 − 8D2 + 20D − 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 14D4 + 79D3 − 224D2 + 316D − 170)]R c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a RabRdeRRbdce ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ 16(D9 − 28D8 + 336D7 − 2286D6 + 9737D5 − 26860D4 + 47638D3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 51642D2 + 30272D − 7024)/[(D − 4)(D − 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D3 − 8D2 + 20D − 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D5 − 14D4 + 79D3 − 224D2 + 316D − 170 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=']RabRcdRR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ac ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='Rbdef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ 32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D8 − 18D7 + 134D6 − 532D5 + 1203D4 − 1534D3 + 1082D2 − 544D + 256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D − 3) (D3 − 8D2 + 20D − 12) (D5 − 14D4 + 79D3 − 224D2 + 316D − 170) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='RabRcdRR e f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a c Rbedf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2D7 − 35D6 + 263D5 − 1098D4 + 2743D3 − 4087D2 + 3352D − 1164 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D − 4) (3D2 − 15D + 16) (D5 − 14D4 + 79D3 − 224D2 + 316D − 170) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='R2R e f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a c RabcdRbedf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ 32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D6 − 14D5 + 82D4 − 254D3 + 433D2 − 380D + 132 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D − 4) (D5 − 14D4 + 79D3 − 224D2 + 316D − 170) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='RabRR cde ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='R f g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='b d Rcfeg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D5 − 10D4 + 39D3 − 74D2 + 68D − 24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='D5 − 14D4 + 79D3 − 224D2 + 316D − 170 RR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ab ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='RabcdR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='gh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='ce ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='Rdfgh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='+ RRabcdRabcdRefghRefgh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3f) B Results of holographic shear viscosity In this section we present the values of the coefficients a, b in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='12) for dimension-generic quartic and quintic quasi-topological terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' The coefficients for quartic case are given in table 5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D−4)(2D8−29D7+145D6−245D5−101D4+226D3+1362D2−1984D+304) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='16(D−2)(D−1)D(D3−9D2+26D−22) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='b1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D−4)(2D7−33D6+188D5−375D4−224D3+1722D2−1744D+304) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='16(D−2)(D−1)D(D3−9D2+26D−22) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D−4)(D8−15D7+98D6−344D5+573D4+9D3−1376D2+1366D−152) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8(D−2)(D−1)D(D3−9D2+26D−22) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D−4)(D7−15D6+108D5−481D4+1303D3−1930D2+1246D−152) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='8(D−2)(D−1)D(D3−9D2+26D−22) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D−4)(D8−13D7+45D6+51D5−466D4+178D3+1828D2−2248D+304) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='16(D−2)(D−1)D(D3−9D2+26D−22) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D−4)(D7−16D6+71D5+38D4−1002D3+2452D2−2008D+304) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='16(D−2)(D−1)D(D3−9D2+26D−22) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D−4)(D5−8D4+17D3+3D2−37D+4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2(D−2)(D−1)D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='b4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D−4)(D4−10D3+32D2−37D+4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2(D−2)(D−1)D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D−4)(D5−8D4+17D3+3D2−37D+4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D−2)(D−1)D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='b5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D−4)(D4−10D3+32D2−37D+4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D−2)(D−1)D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− (D−4)2(D−3)(D+1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4(D−2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='b6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D−4)2(D−3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4(D−2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='– 20 – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D−4)(D−2)2(4D3−9D2−55D+8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='32D(D3−9D2+26D−22) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='b7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D−4)(D−2)2(D3+3D2−40D+8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='32D(D3−9D2+26D−22) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='(D−4)(D−2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='16D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='b8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− (D−4)(D−2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='16D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='a9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3(D−5)(D−4)(D−2)2(D2+5D−2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='32(D3−9D2+26D−22) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='b9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='− 3(D−5)(D−4)(D−2)2(2D−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='16(D3−9D2+26D−22) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='Table 5: Coefficients a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' b of the shear viscosity contribution from quartic quasi-topological terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Each entry corresponds to the solution with the same index in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Entries with zero coefficients are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' For quintic quasi-topological terms, we only present the coefficients of the terms pre- sented in the previous section in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' a3 3(D−5)(D−4)(D−2)2D(D2+8D−4) 16(3D2−15D+16) b3 − 3(D−5)(D−4)(D−2)2D(7D−4) 16(3D2−15D+16) a5 1 64(D − 3)(D − 2)2D b5 − 1 64(D − 3)(D − 2)2D a6 − (D−3)(D−2)2(6D8−81D7+380D6−508D5−1660D4+7593D3−12210D2+8512D−1792) 4(3D2−15D+16)(D5−14D4+79D3−224D2+316D−170) b6 (D−3)(D−2)2(21D7−336D6+2194D5−7582D4+15001D3−16834D2+9472D−1792) 4(3D2−15D+16)(D5−14D4+79D3−224D2+316D−170) Table 6: Coefficients a, b of the shear viscosity contribution from quintic quasi-topological terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Each entry corresponds to the solution with the same indices in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Entries with vanishing coefficients are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Acknowledgments I thank Yi Ling and Hong Lü for helpful discussions, I am also grateful to Ying Chen for supporting my own study projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' References [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Gross and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Sloan, The Quartic Effective Action for the Heterotic String, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' B 291 (1987) 41–89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Grisaru and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Zanon, σ Model Superstring Corrections to the Einstein-hilbert Action, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' B 177 (1986) 347–351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Kovtun, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Son, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 94 (2005) 111601, [hep-th/0405231].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' – 21 – [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Buchel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Escobedo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Myers, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Paulos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Sinha, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Smolkin, Holographic GB gravity in arbitrary dimensions, JHEP 03 (2010) 111, [arXiv:0911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4257].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [5] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Myers, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Paulos, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Sinha, Holographic studies of quasi-topological gravity, JHEP 08 (2010) 035, [arXiv:1004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2055].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [6] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Camanho and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Edelstein, Causality constraints in AdS/CFT from conformal collider physics and Gauss-Bonnet gravity, JHEP 04 (2010) 007, [arXiv:0911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3160].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [7] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Camanho and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Edelstein, Causality in AdS/CFT and Lovelock theory, JHEP 06 (2010) 099, [arXiv:0912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1944].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [8] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Bueno, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Cano, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Min, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Visser, Aspects of general higher-order gravities, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' D 95 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 4 044010, [arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='08519].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [9] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Lovelock, The Einstein tensor and its generalizations, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 12 (1971) 498–501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Lovelock, Divergence-free tensorial concomitants, aequationes mathematicae 4 (Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=', 1970) 127–138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [11] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Myers and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Robinson, Black Holes in Quasi-topological Gravity, JHEP 08 (2010) 067, [arXiv:1003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='5357].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Myers, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Paulos, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Sinha, Holographic studies of quasi-topological gravity, JHEP 08 (2010) 035, [arXiv:1004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2055].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Oliva and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Ray, A new cubic theory of gravity in five dimensions: Black hole, Birkhoff’s theorem and C-function, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 27 (2010) 225002, [arXiv:1003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4773].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Dehghani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Bazrafshan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Mann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Mehdizadeh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Ghanaatian, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Vahidinia, Black Holes in Quartic Quasitopological Gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' D 85 (2012) 104009, [arXiv:1109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4708].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Ahmed, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Hennigar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Mann, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Mir, Quintessential Quartic Quasi-topological Quartet, JHEP 05 (2017) 134, [arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='11007].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Cisterna, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Guajardo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Hassaine, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Oliva, Quintic quasi-topological gravity, JHEP 04 (2017) 066, [arXiv:1702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='04676].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [17] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Hennigar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Kubizňák, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Mann, Generalized quasitopological gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' D 95 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 10 104042, [arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='01631].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [18] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Bueno, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Cano, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Hennigar, (Generalized) quasi-topological gravities at all orders, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 37 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1 015002, [arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='07983].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [19] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Bueno and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Cano, Einsteinian cubic gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' D 94 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 10 104005, [arXiv:1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='06463].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Hennigar and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Mann, Black holes in Einsteinian cubic gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' D 95 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 6 064055, [arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='06675].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [21] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Bueno and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Cano, Four-dimensional black holes in Einsteinian cubic gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' D 94 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 12 124051, [arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='08019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [22] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Bueno, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Cano, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Ruipérez, Holographic studies of Einsteinian cubic gravity, JHEP 03 (2018) 150, [arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='00018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [23] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Mir, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Hennigar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Ahmed, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Mann, Black hole chemistry and holography in generalized quasi-topological gravity, JHEP 08 (2019) 068, [arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='02005].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' – 22 – [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Mir and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Mann, On generalized quasi-topological cubic-quartic gravity: thermodynamics and holography, JHEP 07 (2019) 012, [arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='10906].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [25] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Cano, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Murcia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Rivadulla Sánchez, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Zhang, Higher-derivative holography with a chemical potential, JHEP 07 (2022) 010, [arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='10473].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [26] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Bueno, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Cano, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Moreno, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Murcia, All higher-curvature gravities as Generalized quasi-topological gravities, JHEP 11 (2019) 062, [arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='00987].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [27] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Bueno, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Cano, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Hennigar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Lu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Moreno, Generalized quasi-topological gravities: the whole shebang, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 40 (2023), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 1 015004, [arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='05589].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [28] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Liu, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Lu, Quasi-Topological Ricci Polynomial Gravities, JHEP 02 (2018) 166, [arXiv:1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='07198].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [29] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Hartnoll, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Herzog, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Horowitz, Building a Holographic Superconductor, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 101 (2008) 031601, [arXiv:0803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='3295].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Hartnoll, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Herzog, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Horowitz, Holographic Superconductors, JHEP 12 (2008) 015, [arXiv:0810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='1563].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [31] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Martinez, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Troncoso, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Zanelli, Exact black hole solution with a minimally coupled scalar field, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' D 70 (2004) 084035, [hep-th/0406111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [32] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Dykaar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Hennigar, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Mann, Hairy black holes in cubic quasi-topological gravity, JHEP 05 (2017) 045, [arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='01633].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [33] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Caceres, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Figueroa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Oliva, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Oyarzo, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Stuardo, Quadratic gravity and conformally coupled scalar fields, JHEP 04 (2020) 157, [arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='01478].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [34] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Padmanabhan and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Kothawala, Lanczos-Lovelock models of gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 531 (2013) 115–171, [arXiv:1302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='2151].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [35] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Fulling, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' King, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Wybourne, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Cummins, Normal forms for tensor polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' the riemann tensor, Classical and Quantum Gravity 9 (may, 1992) 1151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [36] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Wald, Black hole entropy is the Noether charge, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' D 48 (1993), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 8 R3427–R3431, [gr-qc/9307038].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [37] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Deruelle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Sasaki, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Sendouda, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Yamauchi, Hamiltonian formulation of f(Riemann) theories of gravity, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 123 (2010) 169–185, [arXiv:0908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='0679].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [38] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Duff, Observations on Conformal Anomalies, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' B 125 (1977) 334–348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [39] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Imbimbo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Schwimmer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Theisen, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Yankielowicz, Diffeomorphisms and holographic anomalies, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 17 (2000) 1129–1138, [hep-th/9910267].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [40] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Lu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Wu, Causality and a-theorem Constraints on Ricci Polynomial and Riemann Cubic Gravities, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' D 97 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' 2 024023, [arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='03650].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [41] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Li, On Thermodynamics of AdS Black Holes with Scalar Hair, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' B 815 (2021) 136123, [arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='05597].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' [42] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' Paulos, Transport coefficients, membrane couplings and universality at extremality, JHEP 02 (2010) 067, [arXiv:0910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content='4602].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} +page_content=' – 23 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQfZ_fS/content/2301.00235v1.pdf'} diff --git a/ttAzT4oBgHgl3EQfBfrk/vector_store/index.faiss b/ttAzT4oBgHgl3EQfBfrk/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..9dd26bd056d8eb6201a03ac70c78d302315d3263 --- /dev/null +++ b/ttAzT4oBgHgl3EQfBfrk/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:96b19e0d693a3595700bfd85db1118de86704b319186d8fe72c45bc7356bcd1c +size 1769517 diff --git a/ttE2T4oBgHgl3EQf2AhI/content/tmp_files/2301.04156v1.pdf.txt b/ttE2T4oBgHgl3EQf2AhI/content/tmp_files/2301.04156v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e94486d10924e2acc8150fae847ed2d069a22107 --- /dev/null +++ b/ttE2T4oBgHgl3EQf2AhI/content/tmp_files/2301.04156v1.pdf.txt @@ -0,0 +1,2125 @@ +Draft version January 12, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +Cosmic Ray Drag and Damping of Compressive Turbulence +Chad Bustard1 and S. Peng Oh2 +1Kavli Institute for Theoretical Physics, University of California - Santa Barbara, Kohn Hall, Santa Barbara, CA 93107, USA +2Department of Physics, University of California - Santa Barbara, Broida Hall, Santa Barbara, CA 93106, USA +Submitted to ApJ +ABSTRACT +While it is well-known that cosmic rays (CRs) can gain energy from turbulence via second order +Fermi acceleration, how this energy transfer affects the turbulent cascade remains largely unexplored. +Here, we show that damping and steepening of the compressive turbulent power spectrum are expected +once the damping time tdamp ∼ ρv2/ ˙ECR ∝ E−1 +CR becomes comparable to the turbulent cascade time. +Magnetohydrodynamic (MHD) simulations of stirred compressive turbulence in a gas-CR fluid with +diffusive CR transport show clear imprints of CR-induced damping, saturating at ˙ECR ∼ ˜ϵ, where ˜ϵ is +the turbulent energy input rate. In that case, almost all the energy in large scale motions is absorbed +by CRs and does not cascade down to grid scale. This “divergence-cleaning” should render small- +scale turbulence largely solenoidal and could suppress fluctuations important for thermal instability. +The lack of small-scale compressive modes is also problematic for hypothesized resonant scattering of +E ∼> 300 GeV CRs, when self-confinement is inefficient. When CR transport is streaming dominated, +CRs also damp large scale motions, with kinetic energy reduced by up to to an order of magnitude +in realistic ECR ∼ Eg scenarios, but turbulence (with a reduced amplitude) still cascades down to +small scales with the same power spectrum. Such large scale damping implies that turbulent velocities +obtained from the observed velocity dispersion may significantly underestimate the turbulent forcing +rate, i.e. ˜ϵ ≫ ρv3/L. These findings motivate future, higher resolution simulations with a mixture of +turbulent driving modes. +1. INTRODUCTION +Cosmic rays (CRs) and magnetized turbulence are +both ubiquitous in the Universe, and their interplay has +long been a fascinating topic of research. Fluctuations +at the small-scale end of a turbulent cascade, on scales of +order the CR gyroscale, are frequently invoked to scat- +ter individual CRs, creating the high degree of observed +CR isotropy and the long residence times of CRs in the +Milky Way disk and its surrounding halo relative to the +light crossing time (Amato & Blasi 2018; Becker Tjus & +Merten 2020). In such a scenario, dubbed the “extrin- +sic turbulence” model (Zweibel 2017), the resulting bulk +CR transport is magnetic field-aligned diffusion, with an +energy-dependent spatial diffusion coefficient κ|| and CR +flux FCR ∝ κ||∇PCR. CRs in this picture can also gain +energy from repeated scattering off gyroscale fluctua- +Corresponding author: Chad Bustard +bustard@ucsb.edu +tions, a second order Fermi mechanism called “resonant +reacceleration.” +Phenomenological models of CR propagation fit to di- +rect and indirect CR observables (Hanasz et al. 2021) +have traditionally assumed a Kolmogorov scaling for tur- +bulence, appropriate for hydrodynamic turbulence; how- +ever, our understanding of CR scattering by turbulence +has been refined over time with new insights into magne- +tohydrodynamic (MHD) turbulence. Most profoundly, +MHD turbulence differs from hydrodynamic turbulence +in that MHD forces and hence turbulence are no longer +isotropic. The resulting anisotropy of slow and Alfv´en +modes (Goldreich & Sridhar 1995) makes them ineffi- +cient CR scatterers, as CRs interact with multiple un- +correlated eddies during one gyro-orbit, essentially can- +celing out gyroresonant contributions from each eddy +(Chandran 2000). +Compressible fast modes, whose velocities are inde- +pendent of magnetic field direction, are more isotropic +(Cho & Lazarian 2003) and therefore considered the best +arXiv:2301.04156v1 [astro-ph.HE] 10 Jan 2023 + +2 +Bustard & Oh +candidate for CR scattering (Yan & Lazarian 2004); al- +though, the degree of isotropy decreases with decreasing +scale due to strong collisionless and viscous damping, +hence the efficacy of CR scattering decreases with de- +creasing CR energy (Kempski & Quataert 2022). Fast +mode scattering, then, is most plausible for higher en- +ergy CRs (E > 300 GeV). +For E < 300 GeV, where most of the CR energy +resides, CRs can largely create scattering perturba- +tions themselves through a resonant streaming insta- +bility (Wentzel 1968; Kulsrud & Pearce 1969). +The +resulting transport is no longer purely diffusive; in- +stead, CRs “stream” down their field-aligned pressure +gradient at the local Alfv´en speed vA = B/√4πρ with +FCR ∝ vAPCR, and additional, energy-dependent CR +diffusivity (FCR ∝ ∇PCR) is introduced by wave damp- +ing, e.g. ion-neutral damping, nonlinear Landau damp- +ing, and turbulent damping (Skilling 1971; Farmer & +Goldreich 2004; Blasi et al. 2012; Wiener et al. 2013; +Zweibel 2017; Bustard & Zweibel 2021). There is also +an important difference regarding energy transfer be- +tween CRs and hydromagnetic waves: whereas extrinsic +turbulence is generated externally, in self-confinement, +the free energy to generate waves comes from the CRs +themselves, and this energy is subsequently dissipated +into the thermal gas via wave damping at a rate H = +−dECR/dt = vA · ∇PCR. We refer to this collisionless +energy transfer as streaming energy loss / gas heating. +While considerable effort has been put towards ex- +ploring resonant-scale interactions between CRs and ei- +ther self-generated (e.g. Skilling 1975; Felice & Kulsrud +2001; Bai et al. 2019; Holcomb & Spitkovsky 2019) or +externally driven (e.g. Giacalone & Jokipii 1999; Yan +& Lazarian 2002; Reichherzer et al. 2020) waves, some- +what less focus has been given to the interplay between +CRs and turbulence on scales much larger than a CR +gyroradius. In this regime, the collective CR population +is well-described as a fluid, and CRs experience com- +pressions and rarefactions in the turbulent flow, leading +to energy transfer between the CRs and turbulence. To +distinguish this from its resonant-scale counterpart, the +flow of energy from turbulence to the bulk CR fluid is +called non-resonant reacceleration (Ptuskin 1988), and +its efficiency depends on CR transport model. +For purely diffusive CR transport, non-resonant reac- +celeration is maximally efficient when CRs are well- +trapped in the turbulent flow (κ < vphL0, where vph +is the phase speed of compressive fluctuations and L0 +is the outer eddy scale). When streaming is taken into +account, the interaction between perturbed CR and gas +variables is fundamentally altered. While CR diffusion +introduces a π/2 phase shift between CR and density +perturbations, leading to a CR force that damps fluctu- +ations much like a damped harmonic oscillator, both +the change in flux (FCR ∝ PCR instead of FCR ∝ +∇PCR) and the associated energy loss that accompany +streaming transport modify the CR force (Tsung et al. +2022). +As we showed in Bustard & Oh (2022) (from +now on referred to as Paper I), CR reacceleration / +turbulent damping rates become dependent on plasma +β = Pg/PB; they remain largely unchanged in high-β +plasmas like the intracluster medium (ICM) where reac- +celeration is a leading explanation for radio halos (e.g. +Brunetti & Lazarian 2011; Brunetti & Jones 2014), but +they are stunted significantly in low-β plasmas. +Despite non-resonant reacceleration being a fairly in- +efficient process compared to diffusive shock acceleration +(a first order Fermi mechanism), with minimum growth +times lengthened even further by streaming transport, +it was pointed out by Thornbury & Drury (2014); Drury +& Strong (2017) that a significant fraction of total CR +power in galaxies could come from reacceleration, conse- +quently creating a large sink for turbulent energy. In this +paper, we present analytical estimates and CR+MHD +simulations suggesting that CRs in very plausible astro- +physical environments can divert significant amounts of +turbulent energy, essentially acting as an unsual form +of viscosity. The outcome is a CR-modified route to gas +heating, rather than the typical conversion to heat at the +dissipation scale, and a damped turbulent energy spec- +trum with decreased small-scale, compressive power. +These changes are, of course, strongest in environ- +ments where CRs are dynamically important such as the +ISM (where CR energy densities are roughly in equipar- +tition with turbulent and magnetic energy densities; +Boulares & Cox 1990) and the Milky Way circumgalac- +tic medium (which may be energetically dominated by +CRs; e.g. Ji et al. 2020), but they would affect any pro- +cess that relies on compressive motions. For instance, +compressions seed thermal instability (Field 1965; Mc- +Court et al. 2012; Mohapatra et al. 2022), which is fre- +quently invoked, for instance, to explain the existence +of cold CGM clouds (Putman et al. 2012). Fluctuations +that scatter CRs are not immune to these modifications +either. Low-energy, self-confined CRs could sap energy +from the turbulent fast mode cascade at large scales, de- +creasing the available small-scale power needed to scat- +ter high energy CRs. +This paper is outlined as follows. In §2, we discuss our +simulation method and setup. In §3, we analytically es- +timate and then quantify in simulations the fractions of +turbulent driving and gas heating that are channeled +through CRs. +We then analytically derive how CR- +induced damping should affect MHD turbulence spectra + +Cosmic Ray Effects on Turbulence +3 +(§4.1) and the conditions under which damping rates +can exceed cascade rates (§4.2). +In §4.3, we present +exploratory simulations strongly suggestive of these an- +alytic estimates and show sensitivities to streaming vs +diffusive CR transport. We discuss regimes of applica- +bility and implications in §5 and conclude in §6. +2. SIMULATION SETUP +We begin by briefly describing the simulation method- +ology and setup, which is described in more detail in +Paper I. Using the Athena++ MHD code (Stone et al. +2020) coupled with an additional CR module that mod- +els CR diffusive and streaming transport in a fluid ap- +proximation using a two-moment method originally de- +veloped for radiation transport (Jiang & Oh 2018), we +numerically solve the ideal MHD equations plus two ad- +ditional equations for the CR energy and energy flux. +We stir turbulence following an Ornstein-Uhlenbeck ran- +dom process (Uhlenbeck & Ornstein 1930; Eswaran & +Pope 1988), randomly generating velocity perturbations +between modes k = 1 and 3 in a cubic box of width +2L. For driving, we set the autocorrelation timescale to +be tcorr = L/cs and drive fluctuations every tdrive = +2 × 10−3(L/cs). For the parameter scans in §3, we use +grids of size 1283 and 2563. We simulate fluids with ei- +ther an isothermal equation of state, where the thermal +energy is fixed, or an adiabatic equation of state. The +latter results in a gradual rise in the gas pressure due to +a combination of CR heating and grid-scale dissipation +of the cascade, which we decompose and quantify. These +simulations all use purely compressive forcing, with two +turbulent driving rates ˜ϵ = dE/dt, resulting in approx- +imately Ms ∼ 0.15 and Ms ∼ 0.5 turbulence with a +weak dependence on plasma β since MHD forces coun- +teract motions. +We avoid solenoidal driving to avoid +turbulent amplification of magnetic fields, so that we +can evolve simulations at approximately fixed plasma β. +To a good approximation, solenoidal driving only ampli- +fies magnetic fields, while compressive driving energizes +CRs. +At our parameter scan resolution of 2L/256, the cas- +cade exhibits only a short inertial range, and in test- +ing we find that the spectral slope in pure MHD runs +(no CRs) is intermediate between E(k) ∼ k−2 and +E(k) ∼ k−3/2 – a shallower slope is expected for com- +pressive fast modes, but the exact exponent has been +highly debated. +In our analytic estimates (§4.1), we +will explore CR-induced deviations to different initial +spectra, but we particularly note significant changes to +Kraichnan turbulence where E(k) ∼ k−3/2 initially. For +§4.3, where we want to test deviations from this spec- +trum due to CR drag, we increase the resolution to +2L/512, though we find that the main trends are well- +recovered even with a resolution of 2L/256 (see Ap- +pendix). Higher resolution simulations giving a larger +inertial range would be preferable, but to ensure an ac- +curate treatment of CR propagation and influence, the +two-moment method has an effective, maximum speed +of light parameter vm that must be much larger than +other propagation speeds in the system and that sets +the Courant-limited timestep. +In Paper I, we found +that vm ∼ 50cs gives seemingly converged CR heating +rates and reacceleration rates. +With this choice, our +MHD+CR simulations are about a factor of 8 more ex- +pensive than pure hydro turbulence sims, prohibiting us +from going to much higher resolution. +3. COSMIC RAY DIVERSION OF TURBULENT +ENERGY +We’ll begin with a short review of non-resonant reac- +celeration (see e.g. Ptuskin 1988; Chandran & Maron +2004; Lynn et al. 2012 and §2 of Paper I for greater +detail) and its relation to the turbulent damping rate. +Variables used in our discussion are summarized in Ta- +ble 1. As discussed in Paper I, “drag” against CRs pro- +vides a frictional force on compressive motions known +as Ptuskin damping (Ptuskin 1981). It is similar to ra- +diative damping of sound waves, which famously leads +to Silk damping of acoustic waves in the early universe +(Silk 1968). In general, since Ek/tdamp ∼ PCR/tgrow, we +have1: +tdamp ∼ ρv2max +�tgrow +PCR +, 1 +˜ϵ +� +∼ max +� +M2 +ctgrow, tinject +� +(1) +where Mc ≡ v/cc is the Mach number in units of the CR +effective sound speed, cc ∼ +� +PCR/ρ, and tinject ≡ ρv2/˜ϵ. +Equation 1 is a general expression for the damping time, +for which one can plug in the appropriate tgrow, the CR +reacceleration (or growth) time. +Working in the limit of purely diffusive spatial CR +transport with isotropic diffusion coefficient κ, the reac- +celeration time can be derived in two limits depending +on the ratio of diffusion time tdiff = L2 +0/κ to compressive +wave crossing time tsc = L0/vph across an eddy of length +L0 in a medium with compressive phase velocity vph ∼ +(Ptot/ρ)1/2 ∼ [Pg+PB+PCR)/ρ]1/2. In the fast diffusion +limit (tdiff ≪ tsc, or equivalently, κ ≫ vphL0), deriving +the CR momentum diffusion coefficient Dpp follows the +textbook argument for second order Fermi acceleration: +Dpp ∼ (∆p)2/τscatter ∼ p2v2/(c2τscatter) ∼ p2v2/κ. The +1 In this paper, we use the notation ˜ϵ ≡ ρv3/L and ϵ ≡ v3/L. + +4 +Bustard & Oh +Table 1. Simulation parameters, CR module settings, and other variable definitions +Parameter +Definition / Setting / Equation +Additional Notes +L +Half box size +k = 2 mode +L0 +Outer eddy scale +k = 3 mode +tdrive +2 × 10−3(L/cs) +Turbulence driven every tdrive +tcorr +L/cs +Autocorrelation time +˜ϵ, ϵ +Input turbulent energy rate, dE/dt +ρv3/L, v3/L in hydro turbulence +vm +50cs +Effective maximum speed of light +κ +CR diffusion coefficient +Assumed to be field-aligned only (κ = κ||) +β +Pg/PB +Plasma beta +cs +� +γPg/ρ +Gas sound speed +vph +� +(γPg + γCRPCR + PB)/ρ +Compressive wave phase speed +vA +B/√4πρ +Alfv´en speed +cc +� +γCRPCR/ρ +Effective CR sound speed +Ms, Mph, MA, Mc +v/cs, v/vph, v/vA, v/cc +Mach numbers +H +vA · ∇PCR +“Collisionless” CR loss rate / gas heating rate +fCR, fth, fCR,heating +˙ECR/˜ϵ, ˙Eth/˜ϵ, < H >/˜ϵ +Fraction of ˜ϵ → CRs, thermal gas, CR heating +E(k) +Kinetic energy spectrum +∝ k−5/3 (Kolmogorov), k−2 (Burgers), k−3/2 (Kraichnan) +tinject +L/v +Energy injection time +tcascade +kE(k)/F(k) +Cascade time (see Equation 10) +tgrow +p2/Dpp +CR reacceleration time (§3 and Paper I) +tdamp +∼ ρv2max +� +tgrow +PCR , 1 +˜ϵ +� +∼ max +� +M2 +ctgrow, tinject +� +Turbulent damping time (Equation 1) +energy growth time, defined as p2/Dpp is +tgrow ∼ κ +v2 ; +κ >> vphL0 +(2) +In the opposite limit of slow diffusion (tdiff ≫ tsc, +or equivalently, κ ≪ vphL0), Dpp ∼ (δp)2/τdiff +∼ +(p2v2/v2 +ph)(κ/L2 +0), and the growth time is +tgrow ∼ p2 +Dpp +∼ +v2 +phL2 +0 +v2κ ; +κ << vphL0 +(3) +Joining the two regimes in the middle, the minimum +growth time is tgrow ∼ (vphL0/v2) when κ ∼ vphL0. +Strictly speaking, these scalings are appropriate if CR +diffusion is isotropic, if streaming is negligible, and if +all reacceleration comes from eddies of a single scale L0. +Relaxing these assumptions introduces further modifi- +cations. In the fast diffusion limit (κ ≫ vphL0), there +are also correction factors that decrease the growth time +if anisotropic rather than isotropic spatial diffusion is +accounted for (Chandran & Maron 2004). Additional +streaming transport, widely applicable for CRs with en- +ergy E ⪅ 300 GeV, introduces a correction factor that +decreases reacceleration rates by fcorr = 1 − +� +2/β and +fcorr = (1 − +� +2/β)1/2 in the slow and fast diffusion +regimes, respectively (Paper I); and in the slow diffusion +limit (κ ≪ vphL0), multiple eddies contribute to reaccel- +eration, with relative contributions dependent upon the +shape of the turbulent power spectrum (see Equation 4 +in Paper I for a more general expression). If the wave +spectrum is Burgers-like (E(k) ∼ k−2), roughly consis- +tent with our simulations, eddies at each logarithmic +interval in the inertial range contribute equally to reac- +celeration, and tgrow has a broad minimum of tgrow ∼ +(vphL0/v2) throughout the entire range of κ|| < vphL0. +If we work in the single eddy limit (i.e., we only con- +sider eddies of size L0), in the fast diffusion (κ ≫ vphL0) +regime, where tgrow ∼ κ/v2 then Equation 1 gives +tdamp ∼ κ/c2 +c, in agreement with the classic (much more +detailed) calculation of this effect by Ptuskin (1981). +Working instead in the broad regime of maximal reac- +celeration, where CRs are well-trapped in the turbulent +flow (when κ < vphL0), the characteristic growth time +is tgrow ∼ (vphL0/v2), which gives: +tdamp ∼ max +�vphL0 +c2c +, tinject +� +(4) +Note that tdamp is velocity independent. +With these reacceleration times in mind, we can now +estimate the fraction of turbulent energy forcing ˜ϵ that +goes toward CRs. It is given by +fCR ∼ +˙ECR +˜ϵ +∼ ECR +˜ϵtgrow +(5) + +Cosmic Ray Effects on Turbulence +5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fi +fCR saturates +fCR follows + analytic expectation +10 +1 +100 +101 +102 +PCR/Pg +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fi +fCR > 0 due to + numerical dissipation +2 +3 +phPCR +v2 +fCR +fth +Figure 1. The average CR energy gain rate and thermal +energy gain rate relative to the turbulent driving rate (fCR = +˙ECR/˜ϵ and fg = ˙Eg/˜ϵ, respectively) for simulations without +streaming, as a function of PCR/Pg. These all are adiabatic, +Ms ∼ 0.5 simulations on a 1283 grid, with β ∼ 1. +Top: +κ|| ∼ 0.15L0vph, where CR energy gain is maximized. The +dashed black curve is the analytic expectation from Equation +6, showing good agreement when PCR/Pg < 1. +Bottom: +κ|| ∼ 0. +For PCR ≫ Pg, even κ ∼ 0 leads to significant +fractions of turbulent energy converted to CR energy, but +this CR reacceleration is due to numerical diffusion caused +by finite resolution. +For example, for Kolmogorov turbulence, where ˜ϵ ∼ +ρv3/L, and for the characteristic growth time tgrow ∼ +9/2vphL/v2 this gives: +fCR ∼ ECR/tgrow +ρv3/L +∼ max +�2 +3Mph +PCR +ρv2 , 1 +� +(6) +The maximum value of 1 reflects energy conservation: +CRs cannot gain more energy than is injected by tur- +bulent forcing, hence fCR ∼ 2 +3Mph PCR +ρv2 is only valid for +˙ECR < ˜ϵ. Within this regime, the fraction of kinetic +energy deposited into CRs is small if PCR ≪ ρv2, in +which case most energy is deposited in the thermal gas; +however, for higher PCR, the fraction increases and can +become quite substantial at close to equipartition values. +Figure 1 compares this expectation to simulations and +is one of the key results of this paper. +The y-axis +shows the partitioning of the input energy rate into +CRs (fCR = ˙ECR/˜ϵ) and thermal energy (fth = ˙Eth/˜ϵ) +for varying PCR/Pg, keeping ˜ϵ fixed, for purely diffusive +CRs. Unlike our previous simulations, which all used an +isothermal equation of state, these simulations have an +adiabatic equation of state, which makes it easier to con- +firm energy conservation. Together, the contributions to +˙ECR and ˙Eth sum to ∼ 80−90% of the driving rate, with +the rest going towards small magnetic and kinetic energy +increases. The top and bottom panels show simulations +each without streaming and with κ = 0.15L0vph and +κ = 0, respectively. For PCR/Pg < 1, fCR follows the +expectation from Equation 6 (shown as a black dashed +line) quite well, an indication that turbulent reacceler- +ation is diverting the driving energy to CRs at the ex- +pense of thermal gas heating. Similar simulations with2 +κ = 0 show far lower fCR, again revealing the depen- +dence of reacceleration on diffusion coefficient. +Note +that while we previously only tested analytic expecta- +tions for the growth time tgrow (on which Equation 6 +depends) when the gas is isothermal in Paper I, they +continue to hold when the gas is adiabatic. +As PCR/Pg increases, fCR deviates from the analytic +expression in Equation 6; fCR increases more slowly to- +wards the asymptotic bound fCR ∼ 1 than in our ansatz. +Nonetheless, for PCR/Pg ∼> 1, what immediately stands +out is the large fraction of energy diverted to CRs, with +fCR as large as 0.8 when PCR/Pg > 1. These large val- +ues of fCR clearly come at the expense of thermal heat- +ing3, with fth decreasing from fth ≈ 1 when PCR ≪ Pg +to fth < 0.2 when PCR > Pg. +In the above, purely diffusive case, turbulent energy +directly accelerates CRs. When streaming is included, +energy is also lost to collisionless heating at a rate +H = vA·∇PCR. In Fig 2, we quantify the partitioning of +turbulent kinetic energy into direct acceleration of CRs +(fCR) and gas heating (fth) in simulations with fixed +˜ϵ producing undamped Ms ∼ 0.15. We distinguish be- +tween collisionless heating by CRs fCR,heating (red bars), +and heating due to turbulence which cascades down to +the grid scale and dissipates fth − fCR,heating (orange +bars). When streaming is included, fCR is a small and +weakly increasing function of β, consistent with Paper +I and evident in Figure 2. Here, we fix the initial state +to have PCR ∼ Pg for each simulation and quantify the +CR energy gain rate as we did in Figure 1. +Despite +the fact that a negligible fraction of energy fCR is ends +2 In practice, κ has a non-zero value because of numerical diffusion, +but here this has little impact up until PCR ≫ Pg. +3 Since we enforce purely compressive driving, magnetic field am- +plification is very weak, and fCR+fth ≈ 1 for an adiabatic setup. + +6 +Bustard & Oh +Diffusion Only +Streaming Only +Diff + Stream +Adiab. Stream Only +Adiab. Diff + Stream +Diffusion Only +Streaming Only +Diff + Stream +Adiab. Stream Only +Adiab. Diff + Stream +Diffusion Only +Streaming Only +Diff + Stream +Adiab. Stream Only +Adiab. Diff + Stream +0 +0.2 +0.4 +0.6 +0.8 +1 +Figure 2. Partitioning of input turbulent energy rate ˜ϵ into three different channels: CR reacceleration fCR, dissipation via CR +collisionless heating fCR,heating (i.e. streaming energy loss), and grid-scale heating fth − fCR,heating. Without CRs, this choice +of ˜ϵ produces Ms ∼ 0.15 turbulence. Each simulation here starts with PCR ∼ Pg but with varying CR transport treatments, +either with diffusion only (all with κ ∼ 0.15vphL0) or diffusion plus additional streaming. For each β, the first three simulations +use an isothermal equation of state, so there is no gas heating. The last two, denoted by “Adiab.”, use an adiabatic equation +of state, in which case the total thermal gas heating rate is the sum of CR heating and grid-scale heating. With diffusion +only, reacceleration is very efficient: most turbulent energy is soaked up by CRs. With streaming, both gas heating and CR +energization are relatively inefficient in the low-β regime, but for β ∼ 10, 100, CR heating is the dominant energy channel. +Instead of turbulent energy cascading to small scales and eventually dissipating into thermal energy at the grid scale, CRs +intercept this energy transfer at large scales; astoundingly, even in these subsonic flows very high fractions of turbulent energy +are channelled through CRs when PCR ∼> Pg. +up in CRs, the latter nonetheless have a strong impact +on the turbulent cascade. In MHD simulations, turbu- +lence cascades to grid scales where numerical diffusion +dominates4 and subsequently dissipates, heating the gas. +Thus, fth is a good barometer of how much kinetic en- +ergy flux makes it to the dissipation scale; however, that +is not the case with turbulence modified by streaming +CRs. +In the right panel of Figure 1, the decrease in +fth with increasing β is marginal, but actually much of +that heating is done by CR streaming energy loss in- +4 In high resolution simulations with explicit viscosity, it would +instead cascade to the viscous scale. +stead of classical small-scale dissipation. Across all β, +only ≈ 10% of driving energy makes it to the grid scale, +with most energy channeled through CRs. +Note that in our estimate of tdamp (Equation 4), we +have not included the effects of CR streaming on tgrow. +If we did, tdamp would be substantially longer in low β +environments. However, as we have seen, this is incor- +rect. When CR streaming is present, the kinetic energy +of compressive motions is still absorbed by CRs at large +scales. This energy is subsequently returned to the gas +in the form of heat via CR streaming, and so streaming +impedes the secular growth of CR energy, resulting in +the lower growth times explored in Paper I. However, +diversion of kinetic energy away from the turbulent cas- + +Cosmic Ray Effects on Turbulence +7 +cade and damping of compressive motions still happens +at a similar rate, even at low β (Figure 1). CR streaming +provides an avenue for gas motions to quickly dissipate +in the form of heat without going through the turbu- +lent cascade. +In this case, CRs can be thought of as +providing an unusual form of viscosity. +To summarize: once Pc/Pg ∼> 1, and for β ∼> 10, +our simulations show that the energy input in turbu- +lent driving appears to be almost completely diverted to +CRs, with only ∼ 10% remaining which cascades down +to grid scales. This is irrespective of whether streaming +is absent (in which case CRs store the energy) or present +(in which case CRs thermalize a significant fraction via +collisionless heating). This is astonishing efficiency, con- +sidering that strong shocks convert at best ∼ 10 − 30% +of kinetic energy to CRs. For β ∼ 1, the fraction of en- +ergy routed through CRs is slightly lower, ∼ 80% in the +diffusion only case, and ∼ 50% with both CR streaming +and diffusion. We now turn to some implications of this +finding. +4. COSMIC RAY IMPRINTS ON KINETIC +ENERGY SPECTRA +In certain regimes, CRs are clearly an important en- +ergy sink for fluid motions. When turbulent energy is +diverted to the CR population, it either +1. Directly accelerates CRs through non-resonant +reacceleration +2. (If CR streaming is significant) Heats the gas at +scales lCR ≫ ldiss through collisionless energy +transfer by self-confined CRs (streaming energy +loss), where ldiss is the Kolmogorov dissipation +scale. +In either case, energy that originally would have cas- +caded to small scales is siphoned out of the turbulent +cascade, and it is interesting to ask what imprint this +might have on the kinetic energy spectrum. In this sec- +tion, we first focus on the effects of purely diffusive CRs, +leaving an initial exploration of streaming CR transport, +the effects of which are less straightforward and deserve +future follow-up, to §4.4. We will first explore CR mod- +ifications to Kolmogorov and Kraichnan spectra analyt- +ically and discuss astrophysical regimes where spectra +could be heavily modified. Of the compressible MHD +modes, it is thought that slow modes have a Kolmogorov +spectrum (E(k) ∝ k−5/3) and fast modes have a Kraich- +nan spectrum (E(k) ∝ k−3/2) (Cho & Lazarian 2003), +though this is still debated. In our simulations, compres- +sive forcing gives rise to something intermediate between +Kraichnan and Burgers turbulence (E(k) ∝ k−2), and +we will see that CR damping also has noticeable effects +in this regime. +4.1. Analytic Theory +We can solve for the turbulent power spectrum by +solving the dynamic equation (Landau & Lifshitz 1987). +If we consider a turbulent energy injection rate ϵ injected +at some outer scale L = k−1 +L +(where ϵ ∼ v2 +l /tcascade ∼ +const in the absence of damping, and tcascade depends on +the form of turbulence), then in steady state the com- +bined effects of the cascade to smaller scales and damp- +ing must balance injection: +ϵ δ(k − kL) = ∂ +∂k F(k) + Γ(k)E(k) +(7) +where E(k) is the power spectrum of turbulence, F(k) is +the turbulent cascade flux in k-space, and Γ(k) ∼ t−1 +damp +is the damping rate. Following Equation 1 and adopting +tgrow ∼ 9/2vphL/v2, it follows that Γ(k) ∼ 2/9c2 +c/vphL, +which is independent of scale. +The cascade flux de- +pends on the type of turbulence: for Kolmogorov turbu- +lence, F(k) ∼ [E(k)]3/2k5/2, while for isotropic Kraich- +nan turbulence, F(k) ∼ k3[E(k)]2/vph. +In the ab- +sence of damping (Γ(k) = 0), integrating both sides of +Equation 7 with respect to k gives E(k) ∼ ϵ2/3k−5/3 +and E(k) ∼ (ϵvph)1/2k−3/2, the power spectra for Kol- +mogorov and Kraichnan turbulence respectively. +The first and second terms on the right hand side of +Equation 7 have units of v2/k × (t−1 +cascade, t−1 +damp) respec- +tively. In Fig. 3, we solve Equation 7 for various values +of tdamp/tcascade. It is easy to understand the asymptotic +behavior. When tcascade ≪ tdamp, the first term on the +RHS dominates: injected energy cascades before it can +damp, and we obtain the usual Kolmogorov/Kraichnan +power spectra. On the other hand, if tdamp ≪ tcascade, +then the second term on the RHS dominates, which gives +ϵ ∼ Γ +� +E(k)dk ∼ Γv2, or +v2 ∼ ϵtdamp ∼ v2 +0 +� tdamp +tcascade +� +(8) +where v2 +0 and tcascade are the velocity and cascade time +at the outer scale in the absence of damping; for a given +energy forcing ϵ, the velocity at the outer scale is re- +duced. However, since tdamp ∼> tinject, the damping time +cannot be made arbitrarily small. We discuss this fur- +ther in §4.2. +To understand the behavior at smaller scales, note +that the cascade time is scale-dependent, while for non- +resonant CR acceleration, tdamp is independent of scale. +We are accustomed to thinking of the cascade time de- +creasing towards small scales (for instance, tcascade ∝ +l2/3, l1/2 for undamped Kolmogorov, Kraichnan tur- +bulence respectively). +However, damping changes the + +8 +Bustard & Oh +1 +10 +100 +1000 k +0.001 +0.01 +0.1 +1 +E(k) k53 +Kolmogorov +1 +10 +100 +1000 k +0.001 +0.01 +0.1 +1 +E(k) k32 +Kraichnan +tcasc +tdamp = 0 +tcasc +tdamp = +1 +2 +tcasc +tdamp = 1 +tcasc +tdamp = 1.25 +tcasc +tdamp = 1.5 +tcasc +tdamp = 2. +1 +k2 +1 +k3 +Figure 3. Modified kinetic energy spectra for a Kolmogorov (left) and Kraichnan (right) cascade with varying levels of CR +damping, all with vph = 2v0. E(k) is in units of the outer-scale, undamped kinetic energy, where k denotes the wavenumber. +Different line colors denote different ratios of the cascade time to the damping time, showing that if damping becomes competitive, +the outer scale velocity decreases, and the slope of the spectrum steepens. Dashed lines show E(k) = k−2 and E(k) = k−3 for +comparison. For tcascade/tdamp ⪆ 1.5, 1 for Kolmogorov and Kraichnan, respectively, the cascade sharply cuts off at progressively +smaller k. For smaller tcascade/tdamp, CRs damp fluctuations, but the cascade returns to its normal scaling at large k. +scale dependence of velocity, further reducing velocities +at small scales, and thus increasing cascade times at +these scales. +If tcascade/tdamp still decreases towards +small scales, then the cascade eventually takes over +and the spectrum rebounds from damping. +However, +if tcascade/tdamp instead increases towards small scales, +then damping becomes increasingly dominant and the +spectrum will cut off precipitously. Since tdamp is inde- +pendent of k, what matters is the scale dependence of +tcascade. +From Equation 7, the cascade time can be written as: +tcascade ∼ kE(k) +F(k) ∼ +1 +[k3E(k)]1/2 (Kolmogorov) +(9) +∼ +vph +k2E(k) (Kraichnan) +(10) +where we have used F(k) ∼ [E(k)]3/2k5/2, F(k) ∼ +k3[E(k)]2/vph for Kolmogorov and Kraichnan turbu- +lence respectively. When damping operates, E(k) will +steepen from standard Kolmogorov/Kraichnan spectra. +From Equation 10, we see that for a power spectrum +E(k) ∝ k−α, tcascade increases with k for α ∼> 3 +(Kolmogorov), α ∼> 2 (Kraichnan). +The steepening +of the power spectrum slope is controlled by the rel- +ative strength of damping, i.e. tcascade/tdamp at large +scales. If this is sufficiently large, it produces a power +spectrum with a slope steeper than the critical value, +and we have a runaway: tcascade/tdamp continually in- +creases towards small scales, producing a rapid cutoff +in the velocity power spectrum. However, if the initial +value of tcascade/tdamp produces a power spectrum with +an index shallower than the critical slope, then damp- +ing initially ‘takes a bite’ out of the turbulent cascade, +but tcascade/tdamp decreases towards small scales, until +damping becomes negligible, the original cascade domi- +nates and the spectrum recovers its original undamped +power law slope. +We clearly see confirmation of this bifurcation in +small scale damping in Fig. +3. +We see that we re- +quire tcascade/tdamp ∼> 1.5 at the outer scale for crit- +ical damping in a Kolmogorov cascade (so that the +power spectrum steepens beyond E(k) ∝ k−3), or +tcascade/tdamp ∼> 1 at the outer scale for critical damp- +ing in a Kraichnan cascade (so that the power spectrum +steepens beyond E(k) ∝ k−2). Indeed, tcascade/tdamp ∼ +1 causes a perfect transformation of the Kraichnan spec- +trum from a E(k) ∝ k−3/2 spectrum to a Burgers-like +E(k) ∝ k−2 spectrum. +This bifurcation in the existence of small scale turbu- +lence is important, so we restate it in simpler terms. +Damping can change the slope of the velocity power +spectrum E(k) ∝ k−α, and hence the scale dependence +of velocity v(k) ∝ k(1−α)/2 (using v2 ∼ kE(k)), but it +does not change the physics of the turbulent cascade. +The latter can be encapsulated in the form of cascade +times tcascade ∼ l/vl (Kolmogorov), tcascade ∼ lvph/v2 +l +(Kraichnan). Using v(k) ∝ k(1−α)/2, these relations im- + +Cosmic Ray Effects on Turbulence +9 +ply tcascade ∝ k(α−3)/2 (Kolmogorov), and tcascade ∝ +kα−2 (Kraichnan), which gives critical slopes α = 3, 2 +respectively, in line with our previous arguments. The +scale dependence of tcascade determines if turbulence is +completely damped at small scales, or recovers with the +original (undamped) power-law scaling. +We briefly note that Equation 7 does not really ap- +ply to Burgers turbulence E(k) ∝ k−2, which is not a +genuine turbulent cascade, but rather an instantaneous +jump from large to small scales via shocks which arise +from non-linear steepening. However, Ptuskin damping +creates friction which can balance non-linear steepening +and prevent shock formation. We can see this by ex- +amining Burgers’ equation in the presence of Ptuskin +damping: +∂v +∂t + v · ∇v = −Γv +(11) +For Γ > ∇v, the damping term exceeds the non-linear +term, so that damping exceeds non-linear steepening +for L > vtdamp. +Unless the nonlinear time tNL ∼ +L/v < tdamp, fluid motions are damped before they can +steepen, and the transfer of power from large to small +scales is delayed or even halted. Thus, tNL/tdamp poten- +tially plays a similar role to tcascade/tdamp. +4.2. What is tcascade/tdamp? +The results of the previous section show that the ra- +tio tcascade/tdamp is the critical parameter determining +the efficacy of small scale damping, and that there is +a critical value (tcascade/tdamp ∼> 1.5, 1 for Kolmogorov +and Kraichnan turbulence, respectively) such that the +turbulence spectrum will show a cutoff. Here, we inves- +tigate the conditions under which these thresholds may +be crossed. +We have previously argued from energy conservation +that ˙ECR ∼< ˜ϵ in steady state, hence tdamp ≥ tinject ∼ +ρv2/˜ϵ ∼ L/v, the timescale on which kinetic energy is in- +jected. In Appendix A, we confirm this expectation and +also show how various scalings, such as δρ/ρ, δv/v, can +be understood as a function of PCR/Pg, or v/cs, v/vph. +When does tdamp reach the minimal value of tinject ∼ +L/v, so that almost all of the injected kinetic energy is +directly dissipated in cosmic rays? Equating the first +and second terms in brackets in Equation 4, tdamp ∼ +tinject when: +Mph ∼< +� Pc +Ptot +� +(12) +Equation 12 is only an order of magnitude estimate; the +exact threshold must come from numerical simulations. +Nonetheless, it illustrates the relevant physics: damping +saturates when the turbulent Mach number is small and +the CR energy density is high. +If tdamp reaches its minimal value of tinject ∼ L/v, +then: +tcascade +tdamp +∼ 1 +(Kolmogorov) +∼ +1 +Mph +(Kraichnan) +(13) +From +Fig +3, +we +see +that +it +is +unclear +whether +damping will be strong enough to enforce a small +scale cutoff in a Kolmogorov cascade (which requires +tcascade/tdamp ∼> 1.5), but any subsonic turbulence in a +Kraichnan cascade which satisfies Equation 12 will au- +tomatically have tcascade/tdamp ∼> 1), the threshold for +critical damping there. The increase in tcascade/tdamp +is not due to a decrease in the damping time (which +has a floor at tinject), but rather the increased cascade +time in MHD turbulence. Longer cascade times are as- +sociated with wave turbulence, where wave-wave inter- +actions produce non-linearities which eventually cause +turbulence to cascade (Nazarenko 2011). Other forms +of wave turbulence can be present, for instance, in sys- +tems with strong stratification (Wang et al. 2022) or +rotation. +Note that even if the threshold for critical damping +(i.e. exponential suppression of small-scale power) is not +met, Figure 3 shows that the damping of gas motions +can still be significant. +4.3. Simulations +The results of §4.1, 4.2 are useful for guiding expec- +tations and driving intuition. +Nonetheless, given the +complex non-linearities, they require validation by nu- +merical simulation – a difficult task, given the limited +inertial range of standard resolution simulations. +We +now present a set of simulations which, to our knowl- +edge, are the first CR hydrodynamics simulations specif- +ically probing CR influence on turbulent kinetic energy +spectra. While a more complete set of simulations with +different driving modes and higher resolution awaits, we +already see that CRs suppress small-scale fluctuations. +We focus first on the case where Ptuskin damping is +maximized, running a series of diffusion-only simulations +near the CR energy gain ‘sweet spot’ κ ∼ 0.15L0vph, +where v2 +ph ∼ (Pc+Pg+PB)/ρ. We vary the input driving +rate ˜ϵ by an order of magnitude to create turbulence with +undamped Ms ∼ 0.5 and Ms ∼ 0.15 , where Ms = v/cs +and cs ∼ +� +Pg/ρ is the gas sound speed (thus, Mph = +v/vph decreases as Pc/Pg increases). Plasma beta, β = +Pg/PB, are denoted in each figure and represent rough +values for the presented suite of simulations; while each +simulation starts with the same β, the saturated value +of β changes by a small amount depending on whether + +10 +Bustard & Oh +100 +101 +102 +k +10 +3 +10 +2 +10 +1 +100 +k2E(k)dk +k +2 +1 +CR-Modified Kinetic Energy Spectra +s +0.5 +MHD +( +PCR +Pg , +tNL +tdamp) + (0.3, 0.12) +( +PCR +Pg , +tNL +tdamp) + (1.5, 0.37) +s +0.15 +MHD +( +PCR +Pg , +tNL +tdamp) + (0.3, 0.25) +( +PCR +Pg , +tNL +tdamp) + (1.5, 0.62) +Figure 4. The turbulent kinetic energy spectrum, multiplied by k2 for a set of Ms ∼ 0.5 and Ms ∼ 0.15 diffusion-only +simulations, keeping κ = 0.15L0vph and β ∼ 1, as we vary PCR/Pg. Spectra are normalized to the k=3 mode for the Ms ∼ 0.5 +MHD run. Ratios of the outer scale nonlinear time to damping time, calculated with tdamp =< ρv2/ ˙PCR > and tNL = L0/v, are +also denoted. Points show the energy in each k-bin averaged over 10 outputs at late times, when turbulence is fully developed, +while shaded regions show the minima and maxima during those time periods. +Each simulation was run on a 5123 grid. +While MHD runs produce spectra shallower than k−2, as expected for compressive fast modes in MHD turbulence, CRs damp +fluctuations, slightly decreasing the power in low k modes while steepening the spectra at high k. +CRs are present, what CR transport model is assumed, +etc. +Figure 4 shows simulation kinetic energy spectra for +both Ms ∼ 0.5 and Ms ∼ 0.15 simulation sets, each +normalized by the k = 3 mode power for the Ms ∼ 0.5 +MHD-only simulation. Different colors denote different +initial PCR/Pg, ranging from 0 to 1.5. +Points denote +average kinetic energies, and the shaded regions denote +the minimum and maximum kinetic energies taken over +10 snapshots at late times when we see converged spec- +tra, typically between 8 and 10 eddy turnover times af- +ter the simulation starts (see Appendix for more about +time convergence). Importantly, we note that the iner- +tial ranges in our MHD-only simulations display some- +thing between a Kraichnan (k−3/2) and a Burgers-like +(k−2) spectrum, where the latter is frequently seen in hy- +drodynamic simulations with compressive driving, due +to non-linear steepening (e.g., Miniati 2015). Thus, the +analytic models of §4.1,4.2 where we assume a Kraich- +nan spectrum do not exactly apply. Nonetheless, we can +look for qualitative agreement. +We can use the non-linear steepening time as a proxy +for the cascade time: tcasc ∼ tNL ∼ L0/vL, where vL +is the outer-scale velocity. +Ratios of cascade time to +damping time, calculated with tdamp =< ρv2/ ˙PCR >, +are noted in the legend. +The trend agrees at least +qualitatively with Figure 3. Power both at large and +small scales is decreased when PCR ≥ Pg, consistent +with mild Ptuskin damping when tcascade ∼ tdamp. As +tcascade/tdamp increases, the spectrum deviates further +and further from the MHD case. +For example, the +tcascade/tdamp ∼ 0.62, Ms ∼ 0.15 simulation has be- +tween 10 and 100 times less power in high-k modes than +the MHD run. +Projections of density, kinetic energy, + +Cosmic Ray Effects on Turbulence +11 +MHD + +PCR ∼ Pg +κ|| ∼ 0.15vphL0 +Density +Kinetic Energy +Magnetic Energy +Figure 5. Projections perpendicular to the initial magnetic field direction of density (left column), kinetic energy (middle +column), and magnetic energy (right column) after ∼ 10 eddy turnover times, normalized by their average values in the MHD +only case. Top: MHD-only simulations with β ∼ 10 and Ms ∼ 0.15. Bottom: Simulations with the same β and forcing rate, +but with PCR ∼ Pg and diffusive CR transport. Density, velocity, and magnetic fluctuations are all suppressed compared to the +MHD case. +and magnetic energy for these Ms ∼ 0.15, β ∼ 10 +simulations vary quite obviously, as seen in Figure 5, +with fluctuations clearly damped in the PCR ∼ Pg case +(bottom row) compared to the MHD case (top row). +Higher Mach number simulations appear to show damp- +ing, as well, but the effect is less obvious. This is in +line with expectations from our previous discussion that +tcascade/tdamp is maximized for smaller values of stirring +velocity. +While our analytic predictions and preliminary simu- +lations suggest that Ptuskin damping could play a role in +suppressing the compressible turbulent cascade at small +scales, it may appear hazardous to draw conclusions +based on moderate resolution simulations with limited +inertial range. We therefore refer the reader to the Ap- +pendix, where we test whether these spectral changes +occur in the absence of CR transport, and also back to +§3 and Figure 1, where we presented a separate, more +robust diagnostic of the suppression of the turbulent cas- +cade by Ptuskin damping: via the heating of adiabatic +gas. In hydrodynamic simulations of adiabatic gas, we +have found that ˙Egas → ˜ϵ, as it should. However, in adi- +abatic simulations with CRs, we have found ˙Egas → 0, +while ˙ECR → ˜ϵ, i.e. almost all of the turbulent energy is +absorbed by the CRs (see Figure 1). Furthermore, all of +this energy is absorbed at large scales, which are well re- +solved. The shift to CRs receiving almost all the energy +of the turbulent cascade is genuine turbulent accelera- +tion, not due to numerical diffusion in the CR module. +We infer this from numerical convergence in our CR ac- +celeration rates, as well as the close conformance to ana- +lytic expectations. Nonetheless, we have tested this ex- +plicitly by checking energy absorption for the two-fluid +case when κ = 0 (bottom panel of Figure 1); in this case +˙Egas/˜ϵ → 0.8 when PCR/Pg ∼ 1, i.e. gas heating is once +again large. +If Ptuskin damping does not allow gas motions to +cascade the ∼ 2 decades to grid scale in our simula- +tions to enable dissipation, this strongly suggests that +real turbulence should not be able to cascade down +the many more decades to e.g. the gyroscale of CRs, +where fast modes are frequently invoked to scatter CRs +with E ⪆ 300 GeV. Of course, it is still imperative to +test these ideas in much higher resolution simulations, +preferably with a spectral code that can better resolve +an MHD Kraichnan cascade. + +t = 9.4 Teddy +1.3 +1.2 +1.1 +1.0 +0.9 +0.8 +0.7t = 9.9 Teddy +1.3 +1.2 +1.1 +1.0 +0.9 +0.8 +0.7t = 9.9 Teddy +2.00 +1.75 +1.50 +1.25 +/EK, +1.00 +Ek/ +0.75 +0.50 +0.25t = 9.9 Teddy +2.00 +1.75 +1.50 +1.25 +2 +B +E +1.00 +0.75 +0.50 +0.25t = 9.4 Teddy +2.00 +1.75 +1.50 +1.25 +1.00 +0.75 +0.50 +0.25t = 9.4 Teddy +2.00 +1.75 +1.50 +1.25 +B +2 +E +1.00 +E +0.75 +0.50 +0.2512 +Bustard & Oh +4.4. Streaming vs Diffusion +In the pure diffusion limit, Γ(k) is well known, and +as we’ve shown analytically and numerically, the result- +ing CR drag damps turbulence at large scales, chang- +ing kinetic energy spectral slopes and even introducing +cut-offs. The functional form for Γ(k) is more uncertain +when streaming transport is introduced. Since we found +in Paper I that streaming stunts reacceleration rates due +to fundamental changes to CR-turbulent interactions, +it’s tempting to append the plasma β-dependent correc- +tion factors from Paper I to Γ(k). If this were true, weak +CR reacceleration should imply very weak changes to +the kinetic energy spectrum; however, we’ve run a num- +ber of simulations with CR streaming, including some +with no diffusive transport where reacceleration is abso- +lutely negligible, that clearly modify the kinetic energy +spectra. We present some simple scalings which match +our simulations, but defer a detailed study to future +work. +All simulations in this section start with PCR ∼ Pg +and assume an isothermal equation of state. Figure 6 +shows the kinetic energy spectra for 2563 simulations of +varying β ∼ 1, 10, 100, each with different CR transport +models but the same turbulent driving rate, which for +simulations without CRs (MHD only) give a sonic Mach +number Ms ∼ 0.15. A partial version of Figure 6, using +a 5123 domain, is included in the Appendix and shows +similar behavior. The left column shows each spectrum +multiplied by k2, normalized to the peak value of the +MHD spectrum at k=3. The right column, in order to +more clearly show differences in the spectral shape and +overall kinetic energy, shows each spectrum divided by +the MHD spectrum. +Note the similarity of the pure +streaming power spectra to the streaming + diffusion +power spectra; in this parameter range, streaming dom- +inates over diffusion. We seek to answer two main ques- +tions about these results: +How does streaming vs diffusive transport affect the +overall kinetic energy in the gas? +It’s important here to note that Alfv´en Mach num- +bers for each run saturate at MA < 1, meaning that +Alfv´en crossing times are faster than eddy turnover +times; hence, streaming transport is relatively fast. Fast +streaming transport leads to small field-aligned CR pres- +sure gradients / large field-aligned CR scale lengths +lCR = PCR/(ˆb · ∇PCR). Compared to CRs with slow +diffusive transport, streaming CRs have comparatively +small pressure gradients and absorb less energy (via the +v ·∇PCR term) in sub-Alfv´enic flows. This qualitatively +explains the behavior seen in Figure 6, where, for in- +stance, β ∼ 1 leads to fast streaming transport, hence +small CR pressure gradients, and little to no change +in the kinetic energy spectrum. At the same time, it +is important to realize that CR transport timescales +are not simply ∼ L/vA, since CR pressure gradients +and magnetic fields are often misaligned; CR bottle- +necks and field line wandering can further alter transport +timescales. Thus, for instance, ∼ L/vA is a poor esti- +mate of CR losses due to collisionless heating, tloss ∼ +Pc/(vA · ∇Pc) > L/vA (Paper I). Over large scales, +CR transport with pure streaming is potentially effec- +tively diffusive (Mertsch 2020), or even super-diffusive +(Hu et al. 2022; Sampson et al. 2022). +The reduction in turbulent kinetic energy is clearly +β-dependent, which we can explain as follows. +Since +streaming-dominated CRs in sub-Alfv´enic turbulence +are in the ‘fast transport’ regime (analogous to the fast +diffusion regime with some effective diffusivity κeff sat- +isfying κeff > vphL0), tdamp ∼ κeff/c2 +cc. +Also, from +Equation 8, v2/v2 +0 ∝ (tdamp/tcascade) ∝ κeff, where v0 +is the undamped turbulent velocity derived by balanc- +ing ˜ϵ = ρv3 +0/L,. While we don’t know the detailed form +of κeff when streaming dominates, it’s tempting to say +κeff ∼ vAL (or at least κeff ∝ vA), in which case v2/v2 +0 ∝ +vA ∝ β−1/2. We should also expect tdamp ∼ κeff/c2 +cc ∝ +P −1 +c +, so that v2/v2 +0 ∝ (tdamp/tcascade) ∝ P −1+α +c +, where +tcascade ∝ P −α +c +. We have not explored the latter scaling, +but it is quite plausible that α ≈ 0. The cosmic ray +pressure only affects cascade times for Kraichnan tur- +bulence tcascade ∼ Lvph/v2 by contributing to the phase +velocity vph ∼ (Pg + Pc + PB/ρ)1/2. However, in the +‘fast transport’ regime, Pc does not contribute to vph +(see Appendix A). We defer a detailed exploration of +these issues to future work. +The top panel of Figure 7 quantifies the partitioning of +turbulent forcing that ends up in CRs (fCR = ˙ECR/˜ϵ), +as well as fE = (ρv3/L)/˜ϵ vs the steady-state plasma +beta, βf. Filled circles denote simulations with stream- +ing and diffusion, while empty circles have just stream- +ing. The streaming plus diffusion results quantify what +we see by eye in the kinetic energy spectra: increasing β +leads to smaller turbulent velocities; in each case, CRs +take only a very small amount of the total energy forc- +ing, with most energy input instead removed from the +system by streaming energy loss. As shown in the bot- +tom panel of Figure 7, which plots v2/v2 +0, our derived +β−1/2 scaling is at least roughly consistent with our sim- +ulations. +Does streaming change the shape of kinetic energy +spectra, as diffusion does? +Streaming CRs, which don’t themselves take an appre- +ciable amount of turbulent energy input, still nonethe- + +Cosmic Ray Effects on Turbulence +13 +10 +2 +10 +1 +100 +1 +k2E(k)dk +10 +2 +10 +1 +100 +E(k)/E(k)MHD +10 +2 +10 +1 +100 +10 +10 +2 +10 +1 +100 +100 +101 +102 +k +10 +2 +10 +1 +100 +100 +100 +101 +102 +k +10 +2 +10 +1 +100 +MHD +|| +0.15vphL0 +|| +0 + Streaming +|| +0.15vphL0 + Streaming +Figure 6. Kinetic energy spectra for Ms ∼ 0.15, 2563 simulations each with PCR/Pg ∼ 1 but varying CR transport and varying +the initial plasma β from 1 (top) to 10 (middle) to 100 (bottom). The magenta-colored lines show the resulting MHD (no CR) +spectra as a reference. The left column shows k2E(k)dk normalized by the MHD value at k = 3, while, to more clearly show +the changes in spectral shape, the right column shows the ratio of each spectrum to the MHD spectrum. For diffusion only, +efficient reacceleration damps the kinetic energy spectrum, resulting in less power at small scales compared to the MHD case. +However, with streaming included, both reacceleration rates and field-aligned CR pressure gradients depend on β. At low β +(low Alfv´en Mach number MA = v/vA), streaming negates reacceleration, and the kinetic energy spectra revert to the MHD +case. For larger β, however, reacceleration becomes somewhat more efficient, causing damping, and a more significant fraction +of turbulent energy is channeled through CRs and lost via streaming energy transfer. This latter effect, most clearly evident +in the streaming only simulations (red curves), decreases the overall kinetic energy in the system but doesn’t appear to induce +cut-offs like the diffusion-only runs. + +14 +Bustard & Oh +100 +101 +102 +f +10 +2 +10 +1 +100 +fE = ( v3/L)/ +|| +0.15vphL0 + Streaming +|| +0 + Streaming +10 +2 +10 +1 +100 +fCR = (ECR)/ +100 +101 +102 +f +10 +1 +100 +2 × 10 +1 +3 × 10 +1 +4 × 10 +1 +6 × 10 +1 +v2/v2 +0 +1/2 +Undamped +s +0.15 +Undamped +s +0.5 +Undamped +s +0.75 +10 +1 +100 +2 × 10 +1 +3 × 10 +1 +4 × 10 +1 +6 × 10 +1 +fCR = (ECR)/ +|| +0.15L0vph + streaming +|| = 0 + streaming +Figure 7. Isothermal, 2563 simulations with streaming and +PCR ∼ Pg. The MHD (undamped) version of these simula- +tions give Ms = v0/cs ∼ 0.15 (black), 0.5 (green), and 0.75 +(cyan). +The top panel shows only the Ms ∼ 0.15 points +and shows the partitioning of turbulent forcing that ends up +in CRs (fCR = +˙ECR/˜ϵ; green points and right y-axis), as +well as fE = (ρv3/L)/˜ϵ (black points and left y-axis) vs the +steady-state plasma beta βf. While for diffusion there was a +clear correlation between fCR and fE, now, fCR is small, and +fE correlates inversely with β, at least in this sub-Alfv´enic +regime studied. The bottom panel shows the turbulent ki- +netic energy relative to the undamped case, where ρv3 +0/L ∼ ˜ϵ. +With CRs, even with streaming only transport where there +is no reacceleration, v2/v2 +0 ∝ β−1/2, at least roughly, in this +sub-Alfv´enic or “fast transport” regime. There is also a weak +trend towards larger overall damping with increasing driving +rate (larger Ms has smaller v2/v2 +0), at fixed β. +less sap kinetic energy from the system. How the kinetic +energy spectra change, however, is fundamentally differ- +ent between streaming and diffusive transport. Chang- +ing β (changing MA) in streaming-dominated simula- +tions effectively changes the ratio of transport time to +eddy turnover time. To glean further insight, it’s inter- +esting to compare to simulations with purely diffusive +transport but varying diffusion coefficients. +Figure 8 +shows kinetic energy spectra for simulations on a 5123 +grid, each with initial β ∼ 10 but diffusion coefficients +varying between κ|| ∼ (0.15 − 15)vphL0. Our fiducial +case of κ|| ∼ 0.15vphL0 shows that damping, in the slow +diffusion regime, exerts meaningful drag on an entire hi- +erarchy of scales, beginning at the outer scale; in other +words, damping and cascade rates are competitive over +a large range of k. Moving to the fast diffusion regime +(κ|| ∼ 15vphL0), this is clearly not the case: the diffu- +sion length scale is larger than the outer eddy scale, and +the damping rate is only competitive with the cascade +time at large scales, leaving the cascade to operate unin- +terrupted after CRs have reduced the outer-scale kinetic +energy. +Following similar logic, we infer that, for streaming- +dominated transport in sub-Alfv´enic turbulence, Γ(k) +must be weighted heavily towards small k, causing an +initial reduction in outer-scale kinetic energy but an +unimpeded cascade at larger k. Thus, we see that the +power spectrum when streaming is included has the +same shape over the effective inertial range of the sim- +ulations k ∼< 30, albeit with a lower normalization (in +the β ∼ 10, 100 cases, when damping is effective). In +the dissipation range, k ∼> 30, there is additional steep- +ening compared to the MHD case, though whether this +is numerical or physical is as yet unclear. +5. DISCUSSION +5.1. Regimes of CR Modified Turbulence +In §4, we showed both analytically and numerically +that CRs can have a significant impact on the power +spectrum of turbulence. In particular, we showed that +as the damping time decreases relative to the turbu- +lent cascade time, the turbulent power spectra will be +steepen and then cut off abruptly at small scales (for +tcascade/tdamp ∼> 1). These results should eventually be +carefully checked by higher resolution numerical simu- +lations. +Nonetheless, we clearly already have seen in +§3 and Fig 1 a situation where CR damping of mo- +tions is stronger than the rate at which energy cascades +to smaller scales, so that little energy reaches the grid +scale. Our analytic estimates can guide expectations as +to which environments these effects might be important. + +Cosmic Ray Effects on Turbulence +15 +100 +101 +102 +k +10 +2 +10 +1 +100 +k2E(k)dk +10 +100 +101 +102 +k +10 +2 +10 +1 +100 +E(k)/E(k)MHD +MHD +|| +0 +|| +0.15L0cs +|| +1.5L0cs +|| +15L0cs +Figure 8. Kinetic energy spectra of 5123 simulations when PCR ∼ Pg but varying the diffusion coefficient from κ|| ∼ 0 (where +the only diffusivity is numerical) to the most efficient reacceleration regime (κ|| ∼ 0.15vphL0 and κ|| ∼ 1.5vphL0) to the fast +diffusion regime (κ|| ∼ 15vphL0). Note how the power spectrum is somewhat different for the two fluid system even in the +absence of CR transport, presumably because of changes to the phase velocity and other adiabatic properties, but deviations +from the MHD spectrum are mild compared to simulations with added diffusion. Left: k2E(k)dk normalized by the k = 3 MHD +value. Right: Ratio of each spectrum to the MHD spectrum. Note how diffusion introduces a characteristic scale lCR where the +kinetic energy is reduced: in the fast diffusion limit, lCR > L0, and the outer-scale kinetic energy drops significantly while the +rest of the spectrum retains the same shape as the MHD case. Going to smaller κ||, overall changes are more drastic because +reacceleration is more efficient but also the scale where the spectrum cuts off most dramatically shifts to lCR < L0. +Intra-cluster medium (ICM) —In the ICM, although sonic +Mach numbers are typically low (Ms ∼ 0.1 − 0.3), the +absence of hadronic γ-ray emission gives an upper bound +on Pc/Ptot << 1 (typically less than a few percent; Ack- +ermann et al. 2014), so that Equation 12 is not satisfied +there. The CR energy density is too small to appreciably +affect gas motions, and it is unlikely that CR reacceler- +ation appreciably damps the turbulent cascade. +Interstellar medium (ISM) —In the ISM, CR damping +could be potentially important: Pc/Ptot ∼ O(1) is rel- +atively large. In the diffusion only case, the main un- +certainty lies in the CR acceleration rate. +The most +efficient reacceleration occurs for diffusivities in the +range κ|| < vphL0 ∼ 3 × 1026cm2/s for ISM-like pa- +rameters (see Table 2 in Paper I). Canonical values of +κ ∼ 1028 −1029cm2/s used in galactic propagation mod- +els are much larger, i.e. we are sufficiently far away from +the ‘sweet spot’ that acceleration and hence damping +times could be long. On the other hand, if CR stream- +ing dominates transport, then since the ISM has β ∼ 1, +damping is small, as we have seen. +Circumgalactic medium (CGM) —Finally, the galactic +halo and CGM are strong candidates for significant CR +damping. For the diffusion only case, these regions oc- +cupy a sweet-spot where κ ∼ vphL0, Mph < 1, and if, +as suggested by simulations of Milky Way mass galaxies +(e.g. Butsky & Quinn 2018; Ji et al. 2020), Pc/Pg ∼> 1, +then Pc/Ptot is order unity. For these conditions, Equa- +tion 12 is satisfied, so that tdamp ∼ tinject. Thus, for +instance, from Equation 13, tcascade/tdamp ∼ M−1 +ph ∼ 2 +for a compressive Kraichnan cascade with Mph ∼ 0.5: +the compressive cascade will be steepened beyond the +critical threshold of E(k) ∝ k−2 and abruptly cut off, so +there is no small scale turbulence. We see hints of this +in Figure 4 for the Ms ∼ 0.5 case, but given our limited +dynamic range, spectral changes are more obvious for +Ms ∼ 0.15, when tNL/tdamp is even larger. +Once streaming is included, we have also seen that +there can be considerable damping in the β ∼ 10 − 100 +cases, with a weak trend towards larger damping with in- +creasing driving rate at fixed β (see how v2/v2 +0 is smaller +for larger Ms), potentially because CRs are more effi- +ciently trapped in turbulent eddies as the Alfven Mach +number approaches unity. This differs from the diffu- +sion only case, where stronger turbulence implies smaller +tNL/tdamp and weaker damping. +For a fixed driving +rate that produces transonic MHD turbulence in a β ∼ +10 environment (reasonable CGM parameters), adding +streaming-dominated CRs up to equipartition PCR ∼ Pg +damps turbulent kinetic energy by a factor of ∼ 5 or +greater (bottom panel of Figure 7). +5.2. Implications of CR-Modified Turbulence +The implications of such CR-modified spectra are pos- +sibly quite intriguing. For instance, a CR-induced cut- + +16 +Bustard & Oh +off could significantly affect the spatial scale of thermal +instability, since there are no small scale compressive +motions, unless there is direct driving at those scales. +Also, since Ptuskin damping only affects compressive +motions, not solenoidal motions, Ptuskin damping can +potentially make turbulence less Burgers-like and more +Kolmogorov-like. It would be interesting to explore this +‘divergence-cleaning’ effect in simulations with a mix- +ture of driving modes. +Perhaps the most interesting consequence of CR +damping of turbulence is its implication for scattering +of high-energy CRs by fast modes in an extrinsically +driven turbulent cascade. This is frequently invoked to +explain the scattering of CRs with E ∼> 300GeV (Yan +& Lazarian 2004), since self-confinement is too weak +to explain observed isotropy and confinement times. +However, the resonant scattering invoked (transit time +damping) requires the turbulence to cascade many or- +ders of magnitude, to the ∼ 300AU gyroscale of such +CRs. Fig 3 shows that for tcasc/tdamp = 1, a Kraich- +nan (E(k) ∝ k−3/2) fast mode spectrum will steepen to +a Burgers (E(k) ∝ k−2) spectrum, which already has +too little small scale power to efficiently scatter CRs +via transit time damping (Miniati 2015; Pinzke et al. +2017), and even higher values of tcasc/tdamp ∼ M−1 +ph will +completely eliminate turbulence at small scales. While +this needs further study, low-energy CRs, by damping +turbulent fluctuations at large scales, could divert tur- +bulent energy that would otherwise scatter high-energy +CRs. This potentially adds to the long list of problems +with ‘standard’ theories of CR scattering in the Milky +Way which have been recently pointed out (Kempski & +Quataert 2022; Hopkins et al. 2022). +Regardless of whether CR drag introduces a cut-off to +kinetic energy spectra, it is clear from our simulations +that CRs in both diffusion-dominated and streaming- +dominated transport regimes can sap a significant frac- +tion of the turbulent forcing rate. This breaks the usual +correspondence between turbulent velocity and turbu- +lent driving rate, i.e. +for hydrodynamic turbulence, +ρv3 +0/L ∼ ˜ϵ. +Now, ρv3/L ∼ fE˜ϵ, where the new cor- +rection factor fE can be ≪ 1. +As derived in Equa- +tion 8, v2/v2 +0 ∝ tdamp/tcascade, which for streaming- +dominated transport in sub-Alfv´enic turbulence gives +v2/v2 +0 ∝ β−1/2 (Figure 7). In the CGM, where we ex- +pect damping to be most significant, turbulent velocities +obtained from the observed velocity dispersion may sig- +nificantly underestimate the turbulent forcing rate, i.e. +˜ϵ ≫ ρv3/L. +6. CONCLUSIONS +In this paper, we present analytical estimates and ac- +companying MHD+CR simulations probing CR effects +on turbulence, namely the damping of turbulence by +large-scale, CR-induced drag on compressive gas mo- +tions. Our main findings are as follows: +• Despite long CR reacceleration times, the damping +time due to CR reacceleration can be very competi- +tive with the turbulent cascade time. +tdamp ∼ ρv2max +�tgrow +PCR +, 1 +˜ϵ +� +∼ max +� +M2 +ctgrow, ρv2 +˜ϵ +� +(14) +where Mc = v/cc is the Mach number with respect +to the CR sound speed cc ∼ +� +Pc/ρ, and ˜ϵ = ρv3/L +is the turbulent energy injection rate. Our key fig- +ures are Figures 1 and 2, where we confirm that CRs +can divert a significant fraction of turbulent energy +that would otherwise dissipate as heat at small scales. +Conditions for strong damping are met under quite +reasonable conditions (Equation 12); the CGM is an +especially strong candidate for this damping. +• If CR diffusion dominates transport, and if the ratio +of the damping time to the cascade time is sufficiently +short, small scale compressive turbulence should be +exponentially suppressed (see Figure 3). This sup- +pression of small scale turbulence has abundant im- +plications for e.g. +thermal instability, “divergence- +cleaning” of turbulence spectra, and suppression of +fast modes at small scales, which have been invoked +to scatter high-energy CRs (see §4.1). We see com- +pelling signatures of damping in our simulation spec- +tra (§4.3; Figure 4), but these effects deserve future +study with higher resolution simulations that capture +a larger turbulent inertial range. +• The effects of streaming transport are more com- +plex and deserve follow-up. +Importantly, tgrow in +Equation 14 does not include the suppression of CR +reacceleration by streaming (the β dependent fac- +tors identified in Bustard & Oh 2022), which would +substantially increase damping times. Instead, from +Figure 2, diversion of turbulent energy through CRs +remains strong even in the presence of CR stream- +ing, for our simulations where MA ∼< 1. +Instead +of introducing spectral cut-offs, streaming uniformly +decreases the normalization of the turbulent power +spectrum, but not its shape, with the turbulent ki- +netic energy scaling as v2 ∝ vA ∝ β−1/2 (Fig 6, +Fig 7). This is likely because damping operates pre- +dominantly at the largest scales in the ‘fast trans- +port’ regime (here, the sub-Alfv´enic regime). Such +large scale damping implies energetic input and tur- +bulent heating rates (much of which gets channeled + +Cosmic Ray Effects on Turbulence +17 +into CR collisionless heating) can be much larger +than standard estimates for Kolmogorov turbulence, +˜ϵ ≫ ρv3/L. +ACKNOWLEDGMENTS +The authors gratefully acknowledge Navin Tsung, +Max Gronke, Yan-Fei Jiang, Christoph Federrath, Hui +Li, and Ellen Zweibel, as well as the organizers and par- +ticipants of the KITP “Fundamentals of Gaseous Halos” +workshop. CB was supported by the National Science +Foundation under Grant No. +NSF PHY-1748958 and +by the Gordon and Betty Moore Foundation through +Grant No. GBMF7392. SPO was supported by NSF +grant AST-1911198, and NASA grant 19-ATP19-0205. +Computations were performed on the Stampede2 and +PSC-Bridges2 supercomputers under allocation TG- +PHY210004 provided by the Extreme Science and Engi- +neering Discovery Environment (XSEDE), which is sup- +ported by National Science Foundation grant number +ACI-1548562 (Towns et al. 2014). +Software: +Athena++ (Stone et al. 2020), yt (Turk +et al. 2011), Matplotlib (Hunter 2007), Mathematica +(Wolfram Research, Inc. 2021) +APPENDIX +A. TURBULENT PROPERTIES AND DAMPING IN A COSMIC RAY DOMINATED MEDIUM +CRs can influence velocity and density perturbations in a turbulent medium, but the extent depends on the relative +partition of CR vs thermal energy, as well as the CR diffusivity / transport speed. Commer¸con et al. (2019) simulate +CRs in a turbulent box with purely diffusive transport and a bi-stable ISM (with radiative cooling). They found +that trapped CRs modify the gas flow, change the density PDF, and provide support against thermal instability, +maintaining the gas in an intermediate temperature state that is classically thermally unstable. It remains to be seen +how these simulations would change when streaming is included. The perturbative heating term from CR streaming +affects thermal instability (Kempski & Quataert 2020), and in low-β plasmas where this heating is most significant, CR +streaming can also drive acoustic waves unstable, generating a “stair-case” cosmic ray pressure profile and additional +multiphase gas (Tsung et al. 2022; Quataert et al. 2022). +Our simulation setup is quite different from that of Commer¸con et al. (2019), most notably because we don’t include +radiative cooling, so we don’t attempt a detailed comparison, but we do find some qualitatively similar behavior. +Figure 9 shows δρ/ρ, δv/v, and δPCR/PCR for diffusion-only simulations with varying PCR/Pg and either κ = 0 +or κ = 0.15L0vph (where vph depends on PCR/Pg). When CRs are dynamically unimportant (PCR/Pg ≪ 1), we +recover the MHD expectation that δρ/ρ ∼ δv/v ∼ Ms = 0.5. Deviations from this relation start when PCR/Pg ⪆ 1. +Interestingly, we find that δρ/ρ is independent of Pc/Pg, while δPc/Pc ∝ 1/Pc (i.e., δPc ∼const is independent of +Pc/Pg). At the same time, we find that the velocity divergence ∇ · v ∝ P −1/2 +c +(not shown), i.e. it does depend on Pc. +This might appear puzzling, since one expects density fluctuations and velocity divergence to be directly related, yet +the former is independent of Pc, while the latter shows dependence. +A key to understanding these results is to realize that the ‘sweet spot’ κ ∼ Lvph is really still in the ‘fast diffusion +regime’. The ratio tdiffuse/tsc ∼ lvph/κ is only unity at the outer scale l ∼ L; at smaller scales, tdiffuse/tsc < 1 and +diffusion dominates. In this diffusion dominated regime, CRs diffuse out of eddies before they contribute significantly +to resisting compression – i.e., they do not provide a significant restoring force (instead, they provide drag). +In +particular, they do not contribute to the phase velocity vph. Thus, δρ/ρ ∼ δPg/Pg ∼ v/cs, where cs is the gas sound +speed, independent of PCR. This is roughly consistent with Commer¸con et al. (2019) (see their Figures 5 and 6), which +finds a similar dependence on κ and a clear decrease in δPCR/PCR as PCR/Pg increases. +Using this information, we can better interpret the lower bound on damping time that we infer from our simulations. +Importantly, as Ptuskin damping saturates (PCR/Pg → ∞, fCR → 1), the maximum rms CR pressure perturbation +is ⟨∆PCR⟩rms ∼ ρv2. +This is a strict upper bound, since the free energy to create CR pressure perturbations is +derived from kinetic energy (similarly, ∆PCR, ∆Pg at a shock cannot exceed the ram pressure ρv2). In this limit, +∆PCR/PCR ∼ ρv2/PCR ∝ 1/PCR. Finally, in the diffusion dominated limit, CR compression is balanced by diffusion, +PCR,0(∇ · v) ∼ −∇ · (κ∇PCR,1) ∼ κρv2/L2, which implies that +∇ · v ∝ +κ +PCR,0 +, +(A1) + +18 +Bustard & Oh +10 +2 +10 +1 +100 +101 +102 +PCR/Pg +10 +2 +10 +1 +/ +v/v +PCR/PCR +v/cs = +s +v/vph = +ph +v/v2 +ph +1/PCR +0.15L0vph += 0 +Figure 9. Fluctuating density (blue symbols), velocity (black symbols), and CR pressure (green symbols) as a function of +PCR/Pg. Open circles denote simulations with non-zero diffusion coefficient κ|| ∼ 0.15L0vph, and open diamonds denote purely +advective CR transport (κ = 0). None of these simulations include additional streaming transport. +where PCR,0, PCR,1 refer to the unperturbed and perturbed CR pressure, respectively. Thus, in the regime where we fix +the sweet-spot diffusion coefficient κ ∼ vphL0 and PCR/Pg ∼> 1 (so that vph ∝ P 1/2 +CR ), then κ ∝ P 1/2 +CR , and ∇·v ∝ P −1/2 +CR . +We have also verified in our simulations that ∇ · v ∝ κ for constant PCR, and ∇ · v ∝ 1/PCR for constant κ. +These results thus indicate that the damping time cannot become arbitrarily small. If drag forces are given by: +˙v ∼ 1 +ρ∇PCR,1 ∼< v2 +L +(A2) +(where PCR,1 ∼< ρv2), this gives a damping time tdamp ∼ v/˙v ∼> L/v ∼ teddy. +Thus, δPCR ∼< ρv2 implies that +tdamp ∼> teddy. This is equivalent to the statement that the work done by CR forces in opposing gas motions cannot +exceed the energy input rate: v · ∇PCR,1 ∼< ˜ϵ ∼ ρv3/L, which implies ∇PCR,1 ∼< ρv2/L, consistent with Equation +A2. Thus, for Kolmogorov turbulence, we expect tcascade/tdamp ∼ 1 for maximally efficient Ptuskin damping. In +the Kraichnan case, however, tcascade/tdamp can be greater than 1 if the Mach number, relative to the velocity of +compressible fluctuations, is small. This is not due to any decrease in the damping time; instead, it is due to cascade +times being lengthened when vph is large. +A.1. Time Convergence +Figure 10 shows the average kinetic energy spectra for 2563 simulations with varying CR transport model, measured +at different time intervals. Most importantly, the diffusion-only simulations show converged, clearly damped spectra +even at early times. Spectra for simulations with CR streaming are also well-converged but at somewhat later times. +Note that these time intervals over which we pull out kinetic energy spectra are much later than the saturation of bulk +turbulent quantities (e.g. kinetic energy, magnetic energy, etc.), which occurs after only a few eddy turnover times. +A.2. Resolution Convergence +A good test of how inherently diffusive our CR module is, and whether that accounts for some observed spectral +changes, is to run simulations with no explicit CR diffusion at various resolutions. Figure 11 compares spectra for +our β = 10 MHD simulations to simulations with PCR ∼ Pg and purely advective CR transport (no streaming and +κ|| = 0). For grid sizes of 2563 and 5123, we see in both cases that, in the inertial range up until k ∼ 20, there is + +Cosmic Ray Effects on Turbulence +19 +100 +101 +102 +k +10 +1 +100 +k2E(k)dk +10 +MHD +|| +0.15vphL0 +Time Interval +6.1 - 7.4 teddy +8.6 - 9.9 teddy +11.1 - 12.3 teddy +100 +101 +102 +k +10 +1 +100 +k2E(k)dk +10 +|| +0 + Streaming +|| +0.15vphL0 + Streaming +Time Interval +6.1 - 7.4 teddy +8.6 - 9.9 teddy +11.1 - 12.3 teddy +Figure 10. Time convergence of select spectra, each run on a 2563 grid. The diffusion-only simulations, which show the most +damping, converge very early. 5123 simulations (not shown) are similarly converged with respect to time. +100 +101 +102 +k +10 +2 +10 +1 +100 +k2E(k)dk +10 +2563 +5123 +100 +101 +102 +k +10 +2 +10 +1 +100 +E(k)/E(k)MHD +2563 +5123 +MHD +|| +0 +Figure 11. Comparison of kinetic energy without CRs (MHD) and with CRs but no transport (κ|| ∼ 0), simulated on grids +with 2563 and 5123 cells. 5123 simulation results are divided by a factor of 10 to separate those curves from the 2563 results. +no appreciable damping due to the presence of CRs, confirming again that CR transport is the cause for clear and +obvious damping seen in Figures 4, 6, 8 beginning at small k. +Figure 12 shows kinetic energy spectra for 5123 simulations when transport is included. These simulations only +comprise part of those on a 2563 grid (compare to Figure 6 in §4.4) because computer resource limits prohibit us +from running the streaming only (κ ∼ 0 + streaming) simulations. In any case, the streaming only simulations and +the streaming + diffusion simulations are both streaming dominated in this sub-Alfv´enic regime, so we expect their +spectra to look very similar, as we saw in Figure 6. +The MHD and diffusion only spectra look qualitatively similar to those on a 2563 grid. Most importantly, diffusive +transport leads to significant damping compared to the MHD case at all β tested (β =1 and 10). As in §4.4, streaming +transport instead appears to uniformly decrease kinetic energy at all scales, and this is β dependent with β ∼ 1 +showing almost no difference between MHD and CR cases. There is some resolution dependence for β = 10, with 5123 +showing less damping compared to the 2563 run, but the difference is mild, especially compared to the heavily damped +diffusion-only simulations. + +20 +Bustard & Oh +10 +2 +10 +1 +100 +1 +k2E(k)dk +10 +2 +10 +1 +100 +E(k)/E(k)MHD +10 +2 +10 +1 +100 +10 +10 +2 +10 +1 +100 +100 +101 +102 +k +10 +2 +10 +1 +100 +100 +100 +101 +102 +k +10 +2 +10 +1 +100 +MHD +|| +0.15vphL0 +|| +0.15vphL0 + Streaming +Figure 12. Kinetic energy spectra for a partial simulation suite run on a 5123 grid instead of a 2563 grid (compare to Figure +6 in §4.4). Computer resource limits prohibit us from running the streaming only (κ ∼ 0 + streaming) simulations of §4.4 on a +5123 domain, but all other spectra look qualitatively similar to those on a 2563 grid; namely, diffusion only transport shows clear +differences in spectral slope at both β = 1 and 10. Streaming simulations instead appear to uniformly decrease kinetic energy +at all scales as β increases. Note there is some resolution dependence for β = 10, with 5123 showing less damping compared to +the 2563, but the difference is mild, especially in comparison to the diffusion-only simulations. + +Cosmic Ray Effects on Turbulence +21 +REFERENCES +Ackermann, M., Ajello, M., Albert, A., et al. 2014, ApJ, +787, 18 +Amato, E., & Blasi, P. 2018, Advances in Space Research, +62, 2731 +Bai, X.-N., Ostriker, E. C., Plotnikov, I., & Stone, J. M. +2019, ApJ, 876, 60 +Becker Tjus, J., & Merten, L. 2020, PhR, 872, 1 +Blasi, P., Amato, E., & Serpico, P. D. 2012, PhRvL, 109, +061101 +Boulares, A., & Cox, D. P. 1990, ApJ, 365, 544 +Brunetti, G., & Jones, T. W. 2014, International Journal of +Modern Physics D, 23, 1430007 +Brunetti, G., & Lazarian, A. 2011, MNRAS, 410, 127 +Bustard, C., & Oh, S. P. 2022, ApJ, 941, 65 +Bustard, C., & Zweibel, E. G. 2021, ApJ, 913, 106 +Butsky, I. S., & Quinn, T. R. 2018, ApJ, 868, 108 +Chandran, B. D. G. 2000, PhRvL, 85, 4656 +Chandran, B. D. G., & Maron, J. L. 2004, ApJ, 603, 23 +Cho, J., & Lazarian, A. 2003, MNRAS, 345, 325 +Commer¸con, B., Marcowith, A., & Dubois, Y. 2019, A&A, +622, A143 +Drury, L. O. C., & Strong, A. W. 2017, A&A, 597, A117 +Eswaran, V., & Pope, S. B. 1988, Computers and Fluids, +16, 257 +Farmer, A. J., & Goldreich, P. 2004, ApJ, 604, 671 +Felice, G. M., & Kulsrud, R. M. 2001, ApJ, 553, 198 +Field, G. B. 1965, ApJ, 142, 531 +Giacalone, J., & Jokipii, J. R. 1999, ApJ, 520, 204 +Goldreich, P., & Sridhar, S. 1995, ApJ, 438, 763 +Hanasz, M., Strong, A. W., & Girichidis, P. 2021, Living +Reviews in Computational Astrophysics, 7, 2 +Holcomb, C., & Spitkovsky, A. 2019, ApJ, 882, 3 +Hopkins, P. F., Squire, J., Butsky, I. S., & Ji, S. 2022, +MNRAS, 517, 5413 +Hu, Y., Lazarian, A., & Xu, S. 2022, MNRAS, 512, 2111 +Hunter, J. D. 2007, Computing in Science & Engineering, 9, +90 +Ji, S., Chan, T. K., Hummels, C. B., et al. 2020, MNRAS, +496, 4221 +Jiang, Y.-F., & Oh, S. P. 2018, ApJ, 854, 5 +Kempski, P., & Quataert, E. 2020, MNRAS, 493, 1801 +—. 2022, MNRAS, 514, 657 +Kulsrud, R., & Pearce, W. P. 1969, ApJ, 156, 445 +Landau, L. D., & Lifshitz, E. M. 1987, Fluid Mechanics +(Butterworth-Heinemann) +Lynn, J. W., Parrish, I. J., Quataert, E., & Chandran, B. +D. G. 2012, ApJ, 758, 78 +McCourt, M., Sharma, P., Quataert, E., & Parrish, I. J. +2012, MNRAS, 419, 3319 +Mertsch, P. 2020, Ap&SS, 365, 135 +Miniati, F. 2015, ApJ, 800, 60 +Mohapatra, R., Federrath, C., & Sharma, P. 2022, +MNRAS, 514, 3139 +Nazarenko, S. 2011, Wave turbulence, Vol. 825 (Springer +Science & Business Media) +Pinzke, A., Oh, S. P., & Pfrommer, C. 2017, MNRAS, 465, +4800 +Ptuskin, V. S. 1981, Ap&SS, 76, 265 +—. 1988, Soviet Astronomy Letters, 14, 255 +Putman, M. E., Peek, J. E. G., & Joung, M. R. 2012, +ARA&A, 50, 491 +Quataert, E., Jiang, F., & Thompson, T. A. 2022, MNRAS, +510, 920 +Reichherzer, P., Becker Tjus, J., Zweibel, E. G., Merten, L., +& Pueschel, M. J. 2020, MNRAS, 498, 5051 +Sampson, M. L., Beattie, J. R., Krumholz, M. R., et al. +2022, MNRAS, arXiv:2205.08174 +Silk, J. 1968, ApJ, 151, 459 +Skilling, J. 1971, ApJ, 170, 265 +—. 1975, MNRAS, 172, 557 +Stone, J. M., Tomida, K., White, C. J., & Felker, K. G. +2020, ApJS, 249, 4 +Thornbury, A., & Drury, L. O. 2014, MNRAS, 442, 3010 +Towns, J., Cockerill, T., Dahan, M., et al. 2014, Computing +in Science & Engineering, 16, 62. +doi.ieeecomputersociety.org/10.1109/MCSE.2014.80 +Tsung, T. H. N., Oh, S. P., & Jiang, Y.-F. 2022, MNRAS, +513, 4464 +Turk, M. J., Smith, B. D., Oishi, J. S., et al. 2011, ApJS, +192, 9 +Uhlenbeck, G. E., & Ornstein, L. S. 1930, Physical Review, +36, 823 +Wang, C., Oh, S. P., & Ruszkowski, M. 2022, arXiv +e-prints, arXiv:2205.01732 +Wentzel, D. G. 1968, ApJ, 152, 987 +Wiener, J., Zweibel, E. G., & Oh, S. P. 2013, ApJ, 767, 87 +Wolfram Research, Inc. 2021, Mathematica, Version 13.0.0, +, . https://www.wolfram.com/mathematica +Yan, H., & Lazarian, A. 2002, PhRvL, 89, 281102 +—. 2004, ApJ, 614, 757 +Zweibel, E. G. 2017, Physics of Plasmas, 24, 055402 + diff --git a/ttE2T4oBgHgl3EQf2AhI/content/tmp_files/load_file.txt b/ttE2T4oBgHgl3EQf2AhI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..38350d0e7dbdc96d5f59eea3a7075d0585dfa7cf --- /dev/null +++ b/ttE2T4oBgHgl3EQf2AhI/content/tmp_files/load_file.txt @@ -0,0 +1,1026 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf,len=1025 +page_content='Draft version January 12, 2023 Typeset using LATEX twocolumn style in AASTeX631 Cosmic Ray Drag and Damping of Compressive Turbulence Chad Bustard1 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Peng Oh2 1Kavli Institute for Theoretical Physics, University of California - Santa Barbara, Kohn Hall, Santa Barbara, CA 93107, USA 2Department of Physics, University of California - Santa Barbara, Broida Hall, Santa Barbara, CA 93106, USA Submitted to ApJ ABSTRACT While it is well-known that cosmic rays (CRs) can gain energy from turbulence via second order Fermi acceleration, how this energy transfer affects the turbulent cascade remains largely unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Here, we show that damping and steepening of the compressive turbulent power spectrum are expected once the damping time tdamp ∼ ρv2/ ˙ECR ∝ E−1 CR becomes comparable to the turbulent cascade time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Magnetohydrodynamic (MHD) simulations of stirred compressive turbulence in a gas-CR fluid with diffusive CR transport show clear imprints of CR-induced damping, saturating at ˙ECR ∼ ˜ϵ, where ˜ϵ is the turbulent energy input rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In that case, almost all the energy in large scale motions is absorbed by CRs and does not cascade down to grid scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This “divergence-cleaning” should render small- scale turbulence largely solenoidal and could suppress fluctuations important for thermal instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The lack of small-scale compressive modes is also problematic for hypothesized resonant scattering of E ∼> 300 GeV CRs, when self-confinement is inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' When CR transport is streaming dominated, CRs also damp large scale motions, with kinetic energy reduced by up to to an order of magnitude in realistic ECR ∼ Eg scenarios, but turbulence (with a reduced amplitude) still cascades down to small scales with the same power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Such large scale damping implies that turbulent velocities obtained from the observed velocity dispersion may significantly underestimate the turbulent forcing rate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' ˜ϵ ≫ ρv3/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' These findings motivate future, higher resolution simulations with a mixture of turbulent driving modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' INTRODUCTION Cosmic rays (CRs) and magnetized turbulence are both ubiquitous in the Universe, and their interplay has long been a fascinating topic of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Fluctuations at the small-scale end of a turbulent cascade, on scales of order the CR gyroscale, are frequently invoked to scat- ter individual CRs, creating the high degree of observed CR isotropy and the long residence times of CRs in the Milky Way disk and its surrounding halo relative to the light crossing time (Amato & Blasi 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Becker Tjus & Merten 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In such a scenario, dubbed the “extrin- sic turbulence” model (Zweibel 2017), the resulting bulk CR transport is magnetic field-aligned diffusion, with an energy-dependent spatial diffusion coefficient κ|| and CR flux FCR ∝ κ||∇PCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' CRs in this picture can also gain energy from repeated scattering off gyroscale fluctua- Corresponding author: Chad Bustard bustard@ucsb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='edu tions, a second order Fermi mechanism called “resonant reacceleration.” Phenomenological models of CR propagation fit to di- rect and indirect CR observables (Hanasz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2021) have traditionally assumed a Kolmogorov scaling for tur- bulence, appropriate for hydrodynamic turbulence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' how- ever, our understanding of CR scattering by turbulence has been refined over time with new insights into magne- tohydrodynamic (MHD) turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Most profoundly, MHD turbulence differs from hydrodynamic turbulence in that MHD forces and hence turbulence are no longer isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The resulting anisotropy of slow and Alfv´en modes (Goldreich & Sridhar 1995) makes them ineffi- cient CR scatterers, as CRs interact with multiple un- correlated eddies during one gyro-orbit, essentially can- celing out gyroresonant contributions from each eddy (Chandran 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Compressible fast modes, whose velocities are inde- pendent of magnetic field direction, are more isotropic (Cho & Lazarian 2003) and therefore considered the best arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='04156v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='HE] 10 Jan 2023 2 Bustard & Oh candidate for CR scattering (Yan & Lazarian 2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' al- though, the degree of isotropy decreases with decreasing scale due to strong collisionless and viscous damping, hence the efficacy of CR scattering decreases with de- creasing CR energy (Kempski & Quataert 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Fast mode scattering, then, is most plausible for higher en- ergy CRs (E > 300 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For E < 300 GeV, where most of the CR energy resides, CRs can largely create scattering perturba- tions themselves through a resonant streaming insta- bility (Wentzel 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Kulsrud & Pearce 1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The resulting transport is no longer purely diffusive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' in- stead, CRs “stream” down their field-aligned pressure gradient at the local Alfv´en speed vA = B/√4πρ with FCR ∝ vAPCR, and additional, energy-dependent CR diffusivity (FCR ∝ ∇PCR) is introduced by wave damp- ing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' ion-neutral damping, nonlinear Landau damp- ing, and turbulent damping (Skilling 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Farmer & Goldreich 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Blasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Wiener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Zweibel 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Bustard & Zweibel 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' There is also an important difference regarding energy transfer be- tween CRs and hydromagnetic waves: whereas extrinsic turbulence is generated externally, in self-confinement, the free energy to generate waves comes from the CRs themselves, and this energy is subsequently dissipated into the thermal gas via wave damping at a rate H = −dECR/dt = vA · ∇PCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We refer to this collisionless energy transfer as streaming energy loss / gas heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' While considerable effort has been put towards ex- ploring resonant-scale interactions between CRs and ei- ther self-generated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Skilling 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Felice & Kulsrud 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Holcomb & Spitkovsky 2019) or externally driven (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Giacalone & Jokipii 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Yan & Lazarian 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Reichherzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2020) waves, some- what less focus has been given to the interplay between CRs and turbulence on scales much larger than a CR gyroradius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In this regime, the collective CR population is well-described as a fluid, and CRs experience com- pressions and rarefactions in the turbulent flow, leading to energy transfer between the CRs and turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' To distinguish this from its resonant-scale counterpart, the flow of energy from turbulence to the bulk CR fluid is called non-resonant reacceleration (Ptuskin 1988), and its efficiency depends on CR transport model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For purely diffusive CR transport, non-resonant reac- celeration is maximally efficient when CRs are well- trapped in the turbulent flow (κ < vphL0, where vph is the phase speed of compressive fluctuations and L0 is the outer eddy scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' When streaming is taken into account, the interaction between perturbed CR and gas variables is fundamentally altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' While CR diffusion introduces a π/2 phase shift between CR and density perturbations, leading to a CR force that damps fluctu- ations much like a damped harmonic oscillator, both the change in flux (FCR ∝ PCR instead of FCR ∝ ∇PCR) and the associated energy loss that accompany streaming transport modify the CR force (Tsung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' As we showed in Bustard & Oh (2022) (from now on referred to as Paper I), CR reacceleration / turbulent damping rates become dependent on plasma β = Pg/PB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' they remain largely unchanged in high-β plasmas like the intracluster medium (ICM) where reac- celeration is a leading explanation for radio halos (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Brunetti & Lazarian 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Brunetti & Jones 2014), but they are stunted significantly in low-β plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Despite non-resonant reacceleration being a fairly in- efficient process compared to diffusive shock acceleration (a first order Fermi mechanism), with minimum growth times lengthened even further by streaming transport, it was pointed out by Thornbury & Drury (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Drury & Strong (2017) that a significant fraction of total CR power in galaxies could come from reacceleration, conse- quently creating a large sink for turbulent energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In this paper, we present analytical estimates and CR+MHD simulations suggesting that CRs in very plausible astro- physical environments can divert significant amounts of turbulent energy, essentially acting as an unsual form of viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The outcome is a CR-modified route to gas heating, rather than the typical conversion to heat at the dissipation scale, and a damped turbulent energy spec- trum with decreased small-scale, compressive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' These changes are, of course, strongest in environ- ments where CRs are dynamically important such as the ISM (where CR energy densities are roughly in equipar- tition with turbulent and magnetic energy densities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Boulares & Cox 1990) and the Milky Way circumgalac- tic medium (which may be energetically dominated by CRs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2020), but they would affect any pro- cess that relies on compressive motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For instance, compressions seed thermal instability (Field 1965;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Mc- Court et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Mohapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2022), which is fre- quently invoked, for instance, to explain the existence of cold CGM clouds (Putman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Fluctuations that scatter CRs are not immune to these modifications either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Low-energy, self-confined CRs could sap energy from the turbulent fast mode cascade at large scales, de- creasing the available small-scale power needed to scat- ter high energy CRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This paper is outlined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In §2, we discuss our simulation method and setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In §3, we analytically es- timate and then quantify in simulations the fractions of turbulent driving and gas heating that are channeled through CRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We then analytically derive how CR- induced damping should affect MHD turbulence spectra Cosmic Ray Effects on Turbulence 3 (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='1) and the conditions under which damping rates can exceed cascade rates (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='3, we present exploratory simulations strongly suggestive of these an- alytic estimates and show sensitivities to streaming vs diffusive CR transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We discuss regimes of applica- bility and implications in §5 and conclude in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' SIMULATION SETUP We begin by briefly describing the simulation method- ology and setup, which is described in more detail in Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Using the Athena++ MHD code (Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2020) coupled with an additional CR module that mod- els CR diffusive and streaming transport in a fluid ap- proximation using a two-moment method originally de- veloped for radiation transport (Jiang & Oh 2018), we numerically solve the ideal MHD equations plus two ad- ditional equations for the CR energy and energy flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We stir turbulence following an Ornstein-Uhlenbeck ran- dom process (Uhlenbeck & Ornstein 1930;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Eswaran & Pope 1988), randomly generating velocity perturbations between modes k = 1 and 3 in a cubic box of width 2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For driving, we set the autocorrelation timescale to be tcorr = L/cs and drive fluctuations every tdrive = 2 × 10−3(L/cs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For the parameter scans in §3, we use grids of size 1283 and 2563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We simulate fluids with ei- ther an isothermal equation of state, where the thermal energy is fixed, or an adiabatic equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The latter results in a gradual rise in the gas pressure due to a combination of CR heating and grid-scale dissipation of the cascade, which we decompose and quantify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' These simulations all use purely compressive forcing, with two turbulent driving rates ˜ϵ = dE/dt, resulting in approx- imately Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15 and Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5 turbulence with a weak dependence on plasma β since MHD forces coun- teract motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We avoid solenoidal driving to avoid turbulent amplification of magnetic fields, so that we can evolve simulations at approximately fixed plasma β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' To a good approximation, solenoidal driving only ampli- fies magnetic fields, while compressive driving energizes CRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' At our parameter scan resolution of 2L/256, the cas- cade exhibits only a short inertial range, and in test- ing we find that the spectral slope in pure MHD runs (no CRs) is intermediate between E(k) ∼ k−2 and E(k) ∼ k−3/2 – a shallower slope is expected for com- pressive fast modes, but the exact exponent has been highly debated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In our analytic estimates (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='1), we will explore CR-induced deviations to different initial spectra, but we particularly note significant changes to Kraichnan turbulence where E(k) ∼ k−3/2 initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='3, where we want to test deviations from this spec- trum due to CR drag, we increase the resolution to 2L/512, though we find that the main trends are well- recovered even with a resolution of 2L/256 (see Ap- pendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Higher resolution simulations giving a larger inertial range would be preferable, but to ensure an ac- curate treatment of CR propagation and influence, the two-moment method has an effective, maximum speed of light parameter vm that must be much larger than other propagation speeds in the system and that sets the Courant-limited timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In Paper I, we found that vm ∼ 50cs gives seemingly converged CR heating rates and reacceleration rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' With this choice, our MHD+CR simulations are about a factor of 8 more ex- pensive than pure hydro turbulence sims, prohibiting us from going to much higher resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' COSMIC RAY DIVERSION OF TURBULENT ENERGY We’ll begin with a short review of non-resonant reac- celeration (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Ptuskin 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Chandran & Maron 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Lynn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2012 and §2 of Paper I for greater detail) and its relation to the turbulent damping rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Variables used in our discussion are summarized in Ta- ble 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' As discussed in Paper I, “drag” against CRs pro- vides a frictional force on compressive motions known as Ptuskin damping (Ptuskin 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' It is similar to ra- diative damping of sound waves, which famously leads to Silk damping of acoustic waves in the early universe (Silk 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In general, since Ek/tdamp ∼ PCR/tgrow, we have1: tdamp ∼ ρv2max �tgrow PCR , 1 ˜ϵ � ∼ max � M2 ctgrow, tinject � (1) where Mc ≡ v/cc is the Mach number in units of the CR effective sound speed, cc ∼ � PCR/ρ, and tinject ≡ ρv2/˜ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Equation 1 is a general expression for the damping time, for which one can plug in the appropriate tgrow, the CR reacceleration (or growth) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Working in the limit of purely diffusive spatial CR transport with isotropic diffusion coefficient κ, the reac- celeration time can be derived in two limits depending on the ratio of diffusion time tdiff = L2 0/κ to compressive wave crossing time tsc = L0/vph across an eddy of length L0 in a medium with compressive phase velocity vph ∼ (Ptot/ρ)1/2 ∼ [Pg+PB+PCR)/ρ]1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In the fast diffusion limit (tdiff ≪ tsc, or equivalently, κ ≫ vphL0), deriving the CR momentum diffusion coefficient Dpp follows the textbook argument for second order Fermi acceleration: Dpp ∼ (∆p)2/τscatter ∼ p2v2/(c2τscatter) ∼ p2v2/κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The 1 In this paper, we use the notation ˜ϵ ≡ ρv3/L and ϵ ≡ v3/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 4 Bustard & Oh Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Simulation parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' CR module settings,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' and other variable definitions Parameter Definition / Setting / Equation Additional Notes L Half box size k = 2 mode L0 Outer eddy scale k = 3 mode tdrive 2 × 10−3(L/cs) Turbulence driven every tdrive tcorr L/cs Autocorrelation time ˜ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' ϵ Input turbulent energy rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' dE/dt ρv3/L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' v3/L in hydro turbulence vm 50cs Effective maximum speed of light κ CR diffusion coefficient Assumed to be field-aligned only (κ = κ||) β Pg/PB Plasma beta cs � γPg/ρ Gas sound speed vph � (γPg + γCRPCR + PB)/ρ Compressive wave phase speed vA B/√4πρ Alfv´en speed cc � γCRPCR/ρ Effective CR sound speed Ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Mph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Mc v/cs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' v/vph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' v/vA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' v/cc Mach numbers H vA · ∇PCR “Collisionless” CR loss rate / gas heating rate fCR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' fth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' fCR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='heating ˙ECR/˜ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' ˙Eth/˜ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' < H >/˜ϵ Fraction of ˜ϵ → CRs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' thermal gas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' CR heating E(k) Kinetic energy spectrum ∝ k−5/3 (Kolmogorov),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' k−2 (Burgers),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' k−3/2 (Kraichnan) tinject L/v Energy injection time tcascade kE(k)/F(k) Cascade time (see Equation 10) tgrow p2/Dpp CR reacceleration time (§3 and Paper I) tdamp ∼ ρv2max � tgrow PCR ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 1 ˜ϵ � ∼ max � M2 ctgrow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' tinject � Turbulent damping time (Equation 1) energy growth time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' defined as p2/Dpp is tgrow ∼ κ v2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' κ >> vphL0 (2) In the opposite limit of slow diffusion (tdiff ≫ tsc, or equivalently, κ ≪ vphL0), Dpp ∼ (δp)2/τdiff ∼ (p2v2/v2 ph)(κ/L2 0), and the growth time is tgrow ∼ p2 Dpp ∼ v2 phL2 0 v2κ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' κ << vphL0 (3) Joining the two regimes in the middle, the minimum growth time is tgrow ∼ (vphL0/v2) when κ ∼ vphL0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Strictly speaking, these scalings are appropriate if CR diffusion is isotropic, if streaming is negligible, and if all reacceleration comes from eddies of a single scale L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Relaxing these assumptions introduces further modifi- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In the fast diffusion limit (κ ≫ vphL0), there are also correction factors that decrease the growth time if anisotropic rather than isotropic spatial diffusion is accounted for (Chandran & Maron 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Additional streaming transport, widely applicable for CRs with en- ergy E ⪅ 300 GeV, introduces a correction factor that decreases reacceleration rates by fcorr = 1 − � 2/β and fcorr = (1 − � 2/β)1/2 in the slow and fast diffusion regimes, respectively (Paper I);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' and in the slow diffusion limit (κ ≪ vphL0), multiple eddies contribute to reaccel- eration, with relative contributions dependent upon the shape of the turbulent power spectrum (see Equation 4 in Paper I for a more general expression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' If the wave spectrum is Burgers-like (E(k) ∼ k−2), roughly consis- tent with our simulations, eddies at each logarithmic interval in the inertial range contribute equally to reac- celeration, and tgrow has a broad minimum of tgrow ∼ (vphL0/v2) throughout the entire range of κ|| < vphL0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' If we work in the single eddy limit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', we only con- sider eddies of size L0), in the fast diffusion (κ ≫ vphL0) regime, where tgrow ∼ κ/v2 then Equation 1 gives tdamp ∼ κ/c2 c, in agreement with the classic (much more detailed) calculation of this effect by Ptuskin (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Working instead in the broad regime of maximal reac- celeration, where CRs are well-trapped in the turbulent flow (when κ < vphL0), the characteristic growth time is tgrow ∼ (vphL0/v2), which gives: tdamp ∼ max �vphL0 c2c , tinject � (4) Note that tdamp is velocity independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' With these reacceleration times in mind, we can now estimate the fraction of turbulent energy forcing ˜ϵ that goes toward CRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' It is given by fCR ∼ ˙ECR ˜ϵ ∼ ECR ˜ϵtgrow (5) Cosmic Ray Effects on Turbulence 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='0 fi fCR saturates fCR follows analytic expectation 10 1 100 101 102 PCR/Pg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='0 fi fCR > 0 due to numerical dissipation 2 3 phPCR v2 fCR fth Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The average CR energy gain rate and thermal energy gain rate relative to the turbulent driving rate (fCR = ˙ECR/˜ϵ and fg = ˙Eg/˜ϵ, respectively) for simulations without streaming, as a function of PCR/Pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' These all are adiabatic, Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5 simulations on a 1283 grid, with β ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Top: κ|| ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15L0vph, where CR energy gain is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The dashed black curve is the analytic expectation from Equation 6, showing good agreement when PCR/Pg < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Bottom: κ|| ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For PCR ≫ Pg, even κ ∼ 0 leads to significant fractions of turbulent energy converted to CR energy, but this CR reacceleration is due to numerical diffusion caused by finite resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For example, for Kolmogorov turbulence, where ˜ϵ ∼ ρv3/L, and for the characteristic growth time tgrow ∼ 9/2vphL/v2 this gives: fCR ∼ ECR/tgrow ρv3/L ∼ max �2 3Mph PCR ρv2 , 1 � (6) The maximum value of 1 reflects energy conservation: CRs cannot gain more energy than is injected by tur- bulent forcing, hence fCR ∼ 2 3Mph PCR ρv2 is only valid for ˙ECR < ˜ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Within this regime, the fraction of kinetic energy deposited into CRs is small if PCR ≪ ρv2, in which case most energy is deposited in the thermal gas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' however, for higher PCR, the fraction increases and can become quite substantial at close to equipartition values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Figure 1 compares this expectation to simulations and is one of the key results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The y-axis shows the partitioning of the input energy rate into CRs (fCR = ˙ECR/˜ϵ) and thermal energy (fth = ˙Eth/˜ϵ) for varying PCR/Pg, keeping ˜ϵ fixed, for purely diffusive CRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Unlike our previous simulations, which all used an isothermal equation of state, these simulations have an adiabatic equation of state, which makes it easier to con- firm energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Together, the contributions to ˙ECR and ˙Eth sum to ∼ 80−90% of the driving rate, with the rest going towards small magnetic and kinetic energy increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The top and bottom panels show simulations each without streaming and with κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15L0vph and κ = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For PCR/Pg < 1, fCR follows the expectation from Equation 6 (shown as a black dashed line) quite well, an indication that turbulent reacceler- ation is diverting the driving energy to CRs at the ex- pense of thermal gas heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Similar simulations with2 κ = 0 show far lower fCR, again revealing the depen- dence of reacceleration on diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Note that while we previously only tested analytic expecta- tions for the growth time tgrow (on which Equation 6 depends) when the gas is isothermal in Paper I, they continue to hold when the gas is adiabatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' As PCR/Pg increases, fCR deviates from the analytic expression in Equation 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' fCR increases more slowly to- wards the asymptotic bound fCR ∼ 1 than in our ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Nonetheless, for PCR/Pg ∼> 1, what immediately stands out is the large fraction of energy diverted to CRs, with fCR as large as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='8 when PCR/Pg > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' These large val- ues of fCR clearly come at the expense of thermal heat- ing3, with fth decreasing from fth ≈ 1 when PCR ≪ Pg to fth < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='2 when PCR > Pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In the above, purely diffusive case, turbulent energy directly accelerates CRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' When streaming is included, energy is also lost to collisionless heating at a rate H = vA·∇PCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In Fig 2, we quantify the partitioning of turbulent kinetic energy into direct acceleration of CRs (fCR) and gas heating (fth) in simulations with fixed ˜ϵ producing undamped Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We distinguish be- tween collisionless heating by CRs fCR,heating (red bars), and heating due to turbulence which cascades down to the grid scale and dissipates fth − fCR,heating (orange bars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' When streaming is included, fCR is a small and weakly increasing function of β, consistent with Paper I and evident in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Here, we fix the initial state to have PCR ∼ Pg for each simulation and quantify the CR energy gain rate as we did in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Despite the fact that a negligible fraction of energy fCR is ends 2 In practice, κ has a non-zero value because of numerical diffusion, but here this has little impact up until PCR ≫ Pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 3 Since we enforce purely compressive driving, magnetic field am- plification is very weak, and fCR+fth ≈ 1 for an adiabatic setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 6 Bustard & Oh Diffusion Only Streaming Only Diff + Stream Adiab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Stream Only Adiab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Diff + Stream Diffusion Only Streaming Only Diff + Stream Adiab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Stream Only Adiab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Diff + Stream Diffusion Only Streaming Only Diff + Stream Adiab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Stream Only Adiab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Diff + Stream 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='8 1 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Partitioning of input turbulent energy rate ˜ϵ into three different channels: CR reacceleration fCR, dissipation via CR collisionless heating fCR,heating (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' streaming energy loss), and grid-scale heating fth − fCR,heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Without CRs, this choice of ˜ϵ produces Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15 turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Each simulation here starts with PCR ∼ Pg but with varying CR transport treatments, either with diffusion only (all with κ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15vphL0) or diffusion plus additional streaming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For each β, the first three simulations use an isothermal equation of state, so there is no gas heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The last two, denoted by “Adiab.”, use an adiabatic equation of state, in which case the total thermal gas heating rate is the sum of CR heating and grid-scale heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' With diffusion only, reacceleration is very efficient: most turbulent energy is soaked up by CRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' With streaming, both gas heating and CR energization are relatively inefficient in the low-β regime, but for β ∼ 10, 100, CR heating is the dominant energy channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Instead of turbulent energy cascading to small scales and eventually dissipating into thermal energy at the grid scale, CRs intercept this energy transfer at large scales;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' astoundingly, even in these subsonic flows very high fractions of turbulent energy are channelled through CRs when PCR ∼> Pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' up in CRs, the latter nonetheless have a strong impact on the turbulent cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In MHD simulations, turbu- lence cascades to grid scales where numerical diffusion dominates4 and subsequently dissipates, heating the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Thus, fth is a good barometer of how much kinetic en- ergy flux makes it to the dissipation scale;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' however, that is not the case with turbulence modified by streaming CRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In the right panel of Figure 1, the decrease in fth with increasing β is marginal, but actually much of that heating is done by CR streaming energy loss in- 4 In high resolution simulations with explicit viscosity, it would instead cascade to the viscous scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' stead of classical small-scale dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Across all β, only ≈ 10% of driving energy makes it to the grid scale, with most energy channeled through CRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Note that in our estimate of tdamp (Equation 4), we have not included the effects of CR streaming on tgrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' If we did, tdamp would be substantially longer in low β environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' However, as we have seen, this is incor- rect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' When CR streaming is present, the kinetic energy of compressive motions is still absorbed by CRs at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This energy is subsequently returned to the gas in the form of heat via CR streaming, and so streaming impedes the secular growth of CR energy, resulting in the lower growth times explored in Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' However, diversion of kinetic energy away from the turbulent cas- Cosmic Ray Effects on Turbulence 7 cade and damping of compressive motions still happens at a similar rate, even at low β (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' CR streaming provides an avenue for gas motions to quickly dissipate in the form of heat without going through the turbu- lent cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In this case, CRs can be thought of as providing an unusual form of viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' To summarize: once Pc/Pg ∼> 1, and for β ∼> 10, our simulations show that the energy input in turbu- lent driving appears to be almost completely diverted to CRs, with only ∼ 10% remaining which cascades down to grid scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This is irrespective of whether streaming is absent (in which case CRs store the energy) or present (in which case CRs thermalize a significant fraction via collisionless heating).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This is astonishing efficiency, con- sidering that strong shocks convert at best ∼ 10 − 30% of kinetic energy to CRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For β ∼ 1, the fraction of en- ergy routed through CRs is slightly lower, ∼ 80% in the diffusion only case, and ∼ 50% with both CR streaming and diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We now turn to some implications of this finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' COSMIC RAY IMPRINTS ON KINETIC ENERGY SPECTRA In certain regimes, CRs are clearly an important en- ergy sink for fluid motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' When turbulent energy is diverted to the CR population, it either 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Directly accelerates CRs through non-resonant reacceleration 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' (If CR streaming is significant) Heats the gas at scales lCR ≫ ldiss through collisionless energy transfer by self-confined CRs (streaming energy loss), where ldiss is the Kolmogorov dissipation scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In either case, energy that originally would have cas- caded to small scales is siphoned out of the turbulent cascade, and it is interesting to ask what imprint this might have on the kinetic energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In this sec- tion, we first focus on the effects of purely diffusive CRs, leaving an initial exploration of streaming CR transport, the effects of which are less straightforward and deserve future follow-up, to §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We will first explore CR mod- ifications to Kolmogorov and Kraichnan spectra analyt- ically and discuss astrophysical regimes where spectra could be heavily modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Of the compressible MHD modes, it is thought that slow modes have a Kolmogorov spectrum (E(k) ∝ k−5/3) and fast modes have a Kraich- nan spectrum (E(k) ∝ k−3/2) (Cho & Lazarian 2003), though this is still debated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In our simulations, compres- sive forcing gives rise to something intermediate between Kraichnan and Burgers turbulence (E(k) ∝ k−2), and we will see that CR damping also has noticeable effects in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Analytic Theory We can solve for the turbulent power spectrum by solving the dynamic equation (Landau & Lifshitz 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' If we consider a turbulent energy injection rate ϵ injected at some outer scale L = k−1 L (where ϵ ∼ v2 l /tcascade ∼ const in the absence of damping,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' and tcascade depends on the form of turbulence),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' then in steady state the com- bined effects of the cascade to smaller scales and damp- ing must balance injection: ϵ δ(k − kL) = ∂ ∂k F(k) + Γ(k)E(k) (7) where E(k) is the power spectrum of turbulence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' F(k) is the turbulent cascade flux in k-space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' and Γ(k) ∼ t−1 damp is the damping rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Following Equation 1 and adopting tgrow ∼ 9/2vphL/v2, it follows that Γ(k) ∼ 2/9c2 c/vphL, which is independent of scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The cascade flux de- pends on the type of turbulence: for Kolmogorov turbu- lence, F(k) ∼ [E(k)]3/2k5/2, while for isotropic Kraich- nan turbulence, F(k) ∼ k3[E(k)]2/vph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In the ab- sence of damping (Γ(k) = 0), integrating both sides of Equation 7 with respect to k gives E(k) ∼ ϵ2/3k−5/3 and E(k) ∼ (ϵvph)1/2k−3/2, the power spectra for Kol- mogorov and Kraichnan turbulence respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The first and second terms on the right hand side of Equation 7 have units of v2/k × (t−1 cascade, t−1 damp) respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 3, we solve Equation 7 for various values of tdamp/tcascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' It is easy to understand the asymptotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' When tcascade ≪ tdamp, the first term on the RHS dominates: injected energy cascades before it can damp, and we obtain the usual Kolmogorov/Kraichnan power spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' On the other hand, if tdamp ≪ tcascade, then the second term on the RHS dominates, which gives ϵ ∼ Γ � E(k)dk ∼ Γv2, or v2 ∼ ϵtdamp ∼ v2 0 � tdamp tcascade � (8) where v2 0 and tcascade are the velocity and cascade time at the outer scale in the absence of damping;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' for a given energy forcing ϵ, the velocity at the outer scale is re- duced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' However, since tdamp ∼> tinject, the damping time cannot be made arbitrarily small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We discuss this fur- ther in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' To understand the behavior at smaller scales, note that the cascade time is scale-dependent, while for non- resonant CR acceleration, tdamp is independent of scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We are accustomed to thinking of the cascade time de- creasing towards small scales (for instance, tcascade ∝ l2/3, l1/2 for undamped Kolmogorov, Kraichnan tur- bulence respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' However, damping changes the 8 Bustard & Oh 1 10 100 1000 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='1 1 E(k) k5\uf00c3 Kolmogorov 1 10 100 1000 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='1 1 E(k) k3\uf00c2 Kraichnan tcasc tdamp = 0 tcasc tdamp = 1 2 tcasc tdamp = 1 tcasc tdamp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='25 tcasc tdamp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5 tcasc tdamp = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 1 k2 1 k3 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Modified kinetic energy spectra for a Kolmogorov (left) and Kraichnan (right) cascade with varying levels of CR damping, all with vph = 2v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' E(k) is in units of the outer-scale, undamped kinetic energy, where k denotes the wavenumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Different line colors denote different ratios of the cascade time to the damping time, showing that if damping becomes competitive, the outer scale velocity decreases, and the slope of the spectrum steepens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Dashed lines show E(k) = k−2 and E(k) = k−3 for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For tcascade/tdamp ⪆ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5, 1 for Kolmogorov and Kraichnan, respectively, the cascade sharply cuts off at progressively smaller k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For smaller tcascade/tdamp, CRs damp fluctuations, but the cascade returns to its normal scaling at large k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' scale dependence of velocity, further reducing velocities at small scales, and thus increasing cascade times at these scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' If tcascade/tdamp still decreases towards small scales, then the cascade eventually takes over and the spectrum rebounds from damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' However, if tcascade/tdamp instead increases towards small scales, then damping becomes increasingly dominant and the spectrum will cut off precipitously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Since tdamp is inde- pendent of k, what matters is the scale dependence of tcascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' From Equation 7, the cascade time can be written as: tcascade ∼ kE(k) F(k) ∼ 1 [k3E(k)]1/2 (Kolmogorov) (9) ∼ vph k2E(k) (Kraichnan) (10) where we have used F(k) ∼ [E(k)]3/2k5/2, F(k) ∼ k3[E(k)]2/vph for Kolmogorov and Kraichnan turbu- lence respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' When damping operates, E(k) will steepen from standard Kolmogorov/Kraichnan spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' From Equation 10, we see that for a power spectrum E(k) ∝ k−α, tcascade increases with k for α ∼> 3 (Kolmogorov), α ∼> 2 (Kraichnan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The steepening of the power spectrum slope is controlled by the rel- ative strength of damping, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' tcascade/tdamp at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' If this is sufficiently large, it produces a power spectrum with a slope steeper than the critical value, and we have a runaway: tcascade/tdamp continually in- creases towards small scales, producing a rapid cutoff in the velocity power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' However, if the initial value of tcascade/tdamp produces a power spectrum with an index shallower than the critical slope, then damp- ing initially ‘takes a bite’ out of the turbulent cascade, but tcascade/tdamp decreases towards small scales, until damping becomes negligible, the original cascade domi- nates and the spectrum recovers its original undamped power law slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We clearly see confirmation of this bifurcation in small scale damping in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We see that we re- quire tcascade/tdamp ∼> 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5 at the outer scale for crit- ical damping in a Kolmogorov cascade (so that the power spectrum steepens beyond E(k) ∝ k−3), or tcascade/tdamp ∼> 1 at the outer scale for critical damp- ing in a Kraichnan cascade (so that the power spectrum steepens beyond E(k) ∝ k−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Indeed, tcascade/tdamp ∼ 1 causes a perfect transformation of the Kraichnan spec- trum from a E(k) ∝ k−3/2 spectrum to a Burgers-like E(k) ∝ k−2 spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This bifurcation in the existence of small scale turbu- lence is important, so we restate it in simpler terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Damping can change the slope of the velocity power spectrum E(k) ∝ k−α, and hence the scale dependence of velocity v(k) ∝ k(1−α)/2 (using v2 ∼ kE(k)), but it does not change the physics of the turbulent cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The latter can be encapsulated in the form of cascade times tcascade ∼ l/vl (Kolmogorov), tcascade ∼ lvph/v2 l (Kraichnan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Using v(k) ∝ k(1−α)/2, these relations im- Cosmic Ray Effects on Turbulence 9 ply tcascade ∝ k(α−3)/2 (Kolmogorov), and tcascade ∝ kα−2 (Kraichnan), which gives critical slopes α = 3, 2 respectively, in line with our previous arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The scale dependence of tcascade determines if turbulence is completely damped at small scales, or recovers with the original (undamped) power-law scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We briefly note that Equation 7 does not really ap- ply to Burgers turbulence E(k) ∝ k−2, which is not a genuine turbulent cascade, but rather an instantaneous jump from large to small scales via shocks which arise from non-linear steepening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' However, Ptuskin damping creates friction which can balance non-linear steepening and prevent shock formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We can see this by ex- amining Burgers’ equation in the presence of Ptuskin damping: ∂v ∂t + v · ∇v = −Γv (11) For Γ > ∇v, the damping term exceeds the non-linear term, so that damping exceeds non-linear steepening for L > vtdamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Unless the nonlinear time tNL ∼ L/v < tdamp, fluid motions are damped before they can steepen, and the transfer of power from large to small scales is delayed or even halted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Thus, tNL/tdamp poten- tially plays a similar role to tcascade/tdamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' What is tcascade/tdamp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The results of the previous section show that the ra- tio tcascade/tdamp is the critical parameter determining the efficacy of small scale damping, and that there is a critical value (tcascade/tdamp ∼> 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5, 1 for Kolmogorov and Kraichnan turbulence, respectively) such that the turbulence spectrum will show a cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Here, we inves- tigate the conditions under which these thresholds may be crossed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We have previously argued from energy conservation that ˙ECR ∼< ˜ϵ in steady state, hence tdamp ≥ tinject ∼ ρv2/˜ϵ ∼ L/v, the timescale on which kinetic energy is in- jected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In Appendix A, we confirm this expectation and also show how various scalings, such as δρ/ρ, δv/v, can be understood as a function of PCR/Pg, or v/cs, v/vph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' When does tdamp reach the minimal value of tinject ∼ L/v, so that almost all of the injected kinetic energy is directly dissipated in cosmic rays?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Equating the first and second terms in brackets in Equation 4, tdamp ∼ tinject when: Mph ∼< � Pc Ptot � (12) Equation 12 is only an order of magnitude estimate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' the exact threshold must come from numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Nonetheless, it illustrates the relevant physics: damping saturates when the turbulent Mach number is small and the CR energy density is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' If tdamp reaches its minimal value of tinject ∼ L/v, then: tcascade tdamp ∼ 1 (Kolmogorov) ∼ 1 Mph (Kraichnan) (13) From Fig 3, we see that it is unclear whether damping will be strong enough to enforce a small scale cutoff in a Kolmogorov cascade (which requires tcascade/tdamp ∼> 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5), but any subsonic turbulence in a Kraichnan cascade which satisfies Equation 12 will au- tomatically have tcascade/tdamp ∼> 1), the threshold for critical damping there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The increase in tcascade/tdamp is not due to a decrease in the damping time (which has a floor at tinject), but rather the increased cascade time in MHD turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Longer cascade times are as- sociated with wave turbulence, where wave-wave inter- actions produce non-linearities which eventually cause turbulence to cascade (Nazarenko 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Other forms of wave turbulence can be present, for instance, in sys- tems with strong stratification (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2022) or rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Note that even if the threshold for critical damping (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' exponential suppression of small-scale power) is not met, Figure 3 shows that the damping of gas motions can still be significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Simulations The results of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='2 are useful for guiding expec- tations and driving intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Nonetheless, given the complex non-linearities, they require validation by nu- merical simulation – a difficult task, given the limited inertial range of standard resolution simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We now present a set of simulations which, to our knowl- edge, are the first CR hydrodynamics simulations specif- ically probing CR influence on turbulent kinetic energy spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' While a more complete set of simulations with different driving modes and higher resolution awaits, we already see that CRs suppress small-scale fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We focus first on the case where Ptuskin damping is maximized, running a series of diffusion-only simulations near the CR energy gain ‘sweet spot’ κ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15L0vph, where v2 ph ∼ (Pc+Pg+PB)/ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We vary the input driving rate ˜ϵ by an order of magnitude to create turbulence with undamped Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5 and Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15 , where Ms = v/cs and cs ∼ � Pg/ρ is the gas sound speed (thus, Mph = v/vph decreases as Pc/Pg increases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Plasma beta, β = Pg/PB, are denoted in each figure and represent rough values for the presented suite of simulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' while each simulation starts with the same β, the saturated value of β changes by a small amount depending on whether 10 Bustard & Oh 100 101 102 k 10 3 10 2 10 1 100 k2E(k)dk k 2 1 CR-Modified Kinetic Energy Spectra s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5 MHD ( PCR Pg , tNL tdamp) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='12) ( PCR Pg , tNL tdamp) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='37) s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15 MHD ( PCR Pg , tNL tdamp) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='25) ( PCR Pg , tNL tdamp) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='62) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The turbulent kinetic energy spectrum, multiplied by k2 for a set of Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5 and Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15 diffusion-only simulations, keeping κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15L0vph and β ∼ 1, as we vary PCR/Pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Spectra are normalized to the k=3 mode for the Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5 MHD run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Ratios of the outer scale nonlinear time to damping time, calculated with tdamp =< ρv2/ ˙PCR > and tNL = L0/v, are also denoted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Points show the energy in each k-bin averaged over 10 outputs at late times, when turbulence is fully developed, while shaded regions show the minima and maxima during those time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Each simulation was run on a 5123 grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' While MHD runs produce spectra shallower than k−2, as expected for compressive fast modes in MHD turbulence, CRs damp fluctuations, slightly decreasing the power in low k modes while steepening the spectra at high k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' CRs are present, what CR transport model is assumed, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Figure 4 shows simulation kinetic energy spectra for both Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5 and Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15 simulation sets, each normalized by the k = 3 mode power for the Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5 MHD-only simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Different colors denote different initial PCR/Pg, ranging from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Points denote average kinetic energies, and the shaded regions denote the minimum and maximum kinetic energies taken over 10 snapshots at late times when we see converged spec- tra, typically between 8 and 10 eddy turnover times af- ter the simulation starts (see Appendix for more about time convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Importantly, we note that the iner- tial ranges in our MHD-only simulations display some- thing between a Kraichnan (k−3/2) and a Burgers-like (k−2) spectrum, where the latter is frequently seen in hy- drodynamic simulations with compressive driving, due to non-linear steepening (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Miniati 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Thus, the analytic models of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='1,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='2 where we assume a Kraich- nan spectrum do not exactly apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Nonetheless, we can look for qualitative agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We can use the non-linear steepening time as a proxy for the cascade time: tcasc ∼ tNL ∼ L0/vL, where vL is the outer-scale velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Ratios of cascade time to damping time, calculated with tdamp =< ρv2/ ˙PCR >, are noted in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The trend agrees at least qualitatively with Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Power both at large and small scales is decreased when PCR ≥ Pg, consistent with mild Ptuskin damping when tcascade ∼ tdamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' As tcascade/tdamp increases, the spectrum deviates further and further from the MHD case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For example, the tcascade/tdamp ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='62, Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15 simulation has be- tween 10 and 100 times less power in high-k modes than the MHD run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Projections of density, kinetic energy, Cosmic Ray Effects on Turbulence 11 MHD PCR ∼ Pg κ|| ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15vphL0 Density Kinetic Energy Magnetic Energy Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Projections perpendicular to the initial magnetic field direction of density (left column), kinetic energy (middle column), and magnetic energy (right column) after ∼ 10 eddy turnover times, normalized by their average values in the MHD only case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Top: MHD-only simulations with β ∼ 10 and Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Bottom: Simulations with the same β and forcing rate, but with PCR ∼ Pg and diffusive CR transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Density, velocity, and magnetic fluctuations are all suppressed compared to the MHD case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' and magnetic energy for these Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15, β ∼ 10 simulations vary quite obviously, as seen in Figure 5, with fluctuations clearly damped in the PCR ∼ Pg case (bottom row) compared to the MHD case (top row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Higher Mach number simulations appear to show damp- ing, as well, but the effect is less obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This is in line with expectations from our previous discussion that tcascade/tdamp is maximized for smaller values of stirring velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' While our analytic predictions and preliminary simu- lations suggest that Ptuskin damping could play a role in suppressing the compressible turbulent cascade at small scales, it may appear hazardous to draw conclusions based on moderate resolution simulations with limited inertial range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We therefore refer the reader to the Ap- pendix, where we test whether these spectral changes occur in the absence of CR transport, and also back to §3 and Figure 1, where we presented a separate, more robust diagnostic of the suppression of the turbulent cas- cade by Ptuskin damping: via the heating of adiabatic gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In hydrodynamic simulations of adiabatic gas, we have found that ˙Egas → ˜ϵ, as it should.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' However, in adi- abatic simulations with CRs, we have found ˙Egas → 0, while ˙ECR → ˜ϵ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' almost all of the turbulent energy is absorbed by the CRs (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Furthermore, all of this energy is absorbed at large scales, which are well re- solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The shift to CRs receiving almost all the energy of the turbulent cascade is genuine turbulent accelera- tion, not due to numerical diffusion in the CR module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We infer this from numerical convergence in our CR ac- celeration rates, as well as the close conformance to ana- lytic expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Nonetheless, we have tested this ex- plicitly by checking energy absorption for the two-fluid case when κ = 0 (bottom panel of Figure 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' in this case ˙Egas/˜ϵ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='8 when PCR/Pg ∼ 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' gas heating is once again large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' If Ptuskin damping does not allow gas motions to cascade the ∼ 2 decades to grid scale in our simula- tions to enable dissipation, this strongly suggests that real turbulence should not be able to cascade down the many more decades to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' the gyroscale of CRs, where fast modes are frequently invoked to scatter CRs with E ⪆ 300 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Of course, it is still imperative to test these ideas in much higher resolution simulations, preferably with a spectral code that can better resolve an MHD Kraichnan cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' t = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='4 Teddy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='7t = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='9 Teddy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='7t = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='9 Teddy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='25 /EK, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='00 Ek/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='25t = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='9 Teddy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='25 2 B E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='25t = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='4 Teddy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='25t = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='4 Teddy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='25 B 2 E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='00 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='2512 Bustard & Oh 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Streaming vs Diffusion In the pure diffusion limit, Γ(k) is well known, and as we’ve shown analytically and numerically, the result- ing CR drag damps turbulence at large scales, chang- ing kinetic energy spectral slopes and even introducing cut-offs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The functional form for Γ(k) is more uncertain when streaming transport is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Since we found in Paper I that streaming stunts reacceleration rates due to fundamental changes to CR-turbulent interactions, it’s tempting to append the plasma β-dependent correc- tion factors from Paper I to Γ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' If this were true, weak CR reacceleration should imply very weak changes to the kinetic energy spectrum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' however, we’ve run a num- ber of simulations with CR streaming, including some with no diffusive transport where reacceleration is abso- lutely negligible, that clearly modify the kinetic energy spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We present some simple scalings which match our simulations, but defer a detailed study to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' All simulations in this section start with PCR ∼ Pg and assume an isothermal equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Figure 6 shows the kinetic energy spectra for 2563 simulations of varying β ∼ 1, 10, 100, each with different CR transport models but the same turbulent driving rate, which for simulations without CRs (MHD only) give a sonic Mach number Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' A partial version of Figure 6, using a 5123 domain, is included in the Appendix and shows similar behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The left column shows each spectrum multiplied by k2, normalized to the peak value of the MHD spectrum at k=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The right column, in order to more clearly show differences in the spectral shape and overall kinetic energy, shows each spectrum divided by the MHD spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Note the similarity of the pure streaming power spectra to the streaming + diffusion power spectra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' in this parameter range, streaming dom- inates over diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We seek to answer two main ques- tions about these results: How does streaming vs diffusive transport affect the overall kinetic energy in the gas?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' It’s important here to note that Alfv´en Mach num- bers for each run saturate at MA < 1, meaning that Alfv´en crossing times are faster than eddy turnover times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' hence, streaming transport is relatively fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Fast streaming transport leads to small field-aligned CR pres- sure gradients / large field-aligned CR scale lengths lCR = PCR/(ˆb · ∇PCR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Compared to CRs with slow diffusive transport, streaming CRs have comparatively small pressure gradients and absorb less energy (via the v ·∇PCR term) in sub-Alfv´enic flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This qualitatively explains the behavior seen in Figure 6, where, for in- stance, β ∼ 1 leads to fast streaming transport, hence small CR pressure gradients, and little to no change in the kinetic energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' At the same time, it is important to realize that CR transport timescales are not simply ∼ L/vA, since CR pressure gradients and magnetic fields are often misaligned;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' CR bottle- necks and field line wandering can further alter transport timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Thus, for instance, ∼ L/vA is a poor esti- mate of CR losses due to collisionless heating, tloss ∼ Pc/(vA · ∇Pc) > L/vA (Paper I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Over large scales, CR transport with pure streaming is potentially effec- tively diffusive (Mertsch 2020), or even super-diffusive (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Sampson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The reduction in turbulent kinetic energy is clearly β-dependent, which we can explain as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Since streaming-dominated CRs in sub-Alfv´enic turbulence are in the ‘fast transport’ regime (analogous to the fast diffusion regime with some effective diffusivity κeff sat- isfying κeff > vphL0), tdamp ∼ κeff/c2 cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Also, from Equation 8, v2/v2 0 ∝ (tdamp/tcascade) ∝ κeff, where v0 is the undamped turbulent velocity derived by balanc- ing ˜ϵ = ρv3 0/L,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' While we don’t know the detailed form of κeff when streaming dominates, it’s tempting to say κeff ∼ vAL (or at least κeff ∝ vA), in which case v2/v2 0 ∝ vA ∝ β−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We should also expect tdamp ∼ κeff/c2 cc ∝ P −1 c , so that v2/v2 0 ∝ (tdamp/tcascade) ∝ P −1+α c , where tcascade ∝ P −α c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We have not explored the latter scaling, but it is quite plausible that α ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The cosmic ray pressure only affects cascade times for Kraichnan tur- bulence tcascade ∼ Lvph/v2 by contributing to the phase velocity vph ∼ (Pg + Pc + PB/ρ)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' However, in the ‘fast transport’ regime, Pc does not contribute to vph (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We defer a detailed exploration of these issues to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The top panel of Figure 7 quantifies the partitioning of turbulent forcing that ends up in CRs (fCR = ˙ECR/˜ϵ), as well as fE = (ρv3/L)/˜ϵ vs the steady-state plasma beta, βf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Filled circles denote simulations with stream- ing and diffusion, while empty circles have just stream- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The streaming plus diffusion results quantify what we see by eye in the kinetic energy spectra: increasing β leads to smaller turbulent velocities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' in each case, CRs take only a very small amount of the total energy forc- ing, with most energy input instead removed from the system by streaming energy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' As shown in the bot- tom panel of Figure 7, which plots v2/v2 0, our derived β−1/2 scaling is at least roughly consistent with our sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Does streaming change the shape of kinetic energy spectra, as diffusion does?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Streaming CRs, which don’t themselves take an appre- ciable amount of turbulent energy input, still nonethe- Cosmic Ray Effects on Turbulence 13 10 2 10 1 100 1 k2E(k)dk 10 2 10 1 100 E(k)/E(k)MHD 10 2 10 1 100 10 10 2 10 1 100 100 101 102 k 10 2 10 1 100 100 100 101 102 k 10 2 10 1 100 MHD || 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15vphL0 || 0 + Streaming || 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15vphL0 + Streaming Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Kinetic energy spectra for Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15, 2563 simulations each with PCR/Pg ∼ 1 but varying CR transport and varying the initial plasma β from 1 (top) to 10 (middle) to 100 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The magenta-colored lines show the resulting MHD (no CR) spectra as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The left column shows k2E(k)dk normalized by the MHD value at k = 3, while, to more clearly show the changes in spectral shape, the right column shows the ratio of each spectrum to the MHD spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For diffusion only, efficient reacceleration damps the kinetic energy spectrum, resulting in less power at small scales compared to the MHD case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' However, with streaming included, both reacceleration rates and field-aligned CR pressure gradients depend on β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' At low β (low Alfv´en Mach number MA = v/vA), streaming negates reacceleration, and the kinetic energy spectra revert to the MHD case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For larger β, however, reacceleration becomes somewhat more efficient, causing damping, and a more significant fraction of turbulent energy is channeled through CRs and lost via streaming energy transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This latter effect, most clearly evident in the streaming only simulations (red curves), decreases the overall kinetic energy in the system but doesn’t appear to induce cut-offs like the diffusion-only runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 14 Bustard & Oh 100 101 102 f 10 2 10 1 100 fE = ( v3/L)/ || 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15vphL0 + Streaming || 0 + Streaming 10 2 10 1 100 fCR = (ECR)/ 100 101 102 f 10 1 100 2 × 10 1 3 × 10 1 4 × 10 1 6 × 10 1 v2/v2 0 1/2 Undamped s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15 Undamped s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5 Undamped s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='75 10 1 100 2 × 10 1 3 × 10 1 4 × 10 1 6 × 10 1 fCR = (ECR)/ || 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15L0vph + streaming || = 0 + streaming Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Isothermal, 2563 simulations with streaming and PCR ∼ Pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The MHD (undamped) version of these simula- tions give Ms = v0/cs ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15 (black), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5 (green), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='75 (cyan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The top panel shows only the Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15 points and shows the partitioning of turbulent forcing that ends up in CRs (fCR = ˙ECR/˜ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' green points and right y-axis), as well as fE = (ρv3/L)/˜ϵ (black points and left y-axis) vs the steady-state plasma beta βf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' While for diffusion there was a clear correlation between fCR and fE, now, fCR is small, and fE correlates inversely with β, at least in this sub-Alfv´enic regime studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The bottom panel shows the turbulent ki- netic energy relative to the undamped case, where ρv3 0/L ∼ ˜ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' With CRs, even with streaming only transport where there is no reacceleration, v2/v2 0 ∝ β−1/2, at least roughly, in this sub-Alfv´enic or “fast transport” regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' There is also a weak trend towards larger overall damping with increasing driving rate (larger Ms has smaller v2/v2 0), at fixed β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' less sap kinetic energy from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' How the kinetic energy spectra change, however, is fundamentally differ- ent between streaming and diffusive transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Chang- ing β (changing MA) in streaming-dominated simula- tions effectively changes the ratio of transport time to eddy turnover time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' To glean further insight, it’s inter- esting to compare to simulations with purely diffusive transport but varying diffusion coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Figure 8 shows kinetic energy spectra for simulations on a 5123 grid, each with initial β ∼ 10 but diffusion coefficients varying between κ|| ∼ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15 − 15)vphL0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Our fiducial case of κ|| ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15vphL0 shows that damping, in the slow diffusion regime, exerts meaningful drag on an entire hi- erarchy of scales, beginning at the outer scale;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' in other words, damping and cascade rates are competitive over a large range of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Moving to the fast diffusion regime (κ|| ∼ 15vphL0), this is clearly not the case: the diffu- sion length scale is larger than the outer eddy scale, and the damping rate is only competitive with the cascade time at large scales, leaving the cascade to operate unin- terrupted after CRs have reduced the outer-scale kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Following similar logic, we infer that, for streaming- dominated transport in sub-Alfv´enic turbulence, Γ(k) must be weighted heavily towards small k, causing an initial reduction in outer-scale kinetic energy but an unimpeded cascade at larger k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Thus, we see that the power spectrum when streaming is included has the same shape over the effective inertial range of the sim- ulations k ∼< 30, albeit with a lower normalization (in the β ∼ 10, 100 cases, when damping is effective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In the dissipation range, k ∼> 30, there is additional steep- ening compared to the MHD case, though whether this is numerical or physical is as yet unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' DISCUSSION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Regimes of CR Modified Turbulence In §4, we showed both analytically and numerically that CRs can have a significant impact on the power spectrum of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In particular, we showed that as the damping time decreases relative to the turbu- lent cascade time, the turbulent power spectra will be steepen and then cut off abruptly at small scales (for tcascade/tdamp ∼> 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' These results should eventually be carefully checked by higher resolution numerical simu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Nonetheless, we clearly already have seen in §3 and Fig 1 a situation where CR damping of mo- tions is stronger than the rate at which energy cascades to smaller scales, so that little energy reaches the grid scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Our analytic estimates can guide expectations as to which environments these effects might be important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Cosmic Ray Effects on Turbulence 15 100 101 102 k 10 2 10 1 100 k2E(k)dk 10 100 101 102 k 10 2 10 1 100 E(k)/E(k)MHD MHD || 0 || 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15L0cs || 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5L0cs || 15L0cs Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Kinetic energy spectra of 5123 simulations when PCR ∼ Pg but varying the diffusion coefficient from κ|| ∼ 0 (where the only diffusivity is numerical) to the most efficient reacceleration regime (κ|| ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15vphL0 and κ|| ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5vphL0) to the fast diffusion regime (κ|| ∼ 15vphL0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Note how the power spectrum is somewhat different for the two fluid system even in the absence of CR transport, presumably because of changes to the phase velocity and other adiabatic properties, but deviations from the MHD spectrum are mild compared to simulations with added diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Left: k2E(k)dk normalized by the k = 3 MHD value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Right: Ratio of each spectrum to the MHD spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Note how diffusion introduces a characteristic scale lCR where the kinetic energy is reduced: in the fast diffusion limit, lCR > L0, and the outer-scale kinetic energy drops significantly while the rest of the spectrum retains the same shape as the MHD case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Going to smaller κ||, overall changes are more drastic because reacceleration is more efficient but also the scale where the spectrum cuts off most dramatically shifts to lCR < L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Intra-cluster medium (ICM) —In the ICM, although sonic Mach numbers are typically low (Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='3), the absence of hadronic γ-ray emission gives an upper bound on Pc/Ptot << 1 (typically less than a few percent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Ack- ermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2014), so that Equation 12 is not satisfied there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The CR energy density is too small to appreciably affect gas motions, and it is unlikely that CR reacceler- ation appreciably damps the turbulent cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Interstellar medium (ISM) —In the ISM, CR damping could be potentially important: Pc/Ptot ∼ O(1) is rel- atively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In the diffusion only case, the main un- certainty lies in the CR acceleration rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The most efficient reacceleration occurs for diffusivities in the range κ|| < vphL0 ∼ 3 × 1026cm2/s for ISM-like pa- rameters (see Table 2 in Paper I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Canonical values of κ ∼ 1028 −1029cm2/s used in galactic propagation mod- els are much larger, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' we are sufficiently far away from the ‘sweet spot’ that acceleration and hence damping times could be long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' On the other hand, if CR stream- ing dominates transport, then since the ISM has β ∼ 1, damping is small, as we have seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Circumgalactic medium (CGM) —Finally, the galactic halo and CGM are strong candidates for significant CR damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For the diffusion only case, these regions oc- cupy a sweet-spot where κ ∼ vphL0, Mph < 1, and if, as suggested by simulations of Milky Way mass galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Butsky & Quinn 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2020), Pc/Pg ∼> 1, then Pc/Ptot is order unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For these conditions, Equa- tion 12 is satisfied, so that tdamp ∼ tinject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Thus, for instance, from Equation 13, tcascade/tdamp ∼ M−1 ph ∼ 2 for a compressive Kraichnan cascade with Mph ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5: the compressive cascade will be steepened beyond the critical threshold of E(k) ∝ k−2 and abruptly cut off, so there is no small scale turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We see hints of this in Figure 4 for the Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5 case, but given our limited dynamic range, spectral changes are more obvious for Ms ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15, when tNL/tdamp is even larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Once streaming is included, we have also seen that there can be considerable damping in the β ∼ 10 − 100 cases, with a weak trend towards larger damping with in- creasing driving rate at fixed β (see how v2/v2 0 is smaller for larger Ms), potentially because CRs are more effi- ciently trapped in turbulent eddies as the Alfven Mach number approaches unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This differs from the diffu- sion only case, where stronger turbulence implies smaller tNL/tdamp and weaker damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For a fixed driving rate that produces transonic MHD turbulence in a β ∼ 10 environment (reasonable CGM parameters), adding streaming-dominated CRs up to equipartition PCR ∼ Pg damps turbulent kinetic energy by a factor of ∼ 5 or greater (bottom panel of Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Implications of CR-Modified Turbulence The implications of such CR-modified spectra are pos- sibly quite intriguing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For instance, a CR-induced cut- 16 Bustard & Oh off could significantly affect the spatial scale of thermal instability, since there are no small scale compressive motions, unless there is direct driving at those scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Also, since Ptuskin damping only affects compressive motions, not solenoidal motions, Ptuskin damping can potentially make turbulence less Burgers-like and more Kolmogorov-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' It would be interesting to explore this ‘divergence-cleaning’ effect in simulations with a mix- ture of driving modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Perhaps the most interesting consequence of CR damping of turbulence is its implication for scattering of high-energy CRs by fast modes in an extrinsically driven turbulent cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This is frequently invoked to explain the scattering of CRs with E ∼> 300GeV (Yan & Lazarian 2004), since self-confinement is too weak to explain observed isotropy and confinement times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' However, the resonant scattering invoked (transit time damping) requires the turbulence to cascade many or- ders of magnitude, to the ∼ 300AU gyroscale of such CRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Fig 3 shows that for tcasc/tdamp = 1, a Kraich- nan (E(k) ∝ k−3/2) fast mode spectrum will steepen to a Burgers (E(k) ∝ k−2) spectrum, which already has too little small scale power to efficiently scatter CRs via transit time damping (Miniati 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Pinzke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2017), and even higher values of tcasc/tdamp ∼ M−1 ph will completely eliminate turbulence at small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' While this needs further study, low-energy CRs, by damping turbulent fluctuations at large scales, could divert tur- bulent energy that would otherwise scatter high-energy CRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This potentially adds to the long list of problems with ‘standard’ theories of CR scattering in the Milky Way which have been recently pointed out (Kempski & Quataert 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Regardless of whether CR drag introduces a cut-off to kinetic energy spectra, it is clear from our simulations that CRs in both diffusion-dominated and streaming- dominated transport regimes can sap a significant frac- tion of the turbulent forcing rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This breaks the usual correspondence between turbulent velocity and turbu- lent driving rate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' for hydrodynamic turbulence, ρv3 0/L ∼ ˜ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Now, ρv3/L ∼ fE˜ϵ, where the new cor- rection factor fE can be ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' As derived in Equa- tion 8, v2/v2 0 ∝ tdamp/tcascade, which for streaming- dominated transport in sub-Alfv´enic turbulence gives v2/v2 0 ∝ β−1/2 (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In the CGM, where we ex- pect damping to be most significant, turbulent velocities obtained from the observed velocity dispersion may sig- nificantly underestimate the turbulent forcing rate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' ˜ϵ ≫ ρv3/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' CONCLUSIONS In this paper, we present analytical estimates and ac- companying MHD+CR simulations probing CR effects on turbulence, namely the damping of turbulence by large-scale, CR-induced drag on compressive gas mo- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Our main findings are as follows: Despite long CR reacceleration times, the damping time due to CR reacceleration can be very competi- tive with the turbulent cascade time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' tdamp ∼ ρv2max �tgrow PCR , 1 ˜ϵ � ∼ max � M2 ctgrow, ρv2 ˜ϵ � (14) where Mc = v/cc is the Mach number with respect to the CR sound speed cc ∼ � Pc/ρ, and ˜ϵ = ρv3/L is the turbulent energy injection rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Our key fig- ures are Figures 1 and 2, where we confirm that CRs can divert a significant fraction of turbulent energy that would otherwise dissipate as heat at small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Conditions for strong damping are met under quite reasonable conditions (Equation 12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' the CGM is an especially strong candidate for this damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' If CR diffusion dominates transport, and if the ratio of the damping time to the cascade time is sufficiently short, small scale compressive turbulence should be exponentially suppressed (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This sup- pression of small scale turbulence has abundant im- plications for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' thermal instability, “divergence- cleaning” of turbulence spectra, and suppression of fast modes at small scales, which have been invoked to scatter high-energy CRs (see §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We see com- pelling signatures of damping in our simulation spec- tra (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Figure 4), but these effects deserve future study with higher resolution simulations that capture a larger turbulent inertial range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The effects of streaming transport are more com- plex and deserve follow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Importantly, tgrow in Equation 14 does not include the suppression of CR reacceleration by streaming (the β dependent fac- tors identified in Bustard & Oh 2022), which would substantially increase damping times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Instead, from Figure 2, diversion of turbulent energy through CRs remains strong even in the presence of CR stream- ing, for our simulations where MA ∼< 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Instead of introducing spectral cut-offs, streaming uniformly decreases the normalization of the turbulent power spectrum, but not its shape, with the turbulent ki- netic energy scaling as v2 ∝ vA ∝ β−1/2 (Fig 6, Fig 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This is likely because damping operates pre- dominantly at the largest scales in the ‘fast trans- port’ regime (here, the sub-Alfv´enic regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Such large scale damping implies energetic input and tur- bulent heating rates (much of which gets channeled Cosmic Ray Effects on Turbulence 17 into CR collisionless heating) can be much larger than standard estimates for Kolmogorov turbulence, ˜ϵ ≫ ρv3/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors gratefully acknowledge Navin Tsung, Max Gronke, Yan-Fei Jiang, Christoph Federrath, Hui Li, and Ellen Zweibel, as well as the organizers and par- ticipants of the KITP “Fundamentals of Gaseous Halos” workshop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' CB was supported by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' NSF PHY-1748958 and by the Gordon and Betty Moore Foundation through Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' GBMF7392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' SPO was supported by NSF grant AST-1911198, and NASA grant 19-ATP19-0205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Computations were performed on the Stampede2 and PSC-Bridges2 supercomputers under allocation TG- PHY210004 provided by the Extreme Science and Engi- neering Discovery Environment (XSEDE), which is sup- ported by National Science Foundation grant number ACI-1548562 (Towns et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Software: Athena++ (Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2020), yt (Turk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2011), Matplotlib (Hunter 2007), Mathematica (Wolfram Research, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2021) APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' TURBULENT PROPERTIES AND DAMPING IN A COSMIC RAY DOMINATED MEDIUM CRs can influence velocity and density perturbations in a turbulent medium, but the extent depends on the relative partition of CR vs thermal energy, as well as the CR diffusivity / transport speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Commer¸con et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' (2019) simulate CRs in a turbulent box with purely diffusive transport and a bi-stable ISM (with radiative cooling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' They found that trapped CRs modify the gas flow, change the density PDF, and provide support against thermal instability, maintaining the gas in an intermediate temperature state that is classically thermally unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' It remains to be seen how these simulations would change when streaming is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The perturbative heating term from CR streaming affects thermal instability (Kempski & Quataert 2020), and in low-β plasmas where this heating is most significant, CR streaming can also drive acoustic waves unstable, generating a “stair-case” cosmic ray pressure profile and additional multiphase gas (Tsung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Quataert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Our simulation setup is quite different from that of Commer¸con et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' (2019), most notably because we don’t include radiative cooling, so we don’t attempt a detailed comparison, but we do find some qualitatively similar behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Figure 9 shows δρ/ρ, δv/v, and δPCR/PCR for diffusion-only simulations with varying PCR/Pg and either κ = 0 or κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15L0vph (where vph depends on PCR/Pg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' When CRs are dynamically unimportant (PCR/Pg ≪ 1), we recover the MHD expectation that δρ/ρ ∼ δv/v ∼ Ms = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Deviations from this relation start when PCR/Pg ⪆ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Interestingly, we find that δρ/ρ is independent of Pc/Pg, while δPc/Pc ∝ 1/Pc (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', δPc ∼const is independent of Pc/Pg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' At the same time, we find that the velocity divergence ∇ · v ∝ P −1/2 c (not shown), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' it does depend on Pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This might appear puzzling, since one expects density fluctuations and velocity divergence to be directly related, yet the former is independent of Pc, while the latter shows dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' A key to understanding these results is to realize that the ‘sweet spot’ κ ∼ Lvph is really still in the ‘fast diffusion regime’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The ratio tdiffuse/tsc ∼ lvph/κ is only unity at the outer scale l ∼ L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' at smaller scales, tdiffuse/tsc < 1 and diffusion dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In this diffusion dominated regime, CRs diffuse out of eddies before they contribute significantly to resisting compression – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', they do not provide a significant restoring force (instead, they provide drag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In particular, they do not contribute to the phase velocity vph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Thus, δρ/ρ ∼ δPg/Pg ∼ v/cs, where cs is the gas sound speed, independent of PCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This is roughly consistent with Commer¸con et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' (2019) (see their Figures 5 and 6), which finds a similar dependence on κ and a clear decrease in δPCR/PCR as PCR/Pg increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Using this information, we can better interpret the lower bound on damping time that we infer from our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Importantly, as Ptuskin damping saturates (PCR/Pg → ∞, fCR → 1), the maximum rms CR pressure perturbation is ⟨∆PCR⟩rms ∼ ρv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This is a strict upper bound, since the free energy to create CR pressure perturbations is derived from kinetic energy (similarly, ∆PCR, ∆Pg at a shock cannot exceed the ram pressure ρv2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In this limit, ∆PCR/PCR ∼ ρv2/PCR ∝ 1/PCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Finally, in the diffusion dominated limit, CR compression is balanced by diffusion, PCR,0(∇ · v) ∼ −∇ · (κ∇PCR,1) ∼ κρv2/L2, which implies that ∇ · v ∝ κ PCR,0 , (A1) 18 Bustard & Oh 10 2 10 1 100 101 102 PCR/Pg 10 2 10 1 / v/v PCR/PCR v/cs = s v/vph = ph v/v2 ph 1/PCR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15L0vph = 0 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Fluctuating density (blue symbols), velocity (black symbols), and CR pressure (green symbols) as a function of PCR/Pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Open circles denote simulations with non-zero diffusion coefficient κ|| ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15L0vph, and open diamonds denote purely advective CR transport (κ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' None of these simulations include additional streaming transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' where PCR,0, PCR,1 refer to the unperturbed and perturbed CR pressure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Thus, in the regime where we fix the sweet-spot diffusion coefficient κ ∼ vphL0 and PCR/Pg ∼> 1 (so that vph ∝ P 1/2 CR ), then κ ∝ P 1/2 CR , and ∇·v ∝ P −1/2 CR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' We have also verified in our simulations that ∇ · v ∝ κ for constant PCR, and ∇ · v ∝ 1/PCR for constant κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' These results thus indicate that the damping time cannot become arbitrarily small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' If drag forces are given by: ˙v ∼ 1 ρ∇PCR,1 ∼< v2 L (A2) (where PCR,1 ∼< ρv2), this gives a damping time tdamp ∼ v/˙v ∼> L/v ∼ teddy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Thus, δPCR ∼< ρv2 implies that tdamp ∼> teddy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This is equivalent to the statement that the work done by CR forces in opposing gas motions cannot exceed the energy input rate: v · ∇PCR,1 ∼< ˜ϵ ∼ ρv3/L, which implies ∇PCR,1 ∼< ρv2/L, consistent with Equation A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Thus, for Kolmogorov turbulence, we expect tcascade/tdamp ∼ 1 for maximally efficient Ptuskin damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In the Kraichnan case, however, tcascade/tdamp can be greater than 1 if the Mach number, relative to the velocity of compressible fluctuations, is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' This is not due to any decrease in the damping time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' instead, it is due to cascade times being lengthened when vph is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Time Convergence Figure 10 shows the average kinetic energy spectra for 2563 simulations with varying CR transport model, measured at different time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Most importantly, the diffusion-only simulations show converged, clearly damped spectra even at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Spectra for simulations with CR streaming are also well-converged but at somewhat later times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Note that these time intervals over which we pull out kinetic energy spectra are much later than the saturation of bulk turbulent quantities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' kinetic energy, magnetic energy, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' ), which occurs after only a few eddy turnover times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Resolution Convergence A good test of how inherently diffusive our CR module is, and whether that accounts for some observed spectral changes, is to run simulations with no explicit CR diffusion at various resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Figure 11 compares spectra for our β = 10 MHD simulations to simulations with PCR ∼ Pg and purely advective CR transport (no streaming and κ|| = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' For grid sizes of 2563 and 5123, we see in both cases that, in the inertial range up until k ∼ 20, there is Cosmic Ray Effects on Turbulence 19 100 101 102 k 10 1 100 k2E(k)dk 10 MHD || 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15vphL0 Time Interval 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='1 - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='4 teddy 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='6 - 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='9 teddy 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='1 - 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='3 teddy 100 101 102 k 10 1 100 k2E(k)dk 10 || 0 + Streaming || 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15vphL0 + Streaming Time Interval 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='1 - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='4 teddy 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='6 - 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='9 teddy 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='1 - 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='3 teddy Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Time convergence of select spectra, each run on a 2563 grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The diffusion-only simulations, which show the most damping, converge very early.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 5123 simulations (not shown) are similarly converged with respect to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 100 101 102 k 10 2 10 1 100 k2E(k)dk 10 2563 5123 100 101 102 k 10 2 10 1 100 E(k)/E(k)MHD 2563 5123 MHD || 0 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Comparison of kinetic energy without CRs (MHD) and with CRs but no transport (κ|| ∼ 0), simulated on grids with 2563 and 5123 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 5123 simulation results are divided by a factor of 10 to separate those curves from the 2563 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' no appreciable damping due to the presence of CRs, confirming again that CR transport is the cause for clear and obvious damping seen in Figures 4, 6, 8 beginning at small k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Figure 12 shows kinetic energy spectra for 5123 simulations when transport is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' These simulations only comprise part of those on a 2563 grid (compare to Figure 6 in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='4) because computer resource limits prohibit us from running the streaming only (κ ∼ 0 + streaming) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' In any case, the streaming only simulations and the streaming + diffusion simulations are both streaming dominated in this sub-Alfv´enic regime, so we expect their spectra to look very similar, as we saw in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' The MHD and diffusion only spectra look qualitatively similar to those on a 2563 grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Most importantly, diffusive transport leads to significant damping compared to the MHD case at all β tested (β =1 and 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' As in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='4, streaming transport instead appears to uniformly decrease kinetic energy at all scales, and this is β dependent with β ∼ 1 showing almost no difference between MHD and CR cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' There is some resolution dependence for β = 10, with 5123 showing less damping compared to the 2563 run, but the difference is mild, especially compared to the heavily damped diffusion-only simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 20 Bustard & Oh 10 2 10 1 100 1 k2E(k)dk 10 2 10 1 100 E(k)/E(k)MHD 10 2 10 1 100 10 10 2 10 1 100 100 101 102 k 10 2 10 1 100 100 100 101 102 k 10 2 10 1 100 MHD || 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15vphL0 || 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='15vphL0 + Streaming Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Kinetic energy spectra for a partial simulation suite run on a 5123 grid instead of a 2563 grid (compare to Figure 6 in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Computer resource limits prohibit us from running the streaming only (κ ∼ 0 + streaming) simulations of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='4 on a 5123 domain, but all other spectra look qualitatively similar to those on a 2563 grid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' namely, diffusion only transport shows clear differences in spectral slope at both β = 1 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Streaming simulations instead appear to uniformly decrease kinetic energy at all scales as β increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Note there is some resolution dependence for β = 10, with 5123 showing less damping compared to the 2563, but the difference is mild, especially in comparison to the diffusion-only simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' Cosmic Ray Effects on Turbulence 21 REFERENCES Ackermann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Ajello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Albert, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2014, ApJ, 787, 18 Amato, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Blasi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2018, Advances in Space Research, 62, 2731 Bai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Ostriker, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Plotnikov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Stone, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2019, ApJ, 876, 60 Becker Tjus, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Merten, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2020, PhR, 872, 1 Blasi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Amato, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Serpico, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2012, PhRvL, 109, 061101 Boulares, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Cox, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 1990, ApJ, 365, 544 Brunetti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Jones, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2014, International Journal of Modern Physics D, 23, 1430007 Brunetti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Lazarian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2011, MNRAS, 410, 127 Bustard, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Oh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2022, ApJ, 941, 65 Bustard, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Zweibel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2021, ApJ, 913, 106 Butsky, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Quinn, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2018, ApJ, 868, 108 Chandran, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2000, PhRvL, 85, 4656 Chandran, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Maron, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2004, ApJ, 603, 23 Cho, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Lazarian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2003, MNRAS, 345, 325 Commer¸con, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Marcowith, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Dubois, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2019, A&A, 622, A143 Drury, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Strong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2017, A&A, 597, A117 Eswaran, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Pope, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 1988, Computers and Fluids, 16, 257 Farmer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Goldreich, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2004, ApJ, 604, 671 Felice, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Kulsrud, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2001, ApJ, 553, 198 Field, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 1965, ApJ, 142, 531 Giacalone, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Jokipii, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 1999, ApJ, 520, 204 Goldreich, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Sridhar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 1995, ApJ, 438, 763 Hanasz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Strong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Girichidis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2021, Living Reviews in Computational Astrophysics, 7, 2 Holcomb, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Spitkovsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2019, ApJ, 882, 3 Hopkins, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Squire, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Butsky, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Ji, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2022, MNRAS, 517, 5413 Hu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Lazarian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Xu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2022, MNRAS, 512, 2111 Hunter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2007, Computing in Science & Engineering, 9, 90 Ji, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Chan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Hummels, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2020, MNRAS, 496, 4221 Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Oh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2018, ApJ, 854, 5 Kempski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Quataert, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2020, MNRAS, 493, 1801 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2022, MNRAS, 514, 657 Kulsrud, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Pearce, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 1969, ApJ, 156, 445 Landau, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Lifshitz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 1987, Fluid Mechanics (Butterworth-Heinemann) Lynn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Parrish, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Quataert, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Chandran, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2012, ApJ, 758, 78 McCourt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Sharma, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Quataert, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Parrish, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2012, MNRAS, 419, 3319 Mertsch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2020, Ap&SS, 365, 135 Miniati, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2015, ApJ, 800, 60 Mohapatra, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Federrath, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Sharma, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2022, MNRAS, 514, 3139 Nazarenko, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2011, Wave turbulence, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 825 (Springer Science & Business Media) Pinzke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Oh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Pfrommer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2017, MNRAS, 465, 4800 Ptuskin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 1981, Ap&SS, 76, 265 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 1988, Soviet Astronomy Letters, 14, 255 Putman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Peek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Joung, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2012, ARA&A, 50, 491 Quataert, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Jiang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Thompson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2022, MNRAS, 510, 920 Reichherzer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Becker Tjus, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Zweibel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Merten, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Pueschel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2020, MNRAS, 498, 5051 Sampson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Beattie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Krumholz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2022, MNRAS, arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='08174 Silk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 1968, ApJ, 151, 459 Skilling, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 1971, ApJ, 170, 265 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 1975, MNRAS, 172, 557 Stone, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Tomida, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', White, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Felker, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2020, ApJS, 249, 4 Thornbury, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Drury, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2014, MNRAS, 442, 3010 Towns, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Cockerill, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Dahan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2014, Computing in Science & Engineering, 16, 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='ieeecomputersociety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='1109/MCSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='80 Tsung, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Oh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2022, MNRAS, 513, 4464 Turk, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Smith, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Oishi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2011, ApJS, 192, 9 Uhlenbeck, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Ornstein, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 1930, Physical Review, 36, 823 Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Oh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Ruszkowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='01732 Wentzel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 1968, ApJ, 152, 987 Wiener, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', Zweibel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Oh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2013, ApJ, 767, 87 Wolfram Research, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2021, Mathematica, Version 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='0, , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='wolfram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content='com/mathematica Yan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=', & Lazarian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2002, PhRvL, 89, 281102 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2004, ApJ, 614, 757 Zweibel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} +page_content=' 2017, Physics of Plasmas, 24, 055402' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE2T4oBgHgl3EQf2AhI/content/2301.04156v1.pdf'} diff --git a/udE2T4oBgHgl3EQf2Qhr/content/2301.04159v1.pdf b/udE2T4oBgHgl3EQf2Qhr/content/2301.04159v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..aa41dc53f4d7bb8fc79e56ce77456c7f6962a976 --- /dev/null +++ b/udE2T4oBgHgl3EQf2Qhr/content/2301.04159v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c1dc15f5c2efef4256236676640144110f95e6565725a31c2580ec18aac0b1e3 +size 1000764 diff --git a/udE2T4oBgHgl3EQf2Qhr/vector_store/index.faiss b/udE2T4oBgHgl3EQf2Qhr/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..c2d7249fe8c529a248ec9c38fbf37b12beacb971 --- /dev/null +++ b/udE2T4oBgHgl3EQf2Qhr/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:029d5a3a7844a1902e8f1b1120414c3e27139010b537de27530c318a70eae305 +size 2687021 diff --git a/udE2T4oBgHgl3EQf2Qhr/vector_store/index.pkl b/udE2T4oBgHgl3EQf2Qhr/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..0382ba85c9d41d7fd047e630d2fcda1bad3f1f99 --- /dev/null +++ b/udE2T4oBgHgl3EQf2Qhr/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0b2ed4d099dec88bb5cbaf08960fa7962ed2d4094fa053928976f26a7d31b962 +size 87818 diff --git a/vtE0T4oBgHgl3EQfbwDA/content/tmp_files/2301.02354v1.pdf.txt b/vtE0T4oBgHgl3EQfbwDA/content/tmp_files/2301.02354v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..311f69821dc400b39992977d427c64e8c44278d2 --- /dev/null +++ b/vtE0T4oBgHgl3EQfbwDA/content/tmp_files/2301.02354v1.pdf.txt @@ -0,0 +1,2155 @@ +arXiv:2301.02354v1 [math.GR] 6 Jan 2023 +KLEIN–MASKIT COMBINATION THEOREM FOR ANOSOV +SUBGROUPS: AMALGAMS +SUBHADIP DEY AND MICHAEL KAPOVICH +Abstract. The classical Klein–Maskit combination theorems provide sufficient conditions +to construct new Kleinian groups using old ones. There are two distinct but closely related +combination theorems: The first deals with amalgamated free products, whereas the second +deals with HNN extensions. This article gives analogs of both combination theorems for +Anosov subgroups. +1. Introduction +In the theory of Kleinian groups (discrete isometry groups of H3), the Klein–Maskit +combination theorems provide techniques to construct new Kleinian groups. The history of +combination theorems dates back to Klein’s 1883 paper [17], which gave sufficient conditions +for a subgroup Γ of G = Isom(H3) generated by two discrete subgroups Γ1 and Γ2 of G to +be discrete and isomorphic to the free product Γ1⋆Γ2. Subsequently, Maskit [25, 26, 27, 29], +dealing with the cases of amalgamated free products and HNN extensions, gave far-reaching +generalizations of the Klein combination theorem. Maskit’s combination theorems also fur- +nish sufficient conditions for the combined group to have certain geometrical properties, such +as convex-cocompactness or geometric-finiteness. See Maskit’s book [28] for an account of +those results. Later on, Ivascu [11] and, more recently, Li–Ohshika–Wang [19, 20] extended +the Klein–Maskit combination theorems to the setting of discrete isometry groups of higher +dimensional hyperbolic spaces. The combination theorems were further generalized in the +context of group actions on Gromov-hyperbolic spaces; see [1, 9, 22, 23, 24]. We refer to +[31] for a recent survey of combination theorems. +Our motivation for this article is to provide suitable analogs of the Klein–Maskit combi- +nation theorems in the context of Anosov subgroups. In recent years, Anosov subgroups of +semisimple Lie groups have emerged as a higher-rank generalization of convex-cocompact +Kleinian groups. In our paper with Bernhard Leeb [5], we gave an earlier form of such +a combination theorem; in that paper, using the local-to-global principle for Morse quasi- +geodesics, we proved a version of the Klein combination theorem for Anosov subgroups. +Moreover, we conjectured a sharper form of the combination theorem in that paper, which +was recently confirmed in [4]. Nevertheless, the discussions in [4, 5] were limited only to +the case of free products. The purpose of the present article is to deal with the case of +amalgams (amalgamated free products and HNN extensions). Building on our work in [4], +we prove versions of both Klein–Maskit combination theorems for Anosov subgroups. +Date: January 5, 2023. +1 + +2 +SUBHADIP DEY AND MICHAEL KAPOVICH +Let G be a real semisimple Lie group of noncompact type and with a finite center. +We will assume some mild conditions on G (see Assumption 3.1). Let X = G/K be the +associated symmetric space, where K is a maximal compact subgroup of G. Let σmod be a +model spherical chamber in the Tits building ∂TitsX of X and let ι : σmod → σmod be the +opposition involution. We consider the class of τmod-Anosov subgroups of G, where τmod +is an ι-invariant face of σmod. For a discrete subgroup Γ of G, the τmod-limit set of Γ in +Flag(τmod), the partial flag manifold associated to the face τmod, is denoted by Λτmod(Γ). +See §3 for the definitions. +The first main result of this paper, which provides an analog of the first Klein–Maskit +combination theorem [28, Theorem VII.C.2], is as follows: +Theorem A. Let ΓA and ΓB be τmod-Anosov subgroups of G. Assume that H := ΓA∩ΓB is +quasiconvex in ΓA or ΓB. Suppose that there exists an interactive pair (see Definition 3.11) +(A, B) in Flag(τmod) for (ΓA, ΓB; H) such that the following conditions are satisfied: +(i) The interiors of A and B are antipodal to each other (see Definition 3.2). +(ii) The pairs of subsets (A, Λτmod(ΓB) \ Λτmod(H)) and (B, Λτmod(ΓA) \ Λτmod(H)) of +Flag(τmod) are antipodal to each other. +Then, the subgroup Γ = ⟨ΓA, ΓB⟩ of G is τmod-Anosov and is naturally isomorphic to the +abstract amalgamated free product ΓA ⋆H ΓB. +Our second main result, which gives an analog of the second Klein–Maskit combination +theorem [28, Theorem VII.E.5], is as follows: +Theorem B. Let M be a τmod-Anosov subgroup of G and let f ∈ G be an element. Assume +that H+ := M ∩ (fMf −1) or H− := (f −1Mf) ∩ M is quasiconvex in M. Suppose that there +exists an interactive triple (see Definition 3.14) (A, B±) in Flag(τmod) for (M; H±; f) such +that the following conditions are satisfied: +(i) The pairs of subsets (A◦, B◦ +±), (B+, B−) of Flag(τmod) are antipodal to each other. +(ii) Λτmod(Γ) \ Λτmod(H±) is antipodal to B±. +Then, the subgroup Γ = ⟨M, f⟩ of G is τmod-Anosov and is naturally isomorphic to the +abstract HNN extension M⋆φ of M by φ, where the isomorphism φ : H− → H+ is given by +φ(η) = fηf −1, for all η ∈ H−. +See §4, resp. §5, for the proof of Theorem A, resp. Theorem B. +To conclude this introduction, let us illustrate the hypotheses and conclusions of the +Theorem A and Theorem B in the following examples: +1.1. Illustrative examples. Let S be a closed, orientable surface of genus g ≥ 2 and +let Γ := π1(S). Labourie’s [18] pioneering work showed that Hitchin representations ρ : +Γ → PSL(d, R) form a rich class of σmod-Anosov representations. Let us denote by ξρ : +∂∞Γ → Flag(σmod) the associated Γ-equivariant boundary map onto the limit set Λρ(Γ) ⊂ +Flag(σmod). According to Fock–Goncharov [8], an important characteristic of the Hitchin +representations is a certain positivity property of the limit map ξρ. +Given an essential separating simple closed curve s ⊂ S, let [η] denote the conjugacy class +in Γ representing s. The curve s cuts the surface into two subsurfaces, SA and SB. The + +COMBINATION THEOREM: AMALGAMS +3 +group Γ can be written as an amalgamated free product +Γ = ΓA ⋆H ΓB, +where ΓA = π1(SA), ΓB = π1(SB), H = ⟨η⟩, for some η ∈ [η], equipped with natural +monomorphisms φA : H → ΓA and φB : H → ΓB induced by the inclusion homeomorphisms +s ֒→ ∂SA and s ֒→ ∂SB. +The image of H = ⟨η⟩ under a Hitchin representation ρ : +Γ → PSL(d, R) is a cyclic σmod-Anosov subgroup of PSL(d, R). Let σ± ∈ Λρ(Γ) = ξρ(∂∞Γ) +denote the attractive/repulsive fixed points of ρ(η) in Flag(σmod). The 2-point set {σ+, σm} +cuts Λρ(Γ) in a pair of closed arcs cA and cB; we chose the names in such a way that +Λρ(ΓA) ⊂ cA and Λρ(ΓB) ⊂ cB. See the left side of Figure 1. +Let Ω be the set of all points in Flag(σmod) antipodal to both σ±; this is the intersection +of a pair of opposite maximal Schubert cells in Flag(σmod): +Ω = C(σ+) ∩ C(σ−). +There are two distinguished connected components of Ω, denoted by A◦ and B◦, whose +closures A and B, respectively, contain cA and cB. Both A and B are preserved by H +because H preserves cA and cB. Using the positivity property of ξ and the fact that, for all +α ∈ ρ(ΓA \ H) and β ∈ ρ(ΓA \ H), αcB ⊂ cA \ {σ±} and βcA ⊂ cB \ {σ±}, it follows that +αB ⊂ A◦ +and +βA ⊂ B◦, +see [10, Corollary 3.9]. Moreover, the interiors A◦ and B◦ are antipodal to each other, see +[10, Proposition 2.5(3)]. Therefore, (A, B) is an interactive pair for (ρ(ΓA), ρ(ΓB); ρ(H)), +which satisfies the hypothesis of Theorem A. +Let us describe a continuous family of Hitchin representations ρt : Γ → PSL(d, R) ob- +tained by bending the given Hitchin representation ρ along s, a type of deformation that +generalizes the classical Dehn twists along simple closed geodesics in a hyperbolic surface. +This family is parametrized by t ∈ Z0(η), where Z0(η) ∼= Rd−1 is the identity component +of the centralizer of η in PSL(d, R), and the σmod-Anosov property of this family can be +verified by a simple application of Theorem A: Each connected component of Ω is also +preserved by Z0(η). In particular, Z0(η) preserves A and B. For t ∈ Z0(η), let +ρt : ΓB → PSL(d, R), +ρt(β) = tρ(β)t−1. +Clearly, ρt|H = ρ|H. Moreover, we observe that (A, B) is still an interactive pair for the +triple (ρ(ΓA), ρt(ΓB); ρ(H)). Therefore, Theorem A directly verifies that +ρt : Γ = ΓA ⋆H ΓB → ⟨ρ(ΓA), tρ(ΓB)t−1⟩ < PSL(d, R), +is injective, and its image is a σmod-Anosov subgroup of G. +Similarly, if s is an essential non-separating simple closed curve in S, then the fundamental +group Γ = π1(S) can be written as an HNN extension +Γ = M⋆φ, +φ : H− +∼ += +−→ H+, +where M is the fundamental group of the surface S′ obtained by cutting S along s, H− = ⟨η⟩ +and H+ = ⟨η′⟩ are the images of the homomorphisms of the fundamental groups induced +by the two inclusion homeomorphisms s ֒→ ∂−S′ and s ֒→ ∂+S′, and φ : H− → H+ is the +isomorphism induced by the conjugation by some element f ∈ Γ. + +4 +SUBHADIP DEY AND MICHAEL KAPOVICH +Let ρ : Γ → PSL(d, R) be a Hitchin representation with associated Γ-equivariant limit +map ξρ : ∂∞Γ → Flag(σmod). +The attractive/repulsive fixed point sets {σ+, σ−} and +{σ′ ++, σ′ +−} of η and η′, respectively, cut the limit set Λρ(Γ) = ξρ(∂∞Γ) in four closed arcs +cA+, cA−, cB+, and cB− (see the right side of Figure 1). +We have ρ(f){σ±} = {σ′ +±}, +ρ(f)(cA+ ∪ cA− ∪ cB+) ⊂ cB+, and ρ(f)−1(cA+ ∪ cA− ∪ cB−) ⊂ cB−. +cB +cA +σ+ +σ− +η +cA+ +cA− +cB− +cB+ +σ+ +σ′ ++ +σ− +σ′ +− +f +η +η′ +Figure 1 +Define B−, B+ ⊂ Flag(σmod) to be the closure of connected component of C(σ+)∩C(σ−) +containing the arc cB− and cB+, respectively. Lastly, define A to be the union of A+ and +A−, where A± are the closures of the connected component of C(σ±) ∩ C(σ±) containing +the arcs cA±, respectively. Then, using the positivity of ξρ, one can show that (A, B±) is +an interactive triple for (ρ(M); ρ(H±); ρ(f)) (compare with the amalgamated free product +case discussed above), thus verifying the hypothesis of Theorem B. +As before, we obtain a continuous family of Hitchin representations ρt : Γ → PSL(d, R), +parametrized by t ∈ Z0(η), by bending ρ along s: Observe that for any t ∈ Z0(η), (A, B±) +is an interactive triple for (ρ(M); ρ(H±); ρ(f) · t). Therefore, Theorem B directly verifies +that +ρt : Γ = M⋆φ → ⟨ρ(M), ρ(f) · t⟩ < PSL(d, R) +is injective with a σmod-Anosov image. +Organization of the paper. In §2, we present some preliminary material and results on +amalgamated free products and HNN extensions of hyperbolic groups. This section presents +some results (e.g., Lemma 2.8, Propositions 2.12 and 2.13) crucial in the proof of our main +results. +Then, in §3 we review some background material on discrete groups acting on +symmetric spaces of noncompact type and present some lemmas (see §3.1), which are also +frequently used in the proof of our main results to verify regularity and flag-convergence of +certain sequences. Finally, in §4, resp. §5, we prove Theorem A, resp. Theorem B. +2. Preliminaries on amalgams of hyperbolic groups +2.1. Amalgamated free products. Let ΓA, ΓB, and H be abstract groups together with +monomorphisms φA : H → ΓA, φB : H → ΓB. The free product of ΓA and ΓB amalgamated +along H, denoted by ΓA ⋆H ΓB, has a presentation +⟨SA, SB | RA, RB, φA(η)φB(η)−1, η ∈ H⟩, + +COMBINATION THEOREM: AMALGAMS +5 +where ⟨SA | RA⟩ and ⟨SB | RB⟩ are presentations of ΓA and ΓB, respectively. We will iden- +tify H with φA(H) < ΓA and φB(H) < ΓB via the monomorphisms φA and φB, respectively. +Let Γ = ΓA ⋆H ΓB. A normal form of an element γ ∈ Γ is an expression +γ = γ1γ2 · · · γl, +(2.1) +such that the following conditions are satisfied: +(i) Each γi lies either in ΓA, or in ΓB. Moreover, if l ≥ 2, then none of the letters γi +belong to H. +(ii) No two successive letters γi, γi+1 above lie in the same group ΓA or ΓB. +Unless H is trivial, the normal form of γ given by (2.1) is not necessarily unique. However, +any other normal form of γ is obtained by a sequence of finite moves consisting of replacing +a consecutive pair γiγi+1 in (2.1) by (γiηi)(η−1 +i +γi+1), where ηi ∈ H. This is a consequence +of the Normal Form Theorem; see [21, Theorem IV.2.6]. It follows that any two normal +forms of γ have the same number of letters. For γ ∈ Γ \ H, the relative length of γ, denoted +by rl(γ), is the number of letters in a(ny) normal form of γ. If γ ∈ H, then define rl(γ) = 0. +Definition 2.1 (Alternating sequences). A sequence (ωk) in Γ is called type A alternating +if there exists a pair of sequences, (αn) in ΓA \ H and (βn) in ΓB \ H, such that +ωk = +� +α1β1 · · · βm−1αm, +k = 2m − 1, +α1β1 · · · αmβm, +k = 2m. +(2.2) +Similarly, a sequence (ωk) in Γ is called type B alternating if there exists a pair of sequences, +(αn) in ΓA \ H and (βn) in ΓB \ H, such that +ωk = +� +β1α1 · · · αm−1βm, +k = 2m − 1, +β1α1 · · · βmαm, +k = 2m. +2.2. HNN extensions. Let M be an abstract group and H± < M be a pair of subgroups. +Suppose that φ : H− → H+ is an isomorphism. The HNN extension of M by φ, denoted by +M⋆φ, has a presentation +⟨SM, f | RM, fηf −1(φ(η))−1, η ∈ H−⟩, +where ⟨SM | RM⟩ is a presentation of M. +Every element γ ∈ Γ = M⋆φ can be written in the form +γ = µ0f ǫ1µ1 · · · f ǫnµn, +(2.3) +for some n ≥ 0, where each µi is in M, and each ǫi is either 1 or −1, such that the following +conditions are satisfied: +(i) For i = 1, . . . , n, if ǫi = −1 and µi ∈ H+, then ǫi+1 = −1. +(ii) For i = 1, . . . , n, if ǫi = 1 and µi ∈ H−, then ǫi+1 = 1. +Such an expression (2.3) is called a normal form of γ. +Although the normal form of an element is not unique, Britton’s Lemma (see [21, p.181]) +establishes certain uniqueness properties of the decomposition (2.3) of γ. In particular, the +relative length rl(γ) of γ, i.e., the total number of letters f and f −1 appearing in (2.3), is +unique. + +6 +SUBHADIP DEY AND MICHAEL KAPOVICH +Definition 2.2 (Alternating sequences). A sequence (ωn) in M⋆φ is called alternating if +there exist sequences (µn) in M and (ǫn) in {±1}, satisfying +ǫn = −1, µn ∈ H+ =⇒ ǫn+1 = −1, +and +ǫn = 1, µn ∈ H− =⇒ ǫn+1 = 1 +(2.4) +for all n ∈ N, and some element µ0 ∈ M such that +ωn = µ0f ǫ1µ1f ǫ2µ2 · · · f ǫn−1µn−1f ǫn. +(2.5) +Remark 2.3. The condition (2.4) simply implies that the word in (2.5) is a normal form. +It is also useful to think about an alternating sequence (ωn) in M⋆φ as an infinite string of +letters: +µ0, f ǫ1, µ1, f ǫ2, µ2, f ǫ3, µ3 . . . , +where µi and ǫi satisfy the condition (2.4). +2.3. Bass-Serre trees. In this paper, we will be using the formalism of Bass–Serre trees T +associated with amalgamated free products and HNN extensions. A detailed treatment of +this material can be found in Serre’s book [32]; see also [6] for a more topological viewpoint +on the construction. +(i) Consider an amalgamated free product Γ = ΓA ⋆H ΓB. The vertex set V (T) of the +tree T is the set of cosets γΓA, γΓB, γ ∈ Γ. Accordingly, the vertex set V (T) is bicolored, +with one color corresponding to the cosets γΓA and the other color corresponding to the +cosets γΓB. The group Γ acts on V (T) by the left multiplication. +Edges of T are defined so that T is a bipartite graph, i.e., vertices of the same color are +never connected by an edge. The cosets γ1ΓA, γ2ΓB are connected by an edge whenever +there exists α ∈ ΓA such that +γ1ΓB = γ2αΓB, +equivalently, there exists β ∈ ΓB such that +γ1βΓA = γ2ΓA. +For instance, the vertices represented by the cosets ΓA, ΓB are connected by an edge +e ∈ E(T) since there exists an element η ∈ H < ΓA such that ΓB = ηΓB. +Moreover, +all elements α ∈ ΓA such that αΓB = ΓB necessarily belong to H. We, thus, label the edge +e by the coset 1Γ · H = H. +The left multiplication by the elements of Γ preserves the incidence relation between the +vertices of T. Accordingly, the edges of T are labeled by the cosets γH, γ ∈ G. The Γ- +stabilizer of the vertex γΓA equals γΓAγ−1, while the Γ-stabilizer of the vertex γΓB equals +γΓBγ−1 and the Γ-stabilizer of an edge labeled γH equals γHγ−1. +(ii) Consider an HNN extension Γ = M⋆φ:H−→H+. The vertex set of T consists of left +cosets of M in Γ. The edge set of T consists of edges of two types: The left H±-cosets in +Γ. The edge gH+ connects γM to γfM, while the edge γH− connects γM and γf −1M. The +Γ-action on T is defined by: γ : γ′M �→ γγ′M. The Γ-action is transitive on the set of +vertices and edges of T. + +COMBINATION THEOREM: AMALGAMS +7 +2.4. Hyperbolic groups. Let Γ be a finitely-generated group equipped with a left-invariant +word metric, denoted by dΓ. We use the notation | · | to denote the word length of elements +of Γ, i.e., |γ| = dΓ(1Γ, γ). Recall that Γ is called (word) hyperbolic if there exists δ ≥ 0 such +that (Γ, dΓ) is a δ-hyperbolic metric space: That is, for all f, g, h, w ∈ Γ, +(f, g)w ≥ min{(f, h)w, (g, h)w} − δ, +where (f, g)w denotes the Gromov product of f and g with respect to w: +(f, g)w = 1 +2(dΓ(f, w) + dΓ(g, w) − dΓ(f, g)). +It is a basic fact that the property of being hyperbolic does not depend on the choice of a +word metric dΓ, but the constant δ possibly depends on the chosen metric dΓ. +Let Γ be a hyperbolic group equipped with a word metric dΓ. A (discrete) geodesic in +Γ is an isometric embedding c : I → Γ of an interval I ⊂ Z. Such a geodesic c : I → Γ +is called a segment, ray, or line when I is bounded, I is only bounded below, or I = Z, +respectively. +The Gromov boundary of Γ, denoted by ∂∞Γ, is the set of equivalence classes of asymptotic +geodesic rays in Γ, which gives a natural compactification of (Γ, dΓ), +Γ = Γ ⊔ ∂∞Γ. +The topology of Γ is understood as follows: A pair of sequences (γn) and (γ′ +n) in Γ are said +to fellow travel if +(γn, γ′ +n)1Γ → ∞, +as n → ∞. +A sequence (γn) in Γ converges to a point ε ∈ ∂∞Γ, which is denoted by +γn +Cay +−−→ ε, +if and only if (γn) fellow-travels the image sequence (c(n)) of a(ny) geodesic ray c : N → Γ +asymptotic to ε. +The following result can be checked easily using the definitions above: +Lemma 2.4. Fellow traveling sequences in Γ have the same accumulation set in ∂∞Γ. +2.5. Nearest point projections to quasiconvex subgroups. Let Γ be a hyperbolic +group equipped with a left-invariant word metric dΓ. A subset Y ⊂ Γ is called quasiconvex +if there exists K ≥ 0 such that, for all y1, y2 ∈ Y , any geodesic segment in Γ connecting +y1 and y2 lies in the K-neighborhood of Y . For a quasiconvex subset Y ⊂ Γ, we choose a +nearest point projection map +prY : Γ → Y. +Note that nearest point projections are not necessarily unique, but since Y is quasiconvex, +any two such maps are a finite distance apart. +Lemma 2.5. Let Y ⊂ Γ be a quasiconvex subset. For every γ ∈ Γ, prY (γ) ∈ Y lies in a +uniform neighborhood of any geodesic segment in Γ connecting γ to Y . + +8 +SUBHADIP DEY AND MICHAEL KAPOVICH +y0 +y′ +y +γ′ = prY (γ) +x +γ +K ≥ +2δ ≥ +≤ 2δ +1 = +Y +Figure 2 +Proof. Let K ≥ 0 be a quasiconvexity constant for Y and δ ≥ 0 be a hyperbolicity constant +for (Γ, dΓ). Let [y0, γ] be a geodesic segment in Γ from y0 ∈ Y to γ. +Consider a geodesic triangle in Γ with vertices y0, γ, and γ′ := prY (γ), whose edge +connecting y0 to γ is [y0, γ]. Since geodesic triangles in Γ are 2δ-thin, we have +[y0, γ] ⊂ N2δ([y0, γ′] ∪ [γ, γ′]), +where Nr(·) denotes the r-neighborhood in (Γ, dΓ). In particular, there exist points x ∈ +[γ, γ′] and y ∈ [y0, γ′] such that dΓ(x, [y0, γ]) ≤ 2δ, dΓ(y, [y0, γ]) ≤ 2δ, and dΓ(x, y) ≤ 4δ +1. +See Figure 2 for an illustration of the points in the Cayley graph of Γ. Since [y0, γ′] lies in +the K-neighborhood of Y , there exists y′ ∈ Y , which is at most K distance away from y. +Since γ′ is also a nearest point in Y to x, dΓ(x, γ′) ≤ dΓ(x, y′) ≤ K + 4δ + 1. Therefore, +γ′ is at most K + 6δ + 1 distance away from [y0, γ]. +□ +For a subset Y ⊂ Γ, let ∂∞Y ⊂ ∂∞Γ denote the set of all accumulation points of Y in +Γ = Γ ⊔ ∂∞Γ. +Corollary 2.6. Let Y ⊂ Γ be a quasiconvex subset. A sequence (γn) in Γ has an accumu- +lation point in ∂∞Y if and only if (prY (γn)) is unbounded in Y . +Proof. For the “if” part, suppose that (prY (γn)) is unbounded. After passing to a subse- +quence, we assume that | prY (γn)| → ∞, as n → ∞. Applying Lemma 2.5, it follows that +(γn) and (prY (γn)) fellow travel. Thus, by Lemma 2.4, (γn) has an accumulation point in +∂∞Y . +For the “only if” part, we prove the contrapositive statement. +Let us assume that +(prY (γn)) is bounded. We show that (γn) has no accumulation points in ∂∞Y . We argue by +contradiction: Suppose that ε ∈ ∂∞Y is an accumulation point of (γn). After extraction, +we may assume that γn +Cay +−−→ ε. Let (yn) be any sequence in Y such that yn +Cay +−−→ ε. Con- +sequently, we must have (γn, yn)1Γ → ∞, as n → ∞. However, Lemma 2.5 can be restated +as +sup +γ∈Γ,y∈Y +(γ, y)prY (γ) < ∞. + +COMBINATION THEOREM: AMALGAMS +9 +Since (prY (γn)) is bounded, the above implies +sup +m,n +(γn, ym)1Γ < ∞, +a contradiction. +□ +A subgroup H < Γ is called a quasiconvex subgroup if H, as a subset of Γ, is quasiconvex. +Lemma 2.7. Let H < Γ be a quasiconvex subgroup. For any sequence (γn) in Γ, consider +the sequence (ˆγn) given by ˆγn = prH(γn)−1γn. +(i) Regarded as a sequence in the compact space Γ = Γ ⊔ ∂∞Γ, (ˆγn) has no accumulation +points in ∂∞H. +(ii) Suppose that (γn) diverges away from H, i.e., dΓ(H, γn) → ∞, as n → ∞. Then, the +accumulation set of (γ−1 +n ) in ∂∞Γ is the same as that of (ˆγ−1 +n ). +Proof. We first observe that, for all γ ∈ Γ, the identity element is a nearest point in H to +ˆγ := prH(γ)−1γ: Indeed, for all η ∈ H, +dΓ(η, ˆγ) = dΓ(prH(γ)η, γ) ≥ dΓ(prH(γ), γ), +(2.6) +where the inequality follows from the fact that prH(γ) is a nearest point in H to γ. Moreover, +dΓ(1Γ, ˆγ) = dΓ(prH(γ), γ). Hence, (2.6) implies that dΓ(H, ˆγ) ≥ dΓ(1Γ, ˆγ). +Therefore, for any sequence (γn) in Γ, {prH(ˆγn) | n ∈ N} ⊂ H is bounded. Part (i) now +follows by applying Corollary 2.6. +We prove part (ii): Since prH(γn) lies uniformly close to any geodesic in Γ connecting 1Γ +to γn (see Lemma 2.5) and dΓ(H, γn) → ∞, it follows that +|γn| − | prH(γn)| → ∞, +as n → ∞. Thus +(γ−1 +n , ˆγ−1 +n )1Γ = 1 +2(|γ−1 +n | + |ˆγ−1 +n | − dΓ(γ−1 +n , ˆγ−1 +n )) += 1 +2(|γn| + |ˆγn| − |γnˆγ−1 +n |) += 1 +2(|γn| + |ˆγn| − | prH(γn)|) → ∞, +as n → ∞. In other words, (γ−1 +n ) and (ˆγ−1 +n ) fellow travel. By Lemma 2.4, (γ−1 +n ) and (ˆγ−1 +n ) +have the same accumulation sets in ∂∞Γ. +□ +An equivalent statement of the above result, which we will often use, is as follows: +Lemma 2.8. Let H < Γ be a quasiconvex subgroup. For any sequence (γn) in Γ, consider +the sequence (ˆγn) given by ˆγn = γn prH(γ−1 +n ). +(i) (ˆγ−1 +n ) has no accumulation points in ∂∞H. +(ii) If (γ−1 +n ) diverges away from H, then the accumulation sets of (γn) and (ˆγn) in ∂∞Γ +coincide. + +10 +SUBHADIP DEY AND MICHAEL KAPOVICH +2.6. Amalgams of hyperbolic groups. For the rest of this section, we restrict our discus- +sion to amalgams (amalgamated free products and HNN extensions) of hyperbolic groups. +The Bestvina–Feighn Combination Theorem, [3], provides some sufficient conditions for the +hyperbolicity of amalgams. We review this theorem in the weakly malnormal case (although +their actual result is much more general): +Definition 2.9. A subgroup H of a group Γ is said to be weakly malnormal if, for every +γ ∈ Γ \ H, the subgroup γHγ−1 ∩ H is finite. +Theorem 2.10 (Bestvina–Feighn, [3]). Let ΓA, ΓB, and M be hyperbolic groups. +(i) If H < ΓA and H < ΓB are quasiconvex, weakly malnormal subgroups, then ΓA ⋆H ΓB +is hyperbolic. +(ii) If H± < M are isomorphic (by φ : H− → H+), quasiconvex, weakly malnormal sub- +groups such that, for all µ ∈ M, H− ∩ µH+µ−1 is finite, then M⋆φ is hyperbolic. +See also [16, Theorems 1 & 2]. +This theorem has several addendums that we will use in what follows: +Theorem 2.11 (Mitra, [30]). Under the assumptions, the subgroups ΓA and ΓB (resp. M) +in Theorem 2.10 are quasiconvex in ΓA ⋆H ΓB (resp. M⋆φ). +2.7. Boundary of an amalgam. Our next goal is to describe the Gromov boundaries +of the amalgamated free products Γ = ΓA ⋆H ΓB and HNN extensions Γ = M⋆φ:H−→H+ +under certain extra assumptions. We will be assuming that the groups ΓA, ΓB, and M are +hyperbolic, H is weakly malnormal and quasiconvex in ΓA, ΓB (in the amalgamated free +product case) and, H± are weakly malnormal and quasiconvex in M and the intersection +H−∩µH+µ−1 is finite, for all µ ∈ M (in the HNN extension case). Under these assumptions, +Γ is hyperbolic (see Theorem 2.10). +Moreover, in the case of HNN extensions, we let +f ∈ Γ denote the stable letter, the element corresponding to the subgroup isomorphism +φ : H− → H+: +fηf −1 = φ(η), +η ∈ H−. +Our description of the boundary follows [15, 7.3], to which we refer the reader for de- +tails and proofs. We will describe the boundary (mainly) in the case of amalgamated free +products since the HNN extension case is similar. +Let T denote the Bass-Serre tree associated with the amalgamated free product Γ = +ΓA ⋆H ΓB (or the HNN extension); see §2.3. The group Γ acts on T with vertex-stabilizers +(the vertex-subgroups Γv) that are conjugates of ΓA, ΓB (or M in the HNN extension case) +and edge-stabilizers (edge-subgroups Γe) which are conjugates of H. +Define a tree of topological spaces as follows: To each v ∈ V (T), e ∈ E(T), we associate the +Gromov boundary ∂∞Γv, ∂∞Γe. Whenever v is a vertex of an edge e, we have the inclusion +homomorphism Γe → Γv, which induces a topological embedding fev : ∂∞Γe → ∂∞Γv. This +data yields a tree of topological spaces. The tree gives rise to a topological space ∂IΓ, the +topological realization of the tree of topological spaces, by taking the push-out of the maps +fev: The topological space ∂IΓ is a union of Gromov boundaries ∂∞Γv, v ∈ V (T). More +precisely, it is the quotient of the disjoint union of these boundaries by the equivalence +relation defined as follows. For every edge e = [v, w] of T, we have ξ ∈ ∂∞Γv is equivalent + +COMBINATION THEOREM: AMALGAMS +11 +to η ∈ ∂∞Γw whenever there exists ζ ∈ ∂∞Γe such that fev(ζ) = ξ, few(ζ) = η. The group +Γ acts on ∂IΓ via the projection of the natural Γ-action on +� +v∈V (T) +∂∞Γv. +In particular, ∂IΓ is either the union of the Γ-orbits of ΓA ∪ ΓB (in the amalgamated free +product case) or ∂∞M (in the HNN extension case). +The weak malnormality assumption on the amalgam implies that whenever e ̸= e′ are +distinct edges of T, ∂∞Γe ∩ ∂∞Γe′ = ∅ in ∂IΓ. Accordingly, whenever the distance between +vertices v, w is > 1, +∂∞Γv ∩ ∂∞Γw = ∅. +(2.7) +Moreover, the weak malnormality assumption also implies that all vertex-subgroups Γv +and edge-subgroups Γe are quasiconvex in Γ. Hence, one obtains a natural Γ-equivariant +inclusion map ∂IΓ → ∂∞Γ. It turns out that this map is injective and continuous. However, +in general, this map need not be a topological embedding. In what follows, we will identify +∂IΓ with its image in ∂∞Γ. +The Gromov boundary ∂∞Γ of Γ is the disjoint union Γ-invariant subsets +∂∞Γ = ∂IΓ ⊔ ∂IIΓ, +where ∂IIΓ := ∂∞Γ\∂IΓ. Elements of ∂IΓ, resp. ∂IIΓ, are called type I, resp. type II, (ideal) +boundary points of Γ. +The second part of the boundary, ∂IIΓ, of ∂∞Γ admits a Γ-equivariant continuous bijection +to ∂∞T. (For instance, ∂∞⟨f⟩ is the 2-point subset of ∂IIΓ corresponding to the fixed-point +set of f in ∂∞T.) +Proposition 2.12. For every point ε ∈ ∂IIΓ, there exists an alternating sequence (ωn) in +Γ converging (in Γ) to ε such that the following holds: If a sequence (γn) in Γ converges to +ε, then there exists a function F : N → N diverging to infinity and n1 ∈ N such that, for +all integers n ≥ n1, there exists a normal form of γn containing ωF (n) as a left subword. +We prove this result in §2.8. +Proposition 2.13. Fix a word metric dΓ on Γ. For all ω ∈ Γ satisfying rl(ω) ≥ 3, there +exists a constant D = D(ω) ≥ 0 such that the following holds: If γ ∈ Γ is any element such +that some normal form of γ contains some normal form of ω as a left subword, then +(i) in the case Γ = ΓA ⋆H ΓB, we have dΓ(prΓA(γ), 1Γ) ≤ D and dΓ(prΓB(γ), 1Γ) ≤ D. +(ii) in the case Γ = M⋆φ, we have dΓ(prM(γ), 1Γ) ≤ D. +Using this result, it can be shown that alternating sequences cannot have type I accumu- +lation points in the boundary of Γ. See §2.9 for a proof of Proposition 2.13. +Notation 2.14. We set up some notation for the rest of this section: We let dΓ denote an +arbitrary word metric on Γ. Moreover, we reserve the notation d to denote a word metric on +Γ induced by a finite symmetric generating set S of Γ, where, (i) in the case of Γ = ΓA⋆HΓB, +S is the union of some chosen finite symmetric generating sets of ΓA and ΓB, and (ii) in +the case of Γ = M⋆φ, S is the union of some chosen finite symmetric generating set of M +and {f, f −1}. Recall that the identity map (Γ, d) → (Γ, dΓ) is a quasiisometry. Finally, we + +12 +SUBHADIP DEY AND MICHAEL KAPOVICH +reserve the notation dT to denote the distance function on the corresponding Bass-Serre +tree T induced by declaring that all the edges of T are of unit length. +2.8. Proof of Proposition 2.12. The following result demonstrates the existence of an +alternating sequence (ωk) converging to ε ∈ ∂IIΓ in the statement of Proposition 2.12. +Lemma 2.15. There exists D0 ≥ 0 such that, for every ε ∈ ∂IIΓ, there exists a D0- +alternating sequence (ωn) in (Γ, dΓ) converging to ε. +In the statement above, “D0-alternating” means that (ωn) is alternating and lies within +distance D0 from any geodesic ray in (Γ, dΓ) emanating from 1Γ asymptotic to ε. +In the proof of Lemma 2.15, we work with the word-metric d; see Notation 2.14. The +general case, i.e., when dΓ is an arbitrary word metric, would then follow by applying the +Morse lemma. +Proof of Lemma 2.15 in the case of amalgamated free products. Let us consider a uniform +quasigeodesic c : N ∪ {0} → (Γ, d) emanating from c(0) = 1Γ asymptotic to ε; such a ray is +described by a sequence (si) in S such that, for all k ∈ N, +c(k) = s1 · · · sk. +Note that the sequence (c(k)) cannot entirely lie in ΓA ∪ ΓB since, otherwise, the sequence +(c(k)) would converge to a type I ideal boundary point. Let i1 be the largest number such +that c(i1) lies in ΓA ∪ ΓB. For the same reason as above, the sequence (c(i1)−1c(k))k∈N +cannot lie in ΓA ∪ ΓB. Thus, let i2 > i1 be the largest number such that c(i1)−1c(i2) lies in +ΓA ∪ ΓB. Similarly, let i3 > i2 be the largest number such that c(i2)−1c(i3) lies in ΓA ∪ ΓB. +Proceeding inductively, we find a sequence (c(ik)−1c(ik+1))k which alternates between ΓA +and ΓB; the elements of that sequence are the letters for our alternating sequence (ωk): +ωk = c(i1)[c(i1)−1c(i2)] · · · [c(ik−1)−1c(ik)] = c(ik). +Since the sequence (ωk) lies in the quasigeodesic ray c, it converges to ε. +□ +Proof of Lemma 2.15 in the case of HNN extensions. For ε ∈ ∂IIΓ, pick any uniform quasi- +geodesic ray c : N ∪ {0} → Γ that emanates from c(0) = 1Γ and is asymptotic to ε. Such a +ray is described by a sequence (si) in S such that, for all k ∈ N, +c(k) = s1 · · · sk. +We find an infinite string +ˆS : +µ0, f ǫ1, µ1, f ǫ1, µ2, . . . , +(2.8) +which has the property that µi ∈ M, ǫi = ±1, and, for every n, +rl(µ0f ǫ1µ1 · · · f ǫn−1µn−1f ǫn) = n. +(2.9) +Let i0 ∈ N ∪ {0} be the largest number such that c(i0) ∈ M; set µ0 = c(i0). Let i1 ≥ i0 +be the largest number such that c(i0)−1c(i1) ∈ {f, f −1}; define ǫ1 in the obvious way. Let +i2 ≥ i1 to be the largest number such that c(i1)−1c(i2) ∈ M; set µ1 = c(i1)−1c(i2). Let +i3 ≥ i2 be the largest number such that c(i2)−1c(i2) ∈ {f, f −1}; define ǫ2 accordingly. +We proceed inductively to obtain the above string. +It is a straightforward check that +rl(µ0f ǫ1µ1 · · · f ǫn−1µn−1f ǫn) = n. + +COMBINATION THEOREM: AMALGAMS +13 +We observe that, if ǫi = −1 and µi ∈ H+ (resp. ǫi = 1 and µi+1 ∈ H−), then (2.9) forces +ǫi = −1 (resp. ǫi = 1). Thus, (ωk), where ωk := µ0f ǫ1µ1 · · · f ǫn−1µn−1f ǫn, is an alternating +sequence. Since the sequence (ωk) lies in the quasigeodesic ray c, it converges to ε. +□ +The same construction used to prove Lemma 2.15 can be applied to “uniform quasi- +geodesic segments” in Γ to show the following result: +Definition 2.16. For D ≥ 0, a D-normal form of γ ∈ Γ satisfying rl(γ) ≥ 1 is a normal +form of γ such that all left subwords in that normal form lie in a D-neighborhood of any +geodesic segment in (Γ, dΓ) connecting 1Γ and γ. +Lemma 2.17. There exists D0 ≥ 0 such that every element γ ∈ Γ satisfying rl(γ) ≥ 1 has +a D0-normal form. +Now, we prove Proposition 2.12. +Proof of Proposition 2.12. We give a proof in the amalgamated free product case; the HNN +extension case is similar. +We continue with the proof of Lemma 2.15 above (the amalgamated free product case). +Let (γn) be any sequence in Γ converging to ε ∈ ∂IIΓ. Since (γn) fellow travels the uniform +quasigeodesic c, we may pick a divergent sequence (tn) in N∪{0} and, for each n, a uniform +quasigeodesic segment +cn : [0, ln] ∩ Z → Γ, +cn(0) = 1Γ, cn(ln) = γn, +such that, for all n ∈ N, cn|[0,tn]∩Z = c|[0,tn]∩Z. See Figure 3 for an illustration of c and cn +in (the Cayley graph) of Γ. We apply the same procedure used in the proof of Lemma 2.15 +on cn to yield a D-normal form for γn. +1Γ +γn = cn(ln) +ε +c(tn) +c +Figure 3 +Lemma 2.18. There exists n1 ∈ N such that, for all n ≥ n1, the prescribed normal form +of γn contains ω1 as a leftmost subword. +Proof. We argue by contradiction: Suppose that the assertion is false. Then, a divergent +sequence (ni) exists in N such that, for all i, the leftmost letter λi of the prescribed normal +form of γni is different from ω1. Since cni|[0,tni]∩Z = c|[0,tni]∩Z, for all i large enough it must +hold that λi = cni(ri), where ri > tni. However, λi ∈ ΓA ∪ ΓB. Since cni are uniform +quasigeodesics with cni(0) = 1Γ ∈ ΓA ∪ ΓB, it follows that cni([0, ri] ∩ Z), which contains +c([0, tni]∩Z) as a subset, lies in a uniform neighborhood of ΓA∪ΓB. However, since tni → ∞, +rl(c(tni)) goes to infinity. Thus, c(tni) cannot stay in a uniform neighborhood of ΓA ∪ ΓB +(cf. Proposition 2.13), yielding a contradiction. +□ + +14 +SUBHADIP DEY AND MICHAEL KAPOVICH +We now finish the proof of Proposition 2.12 by induction. Suppose that c(i1) = ω1. Since +cn|[0,tn]∩Z = c|[0,tn]∩Z, by the claim above, it holds that for all sufficiently large n, say n ≥ n1, +cn(i1) = c(i1) = ω1. Applying the same argument to the quasigeodesic ray/segments +¯c : [0, ∞) ∩ Z → Γ, +c(t) = ω−1 +1 c(t + i1), +¯cn : [0, ln − i1] ∩ Z → Γ, +cn(t) = ω−1 +1 cn(t + i1), +shows that there exists n2 > n1 such that for all n ≥ n2, the prescribed normal form of +γn contains ω2 as a leftmost subword. Arguing inductively, it follows that there exists an +increasing sequence (nk) of natural numbers such that, for all n ≥ nk, the prescribed normal +form of γn contains ωk as a leftmost subword. Thus, the desired function F : N → N in +Proposition 2.12 can be defined as F(n) := k, if n ∈ [nk, nk+1). +□ +2.9. Proof of Proposition 2.13. Let us first consider the case of amalgamated free prod- +ucts: For γ ∈ Γ = ΓA ⋆H ΓB, consider a normal form of γ: +γ = γ1γ2 · · · γl. +If γ1 ∈ ΓA \ H, then normal form given above yields a finite sequence of points in T: +ΓB, +ΓA, +γ1ΓB, +γ1γ2ΓA, +γ1γ2γ3ΓB, +. . . , +γ1γ2 · · · γlΓ∗, +(2.10) +where ∗ = A if l is even, or ∗ = B if l is odd. We observe that any two consecutive points +in the above are adjacent vertices in T (cf. §2.3), and the sequence does not backtrack, i.e., +for any point in (2.10), the vertices on the left and right to it are different: For even i, let +us examine the portion of the path +γ1 · · · γi−1ΓB, +γ1 · · · γiΓA, +γ1 · · · γi+1ΓB. +Note that γi ∈ ΓB \ H and γi+1 ∈ ΓA \ H. Applying (γ1 · · · γi)−1 to the above, we obtain +ΓB = γ−1 +i +ΓB, +ΓA, +γi+1ΓB. +However, since γi+1 ̸∈ ΓB, ΓB ̸= γi+1ΓB. A similar analysis can be done when i is odd. +Therefore, if we connect each pair of consecutive vertices in (2.10) by the unique edge in +T determined by the pair, we obtain a geodesic path in T. +Lemma 2.19. If γ ∈ Γ = ΓA ⋆H ΓB lies in the coset represented by v ∈ V (T), then +|dT (v, ΓA) − rl(γ)| ≤ 1, |dT (v, ΓB) − rl(γ)| ≤ 1. +Proof. This claim is easily checked when γ ∈ H. So, suppose that γ ̸∈ H. Let γ1γ2 · · · γl +be a normal form of γ such that γ1 ∈ ΓA \ H; the case γ1 ∈ ΓB \ H follows by a similar +argument. Note that v is one of the two rightmost entries in the sequence (2.10). Moreover, +the discussion above shows that the sequence (2.10) is a geodesic sequence of vertices in T. +Thus, rl(γ) ≥ dT (v, ΓA) ≥ rl(γ) − 1 and rl(γ) + 1 ≥ dT (v, ΓB) ≥ rl(γ). +□ +Similar discussion holds in the case of HNN extensions: For γ ∈ Γ = M⋆φ, let +γ = µ0f ǫ1µ1 · · · f ǫnµn +be a normal form of γ. The above normal form produces a finite sequence in T: +M, +µ0f ǫ1M, +µ0f ǫ1µ1f ǫ2M, +. . . , +γM = µ0f ǫ1µ1 · · · f ǫnM. + +COMBINATION THEOREM: AMALGAMS +15 +Similarly to the amalgamated free product case discussed above, one can check that the +above sequence does not backtrack and that consecutive vertices in the path are adjacent +in T. This yields the following: +Lemma 2.20. For all γ ∈ Γ = M⋆φ, dT (v, M) = rl(γ). +Proof. The proof is similar to that of Lemma 2.19. We omit the details. +□ +We prove Proposition 2.13. +Proof of Proposition 2.13. We discuss the proof in the case of amalgamated free products +Γ = ΓA ⋆H ΓB; the case of HNN extensions Γ = M⋆φ is similar. +Consider a normal form of γ which contains some normal form of ω as a left subword: +γ = γ1 · · · γm +� +�� +� +=ω +γm+1 · · · γl. +We further assume that γ1 ∈ ΓA \ H; the case γ1 ∈ ΓB \ H follows similarly. The above +normal form induces a geodesic path in T (see the paragraph before Lemma 2.19) from the +vertex ΓB to the vertex vγ := γΓ∗, where ∗ = A if l is even, or ∗ = B if l is odd. This path +also contains the vertex vω := ωΓ∗, where ∗ = A if m is even, or ∗ = B if m is odd. The +vertex ΓA also lies in that path, see (2.10). Thus, any path in T connecting vγ to ΓA or ΓB +must contain vω. +For the rest of the proof, let ∗ denote either A or B. Let c : [0, n] ∩ Z → Γ be a shortest +geodesic in (Γ, d)1 such that c(0) ∈ Γ∗ and c(n) = γ. By definition, c(0) is a closest point +in Γ∗ to c(k), for any k in the domain of c. For k ∈ [0, n] ∩ Z, let ¯c(k) := {c(k)ΓA, c(k)ΓB}. +Claim. For k = 0, . . . , n − 1, we have that ¯c(k) ∩ ¯c(k + 1) ̸= ∅. +Proof. We observe that c(k + 1) = c(k) · sk, for some generator sk ∈ S ⊂ ΓA ∪ ΓB. Let +c(k) = ˜γ1 · · · ˜γr be a normal form of c(k). +Assume that ˜γr ∈ ΓA; the other possibility +˜γr ∈ ΓB can be similarly treated. If sk ∈ ΓA, then c(k)ΓA = ˜γ1 · · · ˜γr−1ΓA ∈ ¯c(k + 1) since +c(k + 1) = ˜γ1 · · · ˜γr−1(˜γrsk) ∈ ˜γ1 · · · ˜γr−1ΓA. Similarly, if sk ∈ ΓB, then c(k)ΓB ∈ ¯c(k + 1). +The claim follows. +□ +Let V ′ = �n +k=0 ¯c(k) ⊂ V (T). Consider the induced subtree of T ′ ⊂ T determined by V ′. +By the claim above, T ′ is connected. Since c(0) ∈ Γ∗, we get that Γ∗ ∈ V ′. Obviously, +vγ ∈ V ′ as well. +Since T ′ is connected, by the first paragraph of this proof, vω ∈ V ′. +Therefore, there exists some k0 in the domain of c such that vω ∈ ¯c(k0). In other words, +c(k0) lies in (the coset represented by) vω. Since c(0) is also a closest point in Γ∗ to c(k0), it +follows that c(0) is uniformly close to prΓ∗(vω). However, since rl(ω) ≥ 3, by Lemma 2.19, +dT (vω, Γ∗) ≥ 2. Thus, by (2.7),2 ∂∞Γ∗ ∩ ∂∞vω = ∅ and, hence, prΓ∗(vω) is bounded in Γ∗ +(see Corollary 2.6). This completes the proof of this result when dΓ = d. +In the general case, i.e., when dΓ is an arbitrary word metric, then the result follows by +the fact that the identity map (Γ, d) → (Γ, dΓ) is a quasiisometry. +□ +1See Notation 2.14 for our notation. +2Note that ∂∞Γvω = ∂∞vω. + +16 +SUBHADIP DEY AND MICHAEL KAPOVICH +3. Preliminaries on discrete isometry groups of symmetric spaces +We recall some preliminary facts on symmetric spaces, mainly to set up some notations. +Then, we refer to [2, Appendix 5] for a quick discussion and to [7] for a more detailed +exposition. +Let G be a real semisimple Lie group of noncompact type with a finite center. +We +impose some mild assumptions on G; see Assumption 3.1 below. Let X = G/K denote the +(globally) symmetric space of G, where K is a maximal compact subgroup of G. Then, X +is a nonpositively curved G-homogeneous space such that X has no compact or Euclidean +de Rham factors. +The ideal boundary ∂∞X is the set of equivalence classes of asymptotic rays in X on which +G acts naturally. The point stabilizers in G of the action G ↷ ∂∞X are called parabolic +subgroups of G. The ideal boundary ∂∞X carries a natural G-invariant spherical building +structure, called the Tits building of X, and denoted by ∂TitsX. +The top-dimensional +simplices in ∂TitsX are called chambers. +In this paper, we impose the following assumption on G (for simplicity, one may only +assume that G is connected), which are standing assumptions in the papers of Kapovich– +Leeb–Porti [12, 13, 14] we rely upon in this work: +Assumption 3.1. The group G has a finitely many connected components such that, for +the associated symmetric space X = G/K, the spherical Tits building ∂TitsX is thick: That +is, every panel (i.e., a codimension one simplex) in ∂TitsX is contained in at least three +distinct chambers. +Each chamber in ∂∞X is also a fundamental domain for the action G ↷ ∂∞X. The +stabilizer in G of a chamber σ (which is the same as the stabilizer in G of any interior +point of σ) is called a minimal parabolic subgroup of G. The group G also acts on the set +of chambers in ∂TitsX transitively, so any two minimal parabolic subgroups are conjugate. +Therefore, the space of all chambers in ∂TitsX can be identified with the analytic manifold +Flag(σmod) := G/Pσmod, +where σmod is a chosen chamber in ∂TitsX and Pσmod is the minimal parabolic subgroup +stabilizing σmod. +More generally, for a face νmod ⊂ σmod, the space of simplices ν in ∂TitsX of type νmod, +i.e., those simplices ν in ∂TitsX which can be brought to νmod by the action G ↷ ∂TitsX, +can be identified with the analytic manifold +Flag(νmod) := G/Pνmod, +where Pνmod is the parabolic subgroup of G stabilizing νmod. +A pair of simplices ν± in ∂TitsX is called antipodal if there is a complete geodesic line in +X, which is forward (resp. backward) asymptotic to some interior point of the simplex ν+ +(resp. ν−). If ν± +mod ⊂ σmod denote the types of ν±, then they satisfy +ν+ +mod = ι(ν− +mod), +where ι : σmod → σmod denotes the opposition involution. For a simplex ν− in ∂TitsX of type +ν− +mod, the set of simplices in ∂TitsX (regarded as points in Flag(ν+ +mod), where ν+ +mod = ι(ν− +mod)) + +COMBINATION THEOREM: AMALGAMS +17 +antipodal to ν−, +C(ν−) := {ν+ ∈ Flag(ν+ +mod) | ν+ is antipodal to ν−} ⊂ Flag(ν+ +mod), +is an open dense Pν− +mod-homogeneous subset of Flag(ν+ +mod). +Definition 3.2 (Antipodality). A pair (A, B), where A ⊂ Flag(ν+ +mod) and B ⊂ Flag(ν− +mod), +is said to be antipodal to each other (or, A is antipodal to B) if, for all ν+ ∈ A and all +ν− ∈ B, ν+ is antipodal to ν−. +3.1. Regular sequences. Let νmod ⊂ σmod be a face. A sequence (gn) in G is said to +νmod-converge to a point ν ∈ Flag(νmod), which is denoted by +gn +flag +−−→ ν, +if every subsequence of (gn) has a further subsequence (gnk) such that there exists ν− ∈ +Flag(ινmod) such that +gnk|C(ν−) → ν, +uniformly on compacts. +In this situation, ν ∈ Flag(νmod) is called a νmod-limit point of (gn). See [13, §4] for more +details. +For a discrete subgroup Γ of G, the set of all νmod-limit points of Γ in Flag(νmod) is called +the νmod-limit set of Γ, which is denoted by +Λνmod(Γ). +The limit set Λνmod(Γ) is a Γ-invariant compact subset of Flag(νmod); however; it may be +empty, even if Γ is infinite. +A sequence (gn) in G is νmod-regular if every subsequence of (gn) contains a νmod-flag- +convergent subsequence. Moreover, if (gn) is νmod-regular, then the inverse sequence (g−1 +n ) +is ινmod-regular. Similarly, a subgroup Γ of G is called νmod-regular if every sequence in Γ +is νmod-regular. Such subgroups are necessarily discrete. Clearly, νmod-regular subgroups +are also ινmod-regular. +The following results help verify the regularity and flag-convergence of certain sequences +considered in this paper. +Let (gn) be a sequence in G. +Lemma 3.3. If there exists a compact subset A ⊂ Flag(νmod) with a nonempty interior +such that the sequence (gnA) of compact subsets of Flag(νmod) converges to a point ν ∈ +Flag(νmod), then (gn) is νmod-regular and gn +flag +−−→ ν. +See [4, Lemma 1.9]. +Let d be a distance function on Flag(νmod) compatible with the manifold topology of +Flag(νmod). +Then, we say that a sequence (An) of subsets of Flag(νmod) shrinks if the +diameter of An converges to zero, as n → ∞. Since Flag(νmod) is compact, this notion does +not depend on the chosen distance function. +Lemma 3.4. If there exists a compact subset A ⊂ Flag(νmod) with a nonempty interior +such that (gnA) shrinks, then (gn) is νmod-regular. + +18 +SUBHADIP DEY AND MICHAEL KAPOVICH +See [4, Corollary 1.10]. +Lemma 3.5. If there exist compact subsets B, B′ ⊂ Flag(νmod) with nonempty interior +such that B′ ⊂ B◦ and, for all n ∈ N, gn+1g−1 +n (B) ⊂ B′, then (gn) is νmod-regular. +The statement above can be extracted from the proof of [4, Lemma 3.4]. For complete- +ness, let us prove this result. +Proof of Lemma 3.5. After replacing the sequence (gn) by (hn), where hn := gng−1 +1 , we +have that +hn(B) ⊂ B +and +hn+1h−1 +n (B) ⊂ B′. +(3.1) +Suppose, to the contrary, that (gn) is not νmod-regular. +Then (hn) is also not νmod- +regular. Therefore, after extraction, (hn) is ηmod-pure for some face ηmod ⊂ σmod such that +νmod ̸⊂ ηmod. By [12, Proposition 9.5], after further extraction, there exist η+ ∈ Flag(ηmod) +and η− ∈ Flag(ιηmod), and a surjective algebraic map φ : CFu(η−) → StFu(η+) such that +hnk|CFu(η−) → φ uniformly on compacts. +By (3.1), we obtain that hnk ˜B ⊂ ˜B, where +˜B = +� +ν∈B +StFu(ν). +Consider any point η ∈ C(η−) ⊂ Flag(ηmod) such that StFu(η) ∩ ( ˜B)◦ is nonempty.3 By [4, +Lemma 1.8], +lim +k→∞ +max +σ∈StFu(hnk η) d(hnkh−1 +nk−1σ, σ) = 0. +(3.2) +Since, for all k ∈ N, hnk( ˜B)◦ ⊂ ( ˜B)◦, StFu(hnkη) ∩ ( ˜B)◦ ̸= ∅. Also, by [4, Corollary 1.4], +StFu(hnkη) ̸⊂ ˜B. Since StFu(hnkη) is connected (by [4, Lemma 1.2]), StFu(hnkη) intersects +∂ ˜B; let σk be any point lying in this intersection. By (3.2), +lim +k→∞ d(hnkh−1 +nk−1σk, σk) = 0. +(3.3) +However, since hn+1h−1 +n (B) ⊂ B′ (by (3.1)), hmh−1 +n (B) ⊂ B′, for all m > n. In particular, +hnkh−1 +nk−1(B) ⊂ B′, for all k ∈ N, which implies that +hnkh−1 +nk−1( ˜B) ⊂ ˜B′ := +� +ν∈B′ +StFu(ν). +Since σk ∈ ˜B, by above, we get hnkh−1 +nk−1(σk) ∈ ˜B′. However, ˜B′ ⊂ ˜B◦, which shows that +d(hnkh−1 +nk−1(σk), σk) ≥ d( ˜B′, ∂ ˜B) > 0, +for all k ∈ N, +contradicting (3.3). +□ +Lemma 3.6. Suppose that (gn) is νmod-regular. +If there exist compact subsets A, B ⊂ +Flag(νmod) with nonempty interior such that, for all n ∈ N, gnB ⊂ A, then all the νmod- +limit-points of (gn) lie in A. +3Note that, since CFu(ν−) is open dense in Flag(σmod) and ˜B has nonempty interior in Flag(σmod), +CFu(ν−) ∩ ˜B is nonempty. + +COMBINATION THEOREM: AMALGAMS +19 +Proof. Let ν ∈ Flag(νmod) be a νmod-limit point of (gn). Then, there exists ν− ∈ Flag(ινmod) +and a subsequence (gnk) such that gnk|C(ν−) converges to the constant map C(ν−) → {ν} +uniformly on compacts, as k → ∞. Let ν1 ∈ C(ν−) ∩ B. Then, the sequence (gnkν1), which +lies in A, converges to ν. Thus, ν ∈ A. +□ +Lemma 3.7. Suppose that gn +flag +−−→ ν ∈ Flag(νmod). Let A− ⊂ Flag(ινmod) be a subset +containing all the ινmod-limit points of (g−1 +n ). If A ⊂ Flag(νmod) is any compact subset +antipodal to A−, then (gnA) converges to {ν}. +Proof. Suppose not. Then, there exists a sequence (νk) in A and a subsequence (gnk) of (gn) +such that (gnkνk) converges to some point ν′ ∈ Flag(νmod) different from ν. After further +extraction of (gnk), we may assume that (g−1 +nk ) ινmod-converges to some point ν− ∈ A−. +Since, by hypothesis, A ⊂ C(ν−), by [14, Lemma 4.18], we get that gnk(A) → ν, as n → ∞. +Since, for each k, νk ∈ A, we also obtain that gnkνk → ν, which implies that ν′ = ν. This +is a contradiction. +□ +The following result follows from Lemma 3.7: +Lemma 3.8. Suppose that (gn) is νmod-regular. Let A− ⊂ Flag(ινmod) be a subset contain- +ing all the ινmod-limit points of (g−1 +n ). If A ⊂ Flag(νmod) is any compact subset antipodal +to A−, then (gnA) shrinks. +3.2. Anosov subgroups. We fix once and for all an ι-invariant face τmod of σmod. We focus +on the special class of discrete subgroups of G, called τmod-Anosov subgroups. This class +of subgroups has several different characterizations. For our purpose, we use the following +characterization of τmod-Anosov subgroups: +Definition 3.9 (Asymptotically embedded). A subgroup Γ of G is τmod-asymptotically +embedded if +(i) Γ is a τmod-regular subgroup of G, +(ii) Γ, as an abstract group, is hyperbolic, and +(iii) there exists a Γ-equivariant antipodal map4 ξ : ∂∞Γ → Flag(τmod), which preserves the +convergence dynamics: That is, for every sequence (γn) in Γ and every point ε ∈ ∂∞Γ, +if γn +Cay +−−→ ε, then γn +flag +−−→ ξ(ε). +(3.4) +Remark 3.10. +(i) The image of the map ξ in the above definition is precisely Λτmod(Γ), the limit set of +Γ in Flag(τmod). +(ii) The above definition of τmod-asymptotically embedded subgroups, which is a minor +variation of the one in [13, Definition 5.12], is more straightforward to verify in this +paper due to a certain “ping-pong” type arguments we use here. The equivalence +between these two definitions can be checked as follows (cf. [4, §5]): +Suppose that Γ < G is τmod-asymptotically embedded in the sense of Definition 3.9. +We first verify that ξ is an embedding. Indeed, since ∂∞Γ is a compact and Flag(τmod) +is Hausdorff, it is enough to establish that ξ is an injective continuous map: Since ξ +4That is, ξ maps every distinct pair of points to a pair of antipodal points. + +20 +SUBHADIP DEY AND MICHAEL KAPOVICH +is an antipodal map, ξ is injective. To verify continuity, pick x ∈ X and consider the +map +φ : Γ → ¯Xτmod = X ⊔ Flag(τmod), +whose restriction to ∂∞Γ is ξ and whose restriction to Γ is the orbit map γ �→ γ·x. The +space X ⊔ Flag(τmod) is topologized with the topology of τmod-flag-convergence. The +restriction of this topology to X or Flag(τmod) coincides with their respective manifold +topologies. By (3.4), φ is continuous and, hence, so is the restriction ξ = φ|∂∞Γ. +Therefore, ξ is a Γ-equivariant homeomorphism between ∂∞Γ and its image, Λτmod(Γ). +Finally, since ξ is an antipodal map, Λτmod(Γ) is an antipodal subset of Flag(τmod). +Therefore, Γ is τmod-antipodal in the sense of [13]. +Hence, Γ is τmod-asymptotically embedded in the sense of [13, Definition 5.12]. +The other direction is straightforward. +3.3. Interactive pairs and triples. Let τmod be an ι-invariant face of σmod. Following +Maskit [28], we define the notion of “interactive pairs” and “interactive triples.” +3.3.1. Interactive pairs. Let ΓA and ΓB be discrete subgroups of G, and let H := ΓA ∩ ΓB. +We denote this data by the triple (ΓA, ΓB; H). +A +B +(ΓB \ H) +(ΓA \ H) +Figure 4. An interactive pair. +Definition 3.11 (Interactive pair). A pair (A, B) of compact subsets of Flag(τmod) = +G/Pτmod is called an interactive pair for (ΓA, ΓB; H) if the following conditions are satisfied: +(i) The interiors A◦ of A and B◦ of B are nonempty and disjoint. +(ii) H leaves the sets A and B precisely invariant, i.e., HA = A, HB = B, and, for all +elements α ∈ ΓA \ H and β ∈ ΓB \ H, we have that αB ⊂ A◦ and βA ⊂ B◦. +See Figure 4 for an illustration. +Proposition 3.12. If (ΓA, ΓB; H) admits an interactive pair (A, B) in Flag(τmod), then +the natural homomorphism +ρ : ΓA ⋆H ΓB → ⟨ΓA, ΓB⟩ < G +from the abstract amalgamated free product ΓA ⋆H ΓB to G in injective. In particular, the +subgroup Γ := ⟨ΓA, ΓB⟩ of G is naturally isomorphic to ΓA ⋆H ΓB. + +COMBINATION THEOREM: AMALGAMS +21 +Proof. Let γ ∈ ΓA⋆HΓB be any nontrivial element. We want to show that ρ(γ) is nontrivial. +Clearly, if γ ∈ (ΓA ∪ ΓB) \ {1Γ}, then ρ(γ) = γ and, hence, ρ(γ) is nontrivial. So, we may +assume that γ ̸∈ ΓA ∪ ΓB. +Choose a normal form γ = γ1 · · · γl of γ (see §2.1). +Since +γ ̸∈ ΓA ∪ ΓB, we have l ≥ 2. Assume that γl ∈ ΓB (the other possibility that γl ∈ ΓA can +be similarly analyzed). Then, ρ(γ)A ⊂ A◦ ∪ B◦ and, in particular, ρ(γ)A ̸= A. Thus, ρ(γ) +is a nontrivial element of G. +□ +Compare with [28, Theorem VII.A.10]. +Remark 3.13. With a little more effort, one can also show that the image of the homomor- +phism ρ in Proposition 3.12 is discrete. However, we do not need this result to prove our +main theorems. +3.3.2. Interactive triples. Let M be a discrete subgroup of G, and let f ∈ G. Define +H− := (f −1Mf) ∩ M, +H+ := M ∩ (fMf −1). +Clearly, fH−f −1 = H+. We denote this data by the quadruple (M; H±; f). +A +B+ +B− +f −1 +f +(M \ H+) +(M \ H−) +Figure 5. An interactive triple. +Definition 3.14 (Interactive triples). A triple (A, B±) of compact subsets of Flag(τmod) is +called an interactive triple for (M; H±; f) if the following conditions are met: +(i) The interiors A◦, B◦ +−, B◦ ++ of A, B−, B+, respectively, are nonempty and pairwise +disjoint. Furthermore, B− ∩ B+ = ∅. +(ii) H± leaves B± precisely invariant, i.e., H±B± = B± and µ(B±) ⊂ A◦, whenever +µ ̸∈ H±. +(iii) f ±1(A) ⊂ B± and f ±1B± ⊂ B◦ +±. +See Figure 5 for an illustration. +Proposition 3.15. If (M; H±; f) admits an interactive triple (A, B±) in Flag(τmod), then +the natural homomorphism +M⋆φ → G +is injective. In particular, the subgroup Γ := ⟨M, f⟩ of G is naturally isomorphic to the +HNN extension of M by the isomorphism φ : H− → H+ is given by φ(η) = fηf −1, for all +η ∈ H−. + +22 +SUBHADIP DEY AND MICHAEL KAPOVICH +Proof. The proof of this result is similar to the one of Proposition 3.12. Hence, we omit the +details. +□ +Compare with [28, Theorem VII.D.12]. +4. A combination theorem for amalgamated free products of Anosov +subgroups +The goal of this section is to prove Theorem A. +In this section, we work under the hypothesis of Theorem A. To simplify our situation, +we may replace A and B by the closure of their respective interiors. It is easy to see that +(cl(A◦), cl(B◦)) is still an interactive pair for (ΓA, ΓB; H). However, the main advantage of +the interactive pair (cl(A◦), cl(B◦)) is a stronger antipodality hypothesis: +Lemma 4.1. cl(A◦), resp. cl(B◦), is antipodal to B◦, resp. A◦. +Proof. Let τ+ ∈ cl(A◦) and τ− ∈ B◦ be any points. We show that τ± are antipodal. +Fix an auxiliary distance function d on Flag(τmod) compatible with its manifold topology. +For τ ∈ Flag(τmod), let E(τ) denote the (compact) subset of Flag(τmod) consisting of all +points which are not antipodal to τ. Let (τn) be a sequence in A◦ converging to τ+ and let +Bǫ(τ−) be a closed metric ball in Flag(τmod) centered at τ− of radius ǫ > 0 small enough such +that Bǫ(τ−) ⊂ B◦. By hypothesis, τn and B◦ are antipodal. Hence, d(E(τn), Bǫ(τ−)) > 0, +implying that d(E(τn), τ−) ≥ ǫ. +Since the Hausdorff distance between E(τn) and E(τ+) +converges to zero as n → ∞, it follows that d(E(τ+), τ−) ≥ ǫ > 0. So, τ− ̸∈ E(τ+). +□ +Assumption 4.2. After replacing (A, B) by (cl(A◦), cl(B◦)) in the hypothesis of Theo- +rem A, in place of (i), we may assume the following: +(i)’ The pairs of subsets (A, B◦) and (A◦, B) of Flag(τmod) are antipodal to each other. +This replacement does not affect the conclusion of Theorem A. +In §4.1, we will take additional advantage given by hypothesis (i)’ above; see, for instance, +the proof of Lemma 4.15. +Lemma 4.3. Under the hypothesis of Theorem A +Λτmod(ΓA) ⊂ A, +Λτmod(ΓB) ⊂ B, +and +Λτmod(H) ⊂ A ∩ B. +Proof. It will be enough to show that Λτmod(ΓA) ⊂ A. +If H = ΓA, then ΓA preserves A, and since A has a nonempty interior, all τmod-limit +points of Γ must lie in A (see Lemma 3.6). +So, we can assume that H is a proper subgroup of ΓA, i.e., there exists some element +α ∈ ΓA which is not an element of H. For any point τ+ ∈ Λτmod(ΓA), consider a sequence +(αn) in ΓA such that αn +flag +−−→ τ+. We may (and will) also assume that no elements of (αn) +lie in H: Indeed, for every n ∈ N, if αn ∈ H, then replace the n-th entry αn in (αn) by +αnα. After replacing all such entries, the resulting sequence, again denoted by (αn), does +not share any elements with H but still satisfies αn +flag +−−→ τ+. +Since for all n ∈ N, αnB ⊂ A, Lemma 3.6 implies that τ+ ∈ A. +□ + +COMBINATION THEOREM: AMALGAMS +23 +Corollary 4.4. Under the hypothesis Theorem A, the subgroup H is weakly malnormal in +both ΓA and ΓB. +Proof. Since Λτmod(H) ⊂ A∩B, the interactive pair assumption implies that for all α ∈ ΓA\H +and β ∈ ΓB \ H, +α(Λτmod(H)) ∩ Λτmod(H) = β(Λτmod(H)) ∩ Λτmod(H) = ∅. +If, say, αHα−1 ∩ H were infinite, it would contain an infinite order element η ∈ H; hence, +α−1(Λτmod(⟨η⟩)) = Λτmod(⟨α−1ηα⟩) +would be nonempty subsets of Λτmod(H). That would be a contradiction. +□ +Corollary 4.5. Under the hypothesis Theorem A, the subgroup Γ of G generated by ΓA and +ΓB in G is hyperbolic. +Proof. If H = ΓA ∩ ΓB is quasiconvex in one of ΓA or ΓB, then it is quasiconvex in both +of them: This follows from the general fact that, if Γ is a τmod-Anosov subgroup of G and +H < Γ is a subgroup, then H is quasiconvex in Γ if and only if H is a τmod-Anosov subgroup +of G. This general fact is a consequence of the τmod-URU characterization of τmod-Anosov +subgroups; see [13, Equivalence Theorem 1.1 & Remark 1.2(i)]. +Thus, under the hypothesis Theorem A, H is quasiconvex in ΓA and ΓB. +Moreover, +by Proposition 3.12, ⟨ΓA, ΓB⟩ is naturally isomorphic to ΓA ⋆H ΓB. Then, together with +Corollary 4.4, Theorem 2.10(i) implies that ⟨ΓA, ΓB⟩ is hyperbolic. +□ +4.1. Regularity. The main result of this subsection is as follows. +Proposition 4.6. Under the hypothesis of Theorem A, the subgroup Γ = ⟨ΓA, ΓB⟩ of G is +τmod-regular. +See §4.1.4 for the proof of this result. We remind our reader that, the subgroup Γ = +⟨ΓA, ΓB⟩ of G is naturally isomorphic to ΓA ⋆H ΓB (see Lemma 4.3) and is hyperbolic (see +Corollary 4.5). +Remark 4.7. The general strategy to prove Proposition 4.6 is similar to the proof of [4, +Theorem 3.2]. However, note that, in [4], we worked under a stronger hypothesis on the +interactive pairs (A, B). Namely, we required them to be antipodal to each other. When +A and B have a nontrivial intersection, which is the situation considered in this paper, +difficulties arise in controlling the dynamics near the intersection A ∩ B. +To show that Γ is τmod-regular, we proceed by showing that every sequence of Γ contains +a τmod-regular subsequence. Let (γn) be an arbitrary sequence in Γ consisting of pairwise +distinct elements. After passing to a subsequence, one of the following alternatives must +hold: Either (γn) is a sequence in H or, for all n ∈ N, γn ̸∈ H. In the former case, (γn) is +clearly τmod-regular. So, for the rest of this subsection, we assume the latter. Under this +assumption, since no elements of (γn) belong to H, rl(γn) ≥ 1, for all n ∈ N. Let us choose +a normal form for each element in the sequence (γn), +γn = βn,lnαn,lnβn,ln−1αn,ln−1 · · · βn,1αn,1. +(4.1) +To simplify our notations, denote βn := βn,ln and αn := αn,1. + +24 +SUBHADIP DEY AND MICHAEL KAPOVICH +Assumption 4.8. Let us further assume that βn and αn, the leftmost and rightmost letters +in (4.1) are in ΓB \ H and ΓA \ H, respectively. This extra assumption can be arranged +by changing the original sequence by a bounded amount; we note that τmod-regularity of +sequences remains unaffected under such perturbations. +4.1.1. Sequences with bounded relative lengths. +Lemma 4.9. Let (βn) be any sequence in ΓB. Then, there exists a sequence (ηn) in H such +that the sequence (ˆβ−1 +n ), where ˆβn := βnηn, has no flag-limit points in Λτmod(H). +Proof. This follows directly from the first part of Lemma 2.8 and the τmod-Anosov property +of ΓB. +□ +Lemma 4.10. Suppose that the leftmost letter sequence (βn) in ΓB corresponding to the +sequence (γn) in Γ has the following property: The inverse sequence (β−1 +n ) diverges away +from H. Then, (γn) is τmod-regular. +Similarly, if the rightmost letter sequence (αn) diverges away from H, then (γn) is τmod- +regular. +Proof. Since (β−1 +n ) diverges away from H, it is an unbounded sequence in ΓB. Hence (β−1 +n ) +is τmod-regular. After extraction, suppose that β±1 +n +flag +−−→ τ± ∈ Λτmod(ΓB) ⊂ B. +We may (and will) assume that τ− ̸∈ Λτmod(H): Indeed, for every sequence (ηn) in H, we +have the freedom to adjust the normal form in (4.1) by +γn = (βn,lnηn)(η−1 +n αn,ln)βn,ln−1αn,ln−1 · · · βn,1αn,1. +(4.2) +Applying Lemma 4.9, we can arrange for a normal form of the elements of (γn) such that +the inverse of the leftmost letter sequence, (β−1 +n ), does not accumulate in Λτmod(H). Note +that, after the above adjustments, (β−1 +n ) still diverges away from H. In particular, any +accumulation points of (β−1 +n ) lie in Λτmod(ΓB) \ Λτmod(H). +Since τ− ̸∈ Λτmod(H), by the antipodality hypothesis (ii) of Theorem A, A is a compact +subset of C(τ−). Hence, βnA +flag +−−→ τ+ as n → ∞ (cf. §3.1). Note that for all n ∈ N, +γnB ⊂ βnA. Thus, γnB +flag +−−→ τ+, as n → ∞. By Lemma 3.3, (γn) is τmod-regular. +□ +Corollary 4.11. If supn{rl(γn)} < ∞, then (γn) is τmod-regular. +Proof. Using induction on the relative length and applying Lemma 4.10, the result follows; +see the proof of [4, Lemma 3.3] for a similar argument. +□ +4.1.2. Special sequences. Next, let us analyze a special type of sequence in Γ: +Definition 4.12 (Special sequences). A sequence (˜γn) in Γ = ΓA ⋆H ΓB is special if there +exist sequences (αn) in ΓA \ H, (βn) in ΓB \ H, and an increasing function F : N → N such +that +˜γn = βF (n)αF (n)βF (n)−1αF (n)−1 · · · β1α1. +(4.3) +Lemma 4.13. Special sequences in Γ are τmod-regular. + +COMBINATION THEOREM: AMALGAMS +25 +Proof. Consider any special sequence (˜γn) in Γ, and assume that the normal form of ˜γn is +given by (4.3). Consider any subsequence of (˜γn), again denoted by (˜γn). +If the inverse sequence (β−1 +F (n)) corresponding to the leftmost letter sequence diverges +away from H, then (˜γn) is τmod-regular (Lemma 4.10). +Otherwise, after passing to a further subsequence (˜γnk) of (˜γn) we can (and will) assume +that, there exists an element ˜β ∈ ΓB \ H and a sequence (ηk) in H such that +βF (nk) = ˜βηk, +∀n ∈ N. +(4.4) +Set B′ = ˜β(A) ⊂ B◦. Note that, for all k ∈ N, ˜γnk+1˜γ−1 +nk B ⊂ B′. So, by Lemma 3.5, (˜γnk) +is τmod-regular. +Therefore, every subsequence of the original sequence contains a τmod-regular subse- +quence, showing that the original sequence is τmod-regular. This concludes the proof. +□ +4.1.3. Alternating sequences. Recall the notion of alternating sequences from Definition 2.1. +Applying Lemma 4.13 to (ω−1 +n ), we obtain the following result. +Corollary 4.14. Alternating sequences in Γ = ΓA ⋆H ΓB are τmod-regular. +Lemma 4.15. If (ωn) is a type A alternating sequence in Γ = ΓA ⋆H ΓB, then the nested +sequence of compact subsets of Flag(τmod), +ω1B ⊃ ω2A ⊃ ω3B ⊃ ω4A ⊃ · · · , +converges to a point τ ∈ Flag(τmod). +Similarly, if (ωn) is a type B alternating sequence in Γ, then the nested sequence of +compact subsets of Flag(τmod), +ω1A ⊃ ω2B ⊃ ω3A ⊃ ω4B ⊃ · · · , +converges to a point τ ∈ Flag(τmod). +For a (type A or B) alternating sequence ω = (ωn), let us denote the point τ ∈ Flag(τmod) +obtained as a limit in the above by τω. As a direct corollary of the above and Lemma 3.3, +we obtain the following: +Corollary 4.16. If ω = (ωn) is an alternating sequence in Γ = ΓA ⋆H ΓB, then, for all +γ ∈ Γ, +γωn +flag +−−→ γτω, +as n → ∞. +Proof of Lemma 4.15. Let us assume that (ωn) is of type A; the other possibility can be +similarly treated. Let (βn) denote the rightmost letter sequence corresponding to (ω2n), +see Definition 2.1. Consider the special sequence (ω−1 +2n ). We first show that there exists +a sequence (ηn) in H such that (ˆω−1 +2n ), where ˆω2n = ω2nη−1 +n , has a subsequence that is +τmod-regular and τmod-flag-converges to some point τ− antipodal to A: By Lemma 2.8, +there exist sequences (ηn) and (ˆηn) in H such that both the sequences (ˆβn) and (ˆβ−1 +n ) +accumulate outside ∂∞H in ¯ΓB = ΓB ⊔ ∂∞ΓB, where ˆβn = ˆηnβnηn. Note that (ˆω2n), where +ˆω2n = ω2nη−1 +n , is still alternating5 and, hence, by Corollary 4.14, is a τmod-regular sequence. +5Indeed, the corresponding sequences in ΓA and ΓB for (ˆωn) can be taken to be (ηn−1αn) and (βnη−1 +n ), +respectively. + +26 +SUBHADIP DEY AND MICHAEL KAPOVICH +Passing to a subsequence, (ˆβnk) is either a constant sequence, ˆβ ∈ ΓB \ H, or +ˆβ±1 +nk +flag +−−→ τ± ∈ Λτmod(ΓB) \ Λτmod(H), +as k → ∞. +(4.5) +If the former holds, then notice that ˆω−1 +2nkB ⊂ ˆβ−1A. In this case, by Lemma 3.6, all the +τmod-limit points of (ˆω−1 +2nk) must lie in ˆβ−1A ⊂ B◦. By Assumption 4.2, ˆβ−1A is antipodal +to A. +If the latter holds, then we observe that ˆω−1 +2nk(B) ⊂ ˆβ−1 +nk (A). Then, the sequence (ˆβ−1 +nk A), +and hence (ˆω−1 +2nkB), converges to the τmod-limit point τ− (see (4.5)) of the sequence ˆβ−1 +nk . +By the antipodality assumption (ii) in the hypothesis of Theorem A, τ− is antipodal to A. +Let (ηn) be a sequence in H as above. By the above two paragraphs, we can (and will) +extract a subsequence (ˆω2kn) of (ˆω2n), where ˆω2n = ω2nη−1 +n , such that the inverse sequence +(ˆω−1 +2kn) is τmod-flag-converges to some point τ−, which is antipodal to A. +After further +extraction, we may assume that (ˆω2kn) τmod-flag-converges to some point τ ∈ Flag(τmod). +Since A ⊂ C(τ−) is compact, the sequence of subsets (ˆω2knA) converges to τ. However, for +all n ∈ N, ˆω2nA = ω2nA. +Thus, the nestedness of the subsets implies limn→∞ ω2nA = {τ}. Again, we also obtain +limn→∞ ω2n−1(B) = {τ} by nestedness. This completes the proof. +□ +4.1.4. Proof of Proposition 4.6. Suppose that (γn) is a sequence in Γ without repeated en- +tries. If the relative length of elements in (γn) is uniformly bounded, then, by Corollary 4.11, +(γn) is τmod-regular. +So, we only need to study sequences in Γ with unbounded relative lengths. We show that +such sequences are also τmod-regular. +Thus, let (γn) be such a sequence. We assume to the contrary that (γn) is τmod-irregular, +i.e., it does not contain any τmod-regular subsequences. Then, we construct subsequences +(ˆγn) and (˜γn) in Γ such that the following hold: (i) (˜γ−1 +n ) is alternating, (ii) (ˆγn) is a +subsequence of (γn), (iii) for each n ∈ N, there exist normal forms of ˆγn such that ˜γn is a +rightmost subword of ˆγn, and (iv) the leftmost letters of the sequence (ˆγn) in the chosen +normal forms all come from the same group, say ΓB. +Choose some normal forms of elements of (γn) as in (4.1) and assume that the right- +most/leftmost letters, αn and βn, respectively, of γn are both nontrivial (Assumption 4.8). +Then, (αn) stays within a bounded distance away from H, because, otherwise, by the second +part of Lemma 4.10, (γn) would contain a τmod-regular subsequence. Hence, after extrac- +tion of (γn), the rightmost letters of γn come from a single right coset of H, say Hˆα1, in +ΓA. Set ˜γ1 = ˆα1. Applying a similar argument to (γnˆα−1 +1 ) and, after further extraction, +we obtain that the rightmost letters of (γnˆα−1 +1 ) all come from a single right coset of H, say +Hˆβ1 in ΓB. Set ˜γ2 = ˆβ1 ˆα1. Proceeding inductively, we obtain a special sequence (˜γn) such +that (˜γ−1 +n ) is alternating. By construction, the original sequence τmod-irregular sequence +(γn) corresponds to a pair of sequences (ˆγn) and (˜γn) with the properties described in the +previous paragraph. +Note that ˆγ−1 +2n A ⊂ ˜γ−1 +2n A. Since, by Lemma 4.15, the sequence of subsets (˜γ−1 +2n A) of +Flag(τmod) converges to a point, (ˆγ−1 +2n A) also converges to that point. Therefore, (ˆγ−1 +2n ) and, +hence, (ˆγ2n) are τmod-regular (Lemma 3.3). However, since (ˆγ2n) is a subsequence of the +τmod-irregular sequence (γn), we arrive at a contradiction. +□ + +COMBINATION THEOREM: AMALGAMS +27 +4.2. The boundary map. Recall from §2.7 that there is a Γ-invariant decomposition of +∂∞Γ as the disjoint union ∂IΓ⊔∂IIΓ, where ∂IIΓ admits an equivariant continuous bijection +to the Gromov boundary of the Bass-Serre tree T associated with the amalgamated free +product Γ = ΓA ⋆H ΓB. +In this subsection, we construct a Γ-equivariant boundary map from the Gromov boundary +∂∞Γ of Γ = ΓA ⋆H ΓB to Flag(τmod), +ξ : ∂∞Γ → Flag(τmod). +(4.6) +We define this map ξ separately on ∂IΓ and ∂IIΓ; see §4.2.1 for the definition of ξ|∂IΓ and +§4.2.2 for the definition of ξ|∂IIΓ. +In §4.2.3, we verify that ξ is an antipodal map. Finally, in §4.2.4, we show that the map +ξ is dynamics preserving. +4.2.1. Definition of the boundary map for type I points. For ε ∈ ∂IΓ, pick γ ∈ Γ such that +γ−1ε ∈ ∂∞ΓA ∪ ∂∞ΓB. If γ−1ε ∈ ∂∞ΓA (resp. γ−1ε ∈ ∂∞ΓB), then define +ξ(ε) := γξA(γ−1ε), +(resp. ξ(ε) := γξB(γ−1ε)). +(4.7) +We first show that the map ξ : ∂IΓ → Flag(τmod) is well-defined: +Lemma 4.17. For γ ∈ Γ, if γ(∂∞ΓA) ∩ ∂∞ΓA ̸= ∅, then γ ∈ ΓA. The same conclusion +holds when A is replaced by B. +Proof. Note that γ(∂∞ΓA) = ∂∞(γΓAγ−1). By (2.7), the nonemptyness of γ(∂∞ΓA)∩∂∞ΓA +implies that dT (γΓA, ΓA) ≤ 1. Thus, γ ∈ ΓA. +□ +By Lemma 4.17, the element γ in the definition of ξ is unique up to the right multiplication +by elements of ΓA (resp. ΓB). Since the maps ξA, ξB are equivariant for ΓA, ΓB, respectively, +it follows that the definition of ξ(ε) in (4.7) does not depend on the choice of γ ∈ Γ. +Finally, note that, by definition, we have the following result: +Lemma 4.18. The map ξ : ∂IΓ → Flag(τmod) in (4.7) is Γ-equivariant. +4.2.2. Definition of the boundary map for type II points. For ε ∈ ∂IIΓ, we choose an alter- +nating sequence (ωn) given by Proposition 2.12 such that ωn +Cay +−−→ ε. If (ωn) is of type A +(resp. type B), then define +ξ(ε) := lim +n→∞ ωnA +(resp. ξ(ε) := lim +n→∞ ωnB) +(4.8) +(cf. Lemma 4.15). +A consequence of the following result is that ξ is well-defined on ∂IIΓ: +Lemma 4.19. For ε ∈ ∂IIΓ and a sequence (γn) in Γ, if γn +Cay +−−→ ε, then γn +flag +−−→ ξ(ε). +Proof. Let (ωn) be an alternating sequence as above, which we suppose to be of type A (the +type B case can be dealt with similarly). Then, by Proposition 2.12, there exists a function +F : N → N diverging to infinity and n0 ∈ N such that, for all n ≥ n0, we may (and will) +choose a normal form of γn containing ωF (n) as a left subword of that form. +We split (γn)n≥n0 into two subsequences: The first subsequence (γkn) contains all el- +ements of (γn)n≥n0 with the rightmost letter contained in ΓA, and the complementary + +28 +SUBHADIP DEY AND MICHAEL KAPOVICH +subsequence (γln) includes all elements of (γn)n≥n0 with rightmost letter contained in ΓB. +Notice that, for all n ∈ N, +γknB ⊂ ωF (kn)A +and +γlnA ⊂ ωF (ln)A. +Since ωnA → ξ(ε), we obtain that both sequences (γknB) and (γlnA) of subsets of Flag(τmod) +converge to ξ(ε). Therefore, by Lemma 3.3, γkn +flag +−−→ ξ(ε) and γln +flag +−−→ ξ(ε), yielding the +conclusion. +□ +Together with Corollary 4.16, the above result implies the following one: +Corollary 4.20. The map in (4.8) is Γ-equivariant. +4.2.3. The boundary map is antipodal. +Proposition 4.21. The map ξ : ∂∞Γ → Flag(τmod) in (4.6) (obtained by combining (4.7) +and (4.8)) is antipodal: That is for every pair of distinct points ε± ∈ ∂∞Γ, the points +τ± := ξ(ε±) ∈ Flag(τmod) are antipodal to each other. +We recall that the action G ↷ G/P preserves antipodality: That is, if ˆτ± ∈ G/P is an +antipodal pair, then, for all g ∈ G, gˆτ± is also an antipodal pair. +Lemma 4.22. Let v, w be any vertices in the Bass-Serre tree T such that dT (v, w) ≥ 2. +Then, ξ(∂∞Γv) = Λτmod(Γv) is antipodal to ξ(∂∞Γw) = Λτmod(Γw). +Proof. By by equivariance, it is enough to assume that w = ΓA or w = ΓB. We assume the +former, i.e., w = ΓA; the possibility w = ΓB can be analyzed similarly. Since dT (v, w) ≥ 2, +we have that ∂∞ΓA ∩ ∂∞Γv = ∅ (see (2.7)). +Suppose first that v is a coset of ΓB; so let γ ∈ Γ be any element such that γΓB = v. It +is easy to see that such an element γ must satisfy rl(γ) ≥ 2. We may also choose γ so that +it has a normal form whose rightmost letter lies in ΓA \ H. If the leftmost letter α of that +normal form also lies in ΓA \ H, then rl(γ) ≥ 3, and Λτmod(Γα−1v) ⊂ βα′A ⊂ B◦, where β +and α′ are the second and third letters from the left in that normal form of γ, respectively. +Since A and B◦ are antipodal to each other (see Assumption 4.2), and Λτmod(ΓA) ⊂ A, +we have that Λτmod(ΓA) and Λτmod(Γα−1v) are antipodal to each other, hence so is the pair +Λτmod(ΓA) = αΛτmod(ΓA) and Λτmod(Γv) = αΛτmod(Γα−1v). +If the leftmost letter of the +normal form of γ is some element β ∈ ΓB \ H, then, by a similar argument, it follows that +Λτmod(Γv) ⊂ B◦, which is antipodal to Λτmod(ΓA) ⊂ A. +Suppose now that v is a coset of ΓA. In this case, we can choose an element γ ∈ Γ such +that γΓA = v, rl(γ) ≥ 1, and the rightmost letter of a normal form of γ is an element of +ΓB \ H. Adapting a similar argument as above, the result follows in this case as well. +□ +Proof of Proposition 4.21. Combining the following cases, it would follow that the map +ξ : ∂∞Γ → Flag(τmod) is antipodal. Recall that ξ is Γ-equivariant (by Lemma 4.18 and +Corollary 4.20). +Case 1. Suppose that both points ε± ∈ ∂∞Γ are of type I. Since ξ is Γ-equivariant, it is +enough to assume that ε− ∈ ∂∞ΓA ∪ ∂∞ΓB. Let us also assume that ε− ∈ ∂∞ΓA; the case +ε− ∈ ∂∞ΓB can be treated similarly. + +COMBINATION THEOREM: AMALGAMS +29 +If ε+ ∈ ∂∞ΓA ∪ (ΓA(∂∞ΓB)), then finding a suitable element α ∈ ΓA, we obtain that +αε+ ∈ ∂∞ΓA ∪ ∂∞ΓB. +Since, by the hypothesis of Theorem A, ξ(∂∞ΓA ∪ ∂∞ΓB) ⊂ +Flag(τmod) is an antipodal subset, ξ(αε±) is an antipodal pair and, hence, so is ξ(ε±). +If ε+ ̸∈ ∂∞ΓA ∪ (ΓA(∂∞ΓB)), then ε+ lies in the boundary of a vertex group Γv of the +Bass-Serre tree T such that dT (ΓA, v) ≥ 2. By Lemma 4.22, ξ(ε±) are antipodal. +Case 2. Suppose that both points ε± ∈ ∂∞Γ are of type II. Consider a pair of alternating +sequences (ω± +n ) converging (in Γ) to ε± (see Lemma 2.15). +Lemma 4.23. There exists n ∈ N such that ω+ +n ̸∈ ω− +n H. +Proof. If this is false, then for all n ∈ N, ω+ +n ∈ ω− +n H. +Consider the sequence (ˆω−1 +n ), +where ˆωn := ω− +n prH((ω− +n )−1). By Lemma 2.8, (ˆω−1 +n ) has no accumulation points in ∂∞H. +Moreover, (ω− +n ) diverges away from H (cf. Proposition 2.13). Thus, (ˆωn(∂∞H)) converges +to ε−, see Lemma 2.8(ii). It follows that the sequence of uniformly quasiconvex subsets +(ˆωn(H)) of Γ converges to ε−. Since we have assumed that ω+ +n ∈ ω− +n H = ˆωnH, we obtain +that ω+ +n +Cay +−−→ ε−, which shows that ε+ = ε−. This is a contradiction. +□ +Let n0 ∈ N be the smallest number such that ω+ +n0 ̸∈ ω− +n0H. Consider the alternating +sequences ((ω+ +n0)−1ω+ +n ) and ((ω+ +n0)−1ω− +n ) converging to (ω+ +n0)−1ε+ and (ω+ +n0)−1ε−, respec- +tively. By choice of n0, it is evident that the first element of those sequences lie in different +groups ΓA and ΓB. +Therefore, one of the points ξ((ω+ +n0)−1ε±) lies in the interior of A +while the other one lies in the interior of B (cf. Lemma 4.15) and thus they are antipodal. +Consequently, ξ(ε±) are antipodal. +Case 3. Suppose that ε− ∈ ∂∞Γ is of type I and ε+ ∈ ∂∞Γ is of type II. By the same +argument as in the third case in [4, §4.2], it follows that ξ(ε±) are antipodal. +□ +4.2.4. The boundary map preserves convergence dynamics. The following result is an analog +of Lemma 4.19 for type I boundary points. +Lemma 4.24. Let ε ∈ ∂IΓ and let (γn) be a sequence in Γ. If γn +Cay +−−→ ε, then γn +flag +−−→ ξ(ε). +Proof. Using the equivariance of the Γ-action, it will be enough to prove the proposition +when ε ∈ ∂∞ΓA ∪ ∂∞ΓB. We argue by contradiction: +Suppose that there exists ε ∈ ∂∞ΓA ∪ ∂∞ΓB and a sequence (γn) in Γ such that +γn +Cay +−−→ ε, +but +γn +flag +−−→ τ ̸= ξ(ε). +(4.9) +By Lemma 2.17, let us choose D-normal forms for each element of (γn), for some D ≥ 0. +After extraction, we may assume that the leftmost and rightmost letters of those forms +come from the same group, say ΓA and Γ∗, respectively, where ∗ is either A or B; let (αn) +be the sequence of those leftmost letters. We also assume that ∗ = A; the other choice can +be similarly analyzed. +Clearly, (α−1 +n ) cannot diverge away from H: Otherwise, since (αn) fellow travels (γn), +αn +Cay +−−→ ε and if (α−1 +n ) diverges away from H, then +γnB ⊂ αnB → τ ′, + +30 +SUBHADIP DEY AND MICHAEL KAPOVICH +where τ ′ = ξ(ε). But, in this case, Lemma 3.3 shows that γn +flag +−−→ τ ′, a disagreement with +(4.9). +Therefore, after passing to a subsequence, we may assume that the elements of (γn) +have normal forms with a common leftmost letter α1. Repeating the same argument to +the sequence (α−1 +1 γn) yields another subsequence whose elements have normal forms with +a common leftmost letter β1. Thus, the original sequence (γn) has a subsequence whose +elements have normal forms with two leftmost common letters. Proceeding inductively, for +every l ∈ N, we can find a subsequence (γ′ +n) of the original sequence (γn) such that the +elements of (γ′ +n) have normal forms with a common leftmost subword of length at least l. +For l = 3, the Proposition 2.13 shows that (γ′ +n) has bounded nearest-point projections to +ΓA and ΓB. Thus, (γ′ +n) cannot have any accumulation points in the boundary of ΓA and +ΓB. This contradicts our initial assumption that ε ∈ ∂∞ΓA ∪ ∂∞ΓB. +□ +Combining Lemmas 4.19 and 4.24, we obtain the following: +Corollary 4.25. The map Γ-equivariant map given by (4.6) preserves the convergence +dynamics: That is, for any sequence (γn) in Γ and any point ε ∈ ∂∞Γ, if γn +Cay +−−→ ε, then +γn +flag +−−→ ξ(ε). +4.3. Proof of Theorem A. By Proposition 3.12, the subgroup Γ = ⟨ΓA, ΓB⟩ of G is nat- +urally isomorphic to ΓA ⋆H ΓB. We show that Γ is a τmod-Anosov subgroup or, equivalently, +a τmod-asymptotically embedded subgroup (see Definition 3.9) of G: +(i) By Proposition 4.6, Γ is a τmod-regular subgroup of G. +(ii) By Corollary 4.5, Γ is hyperbolic. +(iii) Finally, the boundary map ξ : ∂∞Γ → Flag(τmod) in Equation (4.6) is Γ-equivariant +(by Lemma 4.18 and Corollary 4.20), antipodal (by Proposition 4.21), and preserves +convergence dynamics (by Corollary 4.25). +This concludes the proof of the Theorem A. +□ +5. A combination theorem for HNN extensions of Anosov subgroups +The goal of this section is to prove Theorem B and, throughout this section, we work +under the hypothesis of Theorem B. +Assumption 5.1. In the proof of Theorem B, we replace (A, B±) by (cl(A◦), cl(B◦ +±)), which +is again an interactive triple for (M; H±; f). In doing so, we may replace the condition (i) +in the hypothesis of Theorem B by the following stronger one (cf. Lemma 4.1): +(i)’ The pairs of subsets (A, B◦ +±), (A◦, B±) of Flag(τmod) are antipodal. Moreover, B− is +antipodal to B+. +We begin the proof of Theorem B with the following observation: +Lemma 5.2. Under the hypothesis of Theorem B, +(i) Λτmod(⟨f⟩) consists of two points, one of them lies in the interior of B+ and the other +one lies in the interior of B−, and ⟨f⟩ is a cyclic τmod-Anosov subgroup of G. +(ii) Λτmod(M) ⊂ A and Λτmod(H±) ⊂ A ∩ B±. + +COMBINATION THEOREM: AMALGAMS +31 +Proof. Since, for all n ∈ N, f n+1f −n(B+) = f(B+) ⊂ B◦ ++ (by the second condition of +Definition 3.14), applying Lemma 3.5, it follows that ⟨f⟩ is τmod-regular. +Since, for all +n ∈ N, f −n(f −1B−) ⊂ f −1B− ⊂ B◦ +−, all the τmod-limit points of the sequence (f −n)n∈N +lie in B◦ +−. Since B+ is antipodal to B◦ +− (see Assumption 5.1), by Lemma 3.8, (f nB+)n∈N +shrinks. Therefore, since the sequence (f nB+)n∈N is nested, (f nB+)n∈N must converge to +some point τ+ ∈ B◦ ++. Similarly, the nested sequence (f −nB−)n∈N of compact subsets of +Flag(τmod) converges to some point τ− ∈ B◦ +−. In particular, the limit set of ⟨f⟩ is {τ±}, +is antipodal, and has cardinality two. That ⟨f⟩ is τmod-Anosov follows from [13, Lemma +5.38]. This proves (i). +Proof (ii) is similar to that of Lemma 4.3. Hence, we omit the details. +□ +Corollary 5.3. Under the hypothesis Theorem B, the subgroups H± are weakly malnormal +in M, and, for all µ ∈ M, the intersection H− ∩ µH+µ−1 is finite. +Proof. The proof is similar to the one of Corollary 4.4; we omit the details. +□ +Corollary 5.4. Under the hypothesis Theorem B, the subgroup Γ of G generated by M and +f in G is hyperbolic. +Proof. Arguing similarly to the first paragraph of the proof of Corollary 4.5, it follows that +H± are both quasiconvex in M. Then, the claim follows from Theorem 2.10(ii), Propo- +sition 3.15, and Corollary 5.3. Compare this with the second paragraph of the proof of +Corollary 4.5. +□ +For convenience, we introduce the following notation, which are frequently used in this +section. +Notation 5.5. If ǫ = 1, then Hǫ will denote H+ and Bǫ will denote B+. Similarly, if ǫ = −1, +then Hǫ will denote H− and Bǫ will denote B−. +5.1. Regularity. The main result of this subsection is as follows: +Proposition 5.6. Under the hypothesis of Theorem B, the subgroup Γ = ⟨M, f⟩ of G is +τmod-regular. +See §5.1.3 for the proof of this proposition. +5.1.1. Special sequences. +Definition 5.7 (Special sequences). A sequence (˜γn) in Γ = M⋆φ is called special if there +exist sequences (ǫn) in {±1}, (µn) in M satisfying +ǫn = 1, µn ∈ H+ =⇒ ǫn+1 = 1 +and +ǫn = −1, µn ∈ H− =⇒ ǫn+1 = −1, +and an element µ0 ∈ M, and an increasing function F : N → N such that, for all n ∈ N, +˜γn = f ǫF (n)µF (n)−1f ǫF (n)−1 · · · µ1f ǫ1µ0. +(5.1) +Note that special sequences are subsequences of the inverse sequence of some alternating +sequence in Γ. Compare this with Definition 2.2. +Lemma 5.8. Special sequences in Γ are τmod-regular. + +32 +SUBHADIP DEY AND MICHAEL KAPOVICH +Proof. Let (˜γn) be a special sequence as above. +We will assume that µ0 = 1M; such a +change is a bounded perturbation of the original sequence. Hence the property of being +τmod-regular remains unaffected. To show that (˜γn) is τmod-regular, it would be enough to +show that every subsequence of (˜γn) contains a τmod-regular subsequence. +So, consider any subsequence of (˜γn), again denoted by (˜γn). After extraction,6 we may +assume that ǫF (n)−1 are all the same, say +ǫF (n)−1 = 1, +for all n ∈ N. +Case 1. Suppose that, after passing to a subsequence, it holds for all n ∈ N that µF (n)−1 ∈ +H+. Passing to another subsequence, we will also assume that F(n + 1) − F(n) ≥ 10, for +all n ∈ N. Since µF (n)−1 ∈ H+ and ǫF (n)−1 = 1, we have +ǫF (n) = 1, +for all n ∈ N. +(see Definition 5.7). So, +˜γn = fµF (n)−1fµF (n)−2f ǫF (n)−2 · · · µ1f ǫ1 += ff(f −1µF (n)−1fµF (n)−2)f ǫF (n)−2 · · · µ1f ǫ1. +(5.2) +Moreover, +˜γn+1˜γ−1 +n += ff(f −1µF (n)−1fµF (n)−2)f ǫF (n)−1 · · · f ǫF (n−1)+1µF (n−1). +Observe that, if ǫF (n−1)+1 = −1, then µF (n−1) ̸∈ H+, since we observed in the preced- +ing paragraph that ǫF (n−1) = 1. Else, ǫF (n−1)+1 = 1. Consequently, in both cases (i.e., +ǫF (n−1)+1 = 1 or −1), it holds that +˜γn+1˜γ−1 +n (B+) ⊂ f(B+) ⊂ B◦ ++. +Since the above is true for all n ∈ N, Lemma 3.5 applies to show that (˜γn) is τmod-regular. +Case 2. Suppose that, after passing to a subsequence, it holds for all n ∈ N that µF (n)−1 ̸∈ +H+. Consider the sequence (ˆγn), where +ˆγn := µF (n)−1f ǫF (n)−1 · · · µ1f ǫ1 = f −ǫF (n)˜γn. +We show that (ˆγn) is τmod-regular since this would imply that (˜γn) is τmod-regular. +For all n ∈ N, +ˆγnA ⊂ µF (n)−1B+ ⊂ A. +(5.3) +If the sequence (µ−1 +F (n)−1) remains at a bounded distance away from H+, then after further +extraction, we may assume that (µF (n)−1) lies in a single coset µH+, for some µ ̸∈ H+. So, +it holds that +ˆγn+1ˆγ−1 +n (A) ⊂ µB+ ⊂ A◦, +for all n ∈ N. With Lemma 3.5, the above implies that (ˆγn) is τmod-regular. +Otherwise, after further extraction, (µ−1 +F (n)−1) diverges away from H+. +Consider the +sequence (ˆµF (n)−1), where ˆµF (n)−1 := µF (n)−1 prH+(µ−1 +F (n)−1). By Lemma 2.8, (ˆµ−1 +F (n)−1) +accumulates in ∂∞M \ ∂∞H+. +Thus, all τmod-flag accumulation points of (ˆµ−1 +F (n)−1) are +6Note that, by definition, subsequences of special sequences are special. + +COMBINATION THEOREM: AMALGAMS +33 +antipodal to B+ (see Assumption 5.1). By Lemma 3.8, the sequence (ˆµF (n)−1B+) of com- +pact subsets of Flag(τmod) shrinks. Since ˆµF (n)−1B+ = µF (n)−1B+, (µF (n)−1B+) shrinks. +Consequently, it follows from (5.3) that (ˆγnA) shrinks. By Lemma 3.4, (ˆγn) is τmod-regular. +Hence, (˜γn) is τmod-regular. +□ +5.1.2. Alternating sequences. Recall the notion of alternating sequences from Definition 2.2. +By definition, the inverse sequence corresponding to an alternating sequence is special so +that Lemma 5.8 directly implies: +Corollary 5.9. Alternating sequences in Γ = M⋆φ are τmod-regular. +Lemma 5.10. Let (ωn) be an alternating sequence in Γ = M⋆φ equipped with the normal +forms given by (2.5): +ωn = µ0f ǫ1µ1f ǫ2µ2 · · · f ǫn−1µn−1f ǫn. +(5.4) +Then, the nested sequence of compact subsets (ωn(A ∪ Bǫn))n of Flag(τmod) converges to a +point. +We remind our reader that we are using the notation introduced in Notation 5.5. +Proof of Lemma 5.10. It is a straightforward verification that, for all n ∈ N, µn(Bǫn+1) ⊂ +A ∪ Bǫn, yielding that +ωn+1(A ∪ Bǫn+1) = ωnµnf ǫn+1(A ∪ Bǫn+1) ⊂ ωnµnBǫn+1 ⊂ ωn(A ∪ Bǫn). +(5.5) +Therefore, the sequence (ωn(A ∪ Bǫn))n is nested. +Thus, to prove that (ωn(A ∪ Bǫn))n converges to a point, it will be enough to show that +this sequence contains a subsequence that shrinks. This is what we show. +Case 1. Suppose that, for infinitely many n ∈ N, µn−1 ∈ Hǫn. Let P ⊂ N denote an +infinite subset such that for all n ∈ P, µn−1 ∈ Hǫn and, for all n ∈ P, ǫn are the same, say +ǫn = 1. Therefore, for all n ∈ P, ǫn−1 = 1 (see Definition 2.2). +Let us first show that (ωn−1B+)n∈P shrinks: We observe that, for n ∈ P, +ω−1 +n−1(µ0A) ⊂ B−, +which shows that all the τmod-limit points of (ω−1 +n−1)n∈P lie in B− (see Lemma 3.6). By +Corollary 5.9, we know that (ωn−1)n∈P is τmod-regular. Since B+ is antipodal to B−, by +Lemma 3.4 (ωn−1B+)n∈P shrinks. +Since, for all n ∈ P, ǫn = 1, we get that Bǫn = B+. Moreover, since µn−1 ∈ H+, +ωn(A ∪ Bǫn) = ωn−1µn−1f(A ∪ B+) ⊂ ωn−1µn−1B+ = ωn−1B+. +Therefore, by the previous paragraph, (ωnA)n∈P shrinks. +Therefore, we may assume now the complementary case to the above one, which is that, +for at most finitely many n ∈ N, µn−1 ∈ Hǫn. In fact, it would be enough to assume: +Case 2. For all n ∈ N, µn−1 ̸∈ Hǫn. Suppose that, for infinitely many n ∈ N, ǫn = 1 (the +case ǫn = −1 can be dealt with in a similar way). Let P ⊂ N be a subset such that, for all + +34 +SUBHADIP DEY AND MICHAEL KAPOVICH +distinct n, n′ ∈ P, |n − n′| ≥ 10, and ǫn = 1, for all n ∈ P. For n ∈ P, let us consider the +normal form of ωn+1 given by +ωn+1 = µ0f ǫ1µ1 · · · f ǫn−2µn−2f ǫn−1 ˆµn−1f ˆµnf ǫn+1, +(5.6) +where +ˆµn−1 = µn−1 prH+(µ−1 +n−1) +and +ˆµn = f −1(prH+(µ−1 +n−1))−1fµn. +(5.7) +We observe that, since µn−1 ̸∈ H+, ˆµn−1 is also not an element of H+, for all n ∈ P. +For n ∈ P, let +ˆωn := µ0f ǫ1µ1 · · · f ǫn−2µn−2f ǫn−1 ˆµn−1f = ωn+1(ˆµnf ǫn+1)−1. +(5.8) +Since (ˆωn)n∈P is a subsequence of an alternating sequence, by Corollary 5.9, it is τmod- +regular. One directly checks (cf. (5.5)), +ωn+1(A ∪ Bǫn+1) ⊂ ˆωnA, +for all n ∈ P. +(5.9) +After extraction, we may assume that, for all n ∈ P, ǫn−1 is constant, ǫ = ±1. +After +another extraction, we also assume that (ˆµ−1 +n−1)n∈P either (i) diverges away from H−ǫ, or +(ii) remains in a fixed coset ˆµH−ǫ, for some ˆµ ∈ M. So, for n ∈ P, +f ˆω−1 +n (µ0A) = ˆµ−1 +n−1f −ǫµ−1 +n−2 · · · f −ǫ1µ−1 +0 (µ0A). +In the first case (i), since we are assuming that (ˆµn−1)n∈P diverges away from H−ǫ, +by Lemma 2.7, it follows that (˜µn−1)n∈P is τmod-regular and its τmod-limit points lie +only in Λτmod(M) \ Λτmod(H−ǫ), where ˜µn−1 := prH−ǫ(ˆµn−1)−1ˆµn−1. +Since Λτmod(M) \ +Λτmod(H−ǫ) is antipodal to B−ǫ, by Lemma 3.8, (˜µ−1 +n−1B−ǫ)n∈P = (ˆµ−1 +n−1B−ǫ)n∈P shrinks. +So, (f ˆω−1 +n−1(µ0A))n∈P shrinks as well. +By definition of ˆµn−1 in (5.7), all the τmod-limit +points of (ˆµ−1 +n−1)n∈P lie in Λτmod(M) \ Λτmod(H+). Thus, after further extraction, we may +assume that +ˆµ−1 +n−1 +flag +−−→ τ− ∈ Λτmod(M) \ Λτmod(H+), +as n → ∞ in P. +Therefore, it holds that f ˆω−1 +n−1(µ0A) → τ−, as n → ∞ in P. +Thus, by Lemma 3.3 the +sequence (f ˆω−1 +n−1)n∈P τmod-flag converges to τ−. Since τ− is antipodal to B+, by Lemma 3.8, +(ˆωn−1f −1B+)n∈P shrinks. Since fA ⊂ B+, it follows that (ˆωn−1A)n∈P shrinks. Thus, by +(5.9), (ωn+1(A ∪ Bǫn+1))n∈P shrinks. +In the second case (ii), suppose first that ǫ = −1. Therefore, by the assumption of Case +2 and (5.7), we have that ˆµ ̸∈ H+. Thus, +f ˆω−1 +n (µ0A) ⊂ ˆµ−1 +n−1B+ ⊂ ˆµB+ ⊂ A◦, +for all n ∈ P. So, in this case, (f ˆω−1 +n−1)n∈P has no τmod-limit points in B+, since, by above, +all of them lie in the interior of A (cf. Lemma 3.6). Therefore, since (ˆωn−1f −1)n∈P is τmod- +regular,7 by Lemma 3.8, (ˆωn−1f −1B+)n∈P shrinks. Thus, by (5.9), (ωn+1(A ∪ Bǫn+1))n∈P +shrinks. +7This follows by the observation above that (ˆωn−1)n∈P is τmod-regular. + +COMBINATION THEOREM: AMALGAMS +35 +Still assuming (ii), suppose now that ǫ = 1. If ˆµ ̸∈ H−, then proceeding as in the previous +paragraph, it follows that (ωn+1(A ∪ Bǫn+1))n∈P shrinks. Else, we must have ˆµn−1 ∈ H−, +for all n ∈ P. Observing a different normal form of ˆωn, +ˆωn := µ0f ǫ1µ1 · · · f ǫn−2(µn−2f ˆµn−1f −1 +� +�� +� +∈M +)ff. +it follows by Case 1 that (ˆωnA) shrinks. Thus, by (5.9), (ωn+1(A ∪ Bǫn+1))n∈P shrinks. +□ +By the above lemma, it follows that an alternating sequence ω = (ωn) in Γ τmod-flag- +converges to the point +τω := +� +n∈N +ωn(A ∪ Bǫn). +(5.10) +As a corollary, we obtain: +Corollary 5.11. If ω = (ωn) is an alternating sequence in Γ, then the sequence (ωnA) of +compact subsets of Flag(τmod) converges to the τmod-limit point τω of (ωn). +However, we remark that the sequence (ωnA) is possibly not nested. +Corollary 5.12. If ω = (ωn) is an alternating sequence in Γ, then, for all γ ∈ Γ, +γωn +flag +−−→ γτω, +as n → ∞. +Proof. By Corollary 5.11, it follows that (γωnA)n converges to γτω. Then, Lemma 3.3 yields +the conclusion. +□ +5.1.3. Proof of Proposition 5.6. Suppose that, to the contrary, Γ is not τmod-regular. Then, +Γ contains a τmod-irregular sequence (γn), i.e., (γn) has no repeated entries, and it contains +no τmod-regular subsequences. +Let (γn) be such a τmod-irregular sequence. We inductively extract a subsequence (˜γn) of +(γn) so that there exists an alternating sequence (ωn) in Γ such that, under suitable normal +forms of ˜γn, we may write +˜γn = ωnµ′ +n,0f ǫ′ +n,1µ′ +n,1 · · · f ǫ′ +n,lnµ′ +n,ln, +∀n ∈ N. +(5.11) +This would yield a contradiction as follows: Notice that, for all n ∈ N, +˜γnBǫ′ +n,ln ⊂ ωn(A ∪ Bǫn), +However, by Lemma 5.10, the sequence (ωn(A ∪ Bǫn)) converges to a point in Flag(τmod). +Thus, it would follow that (˜γnBǫ′ +n,ln) shrinks, which would imply (by Lemma 3.4) that (˜γn) +is τmod-regular, a contradiction with the τmod-irregularity assumption of (γn). +To finish the proof of the result, let us give a construction of (ωn) and (˜γn): For each +n ∈ N, choose a normal form for γn, +γn = µn,0f ǫn,1µn,1 · · · f ǫn,lnµn,ln. +(5.12) +Let us first show that we may extract a subsequence of (γn) such that, with appropriate +normal forms, the leftmost letters in those normal forms are all the same: After extraction, +we may assume that, for all n, ǫn,1 are the same and, for all n, ǫn,ln are the same, +ǫn,1 = ǫ, +ǫn,ln = ǫ′, +∀n ∈ N. + +36 +SUBHADIP DEY AND MICHAEL KAPOVICH +Note that (µ−1 +n,0) cannot diverge away from Hǫ since, otherwise, +γnBǫ′ ⊂ µn,0Bǫ, +but (µn,0Bǫ) would shrink, which, by above, would show that (γn) is τmod-regular. +So, +after extraction, we may assume that µn,0 ∈ µ0ηn, for some sequence (ηn) in Hǫ and some +µ0 ∈ M. So, we may change the expression in (5.12) by +γn = µ0f ǫ(¯ηnµn,1)f ǫn,2 · · · f ǫn,lnµn,ln, +(5.13) +where ¯ηn ∈ H−ǫ is the element corresponding to ηn ∈ Hǫ given by the isomorphism φ : +H− → H+. +We may now delete the common leftmost letters µ0f ǫ from the normal form of the +elements the extracted subsequence (γn) in the preceding paragraph. Consider the sequence +(γ′ +n) thus obtained: +γ′ +n := (¯ηnµn,1)f ǫn,2 · · · f ǫn,lnµn,ln. +(5.14) +We may now repeat the above procedure to the sequence (γ′ +n) equipped with the normal +form given by (5.14), which is again τmod-irregular, to obtain two more letters µ1 and f ǫ2. +We may proceed in this way inductively so that we obtain an infinite string, +µ0, f ǫ1, µ1, f ǫ2, µ2, f ǫ3, . . . , +producing the desired alternating sequence (ωn) (cf. +Remark 2.3) and a corresponding +subsequence (˜γn) of (γn) such that (5.11) holds. +□ +5.2. The boundary map. We construct a Γ-equivariant map from the Gromov boundary +of Γ = M⋆φ to the flag manifold Flag(τmod): +ξ : ∂∞Γ → Flag(τmod). +(5.15) +Recall that ∂∞Γ decomposes into ∂IΓ ⊔ ∂IIΓ, where ∂IΓ = Γ · (∂∞M). As in the case of +amalgamated free products (§4.2), we define ξ separately on ∂IΓ and ∂IIΓ; see §5.2.1 for the +definition of ξ|∂IΓ and §5.2.2 for the definition of ξ|∂IIΓ. +5.2.1. Definition of the boundary map for type I points. For every point ε ∈ ∂IΓ, we may +pick some element γ ∈ Γ such that γ−1ε ∈ ∂∞M. +Since M is τmod-Anosov, we have a +M-equivariant boundary embedding ξ : ∂∞M → Λτmod(M) ⊂ Flag(τmod). We define +ξ(ε) := γξ(γ−1ε). +(5.16) +We check that ξ(ε) is well-defined, i.e., does not depend on the choice of γ ∈ Γ in (5.16): +If γ1 ∈ Γ is any other element such that γ−1 +1 ε ∈ ∂∞M, then the intersection (γ−1γ1)∂∞M ∩ +∂∞M is nonempty since γ−1ε is a common point. Thus, in the Bass-Serre tree T, M and +(γ−1γ1)M are equal or adjacent vertices, showing that rl(γ−1γ1) ≤ 1 (see Lemma 2.20). If +rl(γ−1γ1) = 0, then γ1 ∈ γM, and, in this case, the well-definedness of (5.16) follows by +M-equivariance of ξ : ∂∞M → Flag(τmod). So, let us assume that rl(γ−1γ1) = 1. In this +case, γ−1γ1 = µ0f ǫµ1, where µ0, µ1 ∈ M and |ǫ| = 1. Let us also assume that ǫ = 1, since +the case ǫ = −1 is similar. So, γ1 = γµ0fµ1 and γ−1ε ∈ µ0fµ1(∂∞M) ∩ ∂∞M = µ0(∂∞H+). +Hence, +µ−1 +0 γ−1ε ∈ ∂∞H+. +(5.17) + +COMBINATION THEOREM: AMALGAMS +37 +Since f ∈ G conjugates H− and H+, i.e., fH−f −1 = H+, for all ε′ ∈ ∂∞H+, +ξ|∂∞H−(fε′) = fξ|∂∞H+(ε′). +(5.18) +Thus, +γ1ξ(γ−1 +1 ε) = γ1ξ(µ−1 +1 f −1µ−1 +0 γ−1ε) += γ1µ−1 +1 ξ(f −1µ−1 +0 γ−1ε) += γ1µ−1 +1 f −1ξ(µ−1 +0 γ−1ε) += γ1µ−1 +1 f −1µ−1 +0 ξ(γ−1ε) = γξ(γ−1ε), +where the second equality is valid because (f −1µ−1 +0 γ−1ε) ∈ ∂∞M, the third equality is +verified by (5.17) and (5.18), and the fourth equality is valid because γ−1ε ∈ ∂∞M. +The following result is immediate from the definition of ξ above: +Lemma 5.13. The map ξ : ∂IΓ → Flag(τmod) defined by (5.16) is Γ-equivariant. +5.2.2. Definition of the boundary map for type II points. For ε ∈ ∂IIΓ, consider an alternat- +ing sequence (see Definition 2.2) ωn +Cay +−−→ ε given by Proposition 2.12. Define +ξ(ε) := lim +n→∞ ωnA, +(5.19) +see Corollary 5.11. The following result shows that ξ(ε) is well-defined. +Lemma 5.14. For ε ∈ ∂IIΓ and for any sequence (γn) in Γ, if γn +Cay +−−→ ε, then γn +flag +−−→ ξ(ε). +Proof. The proof is similar to Lemma 4.19. We omit the details. +□ +Together with Corollary 5.12, the above result implies the following: +Corollary 5.15. The map ξ : ∂IIΓ → Flag(τmod) defined by (5.19) is Γ-equivariant. +5.2.3. The boundary map is antipodal. +Proposition 5.16. The map ξ : ∂∞Γ → Flag(τmod) in (5.15) (obtained by combining (5.16) +and (5.19)) is antipodal: That is, for every pair of distinct points ε± ∈ ∂∞Γ, the points +τ± := ξ(ε±) ∈ Flag(τmod) are antipodal to each other. +Proof. We consider the following cases. Recall that ξ is Γ-equivariant (by Corollary 5.15 +and Lemma 5.13). +Case 1. Suppose that both ε± are type I points. Using the Γ-equivariance, we may assume +that ε− ∈ ∂∞M. If ε+ is also in ∂∞M, then ξ(ε±) are antipodal since ξ : ∂∞M → Flag(τmod) +is an antipodal map. Else, ε+ ̸∈ ∂∞M but there exists γ ∈ Γ such that rl(γ) ≥ 1 and +ε := γ−1ε+ ∈ ∂∞M. Let γ = µ0f ǫ1µ1 · · · f ǫnµn be a normal form. So, +µ−1 +0 ξ(ε+) ∈ µ−1 +0 γ(Λτmod(M)) = µ−1 +0 f ǫ1µ1 · · · f ǫn(Λτmod(M)). +If rl(γ) = 1, then µ−1 +0 ξ(ε+) ∈ f ǫ1(Λτmod(M)). +Moreover, µ−1 +0 ε+ ̸∈ f ǫ1∂∞H−ǫ1, since +we have assumed that ε+ ̸∈ ∂∞M. +Thus, it follows that f −ǫ1µ−1 +0 ξ(ε+) ∈ Λτmod(M) \ +Λτmod(H−ǫ1). Since B−ǫ1 is antipodal to Λτmod(M)\Λτmod(H−ǫ1), f −ǫ1µ−1 +0 ξ(ε−), which is an +element of B−ǫ1, is antipodal to Λτmod(M)\Λτmod(H−ǫ1). Thus, ξ(ε+) is antipodal to ξ(ε−). + +38 +SUBHADIP DEY AND MICHAEL KAPOVICH +If rl(γ) ≥ 2, and µ1 ̸∈ Hǫ2, then +µ−1 +0 ξ(ε+) ∈ µ−1 +0 γA ⊂ f ǫ1µ1Bǫ2 ⊂ B◦ +ǫ1. +If µ1 ∈ Hǫ2, then ǫ1 = ǫ2 and, hence, +µ−1 +0 ξ(ε+) ∈ µ−1 +0 γA ⊂ f ǫ1µ1f ǫ2(A ∪ Bǫ2) = f ǫ1Bǫ1 ⊂ B◦ +ǫ1, +In both cases, µ−1 +0 ξ(ε+) is antipodal to A and in particular, to Λτmod(M). Thus, ξ(ε+) is +antipodal to Λτmod(M) and, in particular, to ξ(ε−). +Case 2. Suppose that both ε± are type II points. Let (ω± +n ) be alternating sequences such +that ω± +n +Cay +−−→ ε± (see Lemma 2.15). +Lemma 5.17. There exists n ∈ N such that ω+ +n ̸∈ ω− +n M. +Proof. The proof is similar to the one of Lemma 4.23. +□ +Let n0 be the smallest natural number such that ω+ +n0 ̸∈ ω− +n0M. Using an argument similar +to the Case 2 in the proof of Proposition 4.21, one can show that ξ((ω+ +n0)−1ε±) are antipodal, +which is equivalent to ξ(ε±) being antipodal. +Case 3. Suppose that ε− is a type I point, but ε+ is a type II point. By Γ-equivariance, we +may assume that ε− ∈ ∂∞M. Let (ωn) alternating sequence, +ωn = µ0f ǫ1µ1f ǫ2µ2 · · · f ǫn−1µn−1f ǫn, +such that ωn +Cay +−−→ ε+. Then, ξ(ε+) = limn→∞ ωnA, see (5.19). +If µ1 ∈ Hǫ2, then ǫ1 = ǫ2, and it follows that +µ−1 +0 ωnA ⊂ f ǫ1Bǫ1 ⊂ B◦ +ǫ1. +Thus, µ−1 +0 ξ(ε+) ∈ B◦ +ǫ1. Else, if µ1 ̸∈ Hǫ2, then (µ0f ǫ1µ1)−1ωnA ⊂ Bǫ2, also showing that +µ−1 +0 ξ(ε+) ∈ f ǫ1µ1Bǫ2 ⊂ B◦ +ǫ1. However, µ−1 +0 ξ(ε−) ∈ A. It follows that ξ(ε−) and ξ(ε+) are +antipodal. +□ +5.2.4. The boundary map preserves convergence dynamics. The following result is reminis- +cent of Lemma 5.13. +Lemma 5.18. Let ε ∈ ∂IΓ and let (γn) be a sequence in Γ. If γn +Cay +−−→ ε, then γn +flag +−−→ ξ(ε). +Proof. Using the action of Γ on ∂IΓ, it will be enough to prove the proposition for ε ∈ ∂∞M. +We argue by contradiction: +Suppose there exists a sequence (γn) in Γ such that +γn +Cay +−−→ ε, +but +γn +flag +−−→ τ ̸= ξ(ε). +(5.20) +Equip each γn with a D-normal form (see Lemma 2.17), +γn = µn,0f ǫn,1µn,1 · · · f ǫn,lnµn,ln. +(5.21) +Passing to a subsequence, we may assume that ǫn,1 are the same, ǫ, for all n. +In this +situation, (µ−1 +n,0) cannot diverge away from Hǫ: Otherwise, after extraction, µn,0 +Cay +−−→ ε′ and +γnBǫn,ln ⊂ µn,0Bǫ → ξ(ε′), + +COMBINATION THEOREM: AMALGAMS +39 +showing that γn +flag +−−→ ξ(ε′). However, since (µn,0) fellow-travels γn, γn +Cay +−−→ ε′. Thus, ε = ε′ +and, therefore, (5.20) is violated. +So, after extraction, µn,0 are all the same, µ0. We may now repeat the same procedure to +the sequence (f −1µ−1 +0 γn)n to show that, after another extraction µn,1 are all the same. We +can continue doing this procedure an arbitrary number of times: Thus, for all l ∈ N, there +exists a subsequence (γ′ +n) of (γn) such that suitable normal forms of the elements of (γ′ +n) +share at least l common leftmost letters. For l = 3, Proposition 2.13 shows that (γ′ +n) has +no accumulation points in the boundary of M. This is a contradiction with the assumption +that γn +Cay +−−→ ε ∈ ∂∞M. +□ +Combining Lemmas 5.14 and 5.18, we obtain: +Corollary 5.19. The boundary map ξ : ∂∞Γ → Flag(τmod) preserves the convergence +dynamics. +5.3. Proof of Theorem B. We show that the subgroup Γ = ⟨M, f⟩ < G in the conclusion +of Theorem B, which is naturally isomorphic to M⋆φ (by Proposition 3.15), is a τmod-Anosov +subgroup; this is equivalent to showing that Γ is a τmod-asymptotically embedded subgroup +(see Definition 3.9) of G: +(i) By Proposition 5.6, Γ is a τmod-regular subgroup of G. +(ii) That Γ is hyperbolic is the content of Corollary 5.4. +(iii) Finally, the boundary map ξ : ∂∞Γ → Flag(τmod) in Equation (5.15) is Γ-equivariant +(by Lemma 5.13 and Corollary 5.15), antipodal (by Proposition 5.16), and preserves +convergence dynamics (by Corollary 5.19). +This concludes the proof of the Theorem B. +□ +References +[1] Mark Baker and Daryl Cooper. A combination theorem for convex hyperbolic manifolds, with applica- +tions to surfaces in 3-manifolds. J. Topol., 1(3):603–642, 2008. +[2] Werner Ballmann, Mikhael Gromov, and Viktor Schroeder. Manifolds of nonpositive curvature, vol- +ume 61 of Progress in Mathematics. Birkh¨auser Boston, Inc., Boston, MA, 1985. +[3] Mladen Bestvina and Mark Feighn. A combination theorem for negatively curved groups. J. Differential +Geom., 35(1):85–101, 1992. +[4] Subhadip Dey and Michael Kapovich. Klein-Maskit combination theorem for Anosov subgroups: Free +products. ArXiv:2205.03919, 2022. +[5] Subhadip Dey, Michael Kapovich, and Bernhard Leeb. A combination theorem for Anosov subgroups. +Math. Z., 293(1-2):551–578, 2019. +[6] Cornelia Drut¸u and Michael Kapovich. Geometric group theory, volume 63 of American Mathemati- +cal Society Colloquium Publications. American Mathematical Society, Providence, RI, 2018. With an +appendix by Bogdan Nica. +[7] Patrick B. Eberlein. Geometry of nonpositively curved manifolds. Chicago Lectures in Mathematics. +University of Chicago Press, Chicago, IL, 1996. +[8] Vladimir Fock and Alexander Goncharov. Moduli spaces of local systems and higher Teichm¨uller theory. +Publ. Math. Inst. Hautes ´Etudes Sci., (103):1–211, 2006. +[9] Rita Gitik. Ping-pong on negatively curved groups. J. Algebra, 217(1):65–72, 1999. +[10] Olivier Guichard, Fran¸cois Labourie, and Anna Wienhard. Positivity and representations of surface +groups. ArXiv:2106.14584, 2021. + +40 +SUBHADIP DEY AND MICHAEL KAPOVICH +[11] Dumitru Ivascu. On Klein–Maskit combination theorem. In Romanian–Finnish Seminar on Complex +Analysis, volume 743, pages 115–124. Springer Lecture Notes in Mathematics, 1976. +[12] Michael Kapovich and Bernhard Leeb. Finsler bordifications of symmetric and certain locally symmetric +spaces. Geom. Topol., 22(5):2533–2646, 2018. +[13] Michael Kapovich, Bernhard Leeb, and Joan Porti. Anosov subgroups: dynamical and geometric char- +acterizations. Eur. J. Math., 3(4):808–898, 2017. +[14] Michael Kapovich, Bernhard Leeb, and Joan Porti. A Morse lemma for quasigeodesics in symmetric +spaces and Euclidean buildings. Geom. Topol., 22(7):3827–3923, 2018. +[15] Michael Kapovich and Pranab Sardar. Trees of hyperbolic spaces. ArXiv:2202.09526, 2022. +[16] Olga Kharlampovich and Alexei Myasnikov. Hyperbolic groups and free constructions. Trans. Amer. +Math. Soc., 350(2):571–613, 1998. +[17] Felix Klein. Neue Beitr¨age zur Riemann’schen Functionentheorie. Math. Ann., 21(2):141–218, 1883. +[18] Fran¸cois Labourie. Anosov flows, surface groups and curves in projective space. Invent. Math., +165(1):51–114, 2006. +[19] Liulan Li, Ken’ichi Ohshika, and Xiantao Wang. On Klein–Maskit combination theorem in space, I. +Osaka J. Math., 46:1097–1141, 2009. +[20] Liulan Li, Ken’ichi Ohshika, and Xiantao Wang. On Klein-Maskit combination theorem in space II. +Kodai Math. J., 38(1):1–22, 2015. +[21] Roger C. Lyndon and Paul E. Schupp. Combinatorial group theory. Classics in Mathematics. Springer- +Verlag, Berlin, 2001. Reprint of the 1977 edition. +[22] Eduardo Mart´ınez Pedroza. Combination of quasiconvex subgroups in relatively hyperbolic groups. Pro- +Quest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–The University of Oklahoma. +[23] Eduardo Mart´ınez-Pedroza. Combination of quasiconvex subgroups of relatively hyperbolic groups. +Groups Geom. Dyn., 3(2):317–342, 2009. +[24] Eduardo Mart´ınez-Pedroza and Alessandro Sisto. Virtual amalgamation of relatively quasiconvex sub- +groups. Algebr. Geom. Topol., 12(4):1993–2002, 2012. +[25] Bernard Maskit. On Klein’s combination theorem. Trans. Amer. Math. Soc., 120:499–509, 1965. +[26] Bernard Maskit. On Klein’s combination theorem. II. Trans. Amer. Math. Soc., 131:32–39, 1968. +[27] Bernard Maskit. On Klein’s combination theorem. III. In Advances in the Theory of Riemann Surfaces +(Proc. Conf., Stony Brook, N.Y., 1969), Ann. of Math. Studies, No. 66, pages 297–316. Princeton Univ. +Press, Princeton, N.J., 1971. +[28] Bernard Maskit. Kleinian groups, volume 287 of Grundlehren der mathematischen Wissenschaften. +Springer-Verlag, Berlin, 1988. +[29] Bernard Maskit. On Klein’s combination theorem. IV. Trans. Amer. Math. Soc., 336(1):265–294, 1993. +[30] Mahan Mitra. Coarse extrinsic geometry: a survey. In The Epstein birthday schrift, volume 1 of Geom. +Topol. Monogr., pages 341–364. Geom. Topol. Publ., Coventry, 1998. +[31] Mahan Mj and Sabyasachi Mukherjee. Combination theorems in groups, geometry and dynamics. In In +the tradition of Thurston II. Geometry and groups, pages 331–383. Springer, 2022. +[32] Jean-Pierre Serre. Trees. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. Trans- +lated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation. +Department of Mathematics, Yale University, 10 Hillhouse Ave, New Haven, CT 06511 +Email address: subhadip.dey@yale.edu +Department of Mathematics, University of California, Davis, One Shields Ave, Davis, CA +95616 +Email address: kapovich@ucdavis.edu + diff --git a/vtE0T4oBgHgl3EQfbwDA/content/tmp_files/load_file.txt b/vtE0T4oBgHgl3EQfbwDA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6df2eb2522ddd955c688bcc9c2a79afc3d377ee8 --- /dev/null +++ b/vtE0T4oBgHgl3EQfbwDA/content/tmp_files/load_file.txt @@ -0,0 +1,1923 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf,len=1922 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='02354v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='GR] 6 Jan 2023 KLEIN–MASKIT COMBINATION THEOREM FOR ANOSOV SUBGROUPS: AMALGAMS SUBHADIP DEY AND MICHAEL KAPOVICH Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The classical Klein–Maskit combination theorems provide sufficient conditions to construct new Kleinian groups using old ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' There are two distinct but closely related combination theorems: The first deals with amalgamated free products, whereas the second deals with HNN extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This article gives analogs of both combination theorems for Anosov subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Introduction In the theory of Kleinian groups (discrete isometry groups of H3), the Klein–Maskit combination theorems provide techniques to construct new Kleinian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The history of combination theorems dates back to Klein’s 1883 paper [17], which gave sufficient conditions for a subgroup Γ of G = Isom(H3) generated by two discrete subgroups Γ1 and Γ2 of G to be discrete and isomorphic to the free product Γ1⋆Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Subsequently, Maskit [25, 26, 27, 29], dealing with the cases of amalgamated free products and HNN extensions, gave far-reaching generalizations of the Klein combination theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Maskit’s combination theorems also fur- nish sufficient conditions for the combined group to have certain geometrical properties, such as convex-cocompactness or geometric-finiteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' See Maskit’s book [28] for an account of those results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Later on, Ivascu [11] and, more recently, Li–Ohshika–Wang [19, 20] extended the Klein–Maskit combination theorems to the setting of discrete isometry groups of higher dimensional hyperbolic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The combination theorems were further generalized in the context of group actions on Gromov-hyperbolic spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' see [1, 9, 22, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We refer to [31] for a recent survey of combination theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Our motivation for this article is to provide suitable analogs of the Klein–Maskit combi- nation theorems in the context of Anosov subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In recent years, Anosov subgroups of semisimple Lie groups have emerged as a higher-rank generalization of convex-cocompact Kleinian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In our paper with Bernhard Leeb [5], we gave an earlier form of such a combination theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' in that paper, using the local-to-global principle for Morse quasi- geodesics, we proved a version of the Klein combination theorem for Anosov subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Moreover, we conjectured a sharper form of the combination theorem in that paper, which was recently confirmed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Nevertheless, the discussions in [4, 5] were limited only to the case of free products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The purpose of the present article is to deal with the case of amalgams (amalgamated free products and HNN extensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Building on our work in [4], we prove versions of both Klein–Maskit combination theorems for Anosov subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Date: January 5, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 1 2 SUBHADIP DEY AND MICHAEL KAPOVICH Let G be a real semisimple Lie group of noncompact type and with a finite center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We will assume some mild conditions on G (see Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let X = G/K be the associated symmetric space, where K is a maximal compact subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let σmod be a model spherical chamber in the Tits building ∂TitsX of X and let ι : σmod → σmod be the opposition involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We consider the class of τmod-Anosov subgroups of G, where τmod is an ι-invariant face of σmod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For a discrete subgroup Γ of G, the τmod-limit set of Γ in Flag(τmod), the partial flag manifold associated to the face τmod, is denoted by Λτmod(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' See §3 for the definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The first main result of this paper, which provides an analog of the first Klein–Maskit combination theorem [28, Theorem VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2], is as follows: Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let ΓA and ΓB be τmod-Anosov subgroups of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Assume that H := ΓA∩ΓB is quasiconvex in ΓA or ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that there exists an interactive pair (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='11) (A, B) in Flag(τmod) for (ΓA, ΓB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' H) such that the following conditions are satisfied: (i) The interiors of A and B are antipodal to each other (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (ii) The pairs of subsets (A, Λτmod(ΓB) \\ Λτmod(H)) and (B, Λτmod(ΓA) \\ Λτmod(H)) of Flag(τmod) are antipodal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, the subgroup Γ = ⟨ΓA, ΓB⟩ of G is τmod-Anosov and is naturally isomorphic to the abstract amalgamated free product ΓA ⋆H ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Our second main result, which gives an analog of the second Klein–Maskit combination theorem [28, Theorem VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5], is as follows: Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let M be a τmod-Anosov subgroup of G and let f ∈ G be an element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Assume that H+ := M ∩ (fMf −1) or H− := (f −1Mf) ∩ M is quasiconvex in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that there exists an interactive triple (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='14) (A, B±) in Flag(τmod) for (M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' H±;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' f) such that the following conditions are satisfied: (i) The pairs of subsets (A◦, B◦ ±), (B+, B−) of Flag(τmod) are antipodal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (ii) Λτmod(Γ) \\ Λτmod(H±) is antipodal to B±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, the subgroup Γ = ⟨M, f⟩ of G is τmod-Anosov and is naturally isomorphic to the abstract HNN extension M⋆φ of M by φ, where the isomorphism φ : H− → H+ is given by φ(η) = fηf −1, for all η ∈ H−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' See §4, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' §5, for the proof of Theorem A, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' To conclude this introduction, let us illustrate the hypotheses and conclusions of the Theorem A and Theorem B in the following examples: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Illustrative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let S be a closed, orientable surface of genus g ≥ 2 and let Γ := π1(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Labourie’s [18] pioneering work showed that Hitchin representations ρ : Γ → PSL(d, R) form a rich class of σmod-Anosov representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let us denote by ξρ : ∂∞Γ → Flag(σmod) the associated Γ-equivariant boundary map onto the limit set Λρ(Γ) ⊂ Flag(σmod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' According to Fock–Goncharov [8], an important characteristic of the Hitchin representations is a certain positivity property of the limit map ξρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Given an essential separating simple closed curve s ⊂ S, let [η] denote the conjugacy class in Γ representing s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The curve s cuts the surface into two subsurfaces, SA and SB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The COMBINATION THEOREM: AMALGAMS 3 group Γ can be written as an amalgamated free product Γ = ΓA ⋆H ΓB, where ΓA = π1(SA), ΓB = π1(SB), H = ⟨η⟩, for some η ∈ [η], equipped with natural monomorphisms φA : H → ΓA and φB : H → ΓB induced by the inclusion homeomorphisms s ֒→ ∂SA and s ֒→ ∂SB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The image of H = ⟨η⟩ under a Hitchin representation ρ : Γ → PSL(d, R) is a cyclic σmod-Anosov subgroup of PSL(d, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let σ± ∈ Λρ(Γ) = ξρ(∂∞Γ) denote the attractive/repulsive fixed points of ρ(η) in Flag(σmod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The 2-point set {σ+, σm} cuts Λρ(Γ) in a pair of closed arcs cA and cB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' we chose the names in such a way that Λρ(ΓA) ⊂ cA and Λρ(ΓB) ⊂ cB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' See the left side of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let Ω be the set of all points in Flag(σmod) antipodal to both σ±;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' this is the intersection of a pair of opposite maximal Schubert cells in Flag(σmod): Ω = C(σ+) ∩ C(σ−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' There are two distinguished connected components of Ω, denoted by A◦ and B◦, whose closures A and B, respectively, contain cA and cB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Both A and B are preserved by H because H preserves cA and cB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Using the positivity property of ξ and the fact that, for all α ∈ ρ(ΓA \\ H) and β ∈ ρ(ΓA \\ H), αcB ⊂ cA \\ {σ±} and βcA ⊂ cB \\ {σ±}, it follows that αB ⊂ A◦ and βA ⊂ B◦, see [10, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Moreover, the interiors A◦ and B◦ are antipodal to each other, see [10, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5(3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, (A, B) is an interactive pair for (ρ(ΓA), ρ(ΓB);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' ρ(H)), which satisfies the hypothesis of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let us describe a continuous family of Hitchin representations ρt : Γ → PSL(d, R) ob- tained by bending the given Hitchin representation ρ along s, a type of deformation that generalizes the classical Dehn twists along simple closed geodesics in a hyperbolic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This family is parametrized by t ∈ Z0(η), where Z0(η) ∼= Rd−1 is the identity component of the centralizer of η in PSL(d, R), and the σmod-Anosov property of this family can be verified by a simple application of Theorem A: Each connected component of Ω is also preserved by Z0(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In particular, Z0(η) preserves A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For t ∈ Z0(η), let ρt : ΓB → PSL(d, R), ρt(β) = tρ(β)t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Clearly, ρt|H = ρ|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Moreover, we observe that (A, B) is still an interactive pair for the triple (ρ(ΓA), ρt(ΓB);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' ρ(H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, Theorem A directly verifies that ρt : Γ = ΓA ⋆H ΓB → ⟨ρ(ΓA), tρ(ΓB)t−1⟩ < PSL(d, R), is injective, and its image is a σmod-Anosov subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' if s is an essential non-separating simple closed curve in S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' then the fundamental group Γ = π1(S) can be written as an HNN extension Γ = M⋆φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' φ : H− ∼ = −→ H+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' where M is the fundamental group of the surface S′ obtained by cutting S along s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' H− = ⟨η⟩ and H+ = ⟨η′⟩ are the images of the homomorphisms of the fundamental groups induced by the two inclusion homeomorphisms s ֒→ ∂−S′ and s ֒→ ∂+S′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' and φ : H− → H+ is the isomorphism induced by the conjugation by some element f ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 4 SUBHADIP DEY AND MICHAEL KAPOVICH Let ρ : Γ → PSL(d, R) be a Hitchin representation with associated Γ-equivariant limit map ξρ : ∂∞Γ → Flag(σmod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The attractive/repulsive fixed point sets {σ+, σ−} and {σ′ +, σ′ −} of η and η′, respectively, cut the limit set Λρ(Γ) = ξρ(∂∞Γ) in four closed arcs cA+, cA−, cB+, and cB− (see the right side of Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We have ρ(f){σ±} = {σ′ ±}, ρ(f)(cA+ ∪ cA− ∪ cB+) ⊂ cB+, and ρ(f)−1(cA+ ∪ cA− ∪ cB−) ⊂ cB−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' cB cA σ+ σ− η cA+ cA− cB− cB+ σ+ σ′ + σ− σ′ − f η η′ Figure 1 Define B−, B+ ⊂ Flag(σmod) to be the closure of connected component of C(σ+)∩C(σ−) containing the arc cB− and cB+, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lastly, define A to be the union of A+ and A−, where A± are the closures of the connected component of C(σ±) ∩ C(σ±) containing the arcs cA±, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, using the positivity of ξρ, one can show that (A, B±) is an interactive triple for (ρ(M);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' ρ(H±);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' ρ(f)) (compare with the amalgamated free product case discussed above), thus verifying the hypothesis of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' As before, we obtain a continuous family of Hitchin representations ρt : Γ → PSL(d, R), parametrized by t ∈ Z0(η), by bending ρ along s: Observe that for any t ∈ Z0(η), (A, B±) is an interactive triple for (ρ(M);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' ρ(H±);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' ρ(f) · t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, Theorem B directly verifies that ρt : Γ = M⋆φ → ⟨ρ(M), ρ(f) · t⟩ < PSL(d, R) is injective with a σmod-Anosov image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Organization of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In §2, we present some preliminary material and results on amalgamated free products and HNN extensions of hyperbolic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This section presents some results (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8, Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='13) crucial in the proof of our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, in §3 we review some background material on discrete groups acting on symmetric spaces of noncompact type and present some lemmas (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1), which are also frequently used in the proof of our main results to verify regularity and flag-convergence of certain sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Finally, in §4, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' §5, we prove Theorem A, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Preliminaries on amalgams of hyperbolic groups 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Amalgamated free products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let ΓA, ΓB, and H be abstract groups together with monomorphisms φA : H → ΓA, φB : H → ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The free product of ΓA and ΓB amalgamated along H, denoted by ΓA ⋆H ΓB, has a presentation ⟨SA, SB | RA, RB, φA(η)φB(η)−1, η ∈ H⟩, COMBINATION THEOREM: AMALGAMS 5 where ⟨SA | RA⟩ and ⟨SB | RB⟩ are presentations of ΓA and ΓB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We will iden- tify H with φA(H) < ΓA and φB(H) < ΓB via the monomorphisms φA and φB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let Γ = ΓA ⋆H ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A normal form of an element γ ∈ Γ is an expression γ = γ1γ2 · · · γl, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1) such that the following conditions are satisfied: (i) Each γi lies either in ΓA, or in ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Moreover, if l ≥ 2, then none of the letters γi belong to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (ii) No two successive letters γi, γi+1 above lie in the same group ΓA or ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Unless H is trivial, the normal form of γ given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1) is not necessarily unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' However, any other normal form of γ is obtained by a sequence of finite moves consisting of replacing a consecutive pair γiγi+1 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1) by (γiηi)(η−1 i γi+1), where ηi ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This is a consequence of the Normal Form Theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' see [21, Theorem IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' It follows that any two normal forms of γ have the same number of letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For γ ∈ Γ \\ H, the relative length of γ, denoted by rl(γ), is the number of letters in a(ny) normal form of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If γ ∈ H, then define rl(γ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1 (Alternating sequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A sequence (ωk) in Γ is called type A alternating if there exists a pair of sequences, (αn) in ΓA \\ H and (βn) in ΓB \\ H, such that ωk = � α1β1 · · · βm−1αm, k = 2m − 1, α1β1 · · · αmβm, k = 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2) Similarly, a sequence (ωk) in Γ is called type B alternating if there exists a pair of sequences, (αn) in ΓA \\ H and (βn) in ΓB \\ H, such that ωk = � β1α1 · · · αm−1βm, k = 2m − 1, β1α1 · · · βmαm, k = 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' HNN extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let M be an abstract group and H± < M be a pair of subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that φ : H− → H+ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The HNN extension of M by φ, denoted by M⋆φ, has a presentation ⟨SM, f | RM, fηf −1(φ(η))−1, η ∈ H−⟩, where ⟨SM | RM⟩ is a presentation of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Every element γ ∈ Γ = M⋆φ can be written in the form γ = µ0f ǫ1µ1 · · · f ǫnµn, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3) for some n ≥ 0, where each µi is in M, and each ǫi is either 1 or −1, such that the following conditions are satisfied: (i) For i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' , n, if ǫi = −1 and µi ∈ H+, then ǫi+1 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (ii) For i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' , n, if ǫi = 1 and µi ∈ H−, then ǫi+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Such an expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3) is called a normal form of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Although the normal form of an element is not unique, Britton’s Lemma (see [21, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='181]) establishes certain uniqueness properties of the decomposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3) of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In particular, the relative length rl(γ) of γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', the total number of letters f and f −1 appearing in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3), is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 6 SUBHADIP DEY AND MICHAEL KAPOVICH Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2 (Alternating sequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A sequence (ωn) in M⋆φ is called alternating if there exist sequences (µn) in M and (ǫn) in {±1}, satisfying ǫn = −1, µn ∈ H+ =⇒ ǫn+1 = −1, and ǫn = 1, µn ∈ H− =⇒ ǫn+1 = 1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4) for all n ∈ N, and some element µ0 ∈ M such that ωn = µ0f ǫ1µ1f ǫ2µ2 · · · f ǫn−1µn−1f ǫn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4) simply implies that the word in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5) is a normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' It is also useful to think about an alternating sequence (ωn) in M⋆φ as an infinite string of letters: µ0, f ǫ1, µ1, f ǫ2, µ2, f ǫ3, µ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' , where µi and ǫi satisfy the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Bass-Serre trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In this paper, we will be using the formalism of Bass–Serre trees T associated with amalgamated free products and HNN extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A detailed treatment of this material can be found in Serre’s book [32];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' see also [6] for a more topological viewpoint on the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (i) Consider an amalgamated free product Γ = ΓA ⋆H ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The vertex set V (T) of the tree T is the set of cosets γΓA, γΓB, γ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Accordingly, the vertex set V (T) is bicolored, with one color corresponding to the cosets γΓA and the other color corresponding to the cosets γΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The group Γ acts on V (T) by the left multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Edges of T are defined so that T is a bipartite graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', vertices of the same color are never connected by an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The cosets γ1ΓA, γ2ΓB are connected by an edge whenever there exists α ∈ ΓA such that γ1ΓB = γ2αΓB, equivalently, there exists β ∈ ΓB such that γ1βΓA = γ2ΓA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For instance, the vertices represented by the cosets ΓA, ΓB are connected by an edge e ∈ E(T) since there exists an element η ∈ H < ΓA such that ΓB = ηΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Moreover, all elements α ∈ ΓA such that αΓB = ΓB necessarily belong to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We, thus, label the edge e by the coset 1Γ · H = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The left multiplication by the elements of Γ preserves the incidence relation between the vertices of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Accordingly, the edges of T are labeled by the cosets γH, γ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The Γ- stabilizer of the vertex γΓA equals γΓAγ−1, while the Γ-stabilizer of the vertex γΓB equals γΓBγ−1 and the Γ-stabilizer of an edge labeled γH equals γHγ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (ii) Consider an HNN extension Γ = M⋆φ:H−→H+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The vertex set of T consists of left cosets of M in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The edge set of T consists of edges of two types: The left H±-cosets in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The edge gH+ connects γM to γfM, while the edge γH− connects γM and γf −1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The Γ-action on T is defined by: γ : γ′M �→ γγ′M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The Γ-action is transitive on the set of vertices and edges of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' COMBINATION THEOREM: AMALGAMS 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Hyperbolic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let Γ be a finitely-generated group equipped with a left-invariant word metric, denoted by dΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We use the notation | · | to denote the word length of elements of Γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', |γ| = dΓ(1Γ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Recall that Γ is called (word) hyperbolic if there exists δ ≥ 0 such that (Γ, dΓ) is a δ-hyperbolic metric space: That is, for all f, g, h, w ∈ Γ, (f, g)w ≥ min{(f, h)w, (g, h)w} − δ, where (f, g)w denotes the Gromov product of f and g with respect to w: (f, g)w = 1 2(dΓ(f, w) + dΓ(g, w) − dΓ(f, g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' It is a basic fact that the property of being hyperbolic does not depend on the choice of a word metric dΓ, but the constant δ possibly depends on the chosen metric dΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let Γ be a hyperbolic group equipped with a word metric dΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A (discrete) geodesic in Γ is an isometric embedding c : I → Γ of an interval I ⊂ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Such a geodesic c : I → Γ is called a segment, ray, or line when I is bounded, I is only bounded below, or I = Z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The Gromov boundary of Γ, denoted by ∂∞Γ, is the set of equivalence classes of asymptotic geodesic rays in Γ, which gives a natural compactification of (Γ, dΓ), Γ = Γ ⊔ ∂∞Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The topology of Γ is understood as follows: A pair of sequences (γn) and (γ′ n) in Γ are said to fellow travel if (γn, γ′ n)1Γ → ∞, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A sequence (γn) in Γ converges to a point ε ∈ ∂∞Γ, which is denoted by γn Cay −−→ ε, if and only if (γn) fellow-travels the image sequence (c(n)) of a(ny) geodesic ray c : N → Γ asymptotic to ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The following result can be checked easily using the definitions above: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Fellow traveling sequences in Γ have the same accumulation set in ∂∞Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Nearest point projections to quasiconvex subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let Γ be a hyperbolic group equipped with a left-invariant word metric dΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A subset Y ⊂ Γ is called quasiconvex if there exists K ≥ 0 such that, for all y1, y2 ∈ Y , any geodesic segment in Γ connecting y1 and y2 lies in the K-neighborhood of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For a quasiconvex subset Y ⊂ Γ, we choose a nearest point projection map prY : Γ → Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Note that nearest point projections are not necessarily unique, but since Y is quasiconvex, any two such maps are a finite distance apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let Y ⊂ Γ be a quasiconvex subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For every γ ∈ Γ, prY (γ) ∈ Y lies in a uniform neighborhood of any geodesic segment in Γ connecting γ to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 8 SUBHADIP DEY AND MICHAEL KAPOVICH y0 y′ y γ′ = prY (γ) x γ K ≥ 2δ ≥ ≤ 2δ 1 = Y Figure 2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let K ≥ 0 be a quasiconvexity constant for Y and δ ≥ 0 be a hyperbolicity constant for (Γ, dΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let [y0, γ] be a geodesic segment in Γ from y0 ∈ Y to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Consider a geodesic triangle in Γ with vertices y0, γ, and γ′ := prY (γ), whose edge connecting y0 to γ is [y0, γ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since geodesic triangles in Γ are 2δ-thin, we have [y0, γ] ⊂ N2δ([y0, γ′] ∪ [γ, γ′]), where Nr(·) denotes the r-neighborhood in (Γ, dΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In particular, there exist points x ∈ [γ, γ′] and y ∈ [y0, γ′] such that dΓ(x, [y0, γ]) ≤ 2δ, dΓ(y, [y0, γ]) ≤ 2δ, and dΓ(x, y) ≤ 4δ +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' See Figure 2 for an illustration of the points in the Cayley graph of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since [y0, γ′] lies in the K-neighborhood of Y , there exists y′ ∈ Y , which is at most K distance away from y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since γ′ is also a nearest point in Y to x, dΓ(x, γ′) ≤ dΓ(x, y′) ≤ K + 4δ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, γ′ is at most K + 6δ + 1 distance away from [y0, γ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ For a subset Y ⊂ Γ, let ∂∞Y ⊂ ∂∞Γ denote the set of all accumulation points of Y in Γ = Γ ⊔ ∂∞Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let Y ⊂ Γ be a quasiconvex subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A sequence (γn) in Γ has an accumu- lation point in ∂∞Y if and only if (prY (γn)) is unbounded in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For the “if” part, suppose that (prY (γn)) is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' After passing to a subse- quence, we assume that | prY (γn)| → ∞, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5, it follows that (γn) and (prY (γn)) fellow travel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4, (γn) has an accumulation point in ∂∞Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For the “only if” part, we prove the contrapositive statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let us assume that (prY (γn)) is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We show that (γn) has no accumulation points in ∂∞Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We argue by contradiction: Suppose that ε ∈ ∂∞Y is an accumulation point of (γn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' After extraction, we may assume that γn Cay −−→ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let (yn) be any sequence in Y such that yn Cay −−→ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Con- sequently, we must have (γn, yn)1Γ → ∞, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' However, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5 can be restated as sup γ∈Γ,y∈Y (γ, y)prY (γ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' COMBINATION THEOREM: AMALGAMS 9 Since (prY (γn)) is bounded, the above implies sup m,n (γn, ym)1Γ < ∞, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ A subgroup H < Γ is called a quasiconvex subgroup if H, as a subset of Γ, is quasiconvex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let H < Γ be a quasiconvex subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For any sequence (γn) in Γ, consider the sequence (ˆγn) given by ˆγn = prH(γn)−1γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (i) Regarded as a sequence in the compact space Γ = Γ ⊔ ∂∞Γ, (ˆγn) has no accumulation points in ∂∞H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (ii) Suppose that (γn) diverges away from H, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', dΓ(H, γn) → ∞, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, the accumulation set of (γ−1 n ) in ∂∞Γ is the same as that of (ˆγ−1 n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We first observe that, for all γ ∈ Γ, the identity element is a nearest point in H to ˆγ := prH(γ)−1γ: Indeed, for all η ∈ H, dΓ(η, ˆγ) = dΓ(prH(γ)η, γ) ≥ dΓ(prH(γ), γ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6) where the inequality follows from the fact that prH(γ) is a nearest point in H to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Moreover, dΓ(1Γ, ˆγ) = dΓ(prH(γ), γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Hence, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6) implies that dΓ(H, ˆγ) ≥ dΓ(1Γ, ˆγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, for any sequence (γn) in Γ, {prH(ˆγn) | n ∈ N} ⊂ H is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Part (i) now follows by applying Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We prove part (ii): Since prH(γn) lies uniformly close to any geodesic in Γ connecting 1Γ to γn (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5) and dΓ(H, γn) → ∞, it follows that |γn| − | prH(γn)| → ∞, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus (γ−1 n , ˆγ−1 n )1Γ = 1 2(|γ−1 n | + |ˆγ−1 n | − dΓ(γ−1 n , ˆγ−1 n )) = 1 2(|γn| + |ˆγn| − |γnˆγ−1 n |) = 1 2(|γn| + |ˆγn| − | prH(γn)|) → ∞, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In other words, (γ−1 n ) and (ˆγ−1 n ) fellow travel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4, (γ−1 n ) and (ˆγ−1 n ) have the same accumulation sets in ∂∞Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ An equivalent statement of the above result, which we will often use, is as follows: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let H < Γ be a quasiconvex subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For any sequence (γn) in Γ, consider the sequence (ˆγn) given by ˆγn = γn prH(γ−1 n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (i) (ˆγ−1 n ) has no accumulation points in ∂∞H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (ii) If (γ−1 n ) diverges away from H, then the accumulation sets of (γn) and (ˆγn) in ∂∞Γ coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 10 SUBHADIP DEY AND MICHAEL KAPOVICH 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Amalgams of hyperbolic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For the rest of this section, we restrict our discus- sion to amalgams (amalgamated free products and HNN extensions) of hyperbolic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The Bestvina–Feighn Combination Theorem, [3], provides some sufficient conditions for the hyperbolicity of amalgams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We review this theorem in the weakly malnormal case (although their actual result is much more general): Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A subgroup H of a group Γ is said to be weakly malnormal if, for every γ ∈ Γ \\ H, the subgroup γHγ−1 ∩ H is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10 (Bestvina–Feighn, [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let ΓA, ΓB, and M be hyperbolic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (i) If H < ΓA and H < ΓB are quasiconvex, weakly malnormal subgroups, then ΓA ⋆H ΓB is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (ii) If H± < M are isomorphic (by φ : H− → H+), quasiconvex, weakly malnormal sub- groups such that, for all µ ∈ M, H− ∩ µH+µ−1 is finite, then M⋆φ is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' See also [16, Theorems 1 & 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This theorem has several addendums that we will use in what follows: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='11 (Mitra, [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Under the assumptions, the subgroups ΓA and ΓB (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' M) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10 are quasiconvex in ΓA ⋆H ΓB (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' M⋆φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Boundary of an amalgam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Our next goal is to describe the Gromov boundaries of the amalgamated free products Γ = ΓA ⋆H ΓB and HNN extensions Γ = M⋆φ:H−→H+ under certain extra assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We will be assuming that the groups ΓA, ΓB, and M are hyperbolic, H is weakly malnormal and quasiconvex in ΓA, ΓB (in the amalgamated free product case) and, H± are weakly malnormal and quasiconvex in M and the intersection H−∩µH+µ−1 is finite, for all µ ∈ M (in the HNN extension case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Under these assumptions, Γ is hyperbolic (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Moreover, in the case of HNN extensions, we let f ∈ Γ denote the stable letter, the element corresponding to the subgroup isomorphism φ : H− → H+: fηf −1 = φ(η), η ∈ H−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Our description of the boundary follows [15, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3], to which we refer the reader for de- tails and proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We will describe the boundary (mainly) in the case of amalgamated free products since the HNN extension case is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let T denote the Bass-Serre tree associated with the amalgamated free product Γ = ΓA ⋆H ΓB (or the HNN extension);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The group Γ acts on T with vertex-stabilizers (the vertex-subgroups Γv) that are conjugates of ΓA, ΓB (or M in the HNN extension case) and edge-stabilizers (edge-subgroups Γe) which are conjugates of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Define a tree of topological spaces as follows: To each v ∈ V (T), e ∈ E(T), we associate the Gromov boundary ∂∞Γv, ∂∞Γe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Whenever v is a vertex of an edge e, we have the inclusion homomorphism Γe → Γv, which induces a topological embedding fev : ∂∞Γe → ∂∞Γv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This data yields a tree of topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The tree gives rise to a topological space ∂IΓ, the topological realization of the tree of topological spaces, by taking the push-out of the maps fev: The topological space ∂IΓ is a union of Gromov boundaries ∂∞Γv, v ∈ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' More precisely, it is the quotient of the disjoint union of these boundaries by the equivalence relation defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For every edge e = [v, w] of T, we have ξ ∈ ∂∞Γv is equivalent COMBINATION THEOREM: AMALGAMS 11 to η ∈ ∂∞Γw whenever there exists ζ ∈ ∂∞Γe such that fev(ζ) = ξ, few(ζ) = η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The group Γ acts on ∂IΓ via the projection of the natural Γ-action on � v∈V (T) ∂∞Γv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In particular, ∂IΓ is either the union of the Γ-orbits of ΓA ∪ ΓB (in the amalgamated free product case) or ∂∞M (in the HNN extension case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The weak malnormality assumption on the amalgam implies that whenever e ̸= e′ are distinct edges of T, ∂∞Γe ∩ ∂∞Γe′ = ∅ in ∂IΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Accordingly, whenever the distance between vertices v, w is > 1, ∂∞Γv ∩ ∂∞Γw = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7) Moreover, the weak malnormality assumption also implies that all vertex-subgroups Γv and edge-subgroups Γe are quasiconvex in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Hence, one obtains a natural Γ-equivariant inclusion map ∂IΓ → ∂∞Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' It turns out that this map is injective and continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' However, in general, this map need not be a topological embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In what follows, we will identify ∂IΓ with its image in ∂∞Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The Gromov boundary ∂∞Γ of Γ is the disjoint union Γ-invariant subsets ∂∞Γ = ∂IΓ ⊔ ∂IIΓ, where ∂IIΓ := ∂∞Γ\\∂IΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Elements of ∂IΓ, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' ∂IIΓ, are called type I, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' type II, (ideal) boundary points of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The second part of the boundary, ∂IIΓ, of ∂∞Γ admits a Γ-equivariant continuous bijection to ∂∞T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (For instance, ∂∞⟨f⟩ is the 2-point subset of ∂IIΓ corresponding to the fixed-point set of f in ∂∞T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=') Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For every point ε ∈ ∂IIΓ, there exists an alternating sequence (ωn) in Γ converging (in Γ) to ε such that the following holds: If a sequence (γn) in Γ converges to ε, then there exists a function F : N → N diverging to infinity and n1 ∈ N such that, for all integers n ≥ n1, there exists a normal form of γn containing ωF (n) as a left subword.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We prove this result in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Fix a word metric dΓ on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For all ω ∈ Γ satisfying rl(ω) ≥ 3, there exists a constant D = D(ω) ≥ 0 such that the following holds: If γ ∈ Γ is any element such that some normal form of γ contains some normal form of ω as a left subword, then (i) in the case Γ = ΓA ⋆H ΓB, we have dΓ(prΓA(γ), 1Γ) ≤ D and dΓ(prΓB(γ), 1Γ) ≤ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (ii) in the case Γ = M⋆φ, we have dΓ(prM(γ), 1Γ) ≤ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Using this result, it can be shown that alternating sequences cannot have type I accumu- lation points in the boundary of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' See §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9 for a proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We set up some notation for the rest of this section: We let dΓ denote an arbitrary word metric on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Moreover, we reserve the notation d to denote a word metric on Γ induced by a finite symmetric generating set S of Γ, where, (i) in the case of Γ = ΓA⋆HΓB, S is the union of some chosen finite symmetric generating sets of ΓA and ΓB, and (ii) in the case of Γ = M⋆φ, S is the union of some chosen finite symmetric generating set of M and {f, f −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Recall that the identity map (Γ, d) → (Γ, dΓ) is a quasiisometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Finally, we 12 SUBHADIP DEY AND MICHAEL KAPOVICH reserve the notation dT to denote the distance function on the corresponding Bass-Serre tree T induced by declaring that all the edges of T are of unit length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The following result demonstrates the existence of an alternating sequence (ωk) converging to ε ∈ ∂IIΓ in the statement of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' There exists D0 ≥ 0 such that, for every ε ∈ ∂IIΓ, there exists a D0- alternating sequence (ωn) in (Γ, dΓ) converging to ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In the statement above, “D0-alternating” means that (ωn) is alternating and lies within distance D0 from any geodesic ray in (Γ, dΓ) emanating from 1Γ asymptotic to ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15, we work with the word-metric d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' see Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The general case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', when dΓ is an arbitrary word metric, would then follow by applying the Morse lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15 in the case of amalgamated free products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let us consider a uniform quasigeodesic c : N ∪ {0} → (Γ, d) emanating from c(0) = 1Γ asymptotic to ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' such a ray is described by a sequence (si) in S such that, for all k ∈ N, c(k) = s1 · · · sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Note that the sequence (c(k)) cannot entirely lie in ΓA ∪ ΓB since, otherwise, the sequence (c(k)) would converge to a type I ideal boundary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let i1 be the largest number such that c(i1) lies in ΓA ∪ ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For the same reason as above, the sequence (c(i1)−1c(k))k∈N cannot lie in ΓA ∪ ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, let i2 > i1 be the largest number such that c(i1)−1c(i2) lies in ΓA ∪ ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Similarly, let i3 > i2 be the largest number such that c(i2)−1c(i3) lies in ΓA ∪ ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proceeding inductively, we find a sequence (c(ik)−1c(ik+1))k which alternates between ΓA and ΓB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' the elements of that sequence are the letters for our alternating sequence (ωk): ωk = c(i1)[c(i1)−1c(i2)] · · · [c(ik−1)−1c(ik)] = c(ik).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since the sequence (ωk) lies in the quasigeodesic ray c, it converges to ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15 in the case of HNN extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For ε ∈ ∂IIΓ, pick any uniform quasi- geodesic ray c : N ∪ {0} → Γ that emanates from c(0) = 1Γ and is asymptotic to ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Such a ray is described by a sequence (si) in S such that, for all k ∈ N, c(k) = s1 · · · sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We find an infinite string ˆS : µ0, f ǫ1, µ1, f ǫ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8) which has the property that µi ∈ M, ǫi = ±1, and, for every n, rl(µ0f ǫ1µ1 · · · f ǫn−1µn−1f ǫn) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9) Let i0 ∈ N ∪ {0} be the largest number such that c(i0) ∈ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' set µ0 = c(i0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let i1 ≥ i0 be the largest number such that c(i0)−1c(i1) ∈ {f, f −1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' define ǫ1 in the obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let i2 ≥ i1 to be the largest number such that c(i1)−1c(i2) ∈ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' set µ1 = c(i1)−1c(i2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let i3 ≥ i2 be the largest number such that c(i2)−1c(i2) ∈ {f, f −1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' define ǫ2 accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We proceed inductively to obtain the above string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' It is a straightforward check that rl(µ0f ǫ1µ1 · · · f ǫn−1µn−1f ǫn) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' COMBINATION THEOREM: AMALGAMS 13 We observe that, if ǫi = −1 and µi ∈ H+ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' ǫi = 1 and µi+1 ∈ H−), then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9) forces ǫi = −1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' ǫi = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, (ωk), where ωk := µ0f ǫ1µ1 · · · f ǫn−1µn−1f ǫn, is an alternating sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since the sequence (ωk) lies in the quasigeodesic ray c, it converges to ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ The same construction used to prove Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15 can be applied to “uniform quasi- geodesic segments” in Γ to show the following result: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For D ≥ 0, a D-normal form of γ ∈ Γ satisfying rl(γ) ≥ 1 is a normal form of γ such that all left subwords in that normal form lie in a D-neighborhood of any geodesic segment in (Γ, dΓ) connecting 1Γ and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' There exists D0 ≥ 0 such that every element γ ∈ Γ satisfying rl(γ) ≥ 1 has a D0-normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Now, we prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We give a proof in the amalgamated free product case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' the HNN extension case is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We continue with the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15 above (the amalgamated free product case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let (γn) be any sequence in Γ converging to ε ∈ ∂IIΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since (γn) fellow travels the uniform quasigeodesic c, we may pick a divergent sequence (tn) in N∪{0} and, for each n, a uniform quasigeodesic segment cn : [0, ln] ∩ Z → Γ, cn(0) = 1Γ, cn(ln) = γn, such that, for all n ∈ N, cn|[0,tn]∩Z = c|[0,tn]∩Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' See Figure 3 for an illustration of c and cn in (the Cayley graph) of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We apply the same procedure used in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15 on cn to yield a D-normal form for γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 1Γ γn = cn(ln) ε c(tn) c Figure 3 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' There exists n1 ∈ N such that, for all n ≥ n1, the prescribed normal form of γn contains ω1 as a leftmost subword.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We argue by contradiction: Suppose that the assertion is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, a divergent sequence (ni) exists in N such that, for all i, the leftmost letter λi of the prescribed normal form of γni is different from ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since cni|[0,tni]∩Z = c|[0,tni]∩Z, for all i large enough it must hold that λi = cni(ri), where ri > tni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' However, λi ∈ ΓA ∪ ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since cni are uniform quasigeodesics with cni(0) = 1Γ ∈ ΓA ∪ ΓB, it follows that cni([0, ri] ∩ Z), which contains c([0, tni]∩Z) as a subset, lies in a uniform neighborhood of ΓA∪ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' However, since tni → ∞, rl(c(tni)) goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, c(tni) cannot stay in a uniform neighborhood of ΓA ∪ ΓB (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='13), yielding a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ 14 SUBHADIP DEY AND MICHAEL KAPOVICH We now finish the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12 by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that c(i1) = ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since cn|[0,tn]∩Z = c|[0,tn]∩Z, by the claim above, it holds that for all sufficiently large n, say n ≥ n1, cn(i1) = c(i1) = ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Applying the same argument to the quasigeodesic ray/segments ¯c : [0, ∞) ∩ Z → Γ, c(t) = ω−1 1 c(t + i1), ¯cn : [0, ln − i1] ∩ Z → Γ, cn(t) = ω−1 1 cn(t + i1), shows that there exists n2 > n1 such that for all n ≥ n2, the prescribed normal form of γn contains ω2 as a leftmost subword.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Arguing inductively, it follows that there exists an increasing sequence (nk) of natural numbers such that, for all n ≥ nk, the prescribed normal form of γn contains ωk as a leftmost subword.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, the desired function F : N → N in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12 can be defined as F(n) := k, if n ∈ [nk, nk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let us first consider the case of amalgamated free prod- ucts: For γ ∈ Γ = ΓA ⋆H ΓB, consider a normal form of γ: γ = γ1γ2 · · · γl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If γ1 ∈ ΓA \\ H, then normal form given above yields a finite sequence of points in T: ΓB, ΓA, γ1ΓB, γ1γ2ΓA, γ1γ2γ3ΓB, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' , γ1γ2 · · · γlΓ∗, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10) where ∗ = A if l is even, or ∗ = B if l is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We observe that any two consecutive points in the above are adjacent vertices in T (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3), and the sequence does not backtrack, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', for any point in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10), the vertices on the left and right to it are different: For even i, let us examine the portion of the path γ1 · · · γi−1ΓB, γ1 · · · γiΓA, γ1 · · · γi+1ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Note that γi ∈ ΓB \\ H and γi+1 ∈ ΓA \\ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Applying (γ1 · · · γi)−1 to the above, we obtain ΓB = γ−1 i ΓB, ΓA, γi+1ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' However, since γi+1 ̸∈ ΓB, ΓB ̸= γi+1ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A similar analysis can be done when i is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, if we connect each pair of consecutive vertices in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10) by the unique edge in T determined by the pair, we obtain a geodesic path in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If γ ∈ Γ = ΓA ⋆H ΓB lies in the coset represented by v ∈ V (T), then |dT (v, ΓA) − rl(γ)| ≤ 1, |dT (v, ΓB) − rl(γ)| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This claim is easily checked when γ ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' So, suppose that γ ̸∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let γ1γ2 · · · γl be a normal form of γ such that γ1 ∈ ΓA \\ H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' the case γ1 ∈ ΓB \\ H follows by a similar argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Note that v is one of the two rightmost entries in the sequence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Moreover, the discussion above shows that the sequence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10) is a geodesic sequence of vertices in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, rl(γ) ≥ dT (v, ΓA) ≥ rl(γ) − 1 and rl(γ) + 1 ≥ dT (v, ΓB) ≥ rl(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Similar discussion holds in the case of HNN extensions: For γ ∈ Γ = M⋆φ, let γ = µ0f ǫ1µ1 · · · f ǫnµn be a normal form of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The above normal form produces a finite sequence in T: M, µ0f ǫ1M, µ0f ǫ1µ1f ǫ2M, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' , γM = µ0f ǫ1µ1 · · · f ǫnM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' COMBINATION THEOREM: AMALGAMS 15 Similarly to the amalgamated free product case discussed above, one can check that the above sequence does not backtrack and that consecutive vertices in the path are adjacent in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This yields the following: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For all γ ∈ Γ = M⋆φ, dT (v, M) = rl(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The proof is similar to that of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ We prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We discuss the proof in the case of amalgamated free products Γ = ΓA ⋆H ΓB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' the case of HNN extensions Γ = M⋆φ is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Consider a normal form of γ which contains some normal form of ω as a left subword: γ = γ1 · · · γm � �� � =ω γm+1 · · · γl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We further assume that γ1 ∈ ΓA \\ H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' the case γ1 ∈ ΓB \\ H follows similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The above normal form induces a geodesic path in T (see the paragraph before Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='19) from the vertex ΓB to the vertex vγ := γΓ∗, where ∗ = A if l is even, or ∗ = B if l is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This path also contains the vertex vω := ωΓ∗, where ∗ = A if m is even, or ∗ = B if m is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The vertex ΓA also lies in that path, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, any path in T connecting vγ to ΓA or ΓB must contain vω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For the rest of the proof, let ∗ denote either A or B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let c : [0, n] ∩ Z → Γ be a shortest geodesic in (Γ, d)1 such that c(0) ∈ Γ∗ and c(n) = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By definition, c(0) is a closest point in Γ∗ to c(k), for any k in the domain of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For k ∈ [0, n] ∩ Z, let ¯c(k) := {c(k)ΓA, c(k)ΓB}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' , n − 1, we have that ¯c(k) ∩ ¯c(k + 1) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We observe that c(k + 1) = c(k) · sk, for some generator sk ∈ S ⊂ ΓA ∪ ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let c(k) = ˜γ1 · · · ˜γr be a normal form of c(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Assume that ˜γr ∈ ΓA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' the other possibility ˜γr ∈ ΓB can be similarly treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If sk ∈ ΓA, then c(k)ΓA = ˜γ1 · · · ˜γr−1ΓA ∈ ¯c(k + 1) since c(k + 1) = ˜γ1 · · · ˜γr−1(˜γrsk) ∈ ˜γ1 · · · ˜γr−1ΓA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Similarly, if sk ∈ ΓB, then c(k)ΓB ∈ ¯c(k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Let V ′ = �n k=0 ¯c(k) ⊂ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Consider the induced subtree of T ′ ⊂ T determined by V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By the claim above, T ′ is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since c(0) ∈ Γ∗, we get that Γ∗ ∈ V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Obviously, vγ ∈ V ′ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since T ′ is connected, by the first paragraph of this proof, vω ∈ V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, there exists some k0 in the domain of c such that vω ∈ ¯c(k0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In other words, c(k0) lies in (the coset represented by) vω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since c(0) is also a closest point in Γ∗ to c(k0), it follows that c(0) is uniformly close to prΓ∗(vω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' However, since rl(ω) ≥ 3, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='19, dT (vω, Γ∗) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7),2 ∂∞Γ∗ ∩ ∂∞vω = ∅ and, hence, prΓ∗(vω) is bounded in Γ∗ (see Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This completes the proof of this result when dΓ = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In the general case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', when dΓ is an arbitrary word metric, then the result follows by the fact that the identity map (Γ, d) → (Γ, dΓ) is a quasiisometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ 1See Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='14 for our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 2Note that ∂∞Γvω = ∂∞vω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 16 SUBHADIP DEY AND MICHAEL KAPOVICH 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Preliminaries on discrete isometry groups of symmetric spaces We recall some preliminary facts on symmetric spaces, mainly to set up some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, we refer to [2, Appendix 5] for a quick discussion and to [7] for a more detailed exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let G be a real semisimple Lie group of noncompact type with a finite center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We impose some mild assumptions on G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' see Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let X = G/K denote the (globally) symmetric space of G, where K is a maximal compact subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, X is a nonpositively curved G-homogeneous space such that X has no compact or Euclidean de Rham factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The ideal boundary ∂∞X is the set of equivalence classes of asymptotic rays in X on which G acts naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The point stabilizers in G of the action G ↷ ∂∞X are called parabolic subgroups of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The ideal boundary ∂∞X carries a natural G-invariant spherical building structure, called the Tits building of X, and denoted by ∂TitsX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The top-dimensional simplices in ∂TitsX are called chambers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In this paper, we impose the following assumption on G (for simplicity, one may only assume that G is connected), which are standing assumptions in the papers of Kapovich– Leeb–Porti [12, 13, 14] we rely upon in this work: Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The group G has a finitely many connected components such that, for the associated symmetric space X = G/K, the spherical Tits building ∂TitsX is thick: That is, every panel (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', a codimension one simplex) in ∂TitsX is contained in at least three distinct chambers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Each chamber in ∂∞X is also a fundamental domain for the action G ↷ ∂∞X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The stabilizer in G of a chamber σ (which is the same as the stabilizer in G of any interior point of σ) is called a minimal parabolic subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The group G also acts on the set of chambers in ∂TitsX transitively, so any two minimal parabolic subgroups are conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, the space of all chambers in ∂TitsX can be identified with the analytic manifold Flag(σmod) := G/Pσmod, where σmod is a chosen chamber in ∂TitsX and Pσmod is the minimal parabolic subgroup stabilizing σmod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' More generally, for a face νmod ⊂ σmod, the space of simplices ν in ∂TitsX of type νmod, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', those simplices ν in ∂TitsX which can be brought to νmod by the action G ↷ ∂TitsX, can be identified with the analytic manifold Flag(νmod) := G/Pνmod, where Pνmod is the parabolic subgroup of G stabilizing νmod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A pair of simplices ν± in ∂TitsX is called antipodal if there is a complete geodesic line in X, which is forward (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' backward) asymptotic to some interior point of the simplex ν+ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' ν−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If ν± mod ⊂ σmod denote the types of ν±, then they satisfy ν+ mod = ι(ν− mod), where ι : σmod → σmod denotes the opposition involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For a simplex ν− in ∂TitsX of type ν− mod, the set of simplices in ∂TitsX (regarded as points in Flag(ν+ mod), where ν+ mod = ι(ν− mod)) COMBINATION THEOREM: AMALGAMS 17 antipodal to ν−, C(ν−) := {ν+ ∈ Flag(ν+ mod) | ν+ is antipodal to ν−} ⊂ Flag(ν+ mod), is an open dense Pν− mod-homogeneous subset of Flag(ν+ mod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2 (Antipodality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A pair (A, B), where A ⊂ Flag(ν+ mod) and B ⊂ Flag(ν− mod), is said to be antipodal to each other (or, A is antipodal to B) if, for all ν+ ∈ A and all ν− ∈ B, ν+ is antipodal to ν−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Regular sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let νmod ⊂ σmod be a face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A sequence (gn) in G is said to νmod-converge to a point ν ∈ Flag(νmod), which is denoted by gn flag −−→ ν, if every subsequence of (gn) has a further subsequence (gnk) such that there exists ν− ∈ Flag(ινmod) such that gnk|C(ν−) → ν, uniformly on compacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In this situation, ν ∈ Flag(νmod) is called a νmod-limit point of (gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' See [13, §4] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For a discrete subgroup Γ of G, the set of all νmod-limit points of Γ in Flag(νmod) is called the νmod-limit set of Γ, which is denoted by Λνmod(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The limit set Λνmod(Γ) is a Γ-invariant compact subset of Flag(νmod);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' however;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' it may be empty, even if Γ is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A sequence (gn) in G is νmod-regular if every subsequence of (gn) contains a νmod-flag- convergent subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Moreover, if (gn) is νmod-regular, then the inverse sequence (g−1 n ) is ινmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Similarly, a subgroup Γ of G is called νmod-regular if every sequence in Γ is νmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Such subgroups are necessarily discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Clearly, νmod-regular subgroups are also ινmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The following results help verify the regularity and flag-convergence of certain sequences considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let (gn) be a sequence in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If there exists a compact subset A ⊂ Flag(νmod) with a nonempty interior such that the sequence (gnA) of compact subsets of Flag(νmod) converges to a point ν ∈ Flag(νmod), then (gn) is νmod-regular and gn flag −−→ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' See [4, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let d be a distance function on Flag(νmod) compatible with the manifold topology of Flag(νmod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, we say that a sequence (An) of subsets of Flag(νmod) shrinks if the diameter of An converges to zero, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since Flag(νmod) is compact, this notion does not depend on the chosen distance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If there exists a compact subset A ⊂ Flag(νmod) with a nonempty interior such that (gnA) shrinks, then (gn) is νmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 18 SUBHADIP DEY AND MICHAEL KAPOVICH See [4, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If there exist compact subsets B, B′ ⊂ Flag(νmod) with nonempty interior such that B′ ⊂ B◦ and, for all n ∈ N, gn+1g−1 n (B) ⊂ B′, then (gn) is νmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The statement above can be extracted from the proof of [4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For complete- ness, let us prove this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' After replacing the sequence (gn) by (hn), where hn := gng−1 1 , we have that hn(B) ⊂ B and hn+1h−1 n (B) ⊂ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1) Suppose, to the contrary, that (gn) is not νmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then (hn) is also not νmod- regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, after extraction, (hn) is ηmod-pure for some face ηmod ⊂ σmod such that νmod ̸⊂ ηmod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By [12, Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5], after further extraction, there exist η+ ∈ Flag(ηmod) and η− ∈ Flag(ιηmod), and a surjective algebraic map φ : CFu(η−) → StFu(η+) such that hnk|CFu(η−) → φ uniformly on compacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1), we obtain that hnk ˜B ⊂ ˜B, where ˜B = � ν∈B StFu(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Consider any point η ∈ C(η−) ⊂ Flag(ηmod) such that StFu(η) ∩ ( ˜B)◦ is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3 By [4, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8], lim k→∞ max σ∈StFu(hnk η) d(hnkh−1 nk−1σ, σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2) Since, for all k ∈ N, hnk( ˜B)◦ ⊂ ( ˜B)◦, StFu(hnkη) ∩ ( ˜B)◦ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Also, by [4, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4], StFu(hnkη) ̸⊂ ˜B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since StFu(hnkη) is connected (by [4, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2]), StFu(hnkη) intersects ∂ ˜B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' let σk be any point lying in this intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2), lim k→∞ d(hnkh−1 nk−1σk, σk) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3) However, since hn+1h−1 n (B) ⊂ B′ (by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1)), hmh−1 n (B) ⊂ B′, for all m > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In particular, hnkh−1 nk−1(B) ⊂ B′, for all k ∈ N, which implies that hnkh−1 nk−1( ˜B) ⊂ ˜B′ := � ν∈B′ StFu(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since σk ∈ ˜B, by above, we get hnkh−1 nk−1(σk) ∈ ˜B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' However, ˜B′ ⊂ ˜B◦, which shows that d(hnkh−1 nk−1(σk), σk) ≥ d( ˜B′, ∂ ˜B) > 0, for all k ∈ N, contradicting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that (gn) is νmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If there exist compact subsets A, B ⊂ Flag(νmod) with nonempty interior such that, for all n ∈ N, gnB ⊂ A, then all the νmod- limit-points of (gn) lie in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 3Note that, since CFu(ν−) is open dense in Flag(σmod) and ˜B has nonempty interior in Flag(σmod), CFu(ν−) ∩ ˜B is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' COMBINATION THEOREM: AMALGAMS 19 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let ν ∈ Flag(νmod) be a νmod-limit point of (gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, there exists ν− ∈ Flag(ινmod) and a subsequence (gnk) such that gnk|C(ν−) converges to the constant map C(ν−) → {ν} uniformly on compacts, as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let ν1 ∈ C(ν−) ∩ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, the sequence (gnkν1), which lies in A, converges to ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, ν ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that gn flag −−→ ν ∈ Flag(νmod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let A− ⊂ Flag(ινmod) be a subset containing all the ινmod-limit points of (g−1 n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If A ⊂ Flag(νmod) is any compact subset antipodal to A−, then (gnA) converges to {ν}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, there exists a sequence (νk) in A and a subsequence (gnk) of (gn) such that (gnkνk) converges to some point ν′ ∈ Flag(νmod) different from ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' After further extraction of (gnk), we may assume that (g−1 nk ) ινmod-converges to some point ν− ∈ A−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since, by hypothesis, A ⊂ C(ν−), by [14, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='18], we get that gnk(A) → ν, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since, for each k, νk ∈ A, we also obtain that gnkνk → ν, which implies that ν′ = ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ The following result follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that (gn) is νmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let A− ⊂ Flag(ινmod) be a subset contain- ing all the ινmod-limit points of (g−1 n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If A ⊂ Flag(νmod) is any compact subset antipodal to A−, then (gnA) shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Anosov subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We fix once and for all an ι-invariant face τmod of σmod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We focus on the special class of discrete subgroups of G, called τmod-Anosov subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This class of subgroups has several different characterizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For our purpose, we use the following characterization of τmod-Anosov subgroups: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9 (Asymptotically embedded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A subgroup Γ of G is τmod-asymptotically embedded if (i) Γ is a τmod-regular subgroup of G, (ii) Γ, as an abstract group, is hyperbolic, and (iii) there exists a Γ-equivariant antipodal map4 ξ : ∂∞Γ → Flag(τmod), which preserves the convergence dynamics: That is, for every sequence (γn) in Γ and every point ε ∈ ∂∞Γ, if γn Cay −−→ ε, then γn flag −−→ ξ(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (i) The image of the map ξ in the above definition is precisely Λτmod(Γ), the limit set of Γ in Flag(τmod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (ii) The above definition of τmod-asymptotically embedded subgroups, which is a minor variation of the one in [13, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12], is more straightforward to verify in this paper due to a certain “ping-pong” type arguments we use here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The equivalence between these two definitions can be checked as follows (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [4, §5]): Suppose that Γ < G is τmod-asymptotically embedded in the sense of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We first verify that ξ is an embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Indeed, since ∂∞Γ is a compact and Flag(τmod) is Hausdorff, it is enough to establish that ξ is an injective continuous map: Since ξ 4That is, ξ maps every distinct pair of points to a pair of antipodal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 20 SUBHADIP DEY AND MICHAEL KAPOVICH is an antipodal map, ξ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' To verify continuity, pick x ∈ X and consider the map φ : Γ → ¯Xτmod = X ⊔ Flag(τmod), whose restriction to ∂∞Γ is ξ and whose restriction to Γ is the orbit map γ �→ γ·x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The space X ⊔ Flag(τmod) is topologized with the topology of τmod-flag-convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The restriction of this topology to X or Flag(τmod) coincides with their respective manifold topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4), φ is continuous and, hence, so is the restriction ξ = φ|∂∞Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, ξ is a Γ-equivariant homeomorphism between ∂∞Γ and its image, Λτmod(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Finally, since ξ is an antipodal map, Λτmod(Γ) is an antipodal subset of Flag(τmod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, Γ is τmod-antipodal in the sense of [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Hence, Γ is τmod-asymptotically embedded in the sense of [13, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The other direction is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Interactive pairs and triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let τmod be an ι-invariant face of σmod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Following Maskit [28], we define the notion of “interactive pairs” and “interactive triples.” 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Interactive pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let ΓA and ΓB be discrete subgroups of G, and let H := ΓA ∩ ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We denote this data by the triple (ΓA, ΓB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A B (ΓB \\ H) (ΓA \\ H) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' An interactive pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='11 (Interactive pair).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A pair (A, B) of compact subsets of Flag(τmod) = G/Pτmod is called an interactive pair for (ΓA, ΓB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' H) if the following conditions are satisfied: (i) The interiors A◦ of A and B◦ of B are nonempty and disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (ii) H leaves the sets A and B precisely invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', HA = A, HB = B, and, for all elements α ∈ ΓA \\ H and β ∈ ΓB \\ H, we have that αB ⊂ A◦ and βA ⊂ B◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' See Figure 4 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If (ΓA, ΓB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' H) admits an interactive pair (A, B) in Flag(τmod), then the natural homomorphism ρ : ΓA ⋆H ΓB → ⟨ΓA, ΓB⟩ < G from the abstract amalgamated free product ΓA ⋆H ΓB to G in injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In particular, the subgroup Γ := ⟨ΓA, ΓB⟩ of G is naturally isomorphic to ΓA ⋆H ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' COMBINATION THEOREM: AMALGAMS 21 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let γ ∈ ΓA⋆HΓB be any nontrivial element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We want to show that ρ(γ) is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Clearly, if γ ∈ (ΓA ∪ ΓB) \\ {1Γ}, then ρ(γ) = γ and, hence, ρ(γ) is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' So, we may assume that γ ̸∈ ΓA ∪ ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Choose a normal form γ = γ1 · · · γl of γ (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since γ ̸∈ ΓA ∪ ΓB, we have l ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Assume that γl ∈ ΓB (the other possibility that γl ∈ ΓA can be similarly analyzed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, ρ(γ)A ⊂ A◦ ∪ B◦ and, in particular, ρ(γ)A ̸= A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, ρ(γ) is a nontrivial element of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Compare with [28, Theorem VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' With a little more effort, one can also show that the image of the homomor- phism ρ in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12 is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' However, we do not need this result to prove our main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Interactive triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let M be a discrete subgroup of G, and let f ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Define H− := (f −1Mf) ∩ M, H+ := M ∩ (fMf −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Clearly, fH−f −1 = H+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We denote this data by the quadruple (M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' H±;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A B+ B− f −1 f (M \\ H+) (M \\ H−) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' An interactive triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='14 (Interactive triples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A triple (A, B±) of compact subsets of Flag(τmod) is called an interactive triple for (M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' H±;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' f) if the following conditions are met: (i) The interiors A◦, B◦ −, B◦ + of A, B−, B+, respectively, are nonempty and pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Furthermore, B− ∩ B+ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (ii) H± leaves B± precisely invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', H±B± = B± and µ(B±) ⊂ A◦, whenever µ ̸∈ H±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (iii) f ±1(A) ⊂ B± and f ±1B± ⊂ B◦ ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' See Figure 5 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If (M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' H±;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' f) admits an interactive triple (A, B±) in Flag(τmod), then the natural homomorphism M⋆φ → G is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In particular, the subgroup Γ := ⟨M, f⟩ of G is naturally isomorphic to the HNN extension of M by the isomorphism φ : H− → H+ is given by φ(η) = fηf −1, for all η ∈ H−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 22 SUBHADIP DEY AND MICHAEL KAPOVICH Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The proof of this result is similar to the one of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Hence, we omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Compare with [28, Theorem VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A combination theorem for amalgamated free products of Anosov subgroups The goal of this section is to prove Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In this section, we work under the hypothesis of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' To simplify our situation, we may replace A and B by the closure of their respective interiors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' It is easy to see that (cl(A◦), cl(B◦)) is still an interactive pair for (ΓA, ΓB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' However, the main advantage of the interactive pair (cl(A◦), cl(B◦)) is a stronger antipodality hypothesis: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' cl(A◦), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' cl(B◦), is antipodal to B◦, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let τ+ ∈ cl(A◦) and τ− ∈ B◦ be any points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We show that τ± are antipodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Fix an auxiliary distance function d on Flag(τmod) compatible with its manifold topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For τ ∈ Flag(τmod), let E(τ) denote the (compact) subset of Flag(τmod) consisting of all points which are not antipodal to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let (τn) be a sequence in A◦ converging to τ+ and let Bǫ(τ−) be a closed metric ball in Flag(τmod) centered at τ− of radius ǫ > 0 small enough such that Bǫ(τ−) ⊂ B◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By hypothesis, τn and B◦ are antipodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Hence, d(E(τn), Bǫ(τ−)) > 0, implying that d(E(τn), τ−) ≥ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since the Hausdorff distance between E(τn) and E(τ+) converges to zero as n → ∞, it follows that d(E(τ+), τ−) ≥ ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' So, τ− ̸∈ E(τ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' After replacing (A, B) by (cl(A◦), cl(B◦)) in the hypothesis of Theo- rem A, in place of (i), we may assume the following: (i)’ The pairs of subsets (A, B◦) and (A◦, B) of Flag(τmod) are antipodal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This replacement does not affect the conclusion of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1, we will take additional advantage given by hypothesis (i)’ above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' see, for instance, the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Under the hypothesis of Theorem A Λτmod(ΓA) ⊂ A, Λτmod(ΓB) ⊂ B, and Λτmod(H) ⊂ A ∩ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' It will be enough to show that Λτmod(ΓA) ⊂ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If H = ΓA, then ΓA preserves A, and since A has a nonempty interior, all τmod-limit points of Γ must lie in A (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' So, we can assume that H is a proper subgroup of ΓA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', there exists some element α ∈ ΓA which is not an element of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For any point τ+ ∈ Λτmod(ΓA), consider a sequence (αn) in ΓA such that αn flag −−→ τ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We may (and will) also assume that no elements of (αn) lie in H: Indeed, for every n ∈ N, if αn ∈ H, then replace the n-th entry αn in (αn) by αnα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' After replacing all such entries, the resulting sequence, again denoted by (αn), does not share any elements with H but still satisfies αn flag −−→ τ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since for all n ∈ N, αnB ⊂ A, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6 implies that τ+ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ COMBINATION THEOREM: AMALGAMS 23 Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Under the hypothesis Theorem A, the subgroup H is weakly malnormal in both ΓA and ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since Λτmod(H) ⊂ A∩B, the interactive pair assumption implies that for all α ∈ ΓA\\H and β ∈ ΓB \\ H, α(Λτmod(H)) ∩ Λτmod(H) = β(Λτmod(H)) ∩ Λτmod(H) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If, say, αHα−1 ∩ H were infinite, it would contain an infinite order element η ∈ H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' hence, α−1(Λτmod(⟨η⟩)) = Λτmod(⟨α−1ηα⟩) would be nonempty subsets of Λτmod(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' That would be a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Under the hypothesis Theorem A, the subgroup Γ of G generated by ΓA and ΓB in G is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If H = ΓA ∩ ΓB is quasiconvex in one of ΓA or ΓB, then it is quasiconvex in both of them: This follows from the general fact that, if Γ is a τmod-Anosov subgroup of G and H < Γ is a subgroup, then H is quasiconvex in Γ if and only if H is a τmod-Anosov subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This general fact is a consequence of the τmod-URU characterization of τmod-Anosov subgroups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' see [13, Equivalence Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1 & Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2(i)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, under the hypothesis Theorem A, H is quasiconvex in ΓA and ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Moreover, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12, ⟨ΓA, ΓB⟩ is naturally isomorphic to ΓA ⋆H ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, together with Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10(i) implies that ⟨ΓA, ΓB⟩ is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The main result of this subsection is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Under the hypothesis of Theorem A, the subgroup Γ = ⟨ΓA, ΓB⟩ of G is τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' See §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4 for the proof of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We remind our reader that, the subgroup Γ = ⟨ΓA, ΓB⟩ of G is naturally isomorphic to ΓA ⋆H ΓB (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3) and is hyperbolic (see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The general strategy to prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6 is similar to the proof of [4, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' However, note that, in [4], we worked under a stronger hypothesis on the interactive pairs (A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Namely, we required them to be antipodal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' When A and B have a nontrivial intersection, which is the situation considered in this paper, difficulties arise in controlling the dynamics near the intersection A ∩ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' To show that Γ is τmod-regular, we proceed by showing that every sequence of Γ contains a τmod-regular subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let (γn) be an arbitrary sequence in Γ consisting of pairwise distinct elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' After passing to a subsequence, one of the following alternatives must hold: Either (γn) is a sequence in H or, for all n ∈ N, γn ̸∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In the former case, (γn) is clearly τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' So, for the rest of this subsection, we assume the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Under this assumption, since no elements of (γn) belong to H, rl(γn) ≥ 1, for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let us choose a normal form for each element in the sequence (γn), γn = βn,lnαn,lnβn,ln−1αn,ln−1 · · · βn,1αn,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1) To simplify our notations, denote βn := βn,ln and αn := αn,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 24 SUBHADIP DEY AND MICHAEL KAPOVICH Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let us further assume that βn and αn, the leftmost and rightmost letters in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1) are in ΓB \\ H and ΓA \\ H, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This extra assumption can be arranged by changing the original sequence by a bounded amount;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' we note that τmod-regularity of sequences remains unaffected under such perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Sequences with bounded relative lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let (βn) be any sequence in ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, there exists a sequence (ηn) in H such that the sequence (ˆβ−1 n ), where ˆβn := βnηn, has no flag-limit points in Λτmod(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This follows directly from the first part of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8 and the τmod-Anosov property of ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that the leftmost letter sequence (βn) in ΓB corresponding to the sequence (γn) in Γ has the following property: The inverse sequence (β−1 n ) diverges away from H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, (γn) is τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Similarly, if the rightmost letter sequence (αn) diverges away from H, then (γn) is τmod- regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since (β−1 n ) diverges away from H, it is an unbounded sequence in ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Hence (β−1 n ) is τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' After extraction, suppose that β±1 n flag −−→ τ± ∈ Λτmod(ΓB) ⊂ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We may (and will) assume that τ− ̸∈ Λτmod(H): Indeed, for every sequence (ηn) in H, we have the freedom to adjust the normal form in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1) by γn = (βn,lnηn)(η−1 n αn,ln)βn,ln−1αn,ln−1 · · · βn,1αn,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2) Applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9, we can arrange for a normal form of the elements of (γn) such that the inverse of the leftmost letter sequence, (β−1 n ), does not accumulate in Λτmod(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Note that, after the above adjustments, (β−1 n ) still diverges away from H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In particular, any accumulation points of (β−1 n ) lie in Λτmod(ΓB) \\ Λτmod(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since τ− ̸∈ Λτmod(H), by the antipodality hypothesis (ii) of Theorem A, A is a compact subset of C(τ−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Hence, βnA flag −−→ τ+ as n → ∞ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Note that for all n ∈ N, γnB ⊂ βnA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, γnB flag −−→ τ+, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3, (γn) is τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If supn{rl(γn)} < ∞, then (γn) is τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Using induction on the relative length and applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10, the result follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' see the proof of [4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3] for a similar argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Special sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Next, let us analyze a special type of sequence in Γ: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12 (Special sequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A sequence (˜γn) in Γ = ΓA ⋆H ΓB is special if there exist sequences (αn) in ΓA \\ H, (βn) in ΓB \\ H, and an increasing function F : N → N such that ˜γn = βF (n)αF (n)βF (n)−1αF (n)−1 · · · β1α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Special sequences in Γ are τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' COMBINATION THEOREM: AMALGAMS 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Consider any special sequence (˜γn) in Γ, and assume that the normal form of ˜γn is given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Consider any subsequence of (˜γn), again denoted by (˜γn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If the inverse sequence (β−1 F (n)) corresponding to the leftmost letter sequence diverges away from H, then (˜γn) is τmod-regular (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Otherwise, after passing to a further subsequence (˜γnk) of (˜γn) we can (and will) assume that, there exists an element ˜β ∈ ΓB \\ H and a sequence (ηk) in H such that βF (nk) = ˜βηk, ∀n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4) Set B′ = ˜β(A) ⊂ B◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Note that, for all k ∈ N, ˜γnk+1˜γ−1 nk B ⊂ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' So, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5, (˜γnk) is τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, every subsequence of the original sequence contains a τmod-regular subse- quence, showing that the original sequence is τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Alternating sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Recall the notion of alternating sequences from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='13 to (ω−1 n ), we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Alternating sequences in Γ = ΓA ⋆H ΓB are τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If (ωn) is a type A alternating sequence in Γ = ΓA ⋆H ΓB, then the nested sequence of compact subsets of Flag(τmod), ω1B ⊃ ω2A ⊃ ω3B ⊃ ω4A ⊃ · · · , converges to a point τ ∈ Flag(τmod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Similarly, if (ωn) is a type B alternating sequence in Γ, then the nested sequence of compact subsets of Flag(τmod), ω1A ⊃ ω2B ⊃ ω3A ⊃ ω4B ⊃ · · · , converges to a point τ ∈ Flag(τmod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For a (type A or B) alternating sequence ω = (ωn), let us denote the point τ ∈ Flag(τmod) obtained as a limit in the above by τω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' As a direct corollary of the above and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3, we obtain the following: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If ω = (ωn) is an alternating sequence in Γ = ΓA ⋆H ΓB, then, for all γ ∈ Γ, γωn flag −−→ γτω, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let us assume that (ωn) is of type A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' the other possibility can be similarly treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let (βn) denote the rightmost letter sequence corresponding to (ω2n), see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Consider the special sequence (ω−1 2n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We first show that there exists a sequence (ηn) in H such that (ˆω−1 2n ), where ˆω2n = ω2nη−1 n , has a subsequence that is τmod-regular and τmod-flag-converges to some point τ− antipodal to A: By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8, there exist sequences (ηn) and (ˆηn) in H such that both the sequences (ˆβn) and (ˆβ−1 n ) accumulate outside ∂∞H in ¯ΓB = ΓB ⊔ ∂∞ΓB, where ˆβn = ˆηnβnηn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Note that (ˆω2n), where ˆω2n = ω2nη−1 n , is still alternating5 and, hence, by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='14, is a τmod-regular sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 5Indeed, the corresponding sequences in ΓA and ΓB for (ˆωn) can be taken to be (ηn−1αn) and (βnη−1 n ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 26 SUBHADIP DEY AND MICHAEL KAPOVICH Passing to a subsequence, (ˆβnk) is either a constant sequence, ˆβ ∈ ΓB \\ H, or ˆβ±1 nk flag −−→ τ± ∈ Λτmod(ΓB) \\ Λτmod(H), as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5) If the former holds, then notice that ˆω−1 2nkB ⊂ ˆβ−1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In this case, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6, all the τmod-limit points of (ˆω−1 2nk) must lie in ˆβ−1A ⊂ B◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2, ˆβ−1A is antipodal to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If the latter holds, then we observe that ˆω−1 2nk(B) ⊂ ˆβ−1 nk (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, the sequence (ˆβ−1 nk A), and hence (ˆω−1 2nkB), converges to the τmod-limit point τ− (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5)) of the sequence ˆβ−1 nk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By the antipodality assumption (ii) in the hypothesis of Theorem A, τ− is antipodal to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let (ηn) be a sequence in H as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By the above two paragraphs, we can (and will) extract a subsequence (ˆω2kn) of (ˆω2n), where ˆω2n = ω2nη−1 n , such that the inverse sequence (ˆω−1 2kn) is τmod-flag-converges to some point τ−, which is antipodal to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' After further extraction, we may assume that (ˆω2kn) τmod-flag-converges to some point τ ∈ Flag(τmod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since A ⊂ C(τ−) is compact, the sequence of subsets (ˆω2knA) converges to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' However, for all n ∈ N, ˆω2nA = ω2nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, the nestedness of the subsets implies limn→∞ ω2nA = {τ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Again, we also obtain limn→∞ ω2n−1(B) = {τ} by nestedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that (γn) is a sequence in Γ without repeated en- tries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If the relative length of elements in (γn) is uniformly bounded, then, by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='11, (γn) is τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' So, we only need to study sequences in Γ with unbounded relative lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We show that such sequences are also τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, let (γn) be such a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We assume to the contrary that (γn) is τmod-irregular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', it does not contain any τmod-regular subsequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, we construct subsequences (ˆγn) and (˜γn) in Γ such that the following hold: (i) (˜γ−1 n ) is alternating, (ii) (ˆγn) is a subsequence of (γn), (iii) for each n ∈ N, there exist normal forms of ˆγn such that ˜γn is a rightmost subword of ˆγn, and (iv) the leftmost letters of the sequence (ˆγn) in the chosen normal forms all come from the same group, say ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Choose some normal forms of elements of (γn) as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1) and assume that the right- most/leftmost letters, αn and βn, respectively, of γn are both nontrivial (Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, (αn) stays within a bounded distance away from H, because, otherwise, by the second part of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10, (γn) would contain a τmod-regular subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Hence, after extrac- tion of (γn), the rightmost letters of γn come from a single right coset of H, say Hˆα1, in ΓA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Set ˜γ1 = ˆα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Applying a similar argument to (γnˆα−1 1 ) and, after further extraction, we obtain that the rightmost letters of (γnˆα−1 1 ) all come from a single right coset of H, say Hˆβ1 in ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Set ˜γ2 = ˆβ1 ˆα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proceeding inductively, we obtain a special sequence (˜γn) such that (˜γ−1 n ) is alternating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By construction, the original sequence τmod-irregular sequence (γn) corresponds to a pair of sequences (ˆγn) and (˜γn) with the properties described in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Note that ˆγ−1 2n A ⊂ ˜γ−1 2n A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15, the sequence of subsets (˜γ−1 2n A) of Flag(τmod) converges to a point, (ˆγ−1 2n A) also converges to that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, (ˆγ−1 2n ) and, hence, (ˆγ2n) are τmod-regular (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' However, since (ˆγ2n) is a subsequence of the τmod-irregular sequence (γn), we arrive at a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ COMBINATION THEOREM: AMALGAMS 27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The boundary map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Recall from §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7 that there is a Γ-invariant decomposition of ∂∞Γ as the disjoint union ∂IΓ⊔∂IIΓ, where ∂IIΓ admits an equivariant continuous bijection to the Gromov boundary of the Bass-Serre tree T associated with the amalgamated free product Γ = ΓA ⋆H ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In this subsection, we construct a Γ-equivariant boundary map from the Gromov boundary ∂∞Γ of Γ = ΓA ⋆H ΓB to Flag(τmod), ξ : ∂∞Γ → Flag(τmod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6) We define this map ξ separately on ∂IΓ and ∂IIΓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' see §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1 for the definition of ξ|∂IΓ and §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2 for the definition of ξ|∂IIΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3, we verify that ξ is an antipodal map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Finally, in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4, we show that the map ξ is dynamics preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Definition of the boundary map for type I points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For ε ∈ ∂IΓ, pick γ ∈ Γ such that γ−1ε ∈ ∂∞ΓA ∪ ∂∞ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If γ−1ε ∈ ∂∞ΓA (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' γ−1ε ∈ ∂∞ΓB), then define ξ(ε) := γξA(γ−1ε), (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' ξ(ε) := γξB(γ−1ε)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7) We first show that the map ξ : ∂IΓ → Flag(τmod) is well-defined: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For γ ∈ Γ, if γ(∂∞ΓA) ∩ ∂∞ΓA ̸= ∅, then γ ∈ ΓA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The same conclusion holds when A is replaced by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Note that γ(∂∞ΓA) = ∂∞(γΓAγ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7), the nonemptyness of γ(∂∞ΓA)∩∂∞ΓA implies that dT (γΓA, ΓA) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, γ ∈ ΓA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='17, the element γ in the definition of ξ is unique up to the right multiplication by elements of ΓA (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' ΓB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since the maps ξA, ξB are equivariant for ΓA, ΓB, respectively, it follows that the definition of ξ(ε) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7) does not depend on the choice of γ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Finally, note that, by definition, we have the following result: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The map ξ : ∂IΓ → Flag(τmod) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7) is Γ-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Definition of the boundary map for type II points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For ε ∈ ∂IIΓ, we choose an alter- nating sequence (ωn) given by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12 such that ωn Cay −−→ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If (ωn) is of type A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' type B), then define ξ(ε) := lim n→∞ ωnA (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' ξ(ε) := lim n→∞ ωnB) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A consequence of the following result is that ξ is well-defined on ∂IIΓ: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For ε ∈ ∂IIΓ and a sequence (γn) in Γ, if γn Cay −−→ ε, then γn flag −−→ ξ(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let (ωn) be an alternating sequence as above, which we suppose to be of type A (the type B case can be dealt with similarly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12, there exists a function F : N → N diverging to infinity and n0 ∈ N such that, for all n ≥ n0, we may (and will) choose a normal form of γn containing ωF (n) as a left subword of that form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We split (γn)n≥n0 into two subsequences: The first subsequence (γkn) contains all el- ements of (γn)n≥n0 with the rightmost letter contained in ΓA, and the complementary 28 SUBHADIP DEY AND MICHAEL KAPOVICH subsequence (γln) includes all elements of (γn)n≥n0 with rightmost letter contained in ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Notice that, for all n ∈ N, γknB ⊂ ωF (kn)A and γlnA ⊂ ωF (ln)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since ωnA → ξ(ε), we obtain that both sequences (γknB) and (γlnA) of subsets of Flag(τmod) converge to ξ(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3, γkn flag −−→ ξ(ε) and γln flag −−→ ξ(ε), yielding the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Together with Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='16, the above result implies the following one: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The map in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8) is Γ-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The boundary map is antipodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The map ξ : ∂∞Γ → Flag(τmod) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6) (obtained by combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8)) is antipodal: That is for every pair of distinct points ε± ∈ ∂∞Γ, the points τ± := ξ(ε±) ∈ Flag(τmod) are antipodal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We recall that the action G ↷ G/P preserves antipodality: That is, if ˆτ± ∈ G/P is an antipodal pair, then, for all g ∈ G, gˆτ± is also an antipodal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let v, w be any vertices in the Bass-Serre tree T such that dT (v, w) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, ξ(∂∞Γv) = Λτmod(Γv) is antipodal to ξ(∂∞Γw) = Λτmod(Γw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By by equivariance, it is enough to assume that w = ΓA or w = ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We assume the former, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', w = ΓA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' the possibility w = ΓB can be analyzed similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since dT (v, w) ≥ 2, we have that ∂∞ΓA ∩ ∂∞Γv = ∅ (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose first that v is a coset of ΓB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' so let γ ∈ Γ be any element such that γΓB = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' It is easy to see that such an element γ must satisfy rl(γ) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We may also choose γ so that it has a normal form whose rightmost letter lies in ΓA \\ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If the leftmost letter α of that normal form also lies in ΓA \\ H, then rl(γ) ≥ 3, and Λτmod(Γα−1v) ⊂ βα′A ⊂ B◦, where β and α′ are the second and third letters from the left in that normal form of γ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since A and B◦ are antipodal to each other (see Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2), and Λτmod(ΓA) ⊂ A, we have that Λτmod(ΓA) and Λτmod(Γα−1v) are antipodal to each other, hence so is the pair Λτmod(ΓA) = αΛτmod(ΓA) and Λτmod(Γv) = αΛτmod(Γα−1v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If the leftmost letter of the normal form of γ is some element β ∈ ΓB \\ H, then, by a similar argument, it follows that Λτmod(Γv) ⊂ B◦, which is antipodal to Λτmod(ΓA) ⊂ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose now that v is a coset of ΓA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In this case, we can choose an element γ ∈ Γ such that γΓA = v, rl(γ) ≥ 1, and the rightmost letter of a normal form of γ is an element of ΓB \\ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Adapting a similar argument as above, the result follows in this case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Combining the following cases, it would follow that the map ξ : ∂∞Γ → Flag(τmod) is antipodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Recall that ξ is Γ-equivariant (by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='18 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that both points ε± ∈ ∂∞Γ are of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since ξ is Γ-equivariant, it is enough to assume that ε− ∈ ∂∞ΓA ∪ ∂∞ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let us also assume that ε− ∈ ∂∞ΓA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' the case ε− ∈ ∂∞ΓB can be treated similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' COMBINATION THEOREM: AMALGAMS 29 If ε+ ∈ ∂∞ΓA ∪ (ΓA(∂∞ΓB)), then finding a suitable element α ∈ ΓA, we obtain that αε+ ∈ ∂∞ΓA ∪ ∂∞ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since, by the hypothesis of Theorem A, ξ(∂∞ΓA ∪ ∂∞ΓB) ⊂ Flag(τmod) is an antipodal subset, ξ(αε±) is an antipodal pair and, hence, so is ξ(ε±).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If ε+ ̸∈ ∂∞ΓA ∪ (ΓA(∂∞ΓB)), then ε+ lies in the boundary of a vertex group Γv of the Bass-Serre tree T such that dT (ΓA, v) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='22, ξ(ε±) are antipodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that both points ε± ∈ ∂∞Γ are of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Consider a pair of alternating sequences (ω± n ) converging (in Γ) to ε± (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' There exists n ∈ N such that ω+ n ̸∈ ω− n H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If this is false, then for all n ∈ N, ω+ n ∈ ω− n H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Consider the sequence (ˆω−1 n ), where ˆωn := ω− n prH((ω− n )−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8, (ˆω−1 n ) has no accumulation points in ∂∞H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Moreover, (ω− n ) diverges away from H (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, (ˆωn(∂∞H)) converges to ε−, see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' It follows that the sequence of uniformly quasiconvex subsets (ˆωn(H)) of Γ converges to ε−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since we have assumed that ω+ n ∈ ω− n H = ˆωnH, we obtain that ω+ n Cay −−→ ε−, which shows that ε+ = ε−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Let n0 ∈ N be the smallest number such that ω+ n0 ̸∈ ω− n0H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Consider the alternating sequences ((ω+ n0)−1ω+ n ) and ((ω+ n0)−1ω− n ) converging to (ω+ n0)−1ε+ and (ω+ n0)−1ε−, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By choice of n0, it is evident that the first element of those sequences lie in different groups ΓA and ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, one of the points ξ((ω+ n0)−1ε±) lies in the interior of A while the other one lies in the interior of B (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15) and thus they are antipodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Consequently, ξ(ε±) are antipodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that ε− ∈ ∂∞Γ is of type I and ε+ ∈ ∂∞Γ is of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By the same argument as in the third case in [4, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2], it follows that ξ(ε±) are antipodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The boundary map preserves convergence dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The following result is an analog of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='19 for type I boundary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let ε ∈ ∂IΓ and let (γn) be a sequence in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If γn Cay −−→ ε, then γn flag −−→ ξ(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Using the equivariance of the Γ-action, it will be enough to prove the proposition when ε ∈ ∂∞ΓA ∪ ∂∞ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We argue by contradiction: Suppose that there exists ε ∈ ∂∞ΓA ∪ ∂∞ΓB and a sequence (γn) in Γ such that γn Cay −−→ ε, but γn flag −−→ τ ̸= ξ(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9) By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='17, let us choose D-normal forms for each element of (γn), for some D ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' After extraction, we may assume that the leftmost and rightmost letters of those forms come from the same group, say ΓA and Γ∗, respectively, where ∗ is either A or B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' let (αn) be the sequence of those leftmost letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We also assume that ∗ = A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' the other choice can be similarly analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Clearly, (α−1 n ) cannot diverge away from H: Otherwise, since (αn) fellow travels (γn), αn Cay −−→ ε and if (α−1 n ) diverges away from H, then γnB ⊂ αnB → τ ′, 30 SUBHADIP DEY AND MICHAEL KAPOVICH where τ ′ = ξ(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' But, in this case, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3 shows that γn flag −−→ τ ′, a disagreement with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, after passing to a subsequence, we may assume that the elements of (γn) have normal forms with a common leftmost letter α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Repeating the same argument to the sequence (α−1 1 γn) yields another subsequence whose elements have normal forms with a common leftmost letter β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, the original sequence (γn) has a subsequence whose elements have normal forms with two leftmost common letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proceeding inductively, for every l ∈ N, we can find a subsequence (γ′ n) of the original sequence (γn) such that the elements of (γ′ n) have normal forms with a common leftmost subword of length at least l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For l = 3, the Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='13 shows that (γ′ n) has bounded nearest-point projections to ΓA and ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, (γ′ n) cannot have any accumulation points in the boundary of ΓA and ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This contradicts our initial assumption that ε ∈ ∂∞ΓA ∪ ∂∞ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Combining Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='19 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='24, we obtain the following: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The map Γ-equivariant map given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6) preserves the convergence dynamics: That is, for any sequence (γn) in Γ and any point ε ∈ ∂∞Γ, if γn Cay −−→ ε, then γn flag −−→ ξ(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12, the subgroup Γ = ⟨ΓA, ΓB⟩ of G is nat- urally isomorphic to ΓA ⋆H ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We show that Γ is a τmod-Anosov subgroup or, equivalently, a τmod-asymptotically embedded subgroup (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9) of G: (i) By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6, Γ is a τmod-regular subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (ii) By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5, Γ is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (iii) Finally, the boundary map ξ : ∂∞Γ → Flag(τmod) in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6) is Γ-equivariant (by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='18 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='20), antipodal (by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='21), and preserves convergence dynamics (by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This concludes the proof of the Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A combination theorem for HNN extensions of Anosov subgroups The goal of this section is to prove Theorem B and, throughout this section, we work under the hypothesis of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In the proof of Theorem B, we replace (A, B±) by (cl(A◦), cl(B◦ ±)), which is again an interactive triple for (M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' H±;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In doing so, we may replace the condition (i) in the hypothesis of Theorem B by the following stronger one (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1): (i)’ The pairs of subsets (A, B◦ ±), (A◦, B±) of Flag(τmod) are antipodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Moreover, B− is antipodal to B+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We begin the proof of Theorem B with the following observation: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Under the hypothesis of Theorem B, (i) Λτmod(⟨f⟩) consists of two points, one of them lies in the interior of B+ and the other one lies in the interior of B−, and ⟨f⟩ is a cyclic τmod-Anosov subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (ii) Λτmod(M) ⊂ A and Λτmod(H±) ⊂ A ∩ B±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' COMBINATION THEOREM: AMALGAMS 31 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since, for all n ∈ N, f n+1f −n(B+) = f(B+) ⊂ B◦ + (by the second condition of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='14), applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5, it follows that ⟨f⟩ is τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since, for all n ∈ N, f −n(f −1B−) ⊂ f −1B− ⊂ B◦ −, all the τmod-limit points of the sequence (f −n)n∈N lie in B◦ −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since B+ is antipodal to B◦ − (see Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1), by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8, (f nB+)n∈N shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, since the sequence (f nB+)n∈N is nested, (f nB+)n∈N must converge to some point τ+ ∈ B◦ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Similarly, the nested sequence (f −nB−)n∈N of compact subsets of Flag(τmod) converges to some point τ− ∈ B◦ −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In particular, the limit set of ⟨f⟩ is {τ±}, is antipodal, and has cardinality two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' That ⟨f⟩ is τmod-Anosov follows from [13, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This proves (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof (ii) is similar to that of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Hence, we omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Under the hypothesis Theorem B, the subgroups H± are weakly malnormal in M, and, for all µ ∈ M, the intersection H− ∩ µH+µ−1 is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The proof is similar to the one of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' we omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Under the hypothesis Theorem B, the subgroup Γ of G generated by M and f in G is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Arguing similarly to the first paragraph of the proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5, it follows that H± are both quasiconvex in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, the claim follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10(ii), Propo- sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15, and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Compare this with the second paragraph of the proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ For convenience, we introduce the following notation, which are frequently used in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Notation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If ǫ = 1, then Hǫ will denote H+ and Bǫ will denote B+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Similarly, if ǫ = −1, then Hǫ will denote H− and Bǫ will denote B−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The main result of this subsection is as follows: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Under the hypothesis of Theorem B, the subgroup Γ = ⟨M, f⟩ of G is τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' See §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3 for the proof of this proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Special sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7 (Special sequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A sequence (˜γn) in Γ = M⋆φ is called special if there exist sequences (ǫn) in {±1}, (µn) in M satisfying ǫn = 1, µn ∈ H+ =⇒ ǫn+1 = 1 and ǫn = −1, µn ∈ H− =⇒ ǫn+1 = −1, and an element µ0 ∈ M, and an increasing function F : N → N such that, for all n ∈ N, ˜γn = f ǫF (n)µF (n)−1f ǫF (n)−1 · · · µ1f ǫ1µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1) Note that special sequences are subsequences of the inverse sequence of some alternating sequence in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Compare this with Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Special sequences in Γ are τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 32 SUBHADIP DEY AND MICHAEL KAPOVICH Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let (˜γn) be a special sequence as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We will assume that µ0 = 1M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' such a change is a bounded perturbation of the original sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Hence the property of being τmod-regular remains unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' To show that (˜γn) is τmod-regular, it would be enough to show that every subsequence of (˜γn) contains a τmod-regular subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' So, consider any subsequence of (˜γn), again denoted by (˜γn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' After extraction,6 we may assume that ǫF (n)−1 are all the same, say ǫF (n)−1 = 1, for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that, after passing to a subsequence, it holds for all n ∈ N that µF (n)−1 ∈ H+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Passing to another subsequence, we will also assume that F(n + 1) − F(n) ≥ 10, for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since µF (n)−1 ∈ H+ and ǫF (n)−1 = 1, we have ǫF (n) = 1, for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (see Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' So, ˜γn = fµF (n)−1fµF (n)−2f ǫF (n)−2 · · · µ1f ǫ1 = ff(f −1µF (n)−1fµF (n)−2)f ǫF (n)−2 · · · µ1f ǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2) Moreover, ˜γn+1˜γ−1 n = ff(f −1µF (n)−1fµF (n)−2)f ǫF (n)−1 · · · f ǫF (n−1)+1µF (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Observe that, if ǫF (n−1)+1 = −1, then µF (n−1) ̸∈ H+, since we observed in the preced- ing paragraph that ǫF (n−1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Else, ǫF (n−1)+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Consequently, in both cases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', ǫF (n−1)+1 = 1 or −1), it holds that ˜γn+1˜γ−1 n (B+) ⊂ f(B+) ⊂ B◦ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since the above is true for all n ∈ N, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5 applies to show that (˜γn) is τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that, after passing to a subsequence, it holds for all n ∈ N that µF (n)−1 ̸∈ H+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Consider the sequence (ˆγn), where ˆγn := µF (n)−1f ǫF (n)−1 · · · µ1f ǫ1 = f −ǫF (n)˜γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We show that (ˆγn) is τmod-regular since this would imply that (˜γn) is τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For all n ∈ N, ˆγnA ⊂ µF (n)−1B+ ⊂ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3) If the sequence (µ−1 F (n)−1) remains at a bounded distance away from H+, then after further extraction, we may assume that (µF (n)−1) lies in a single coset µH+, for some µ ̸∈ H+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' So, it holds that ˆγn+1ˆγ−1 n (A) ⊂ µB+ ⊂ A◦, for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' With Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5, the above implies that (ˆγn) is τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Otherwise, after further extraction, (µ−1 F (n)−1) diverges away from H+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Consider the sequence (ˆµF (n)−1), where ˆµF (n)−1 := µF (n)−1 prH+(µ−1 F (n)−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8, (ˆµ−1 F (n)−1) accumulates in ∂∞M \\ ∂∞H+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, all τmod-flag accumulation points of (ˆµ−1 F (n)−1) are 6Note that, by definition, subsequences of special sequences are special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' COMBINATION THEOREM: AMALGAMS 33 antipodal to B+ (see Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8, the sequence (ˆµF (n)−1B+) of com- pact subsets of Flag(τmod) shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since ˆµF (n)−1B+ = µF (n)−1B+, (µF (n)−1B+) shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Consequently, it follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3) that (ˆγnA) shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4, (ˆγn) is τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Hence, (˜γn) is τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Alternating sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Recall the notion of alternating sequences from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By definition, the inverse sequence corresponding to an alternating sequence is special so that Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8 directly implies: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Alternating sequences in Γ = M⋆φ are τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let (ωn) be an alternating sequence in Γ = M⋆φ equipped with the normal forms given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5): ωn = µ0f ǫ1µ1f ǫ2µ2 · · · f ǫn−1µn−1f ǫn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4) Then, the nested sequence of compact subsets (ωn(A ∪ Bǫn))n of Flag(τmod) converges to a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We remind our reader that we are using the notation introduced in Notation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' It is a straightforward verification that, for all n ∈ N, µn(Bǫn+1) ⊂ A ∪ Bǫn, yielding that ωn+1(A ∪ Bǫn+1) = ωnµnf ǫn+1(A ∪ Bǫn+1) ⊂ ωnµnBǫn+1 ⊂ ωn(A ∪ Bǫn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5) Therefore, the sequence (ωn(A ∪ Bǫn))n is nested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, to prove that (ωn(A ∪ Bǫn))n converges to a point, it will be enough to show that this sequence contains a subsequence that shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This is what we show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that, for infinitely many n ∈ N, µn−1 ∈ Hǫn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let P ⊂ N denote an infinite subset such that for all n ∈ P, µn−1 ∈ Hǫn and, for all n ∈ P, ǫn are the same, say ǫn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, for all n ∈ P, ǫn−1 = 1 (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let us first show that (ωn−1B+)n∈P shrinks: We observe that, for n ∈ P, ω−1 n−1(µ0A) ⊂ B−, which shows that all the τmod-limit points of (ω−1 n−1)n∈P lie in B− (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9, we know that (ωn−1)n∈P is τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since B+ is antipodal to B−, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4 (ωn−1B+)n∈P shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since, for all n ∈ P, ǫn = 1, we get that Bǫn = B+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Moreover, since µn−1 ∈ H+, ωn(A ∪ Bǫn) = ωn−1µn−1f(A ∪ B+) ⊂ ωn−1µn−1B+ = ωn−1B+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, by the previous paragraph, (ωnA)n∈P shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, we may assume now the complementary case to the above one, which is that, for at most finitely many n ∈ N, µn−1 ∈ Hǫn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In fact, it would be enough to assume: Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For all n ∈ N, µn−1 ̸∈ Hǫn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that, for infinitely many n ∈ N, ǫn = 1 (the case ǫn = −1 can be dealt with in a similar way).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let P ⊂ N be a subset such that, for all 34 SUBHADIP DEY AND MICHAEL KAPOVICH distinct n, n′ ∈ P, |n − n′| ≥ 10, and ǫn = 1, for all n ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For n ∈ P, let us consider the normal form of ωn+1 given by ωn+1 = µ0f ǫ1µ1 · · · f ǫn−2µn−2f ǫn−1 ˆµn−1f ˆµnf ǫn+1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6) where ˆµn−1 = µn−1 prH+(µ−1 n−1) and ˆµn = f −1(prH+(µ−1 n−1))−1fµn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7) We observe that, since µn−1 ̸∈ H+, ˆµn−1 is also not an element of H+, for all n ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For n ∈ P, let ˆωn := µ0f ǫ1µ1 · · · f ǫn−2µn−2f ǫn−1 ˆµn−1f = ωn+1(ˆµnf ǫn+1)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8) Since (ˆωn)n∈P is a subsequence of an alternating sequence, by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9, it is τmod- regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' One directly checks (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='5)), ωn+1(A ∪ Bǫn+1) ⊂ ˆωnA, for all n ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9) After extraction, we may assume that, for all n ∈ P, ǫn−1 is constant, ǫ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' After another extraction, we also assume that (ˆµ−1 n−1)n∈P either (i) diverges away from H−ǫ, or (ii) remains in a fixed coset ˆµH−ǫ, for some ˆµ ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' So, for n ∈ P, f ˆω−1 n (µ0A) = ˆµ−1 n−1f −ǫµ−1 n−2 · · · f −ǫ1µ−1 0 (µ0A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In the first case (i), since we are assuming that (ˆµn−1)n∈P diverges away from H−ǫ, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7, it follows that (˜µn−1)n∈P is τmod-regular and its τmod-limit points lie only in Λτmod(M) \\ Λτmod(H−ǫ), where ˜µn−1 := prH−ǫ(ˆµn−1)−1ˆµn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since Λτmod(M) \\ Λτmod(H−ǫ) is antipodal to B−ǫ, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8, (˜µ−1 n−1B−ǫ)n∈P = (ˆµ−1 n−1B−ǫ)n∈P shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' So, (f ˆω−1 n−1(µ0A))n∈P shrinks as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By definition of ˆµn−1 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7), all the τmod-limit points of (ˆµ−1 n−1)n∈P lie in Λτmod(M) \\ Λτmod(H+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, after further extraction, we may assume that ˆµ−1 n−1 flag −−→ τ− ∈ Λτmod(M) \\ Λτmod(H+), as n → ∞ in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, it holds that f ˆω−1 n−1(µ0A) → τ−, as n → ∞ in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3 the sequence (f ˆω−1 n−1)n∈P τmod-flag converges to τ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since τ− is antipodal to B+, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8, (ˆωn−1f −1B+)n∈P shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since fA ⊂ B+, it follows that (ˆωn−1A)n∈P shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9), (ωn+1(A ∪ Bǫn+1))n∈P shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In the second case (ii), suppose first that ǫ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, by the assumption of Case 2 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='7), we have that ˆµ ̸∈ H+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, f ˆω−1 n (µ0A) ⊂ ˆµ−1 n−1B+ ⊂ ˆµB+ ⊂ A◦, for all n ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' So, in this case, (f ˆω−1 n−1)n∈P has no τmod-limit points in B+, since, by above, all of them lie in the interior of A (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Therefore, since (ˆωn−1f −1)n∈P is τmod- regular,7 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='8, (ˆωn−1f −1B+)n∈P shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9), (ωn+1(A ∪ Bǫn+1))n∈P shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 7This follows by the observation above that (ˆωn−1)n∈P is τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' COMBINATION THEOREM: AMALGAMS 35 Still assuming (ii), suppose now that ǫ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If ˆµ ̸∈ H−, then proceeding as in the previous paragraph, it follows that (ωn+1(A ∪ Bǫn+1))n∈P shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Else, we must have ˆµn−1 ∈ H−, for all n ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Observing a different normal form of ˆωn, ˆωn := µ0f ǫ1µ1 · · · f ǫn−2(µn−2f ˆµn−1f −1 � �� � ∈M )ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' it follows by Case 1 that (ˆωnA) shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9), (ωn+1(A ∪ Bǫn+1))n∈P shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ By the above lemma, it follows that an alternating sequence ω = (ωn) in Γ τmod-flag- converges to the point τω := � n∈N ωn(A ∪ Bǫn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10) As a corollary, we obtain: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If ω = (ωn) is an alternating sequence in Γ, then the sequence (ωnA) of compact subsets of Flag(τmod) converges to the τmod-limit point τω of (ωn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' However, we remark that the sequence (ωnA) is possibly not nested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If ω = (ωn) is an alternating sequence in Γ, then, for all γ ∈ Γ, γωn flag −−→ γτω, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='11, it follows that (γωnA)n converges to γτω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3 yields the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that, to the contrary, Γ is not τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, Γ contains a τmod-irregular sequence (γn), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', (γn) has no repeated entries, and it contains no τmod-regular subsequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let (γn) be such a τmod-irregular sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We inductively extract a subsequence (˜γn) of (γn) so that there exists an alternating sequence (ωn) in Γ such that, under suitable normal forms of ˜γn, we may write ˜γn = ωnµ′ n,0f ǫ′ n,1µ′ n,1 · · · f ǫ′ n,lnµ′ n,ln, ∀n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='11) This would yield a contradiction as follows: Notice that, for all n ∈ N, ˜γnBǫ′ n,ln ⊂ ωn(A ∪ Bǫn), However, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='10, the sequence (ωn(A ∪ Bǫn)) converges to a point in Flag(τmod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, it would follow that (˜γnBǫ′ n,ln) shrinks, which would imply (by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4) that (˜γn) is τmod-regular, a contradiction with the τmod-irregularity assumption of (γn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' To finish the proof of the result, let us give a construction of (ωn) and (˜γn): For each n ∈ N, choose a normal form for γn, γn = µn,0f ǫn,1µn,1 · · · f ǫn,lnµn,ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12) Let us first show that we may extract a subsequence of (γn) such that, with appropriate normal forms, the leftmost letters in those normal forms are all the same: After extraction, we may assume that, for all n, ǫn,1 are the same and, for all n, ǫn,ln are the same, ǫn,1 = ǫ, ǫn,ln = ǫ′, ∀n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 36 SUBHADIP DEY AND MICHAEL KAPOVICH Note that (µ−1 n,0) cannot diverge away from Hǫ since, otherwise, γnBǫ′ ⊂ µn,0Bǫ, but (µn,0Bǫ) would shrink, which, by above, would show that (γn) is τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' So, after extraction, we may assume that µn,0 ∈ µ0ηn, for some sequence (ηn) in Hǫ and some µ0 ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' So, we may change the expression in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12) by γn = µ0f ǫ(¯ηnµn,1)f ǫn,2 · · · f ǫn,lnµn,ln, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='13) where ¯ηn ∈ H−ǫ is the element corresponding to ηn ∈ Hǫ given by the isomorphism φ : H− → H+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We may now delete the common leftmost letters µ0f ǫ from the normal form of the elements the extracted subsequence (γn) in the preceding paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Consider the sequence (γ′ n) thus obtained: γ′ n := (¯ηnµn,1)f ǫn,2 · · · f ǫn,lnµn,ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='14) We may now repeat the above procedure to the sequence (γ′ n) equipped with the normal form given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='14), which is again τmod-irregular, to obtain two more letters µ1 and f ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We may proceed in this way inductively so that we obtain an infinite string, µ0, f ǫ1, µ1, f ǫ2, µ2, f ǫ3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' , producing the desired alternating sequence (ωn) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3) and a corresponding subsequence (˜γn) of (γn) such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='11) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The boundary map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We construct a Γ-equivariant map from the Gromov boundary of Γ = M⋆φ to the flag manifold Flag(τmod): ξ : ∂∞Γ → Flag(τmod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15) Recall that ∂∞Γ decomposes into ∂IΓ ⊔ ∂IIΓ, where ∂IΓ = Γ · (∂∞M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' As in the case of amalgamated free products (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2), we define ξ separately on ∂IΓ and ∂IIΓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' see §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1 for the definition of ξ|∂IΓ and §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2 for the definition of ξ|∂IIΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Definition of the boundary map for type I points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For every point ε ∈ ∂IΓ, we may pick some element γ ∈ Γ such that γ−1ε ∈ ∂∞M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since M is τmod-Anosov, we have a M-equivariant boundary embedding ξ : ∂∞M → Λτmod(M) ⊂ Flag(τmod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We define ξ(ε) := γξ(γ−1ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='16) We check that ξ(ε) is well-defined, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', does not depend on the choice of γ ∈ Γ in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='16): If γ1 ∈ Γ is any other element such that γ−1 1 ε ∈ ∂∞M, then the intersection (γ−1γ1)∂∞M ∩ ∂∞M is nonempty since γ−1ε is a common point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, in the Bass-Serre tree T, M and (γ−1γ1)M are equal or adjacent vertices, showing that rl(γ−1γ1) ≤ 1 (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If rl(γ−1γ1) = 0, then γ1 ∈ γM, and, in this case, the well-definedness of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='16) follows by M-equivariance of ξ : ∂∞M → Flag(τmod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' So, let us assume that rl(γ−1γ1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In this case, γ−1γ1 = µ0f ǫµ1, where µ0, µ1 ∈ M and |ǫ| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let us also assume that ǫ = 1, since the case ǫ = −1 is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' So, γ1 = γµ0fµ1 and γ−1ε ∈ µ0fµ1(∂∞M) ∩ ∂∞M = µ0(∂∞H+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Hence, µ−1 0 γ−1ε ∈ ∂∞H+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='17) COMBINATION THEOREM: AMALGAMS 37 Since f ∈ G conjugates H− and H+, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', fH−f −1 = H+, for all ε′ ∈ ∂∞H+, ξ|∂∞H−(fε′) = fξ|∂∞H+(ε′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='18) Thus, γ1ξ(γ−1 1 ε) = γ1ξ(µ−1 1 f −1µ−1 0 γ−1ε) = γ1µ−1 1 ξ(f −1µ−1 0 γ−1ε) = γ1µ−1 1 f −1ξ(µ−1 0 γ−1ε) = γ1µ−1 1 f −1µ−1 0 ξ(γ−1ε) = γξ(γ−1ε), where the second equality is valid because (f −1µ−1 0 γ−1ε) ∈ ∂∞M, the third equality is verified by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='17) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='18), and the fourth equality is valid because γ−1ε ∈ ∂∞M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The following result is immediate from the definition of ξ above: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The map ξ : ∂IΓ → Flag(τmod) defined by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='16) is Γ-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Definition of the boundary map for type II points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For ε ∈ ∂IIΓ, consider an alternat- ing sequence (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2) ωn Cay −−→ ε given by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Define ξ(ε) := lim n→∞ ωnA, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='19) see Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The following result shows that ξ(ε) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For ε ∈ ∂IIΓ and for any sequence (γn) in Γ, if γn Cay −−→ ε, then γn flag −−→ ξ(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The proof is similar to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Together with Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='12, the above result implies the following: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The map ξ : ∂IIΓ → Flag(τmod) defined by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='19) is Γ-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The boundary map is antipodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The map ξ : ∂∞Γ → Flag(τmod) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15) (obtained by combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='16) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='19)) is antipodal: That is, for every pair of distinct points ε± ∈ ∂∞Γ, the points τ± := ξ(ε±) ∈ Flag(τmod) are antipodal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We consider the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Recall that ξ is Γ-equivariant (by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that both ε± are type I points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Using the Γ-equivariance, we may assume that ε− ∈ ∂∞M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If ε+ is also in ∂∞M, then ξ(ε±) are antipodal since ξ : ∂∞M → Flag(τmod) is an antipodal map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Else, ε+ ̸∈ ∂∞M but there exists γ ∈ Γ such that rl(γ) ≥ 1 and ε := γ−1ε+ ∈ ∂∞M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let γ = µ0f ǫ1µ1 · · · f ǫnµn be a normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' So, µ−1 0 ξ(ε+) ∈ µ−1 0 γ(Λτmod(M)) = µ−1 0 f ǫ1µ1 · · · f ǫn(Λτmod(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If rl(γ) = 1, then µ−1 0 ξ(ε+) ∈ f ǫ1(Λτmod(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Moreover, µ−1 0 ε+ ̸∈ f ǫ1∂∞H−ǫ1, since we have assumed that ε+ ̸∈ ∂∞M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, it follows that f −ǫ1µ−1 0 ξ(ε+) ∈ Λτmod(M) \\ Λτmod(H−ǫ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Since B−ǫ1 is antipodal to Λτmod(M)\\Λτmod(H−ǫ1), f −ǫ1µ−1 0 ξ(ε−), which is an element of B−ǫ1, is antipodal to Λτmod(M)\\Λτmod(H−ǫ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, ξ(ε+) is antipodal to ξ(ε−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 38 SUBHADIP DEY AND MICHAEL KAPOVICH If rl(γ) ≥ 2, and µ1 ̸∈ Hǫ2, then µ−1 0 ξ(ε+) ∈ µ−1 0 γA ⊂ f ǫ1µ1Bǫ2 ⊂ B◦ ǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If µ1 ∈ Hǫ2, then ǫ1 = ǫ2 and, hence, µ−1 0 ξ(ε+) ∈ µ−1 0 γA ⊂ f ǫ1µ1f ǫ2(A ∪ Bǫ2) = f ǫ1Bǫ1 ⊂ B◦ ǫ1, In both cases, µ−1 0 ξ(ε+) is antipodal to A and in particular, to Λτmod(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, ξ(ε+) is antipodal to Λτmod(M) and, in particular, to ξ(ε−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that both ε± are type II points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let (ω± n ) be alternating sequences such that ω± n Cay −−→ ε± (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' There exists n ∈ N such that ω+ n ̸∈ ω− n M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The proof is similar to the one of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Let n0 be the smallest natural number such that ω+ n0 ̸∈ ω− n0M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Using an argument similar to the Case 2 in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='21, one can show that ξ((ω+ n0)−1ε±) are antipodal, which is equivalent to ξ(ε±) being antipodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Suppose that ε− is a type I point, but ε+ is a type II point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' By Γ-equivariance, we may assume that ε− ∈ ∂∞M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let (ωn) alternating sequence, ωn = µ0f ǫ1µ1f ǫ2µ2 · · · f ǫn−1µn−1f ǫn, such that ωn Cay −−→ ε+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Then, ξ(ε+) = limn→∞ ωnA, see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If µ1 ∈ Hǫ2, then ǫ1 = ǫ2, and it follows that µ−1 0 ωnA ⊂ f ǫ1Bǫ1 ⊂ B◦ ǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, µ−1 0 ξ(ε+) ∈ B◦ ǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Else, if µ1 ̸∈ Hǫ2, then (µ0f ǫ1µ1)−1ωnA ⊂ Bǫ2, also showing that µ−1 0 ξ(ε+) ∈ f ǫ1µ1Bǫ2 ⊂ B◦ ǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' However, µ−1 0 ξ(ε−) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' It follows that ξ(ε−) and ξ(ε+) are antipodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The boundary map preserves convergence dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The following result is reminis- cent of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Let ε ∈ ∂IΓ and let (γn) be a sequence in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' If γn Cay −−→ ε, then γn flag −−→ ξ(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Using the action of Γ on ∂IΓ, it will be enough to prove the proposition for ε ∈ ∂∞M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We argue by contradiction: Suppose there exists a sequence (γn) in Γ such that γn Cay −−→ ε, but γn flag −−→ τ ̸= ξ(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='20) Equip each γn with a D-normal form (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='17), γn = µn,0f ǫn,1µn,1 · · · f ǫn,lnµn,ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='21) Passing to a subsequence, we may assume that ǫn,1 are the same, ǫ, for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In this situation, (µ−1 n,0) cannot diverge away from Hǫ: Otherwise, after extraction, µn,0 Cay −−→ ε′ and γnBǫn,ln ⊂ µn,0Bǫ → ξ(ε′), COMBINATION THEOREM: AMALGAMS 39 showing that γn flag −−→ ξ(ε′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' However, since (µn,0) fellow-travels γn, γn Cay −−→ ε′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thus, ε = ε′ and, therefore, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='20) is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' So, after extraction, µn,0 are all the same, µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We may now repeat the same procedure to the sequence (f −1µ−1 0 γn)n to show that, after another extraction µn,1 are all the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We can continue doing this procedure an arbitrary number of times: Thus, for all l ∈ N, there exists a subsequence (γ′ n) of (γn) such that suitable normal forms of the elements of (γ′ n) share at least l common leftmost letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' For l = 3, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='13 shows that (γ′ n) has no accumulation points in the boundary of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This is a contradiction with the assumption that γn Cay −−→ ε ∈ ∂∞M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ Combining Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='14 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='18, we obtain: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' The boundary map ξ : ∂∞Γ → Flag(τmod) preserves the convergence dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' We show that the subgroup Γ = ⟨M, f⟩ < G in the conclusion of Theorem B, which is naturally isomorphic to M⋆φ (by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15), is a τmod-Anosov subgroup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' this is equivalent to showing that Γ is a τmod-asymptotically embedded subgroup (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='9) of G: (i) By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='6, Γ is a τmod-regular subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (ii) That Γ is hyperbolic is the content of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' (iii) Finally, the boundary map ξ : ∂∞Γ → Flag(τmod) in Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15) is Γ-equivariant (by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='13 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='15), antipodal (by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='16), and preserves convergence dynamics (by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' This concludes the proof of the Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' □ References [1] Mark Baker and Daryl Cooper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A combination theorem for convex hyperbolic manifolds, with applica- tions to surfaces in 3-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', 1(3):603–642, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [2] Werner Ballmann, Mikhael Gromov, and Viktor Schroeder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Manifolds of nonpositive curvature, vol- ume 61 of Progress in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Birkh¨auser Boston, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', Boston, MA, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [3] Mladen Bestvina and Mark Feighn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A combination theorem for negatively curved groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Differential Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', 35(1):85–101, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [4] Subhadip Dey and Michael Kapovich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Klein-Maskit combination theorem for Anosov subgroups: Free products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' ArXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='03919, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [5] Subhadip Dey, Michael Kapovich, and Bernhard Leeb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A combination theorem for Anosov subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', 293(1-2):551–578, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [6] Cornelia Drut¸u and Michael Kapovich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Geometric group theory, volume 63 of American Mathemati- cal Society Colloquium Publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' American Mathematical Society, Providence, RI, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' With an appendix by Bogdan Nica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [7] Patrick B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Eberlein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Geometry of nonpositively curved manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Chicago Lectures in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' University of Chicago Press, Chicago, IL, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [8] Vladimir Fock and Alexander Goncharov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Moduli spaces of local systems and higher Teichm¨uller theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Hautes ´Etudes Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', (103):1–211, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [9] Rita Gitik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Ping-pong on negatively curved groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Algebra, 217(1):65–72, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [10] Olivier Guichard, Fran¸cois Labourie, and Anna Wienhard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Positivity and representations of surface groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' ArXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='14584, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 40 SUBHADIP DEY AND MICHAEL KAPOVICH [11] Dumitru Ivascu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' On Klein–Maskit combination theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In Romanian–Finnish Seminar on Complex Analysis, volume 743, pages 115–124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Springer Lecture Notes in Mathematics, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [12] Michael Kapovich and Bernhard Leeb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Finsler bordifications of symmetric and certain locally symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', 22(5):2533–2646, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [13] Michael Kapovich, Bernhard Leeb, and Joan Porti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Anosov subgroups: dynamical and geometric char- acterizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', 3(4):808–898, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [14] Michael Kapovich, Bernhard Leeb, and Joan Porti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' A Morse lemma for quasigeodesics in symmetric spaces and Euclidean buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', 22(7):3827–3923, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [15] Michael Kapovich and Pranab Sardar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Trees of hyperbolic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' ArXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='09526, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [16] Olga Kharlampovich and Alexei Myasnikov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Hyperbolic groups and free constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', 350(2):571–613, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [17] Felix Klein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Neue Beitr¨age zur Riemann’schen Functionentheorie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', 21(2):141–218, 1883.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [18] Fran¸cois Labourie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Anosov flows, surface groups and curves in projective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', 165(1):51–114, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [19] Liulan Li, Ken’ichi Ohshika, and Xiantao Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' On Klein–Maskit combination theorem in space, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Osaka J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', 46:1097–1141, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [20] Liulan Li, Ken’ichi Ohshika, and Xiantao Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' On Klein-Maskit combination theorem in space II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Kodai Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', 38(1):1–22, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [21] Roger C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Lyndon and Paul E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Schupp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Combinatorial group theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Classics in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Springer- Verlag, Berlin, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Reprint of the 1977 edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [22] Eduardo Mart´ınez Pedroza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Combination of quasiconvex subgroups in relatively hyperbolic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Pro- Quest LLC, Ann Arbor, MI, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Thesis (Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=')–The University of Oklahoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [23] Eduardo Mart´ınez-Pedroza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Combination of quasiconvex subgroups of relatively hyperbolic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Groups Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', 3(2):317–342, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [24] Eduardo Mart´ınez-Pedroza and Alessandro Sisto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Virtual amalgamation of relatively quasiconvex sub- groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Algebr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', 12(4):1993–2002, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [25] Bernard Maskit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' On Klein’s combination theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', 120:499–509, 1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [26] Bernard Maskit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' On Klein’s combination theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', 131:32–39, 1968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [27] Bernard Maskit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' On Klein’s combination theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In Advances in the Theory of Riemann Surfaces (Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', Stony Brook, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', 1969), Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Studies, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' 66, pages 297–316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Princeton Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Press, Princeton, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [28] Bernard Maskit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Kleinian groups, volume 287 of Grundlehren der mathematischen Wissenschaften.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Springer-Verlag, Berlin, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [29] Bernard Maskit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' On Klein’s combination theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', 336(1):265–294, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [30] Mahan Mitra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Coarse extrinsic geometry: a survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In The Epstein birthday schrift, volume 1 of Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Monogr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', pages 341–364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=', Coventry, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [31] Mahan Mj and Sabyasachi Mukherjee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Combination theorems in groups, geometry and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' In In the tradition of Thurston II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Geometry and groups, pages 331–383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Springer, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' [32] Jean-Pierre Serre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Springer Monographs in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Springer-Verlag, Berlin, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Trans- lated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content=' Department of Mathematics, Yale University, 10 Hillhouse Ave, New Haven, CT 06511 Email address: subhadip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='dey@yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='edu Department of Mathematics, University of California, Davis, One Shields Ave, Davis, CA 95616 Email address: kapovich@ucdavis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE0T4oBgHgl3EQfbwDA/content/2301.02354v1.pdf'} diff --git a/wtAzT4oBgHgl3EQfCPrg/vector_store/index.faiss b/wtAzT4oBgHgl3EQfCPrg/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..c29af84b8dc444ad056a765d8c6b82cd01b3ee1e --- /dev/null +++ b/wtAzT4oBgHgl3EQfCPrg/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d8102d205b2176455c786cf70d93369a105bfe9909dd99173b04d3f4f85e1e81 +size 4194349 diff --git a/x9E4T4oBgHgl3EQfxg23/content/tmp_files/2301.05259v1.pdf.txt b/x9E4T4oBgHgl3EQfxg23/content/tmp_files/2301.05259v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cafccc75c5653d906f669f65132e5000bc43d7f3 --- /dev/null +++ b/x9E4T4oBgHgl3EQfxg23/content/tmp_files/2301.05259v1.pdf.txt @@ -0,0 +1,2479 @@ +Detection of HCN and diverse redox chemistry in the plume of Enceladus +Jonah S. Peter,1, 2, ∗ Tom A. Nordheim,1 and Kevin P. Hand1 +1Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA +2Biophysics Program, Harvard University, Boston, Massachusetts 02115, USA +The Cassini spacecraft discovered that Saturn’s moon Enceladus possesses a series of jets erupting +from its South Polar Terrain. Previous studies of in situ data collected by Cassini’s Ion and Neutral +Mass Spectrometer (INMS) have identified H2O, CO2, CH4, H2, and NH3 within the plume of ejected +material. Identification of minor species in the plume remains an ongoing challenge, owing to the +large number of possible combinations that can be used to fit the INMS data. Here, we present +the discovery of several new compounds of strong importance to the habitability of Enceladus, +including HCN, CH2O, C2H2, and C3H6. +Our analyses of the low velocity INMS data coupled +with our detailed statistical framework enable discriminating between previously ambiguous species +in the plume by alleviating the effects of high-dimensional model fitting. Together with plausible +mineralogical catalysts and redox gradients derived from surface radiolysis, these compounds could +potentially support extant microbial communities or drive complex organic synthesis leading to the +origin of life. +Shortly after its arrival in the Saturn system, the +Cassini spacecraft discovered intense plume activity at +the mid-sized moon, Enceladus [1, 2]. +Cassini in situ +and remote sensing observations have confirmed that the +plume consists primarily of H2O gas [3–6] as well as +H2O-ice grains that feed Saturn’s E-ring [7–10]. +CO2 +has also been detected in both the gaseous phase within +the plume itself [4–6] and as a condensate in plume de- +posits on Enceladus’ surface [11]. In situ measurements +of the plume’s neutral gas component made by Cassini’s +Ion and Neutral Mass Spectrometer (INMS) further in- +dicate the presence of CH4, H2, and NH3 within Ence- +ladus’ subsurface ocean [4]. Although early publications +identified several additional species within the plume [5], +more recent work suggests that many of these compounds +resulted from incidental high velocity impact fragmenta- +tion of larger molecules within the instrument antecham- +ber [4, 12]. Analyses of plume material sampled during +the lower velocity flybys of Enceladus (for which this frag- +mentation was less significant) do imply the existence of +additional plume species, however no study to date has +been able to verify the identity of any other intrinsic com- +pounds. +Difficulty in resolving minor plume constituents stems +from the large number of plausible compounds relative to +the low mass resolution of INMS. When training statis- +tical models in this high dimensional regime, simple re- +gression techniques tend to form overly complex models +that produce specious results [13–16]. Models of INMS +spectra suffer from an additional complexity in that the +signals produced by individual molecules are not neces- +sarily linearly independent. As such, there may be mul- +tiple different combinations of species that appear to fit +the data equally well. The resulting large correlations +between model components generally reduce model per- +formance and can mask the importance of any particular +component by limiting statistical power [13, 17]. Studies +that leave these issues unaddressed are likely to encounter +model ambiguities that preclude reliable statistical infer- +ence about the plume’s composition. +In this work, we seek to resolve the apparent composi- +tional ambiguity of Enceladus’ plume. By characterizing +the information content of the average low velocity spec- +trum obtained during the E14, E17, and E18 flybys, we +determine constraints on the number of species that can +reliably be extracted from the INMS dataset under opti- +mal spacecraft conditions. We then use relative entropy +minimization to assess the likelihood of tens of billions of +potential models and show that multi-model inference al- +lows for the identification of several new compounds not +previously confirmed at Enceladus. Our results indicate +the presence of a rich, chemically diverse environment +that could support complex organic synthesis and possi- +bly even the origin of life (Fig. 1). +CONSTRAINTS ON THE COMPLEXITY OF +PLUME MODELS +Deconvolving the overlapping signals of each species in +the plume requires comparing features in the INMS flight +data to a library of known mass spectra. Consequently, +species in the plume cannot be identified unless explicitly +included within the models used for comparison. Previ- +ous studies have constructed model INMS spectra using +a variety of methods, including singular value decom- +position [18] and the application of custom-defined fit +statistics [19–22] (see Supplementary Information). +In +all cases, these fitting procedures seek to minimize the +training error associated with the residual counts be- +tween the INMS spectrum and the reconstructed model +fit. Crucially, however, the training error is not an ap- +propriate metric for evaluating model performance. In +fact, it is a well-known concept in statistical modeling +that the training error will continue to decrease with the +inclusion of additional model components, regardless of +arXiv:2301.05259v1 [astro-ph.EP] 12 Jan 2023 + +2 +0 ++ 4 +- 4 ++ 3 ++ 2 ++ 1 +- 3 +- 2 +- 1 +CH2O, C2H6N2 +CH4 +CO2 +HCN +C3H6, CH3OH +C2H2, C2H6O2 +Oxidation +state +Plume +species +H2 +O2 +H2 +HCN +O2 +40Ar +C3H6 +CH2O +C2H2 +43 u fragments +Alcohols +a +b +Fig. 1. New compounds identified in the Enceladus plume indicate a potentially habitable environment. (a) Jets emanating +from ice fissures in Enceladus’ South Polar Terrain feed a plume of ejected material containing organic molecules with varying +oxidation states. Electron bombardment of the surface might help facilitate the production of oxidants and prebiotic feedstock +molecules observed in the plume. +These compounds could potentially support biologically-mediated redox metabolisms or +polymerize to form nucleic and amino acid precursors leading to the origin of life. (b) The average oxidation state of carbon for +organic compounds confirmed or suspected in the plume. Plume-derived H2 and O2 could act as strong reducing and oxidizing +agents, respectively, and may be responsible for the diverse redox chemistry seen at Enceladus. +whether those components are genuinely related to the +observed data. Instead, it is necessary to approximate +how each model would perform on a set of independent +observations. This is the preferred method of model val- +idation when additional data sets are unavailable for ex- +plicit model testing [13]. Analysis techniques that utilize +only the training error risk developing overly complex +models and claiming false species detections. +Here, +we evaluate model performance using the +small sample bias-corrected Akaike Information Crite- +rion (AICc) [23–27] which estimates the relative entropy +between a given model and the unknown, true distribu- +tion that produced the observed data. We construct each +candidate model as a linear combination of end-member +spectra representing the different cracking patterns of in- +dividual molecules. A model M is given by, +M : +ˆyi = +d +� +k=1 +βkxk,i +(1) +where ˆyi is the total modeled counts in mass channel i, +xk,i is the cracking pattern of species k at mass channel i, +βk ≥ 0 is the regression coefficient for species k, and d is +the total number of species in the model. The observed +INMS counts at each mass channel, yi, are treated as +independent data points with unequal Gaussian uncer- +tainties, σi (see Methods). The AICc for each model is +then given by, +AICc = 2d − 2 ln[L(M0|y)] + 2d(d + 1) +n − d − 1 +(2) +where n is the total number of mass channels and L(M|y) +is the model’s likelihood function, +L(M|y) = +n +� +j=1 +� +1 +σj +√ +2π +� +exp +� n +� +i=1 +�yi − ˆyi +σi +�2� +(3) +evaluated at the maximum likelihood estimate, M0, ob- +tained by optimizing the set of {βk}. +The last term +in Eq. (2) is a correction factor that penalizes overly +complex models when the sample size is small (n/d ≲ +40) [27]. The model with the minimium AICc (AICcmin) +asymptotically approximates the model with the lowest +Kullback-Leibler information loss relative to the observed +data [27–29]. +Whereas standard maximum likelihood +estimation seeks to minimize the training error quanti- +fied by the variance-weighted sum of squared residuals, +�n +i=1((yi − ˆyi)/σi)2, minimization of the AICc accounts +for the bias introduced by estimating the regression co- +efficients via the training data. The relative likelihood, +or evidence ratio, of each model follows as [30], +λ = exp {−(AICc − AICcmin)/2} +(4) +and can be used to compare models of differing complex- +ity. Generally, models with λ < 1/e exhibit little to no + +3889QQQ3 +more +redundancy +less +redundancy +underfit +overfit +a +b +Fig. 2. Model performance as a function of model complexity. (a) Maximum relative likelihoods across all models with d species. +Blue circles indicate best-fit (highest λ) models identified via an exhaustive search for d < 15. Orange circles represent models +constructed using a benchmark forward modeling procedure extending to d = 50 (see Methods). There is good agreement +between the statistics obtained with forward modeling and the exhaustive search. Models with λ > 1/e (dashed line) exhibit +strong predictive power. All models with d < 10 underfit the data, while those with d > 13 are overfitting. (b) The maximum +magnitude of pairwise correlations rij = cov(βiβj)/ +� +var(βi)var(βj) between regression coefficients in the best-fitting models +at each value of d. Blue and orange circles represent the same models as in panel (a). Correlations follow a logistic curve +demonstrating model redundancy for d > 13. The inflection point occurs within the optimal performance interval d ∈ [10, 13] +characterized by λ > 1/e (green shaded area). +predictive power while those above this threshold form a +family of most probable models suitable for multi-model +inference [27]. +To capture the full range of possible plume con- +stituents, we composed a large spectral library contain- +ing the most recently published list of plausible INMS- +detectable plume species [31] as well as several addi- +tional compounds found in organic synthesis and labo- +ratory experiments simulating icy satellites (see Meth- +ods and Extended Data Table 1). Using this library, we +performed an exhaustive search up to d = 14 of tens +of billions of potential models for the plume’s composi- +tion. Fig. 2a demonstrates that optimal model perfor- +mance is achieved with only 10-13 species. The maxi- +mum relative likelihood across all models peaks sharply +at 11 species and drops precipitously for more complex +models. A benchmark forward modeling procedure ex- +tending to d = 50 confirms this trend for large d (see +Methods). +Models with greater than 13 species dras- +tically overfit the INMS data and incorrectly extract +false signal from statistical noise. +Large correlations +rij = cov(βiβj)/ +� +var(βi)var(βj) between regression co- +efficients in these models indicate the erroneous inclusion +of redundant species with similar mass spectra (Fig. 2b). +This leads to overfitting and poor model performance, +despite a monotonic decrease in the training error with +increasing complexity. By contrast, models with fewer +than 10 species exhibit low correlations between model +parameters, indicating the presence of additional features +within the INMS spectrum that have not yet been fit by +a candidate species (i.e., underfitting). Optimal perfor- +mance occurs near the inflection point after all major +data features have been explained but before redundant +species are incorporated. +NEW SPECIES DETECTED IN THE PLUME +The most recently published list of neutral gas species +confirmed in the plume consists only of H2O, CO2, CH4, +H2, and NH3 [4, 31]. According to that work, no other +compounds could be definitively detected by INMS dur- +ing the low velocity flybys due to model ambiguities at +low mixing ratios. Here, we account for those ambigu- +ities via a multi-model averaging procedure that treats +the minimum Kullback-Leibler divergence as a statistical +random variable and weights each model by its probabil- +ity of minimizing this information loss (see Methods). +Whereas previous studies have relied on ad-hoc assess- +ments of model ambiguity, the procedure here is rooted in +fundamental concepts of information theory and explic- +itly incorporates uncertainties resulting from the model- +ing process into the standard error (SE) of each mixing +ratio. +In addition to H2O, CO2, and CH4, we find signifi- +cant evidence—beyond 2 SE precision—for HCN, CH2O, +C2H2, and C3H6 in the plume (Table 1). Our results are +agnostic to the presence of H2 which requires analysis of +additional INMS data not examined here (see Methods). +We also demonstrate strong evidence for 40Ar and O2 +at 1 SE precision, as well as the probable detection of +an alcohol (likely C2H6O2 or CH3OH), and an unidenti- + +4 +Table 1. Volume mixing ratios for the Enceladus plume. Probabilities and mixing ratios are shown for all species with upper +limits above the estimated INMS noise floor. Values are calculated using a multi-model averaging procedure that incorporates +uncertainties due to model ambiguities (see Methods). Mixing ratios are given as mean +/- SE and are scaled to incorporate the +0.9% H2 number abundance reported in ref. [4] (see Methods). Upper limits (< 3 SE) are reported for species with mean values +below their corresponding SE. The minimal model is comprised of “confirmed” species with > 2 SE precision and represents +the most conservative model of the plume. Species listed in brackets are included in the alcohol mixing ratio. Species listed in +parentheses are included in the 43 u fragment mixing ratio. +Evidence +Species +Probabilityd Mixing Ratio (%)d Previous Limit (%)e +Confirmed H2O +1 +98.3 ± 0.3 +96 − 99 +CO2 +1 +0.29 ± 0.04 +0.3 − 0.8 +CH4 +1 +0.23 ± 0.05 +0.1 − 0.3 +HCN +> 0.99 +0.12 ± 0.04 +0.01 − 0.2 +C2H2 +> 0.99 +0.025 ± 0.005 +0.01 − 0.2 +CH2O +> 0.99 +0.026 ± 0.010 +0.01 − 0.2 +C3H6 +0.88 +0.0041 ± 0.0020 +< 0.01 +H2 +a +− +− +0.4 − 1.4, 0.9f +Strong +40Ar +0.85 +0.0014 ± 0.0009 +< 0.01 +Alcoholsb +0.81 +< 0.021 +− +[C2H6O2] +0.38 +< 0.021 +< 0.01 +[CH3OH] +0.31 +< 0.0041 +< 0.01 +43 u fragmentsc 0.81 +< 0.0016 +− +(C3H6O) +0.27 +< 0.0015 +< 0.01 +(C2H6N2) +0.26 +< 0.0016 +< 0.01 +O2 +0.79 +0.0027 ± 0.0020 +< 0.01, < 0.004f +Moderate +H2S +0.44 +< 0.0031 +< 0.01 +PH3 +0.36 +< 0.0025 +< 0.01 +Poor +C3H5Cl +0.17 +< 0.0012 +< 0.01 +CH3CN +0.15 +< 0.0028 +< 0.01 +NH3 +0.10 +< 0.58 +0.4 − 1.3 +C3H7NO2 +0.08 +< 0.0014 +< 0.01 +a Determination of the H2 mixing ratio requires analysis of additional INMS data not examined here (see Methods). +b Other alcohols such as C3H8O, C2H6O, or C4H10O might also contribute to this mixing ratio (see Supplementary Material). +c Other mass 43 fragments produced by C4H10, C4H6O2, C4H9N, C5H9N, C8H18, C5H12, or C2H4O2 might also contribute +to this mixing ratio. +d Denotes values calculated in this work. +e Denotes values presented in ref. [31] unless otherwise specified. +f Denotes values presented in ref. [4]. +fied 43 u fragment. Because the spectrum analyzed here +is free of the major fragmentation signatures produced +during the high velocity flybys, these species are most +likely intrinsic to the plume itself. In general, our mixing +ratios are in excellent agreement with the most recently +published limits [4, 31]. The reconstructed model fit is +plotted alongside the INMS spectrum in Fig. 3. +Interestingly, our results suggest that NH3 is not re- +quired to fit the INMS data. Although our upper limit +of < 0.58% is consistent with the lower end of the range +reported by Waite et al. [4, 31], we conclude that high +inter-model uncertainty stemming from large contribu- +tions of CH4 and H2O at overlapping mass channels +precludes the firm detection of NH3 (Fig. 4a). +Some +amount of NH3 is likely required to explain the pres- +ence of nitrogen-bearing ions observed in Saturn’s mag- +netosphere [32], but a stricter bound on the mass 16 +count rate is needed before a percent-level concentration +of NH3 from Enceladus can be presumed (see Supple- +mentary Information). Notably, it has been shown that +even a significantly smaller concentration of NH3 in the +plume could still source the observed nitrogen abundance +on Titan [33]. +We find that nitrogen is, however, definitively present +at Enceladus in the form of HCN. Previous studies have +been unable to resolve the HCN abundance due to con- +founding signals from fragmentation products at mass +28. +In their analysis of the E2 flyby, Waite et al. [6] +quote an upper limit of < 0.5%, whereas upper limits +of < 0.58% and < 0.74% are given by Waite et al. [5], +based on the high velocity E3 and E5 flybys. For the +faster flybys in particular, the authors note that the ele- +vated count rate at 28 u introduces a model ambiguity at +masses 27 and 28 (N2+HCN versus C2H4) that precludes +the identification of HCN. Here we correct the counts at +mass 28 based on the INMS Open Source Neutral Beam +(OSNB) data [4] to reveal that HCN is required at 27 u +(see Methods, Fig. 4b, and Extended Data Fig. 1). The +mixing ratio reported here (0.12 ± 0.04%) is consistent +with that observed in cometary comae [34–36]. + +5 +Fig. 3. +Average low velocity INMS spectrum and recon- +structed model fit. The black silhouette shows the 12-47 u +range of the average INMS spectrum obtained during the +E14, E17, and E18 flybys adapted from ref. [12] with cor- +rection for minor artifacts (see Methods and Extended Data +Fig. 1). Counts at all other mass channels are at or below +the estimated noise floor (dashed line). Error bars show 1σ +Gaussian uncertainty in the observed count rates. Red circles +indicate the model fit based on the mixing ratios in Table 1. +We also report the first strong evidence for native O2 +in the plume. In their analysis of the low velocity fly- +bys, Waite et al. [4] noted that mass 32 exhibits an el- +evated count rate, likely in part due to surface process- +ing between H2O and the Ti instrument antechamber. +Accounting for this instrument effect, they estimated a +corrected mass 32 signal equal to (100/45) × 0.004% = +0.0089% of the counts measured at mass 18. In account- +ing for additional species, our statistical analysis yields +an excess at mass channel 32 that is best explained by +an O2 mixing ratio of 0.0027%. The value of mass 32 +counts used in this work (Fig. 4c) is well within the limit +imposed by Waite et al., and thus we argue that it is a +true measure of native O2 from Enceladus. +Our analysis also indicates moderate (Table 1) evi- +dence for either H2S or PH3 in the plume, with probabil- +ities of 0.44 and 0.36, respectively. This conclusion stems +primarily from masses 33 and 34, where the omission of +these species leaves a small residual signal that cannot be +explained by other library spectra (Fig. 4c). However, +this signal alone is not enough to unambiguously con- +clude their presence within the INMS data. Both species +improve the model fit when added to the minimal model +(“confirmed” species in Table 1), but not when additional +higher likelihood species are included (“strong” species in +Table 1). Our results are therefore only suggestive of a +34 u compound in the plume. Given the profound astro- +biological implications of finding sulfur or phosphorous +compounds at Enceladus, mass spectra for alternative 34 +u species (such as H2O2) should be characterized exper- +imentally and investigated in future studies. +a +b +c +d +Fig. 4. Contributions of individual species to the model fit. +Black silhouettes show different mass ranges for the average +low velocity INMS spectrum plotted in Fig. 3. +Error bars +show 1σ Gaussian uncertainty in the observed count rates +and dashed lines denote the INMS noise floor. (a) Blue cir- +cles show the total contribution of H2O+CO2+CH4. Green +circles show the contribution of NH3, which is not required +for satisfactory model performance. (b) Model contributions +from CO2, HCN, CH2O, and C2H2 (magenta, cyan, orange, +and purple circles, respectively). (c) Navy circles show the +contribution from O2 added to the total contribution from +alcohols. Pink circles show the tentatively proposed contri- +bution of H2S+PH3. Low signal-to-noise ratios at these mass +channels preclude the firm identification of H2S or PH3. (d) +Brown and indigo circles show contributions from C3H6 and +40Ar, respectively. +Teal circles show the total contribution +from 43 u fragments. +DISCUSSION +In aggregate, the results presented here indicate that +Enceladus is host to a multiphasic and compositionally +diverse chemical environment that is consistent with a +habitable subsurface ocean. The new species identified +in this work also suggest that this ocean may contain +the necessary building blocks required to synthesize com- +pounds important to the origin of life. +The detection of CH2O, O2, and an alcohol is par- +ticularly interesting as these species potentially imply +a diverse redox environment within the ocean (Fig. 1). +Past measurements of CH4 and H2 in the plume [4–6] +have supported the hypothesis that Enceladus may be +hydrothermally active and could be a source of biologi- +cally useful reductants. In particular, methanogenesis via +the reduction of CO2 has been proposed as a potential +pathway that could support extant microbial communi- +ties near the sea floor [37, 38]. However, without addi- +tional oxidants, reductants in the ocean would be of lit- +tle biochemical utility, as no electron transfer mechanism +beyond methanogenesis would be available to yield a neg- + +6 +ative change in Gibbs free energy. The detection of O2 +and partially oxidized carbon compounds may solve this +problem, as they provide a multitude of highly exergonic +redox pathways that could help power life in Enceladus’ +subsurface ocean (see e.g., refs. [39, 40]). +Our results also provide the first conclusive evidence +for HCN, CH2O, C2H2, and C3H6 in the plume, which, in +addition to their significance for habitability, are particu- +larly intriguing for their relevance to prebiotic chemistry. +HCN polymerization is implicated in a number of poten- +tial pathways for the formation of nucleobases and amino +acids [41–44]. The autocatalysis of CH2O to form simple +sugars and RNA precursors is also well-documented [45], +as is its production on the early Earth [46]. Although +these reactions might be limited within a dilute subsur- +face ocean, concentrated conditions favorable for poly- +merization could be achieved within the ice shell via eu- +tectic freezing [42, 47, 48]. Indeed, Levy et al. [49] demon- +strated the production of alanine, glycine, aspartic acid, +and adenine from HCN and NH3 under conditions similar +to those of icy satellites. Repeated freezing and thawing +caused by the cycling of material between the ice shell +and the warmer interior fissures of the plume vents may +provide conditions favorable for organic synthesis. +In accordance with this scenario, there does exist ev- +idence that the plume contains at least some material +that has not been diluted by the ocean. Liquid water fa- +cilitates the rapid hydrolysis of HCN into CH3NO, which +subsequently decays into CH2O2 and NH3 [47]. The si- +multaneous presence of ppt amounts of HCN and absence +of CH3NO and CH2O2 suggests that the plume may be +sourced, in part, by solid-phase material residing in the +ice shell [5]. HCN at Enceladus could be primordial in +nature or produced via the radiolysis of nitrogen-bearing +surface ices by magnetospheric electrons. Laboratory ex- +periments of hydrocarbon-rich ices simulating the tem- +perature and surface radiolysis conditions of Enceladus +support the evidence for cyano group-containing species. +Hand [50] and Hand et al. [51] found that the warming +of H2O+NH3+C3H6 ice films from 70 K to 300 K after +irradiation by 10 keV electrons produces HCN and nitrile +species, including CH3CN. Irradiation of H2O+CO2 ice +in the presence of hydrocarbons can also produce CH2O +and alcohols [50]. Thus, the detection of HCN, CH2O, +and an alcohol (as well as O2 and possibly CH3CN) in the +plume could potentially be explained by the aerosoliza- +tion of radiolytically processed material on or near the +surface. Additional evidence from organic-rich ice grains +detected in the plume further suggests the presence of +solid-phase organic films that exist above the water ta- +ble and are transported to the surface via the plume +vents [12]. +The macromolecular nature of these com- +pounds, some in excess of 200 u, might be evidence of +ongoing synthetic chemistry. Accumulation and isolation +of buoyant organic compounds at the cold ice-water in- +terface between the ocean and ice shell may also promote +longer lifetimes and inhibit hydrolysis [52]. +Of course, the availability of organic compounds at +Enceladus to support or facilitate the origin of life likely +depends strongly on the geochemistry of the subsurface +ocean. Although the mineral composition of the ocean +floor is unknown, the simultaneous detections of SiO2 +particles by CDA [53] and gaseous H2 by INMS [4] indi- +cate the presence of a complex hydrothermal environ- +ment [54]. +Similarities between the inferred tempera- +ture and pH of Enceladus’ ocean and those of the Lost +City hydrothermal vents point towards serpentinization +reactions as a possible source for the observed H2 abun- +dance [4, 55, 56]. +Such an explanation would also be +consistent with the tentative evidence for H2S presented +here. If ferrous iron is present at Enceladus (as it is at +the Lost City sites) along with H2S, sufficient reducing +power may be available for a metabolic pathway to abio- +genesis [42, 57, 58]. Laboratory evidence suggests that +confirmation of C3H6 in the plume might allow for the +formation of vesicle-type structures at Enceladus, which +could then shelter the burgeoning proto-metabolism [50]. +An alternative scenario for complex organic synthesis +could be realized through the photochemical processing +of ocean material ejected by the plume onto the surface. +HCN might be sequestered as ferrocyanide by sodium +or potassium salts, both of which have been found in +plume ice grains [8, 9]. +Ferrocyanide is an important +feedstock molecule for the cyanosulfidic prebiotic chem- +istry paradigm, which relies on reductive homologation +of HCN to form the sugars needed for ribonucleotide as- +sembly [44, 59, 60]. If there exists sufficient UV photon +processing of organic material on Enceladus’ surface, the +selective production of RNA and amino acid precursors +from plume-derived HCN, CH2O, C2H2, (and possibly +H2S or PH3) together with plausible mineralogical cata- +lysts might occur [44, 61]. This surface-based production +could then feed back into the plume or ocean via down- +ward transport through the ice shell. Whether this type +of chemistry is efficient under Enceladus-like conditions +could be explored in future experimental studies, while +more detailed examination of Enceladus’ oceanic mate- +rial will require future robotic missions. +METHODS +Instrument overview +The Cassini +Ion and Neutral Mass Spectrometer +(INMS) was a quadrupole mass spectrometer designed +primarily for neutral gas analysis [62]. When INMS was +operating in its Closed Source Neutral (CSN) mode, neu- +tral molecules were accumulated in the instrument an- +techamber before being directed towards a 70 eV electron +source for subsequent ionization. Detections of ionized +molecules at the instrument target were then recorded + +7 +sequentially at individual mass channels corresponding +to mass-to-charge ratios (m/z), where z is typically as- +sumed to be 1. The INMS mass channels ranged from 1 +to 99 u at a resolution of 1 u, excluding 9, 10, and 11 u. +When neutral molecules enter the ionization region of +a mass spectrometer such as INMS, emitted electrons +from the electron source both ionize incoming parent +molecules and produce dissociated, ionized fragments. +For a given electron energy (e.g., 70 eV), each incoming +parent molecule fragments according to a specific crack- +ing pattern that describes the relative proportions of the +dissociated products. As such, a mass spectrum obtained +from a single molecular species will exhibit counts at mass +channels pertaining to the molecular weight of the ionized +parent molecule, as well as those of each of the ionized +dissociation products. INMS spectra therefore consist of +a combination of overlapping spectral features resulting +from each parent molecule’s cracking pattern. +INMS was not able to detect parent molecules or frag- +ments with masses above the maximum mass of 99 u. +However, larger molecules may have impacted the instru- +ment walls and fragmented into smaller molecules that +were within the detectable mass range [5]. The rate of +these impact fragmentation events depends heavily on +the kinetic energy of the spacecraft [31]. As such, spec- +tra generated from faster flybys (> 10-15 km/s) exhibit +a suite of spectral features (associated with impact frag- +ments of higher mass molecules) that are not observed +at lower speeds [4, 12, 31]. As a result, measurements +obtained during the lower velocity flybys of Enceladus +provide the best opportunity to study the intrinsic plume +composition in the absence of fragmentation effects. +Data set selection +During the E14, E17, and E18 flybys of Enceladus, +Cassini made a sequence of three low velocity (< 7.5 +km/s) passages through the plume along nearly identi- +cal trajectories. These flybys, referred to as the “slow” +flybys [4], produced the most consistent INMS data with +which to analyze the plume’s intrinsic neutral gas com- +position. Here, we use the averaged CSN spectrum ob- +tained across these three flybys, as done by Waite et al. +in ref. [4]. This averaged slow flyby spectrum is identi- +cal to that presented in ref. [12], with minor differences +described below. +In their supplementary material, Waite et al. [4] deduce +that the majority of counts at mass 28 are attributable to +fragmentation products (possibly of CO2) that are not re- +flective of the plume’s intrinsic composition. Fragments +from higher mass hydrocarbon species likely also con- +tribute to this mass channel [12]. Measurements taken +during the E21 flyby using the INMS Open Source Neu- +tral Beam (OSNB) mode place an upper limit on intrinsic +mass 28 counts at 0.05% of the signal at mass 18 [4]. As +this study is focused on identifying native species in the +plume, we account for this effect by correcting the counts +at mass 28 accordingly (Extended Data Fig. 1). +In order to quantify variability in the INMS data be- +tween flybys, we assume a 30% Gaussian uncertainty on +all mass channels, similar to that described by Waite et +al. in their own analysis of the slow flyby spectrum [4]. +However, given the strong signal at mass 18, we use a +standard Poisson uncertainty at this mass channel, equal +to the square root of the number of counts. This choice is +warranted given that mass 18 is the anchor mass used to +standardize the E14, E17, and E18 flybys. +Moreover, +mass channels beyond 46 u exhibit significantly lower +signal-to-noise ratios than lower mass channels. To avoid +erroneously fitting to these data points, we implement a +larger fractional uncertainty on peaks that lie below the +count levels at these mass channels. This larger uncer- +tainty is equal to the typical count value measured at +these mass channels (2 counts) and serves as the global +noise floor for the spectrum. +When INMS was operating in the CSN mode, the vast +majority of counts at 1 and 2 u arose due to interactions +between gas phase H2O molecules and the walls of the +instrument antechamber [4]. Proper assessment of these +mass channels requires a careful analysis of data from +the instrument’s OSNB mode. This problem has been +explored in detail by Waite et al. [4] who attribute ∼98% +of the total signal at 1 and 2 u to H2 (at a mixing ratio of +∼0.9%) and H2O. The cracking pattern for H2 does not +contribute to any other mass channels and is therefore +independent from the deconvolution of the remainder of +the spectrum. Furthermore, the mixing ratio of H2O is +predominantly dictated by high leverage points at masses +18 and 17 and not significantly affected by these lower +mass channels. For these reasons, we chose to adopt the +H2 mixing ratio of ref. [4] and exclude masses 1 and 2 +from our analysis (Extended Data Fig. 1). +Spectral library +As discussed in the main text, species that are present +within the INMS spectrum cannot be identified unless +explicitly included within a reference library used for +comparison. However, large spectral libraries suffer from +high dimensionality; the inclusion of too many library +species leads to models that are unnecessarily complex +and prone to overfitting. +This phenomenon is similar +to how any univariate relationship can be fit with zero +residual error using a sufficiently high order polynomial. +Although overfitting can be ameliorated by using bias +correcting heuristics such as the AICc, chemically im- +plausible species may still be incorporated into the final +model if they are included within the initial library. As +an additional hindrance, the inclusion of many (possibly +collinear) model parameters reduces statistical power and + +8 +raises the computational intensity of the analysis. It is +therefore desirable to constrain the library of candidate +species as much as possible based on prior knowledge and +the results of previous studies. +Our library consists of a carefully chosen set of 50 +compounds, including the most recent list of INMS- +detectable candidate plume species presented in ref. [31]. +All library species and their reasons for inclusion are pre- +sented in Extended Data Table 1. Because all counts ob- +served above 46 u during the slow flybys are below the +estimated noise floor, no species with base peaks above +this threshold were included in this analysis. The spectra +in our library are 70 eV electron ionization mass spectra +collected from the INMS Refurbished Engineering Unit +(REU) and the National Institute of Standards and Tech- +nology (NIST) online database. We adopt the method +used in refs. [19, 63] to prioritize the REU spectra when +available. +REU spectra were obtained from the most +recent INMS calibration file provided on the Planetary +Data System. All spectra were analyzed at 1 u resolu- +tion and matched to the mass detection range of INMS. +All spectra were base peak normalized to ensure a uni- +form method of comparison. +Benchmarking model complexity +In deducing limits on INMS model complexity, we con- +structed all possible combinations of plume species up to +d = 14 from a candidate library of 50 plausible com- +pounds (Extended Data Table 1). To reduce the com- +putational intensity associated with this analysis, we as- +sumed H2O, CO2, and CH4 to be present in each model. +H2O and CO2 have been verified at Enceladus by inde- +pendent Cassini instruments [64, 65], and CH4 is among +the most consistent INMS observations, having been de- +tected on every flyby that sampled the plume [4–6]. This +resulted in a total of �14 +d=3 +47! +(d−3)!(50−d)! ≈ 2.4×1010 mod- +els for direct comparison. In order to efficiently sample +the parameter space of more complex models, we im- +plemented a forward modeling stepwise selection proce- +dure [13]. Starting from the three-component model of +H2O+CO2+CH4, the next most complex model was con- +structed by including an additional species that, when +added to the model, resulted in the greatest decrease +in the variance-weighted sum of squared residuals. This +selection process was repeated—sequentially incorporat- +ing new species based on their influence on the sum of +squared residuals—to produce a set of nested models, +each containing one more species than the last. Notably, +the sum of squared residuals (i.e., the training error) was +only used to compare models of equal complexity and +therefore selects the same species as would comparisons +based on the AICc. As shown in Fig. 2 of the main text, +the results of this forward modeling algorithm closely re- +semble those of the exhaustive model search for d < 15. +Extrapolation of these results to more complex models +confirms a monotonic decrease in relative model likeli- +hood for d > 11. +Multi-model inference +In order to determine mixing ratios that properly ac- +count for the ambiguity induced by model degeneracy, we +computed model-averaged regression coefficients based +on the relative likelihood of each model, +¯βk = +R +� +j=1 +wjβk,j. +(5) +Here, βk,j is the regression coefficient for species k in +model j, R is the total number of most probable mod- +els (those with λ > 1/e), and wj = λj/ �R +m λm is the +Akaike weight [27] of each model. Because the observed +INMS spectrum consists of sample data taken from the +unknown population distribution of plume contents, the +minimum AICc model is a statistical random variable +that estimates the actual (also unknown) model that +minimizes the Kullback-Leibler divergence with the true +distribution. Therefore, the Akaike weight may be inter- +preted as the probability that a given model minimizes +this information loss between the model and the un- +known, true distribution from which the observed INMS +data were sampled. The probability that each species is +present in the true minimum AICc model follows accord- +ingly as, +Pk = +R +� +j=1 +wjΘ(βk,j) +(6) +where Θ(βk,j) is the Heaviside step-function that equals +1 when βk,j > 0 and is zero otherwise. We then com- +puted uncertainties in the model parameters using the +unconditional standard error estimator [27], +SE(¯βk) = +R +� +j=1 +wj +� +var(βk,j) + (βk,j − ¯βk)2 +(7) +where var(βk,j) is the variance of the regression coeffi- +cient conditional on model Mj. Here, var(βk,j) charac- +terizes the intra-model uncertainty associated with maxi- +mum likelihood estimation, whereas the term (βk,j − ¯βk)2 +quantifies inter-model variability due to the presence of +additional high likelihood models. Importantly, the re- +sults presented in this work incorporate the relative prob- +abilities of each model and account for ambiguities in the +model selection process. + +9 +Conversion to mixing ratios +The formula for converting INMS counts to ambient +densities is described in ref. [66] and given by, +nk = +�T0 +Ta +�1/2 1 +Dk +Xk +sk +(8) +where Xk is the count rate of species k (measured at +the base peak), sk is the INMS sensitivity, Dk is the +ram enhancement factor, Ta is the ambient temperature, +and T0 is room temperature (293 K). When available, +INMS sensitivities were obtained from refs. [18, 19, 63] +and otherwise estimated based on the electron impact +ionization cross section procedure of Fitch and Sauter +(see Eq. 2 of ref. [67]) adapted to INMS data in refs. [19, +63]. For one species (PH3), cross section data was taken +from ref. [68]. As done in ref. [63], a 30% uncertainty +was implemented on all sensitivities estimated from NIST +spectra. +The count rate is related to the model-averaged re- +gression coefficient of Equation (5) by Xk = ¯βkc0 where +c0 is the standardized (species-independent) base peak +count rate of each library spectrum. The mixing ratio +for each species mk (scaled to include the H2 mixing ra- +tio mH2 = 0.9% given in ref. [4]) is then, +mk = (100 − mH2) nk +� +l nl += (100 − mH2) +¯βk +Dksk +� +l +Dlsl +¯βl +. +(9) +At suprathermal spacecraft speed and small ram angle +(conditions valid during the E14, E17, and E18 flybys), +the ram enhancement factor is approximately [4, 66], +Dk ∼ 0.7u +� +2πµk +kBTa +(10) +for spacecraft speed u and molecular mass µk. The final +expression for the mixing ratio of each species is there- +fore, +mk = (100 − mH2) +¯βk +√µksk +� +l +√µlsl +¯βl +. +(11) +Future updates to INMS sensitivity coefficients or ram +enhancement factors may change the relative mixing ra- +tios presented here but will not affect which species have +been identified—or the evidence for their detection—as +these depend only on the set of {¯βk}. +ACKNOWLEDGEMENTS +The authors thank Dr. J. Hunter Waite and Dr. Brian +A. Magee for their help on interpreting INMS instrument +effects. +J.S.P. thanks Dr. +Masao Sako and Dr. +Kiri +L. Wagstaff for useful discussions and statistical insight. +J.S.P. also thanks Dr. Dimitar D. Sasselov for helpful +discussions on prebiotic chemistry. All authors acknowl- +edge the support of the Cassini Data Analysis Program +(NNN13D466T) and the Jet Propulsion Laboratory, Cal- +ifornia Institute of Technology, under a contract with +NASA. T.A.N. was also supported by an appointment +to the NASA Postdoctoral Fellowship Program at the +Jet Propulsion Laboratory administered by Oak Ridge +Associated Universities and Universities Space Research +Association through a contract with NASA. K.P.H. also +acknowledges support from the NASA Astrobiology Pro- +gram (80NSSC19K1427) and the Europa Lander Pre- +Project, managed by the Jet Propulsion Laboratory, Cal- +ifornia Institute of Technology, under a contract with +NASA. + +10 +Extended Data Table 1. Complete list of species included in the analysis. Those taken from ref. [31] comprise the most recently +published list of INMS-detectable species in the plume. REU: INMS Refurbished Engineering Unit; NIST: National Institute +of Standards and Technology online database. +Species +Name +Mass (u) Source sk +sk Ref. Reason/Ref. +CH4 +Methane +16 +REU +6.01×104 +[63] +[4, 31] +NH3 +Ammonia +17 +REU +4.77×104 +[63] +[4, 31] +H2O +Water +18 +REU +4.34×104 +[63] +[4, 31] +C2H2 +Acetylene +26 +REU +8.81×104 +[63] +[6, 31] +HCN +Hydrogen Cyanide 27 +REU +5.20×104 +[63] +[5, 31] +C2H4 +Ethylene +28 +REU +6.21×104 +[63] +[4, 31] +CO +Carbon Monoxide +28 +REU +6.60×104 +[63] +[4, 31] +N2 +Nitrogen +28 +REU +6.29×104 +[63] +[4, 31] +CH2O +Formaldehyde +30 +NIST +3.21×104 +[63] +[5, 31] +NO +Nitric Oxide +30 +NIST +5.28×104 − +[31] +C2H6 +Ethane +30 +REU +6.94×104 +[63] +[5, 31] +CH5N +Methylamine +31 +NIST +4.07×104 +[63] +[31] +CH3OH +Methanol +32 +NIST +4.17×104 +[63] +[4, 31] +H2S +Hydrogen Sulfide +34 +NIST +5.38×104 +[63] +[5, 31] +PH3 +Phospine +34 +NIST +5.98×104 − +[31] +O2 +Oxygen +36 +NIST +5.03×104 +[63] +[4, 31] +36Ar +Argon 36 +36 +REU +7.87×104 +[18] +[4, 31] +C3H4 +Propyne +40 +REU +4.19×104 +[63] +[5, 31] +40Ar +Argon 40 +40 +REU +7.29×104 +[19] +[5, 31] +CH3CN +Acetonitrile +41 +REU +5.28×104 +[63] +[31, 50] +C2H2O +Ketene +42 +NIST +4.37×104 +[63] +[31] +C3H6 +Propene +42 +NIST +4.06×104 +[63] +[5, 31] +CH2N2 +Cyanamide +42 +NIST +7.55×104 − +Organic synthesis [59, 61] +C2H4O +Acetaldehyde +44 +NIST +4.42×104 +[63] +[5, 31] +C3H8 +Propane +44 +REU +5.11×104 +[63] +[6, 31] +CO2 +Carbon Dioxide +44 +REU +7.09×104 +[63] +[4, 31] +C2H7N +Ethylamine +45 +NIST +1.71×104 +[63] +[31] +CH3NO +Formamide +45 +NIST +6.22×104 +[63] +HCN decomposition [47] +C2H6O +Ethanol +46 +NIST +1.62×104 +[63] +[5, 31] +CH2O2 +Formic Acid +46 +NIST +2.94×104 +[63] +HCN decomposition [47] +C4H8 +1-Butene +56 +NIST +3.73×104 +[63] +[5, 31] +C2H6N2 +Azomethane +58 +NIST +8.46×104 − +[31] +C3H6O +Acetone +58 +NIST +6.70×104 +[63] +[5, 31] +C4H10 +Isobutane +58 +NIST +3.24×105 +[63] +[5, 31] +C2H4O2 +Acetic Acid +60 +NIST +4.51×104 +[63] +[5, 31] +C3H8O +1-Propanol +60 +NIST +8.80×105 +[63] +[5, 31] +C2H7NO +Monoethanolamine 61 +NIST +1.67×103 − +[31] +C2H6O2 +1,2-Ethanediol +62 +NIST +5.77×105 +[63] +[5, 31] +C5H10 +Cyclopentane +70 +NIST +4.39×104 +[63] +[31] +C4H9N +Pyrrolidine +71 +NIST +1.18×103 − +[31] +C5H12 +Pentane +72 +NIST +1.53×104 +[63] +[5, 31] +C4H10O +1-Butanol +74 +NIST +6.14×105 − +[31] +C2H5NO2 +Glycine +75 +NIST +6.14×105 +[63] +[5, 31] +C3H5Cl +Allyl Chloride +76 +NIST +9.63×104 − +[31] +C5H9N +Butyl Isocyanide +83 +NIST +8.34×104 − +Hydrocarbon irradiation [50] +C4H6O2 +2,3-Butanedione +86 +NIST +3.91×104 +[63] +[5, 31] +C3H7NO2 +Alanine +89 +NIST +1.73×103 − +[31] +C8H18 +Octane +114 +NIST +1.65×103 − +[31] +C6H12N4 +Methenamine +140 +NIST +3.86×103 − +[31] +C6H14N2O2 Lysine +146 +NIST +1.34×103 − +Amino acid [69] + +11 +a +b +c +H2 +correction for 28 u fragments +Extended Data Fig. 1. Dataset corrections and minimal model fit. (a) The black silhouette shows the full mass range of +the INMS spectrum used in this work. Shaded gray bars indicate count values that differ from the spectrum presented in +ref. [12] (see Methods). The noise floor (dashed line) is estimated from the count rates of noisy mass channels > 46 u as ϵ = 2. +Blue circles show the results of fitting the minimal model consisting of the “confirmed” species in Table 1. (b) Scatterplot +of the standardized residuals produced by fitting the minimal model. There is no discernable pattern amongst the residuals +or evidence of heteroscedasticity. (c) Histogram of the standardized residuals (blue bars) compared to a reference Gaussian +distribution with zero mean (black curve). The residuals show good agreement with the Gaussian distribution, indicating a +robust model fit. + +12 +SUPPLEMENTARY RESULTS +Analysis of pairwise collinearity +As discussed in the main text, compositional ambigui- +ties within INMS spectra arise due to the large number of +candidate species combined with the relatively low mass +resolution of the instrument. In other words, models of +the plume’s composition contain many possible model +components with comparatively few data points avail- +able to constrain them. +This difficulty is accentuated +when there exist combinations of multiple species that +can reproduce the signal produced by another. If this +collinearity is exact, discriminating between such singu- +lar species in the composite INMS spectrum is impossi- +ble. In practice, even approximately collinear mass spec- +tra can present a significant obstacle when interpreting +data with a finite mass resolution. +These issues have +been encountered in previous studies [5, 6, 31] and have +greatly hindered efforts to identify trace compounds in +the plume. +Here, we quantify the extent of pairwise collinearity +amongst library spectra by computing the correlation +matrix for each species’ cracking pattern. We use ρ to de- +note correlation coefficients of cracking patterns, in con- +trast to r (defined in the main text), which signifies cor- +relations between regression coefficients. Whereas r is a +model dependent quantity (see Fig. 2 of the main text), +ρ is a static property of the spectral library. +Supplementary Fig. 2 demonstrates that large positive +correlations are common and manifest between molecules +with similar cracking patterns. A correlation coefficient +between two species of ρ = 1 would imply identical mass +spectra. Pairs of species with correlation coefficients close +to 1 are approximately collinear and may still be indis- +tinguishable in practice. +By contrast, pairs of species +with correlation coefficients close to 0 lack a consistent +pattern of strong overlapping features in their cracking +patterns. Values near 0.5 represent the intermediate case +where two species share certain features but also possess +additional large mass peaks that are not shared. NH3 +and CH4, for example, have a correlation coefficient of +0.47, owing to their shared peak at mass 16 and unshared +peaks at masses 17 (NH3) and 15 (CH4). Of course, large +negative values are also possible, though they are effec- +tively absent from the spectral library due to the inher- +ent structural similarities between organic compounds. +Large negative values would signify that one species has +significant mass peaks predominantly at mass channels +where another species does not. +For compounds with cracking patterns dominated by +only a few major peaks, sharing as few as one of these +peaks can lead to high correlation coefficients. A notable +example is CO2 and alanine (C3H7NO2), which exhibit +a correlation coefficient of ρ = 0.97. Based on the results +of the main text, it is tempting to view the presence of +alanine in 13 of 157 high likelihood models as the first ten- +tative evidence for amino acids at Enceladus. However, +at such low concentrations, alanine is virtually indistin- +guishable from CO2. Although the cracking pattern for +alanine contains counts at many different mass channels, +the peak at 44 u is by far the dominant feature. As a re- +sult, this mass channel acts as a high leverage point and +drags up correlations between alanine and other species +with large peaks at 44 u. This effect is particularly pro- +nounced when there are no other major peaks present, +as is the case for CO2. Supplementary Fig. 3 shows the +mean contribution of alanine to the INMS spectrum cal- +culated based on the multi-model averaging procedure +presented in the main text. An equal amount of CO2 is +shown for comparison. All peaks are well within the as- +sociated 1σ uncertainty for each mass channel. The base +peak is the only feature that extends above the noise floor +and mimics the signal for CO2 at similar concentrations. +This ambiguity precludes the detection of trace amounts +of alanine in the plume. +Similar effects underlie the ambiguities amongst the al- +cohols and mass 43 fragments discussed in the main text. +Large positive correlations amongst the alcohols (as high +as ρ = 0.91) manifest via high count rates at 31 u, corre- +sponding to the hydroxymethyl group CH2OH+. Species +with prominent 43 u fragments exhibit correlations as +high as ρ = 0.95 and could be attributable to a number +of different structures (see Supplementary Fig. 4 and the +Supplementary Discussion section below). +Ambiguities of this nature might have important impli- +cations for the detection of amino acids on future space- +craft missions. +The high pairwise correlation between +CO2 and alanine suggests that alanine likely cannot be +independently detected at Enceladus using a 1 u resolu- +tion mass spectrometer (such as INMS). Instead, an inde- +pendent measurement of the CO2 mixing ratio with pre- +cision at least as great as the instrument used to detect +alanine would be necessary. A similar argument would +apply to glycine (C2H5NO2) in an environment with high +NO abundance (ρ = 0.99) due to the large peak at mass +30. +Ramifications for hypothesis testing +Extensive amounts of collinearity between model com- +ponents can have profound consequences on statistical in- +ference. Although one might expect traditional hypothe- +sis testing to identify important model parameters, high- +dimensionality and collinearity of the spectral library +limit statistical power and prevent individual regression +coefficients from achieving statistical significance. Sup- +plementary Table 2 demonstrates this phenomenon for +the low velocity INMS data set. An F-test for overall sig- +nificance indicates that the spectral library, in aggregate, + +13 +36Ar +40Ar +C2H2 +C2H2O +C2H4 +C2H4O +C2H4O2 +C2H5NO2 +C2H6 +C2H6N2 +C2H6O +C2H6O2 +C2H7N +C2H7NO +C3H4 +C3H5Cl +C3H6 +C3H6O +C3H7NO2 +C3H8 +C3H8O +C4H10 +C4H10O +C4H6O2 +C4H8 +C4H9N +C5H10 +C5H12 +C5H9N +C6H12N4 +C6H14N2O2 +C8H18 +CH2N2 +CH2O +CH2O2 +CH3CN +CH3NO +CH3OH +CH4 +CH5N +CO +CO2 +H2O +H2S +HCN +N2 +NH3 +NO +O2 +40Ar +C2H2 +C2H2O +C2H4 +C2H4O +C2H4O2 +C2H5NO2 +C2H6 +C2H6N2 +C2H6O +C2H6O2 +C2H7N +C2H7NO +C3H4 +C3H5Cl +C3H6 +C3H6O +C3H7NO2 +C3H8 +C3H8O +C4H10 +C4H10O +C4H6O2 +C4H8 +C4H9N +C5H10 +C5H12 +C5H9N +C6H12N4 +C6H14N2O2 +C8H18 +CH2N2 +CH2O +CH2O2 +CH3CN +CH3NO +CH3OH +CH4 +CH5N +CO +CO2 +H2O +H2S +HCN +N2 +NH3 +NO +O2 +PH3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Supplementary Fig. 2. Correlation matrix for the spectral library. The correlation coefficient (−1 ≤ ρ ≤ 1) measures the +extent of collinearity between two cracking patterns. Large positive values (brighter colors) indicate high levels of collinearity, +whereas smaller values signify less collinearity. Large negative values (darker colors) are essentially absent, indicating that most +collinear species are positively correlated. +does indeed better explain the INMS data than does the +“null model,” which consists of fitting only the regression +coefficient for H2O (F = 10; p = 9.1 × 10−13). However, +one-tailed t-tests indicate that only the two strongest +spectral features, H2O and CO2, are individually sta- +tistically significant (p = 3.2 × 10−79 and 1.9 × 10−4, +respectively) in the presence of the entire spectral li- +brary. These species account for the majority of counts at +masses 18, 28, and 44 and produce signals that are large +enough to stand out amongst the majority of collinear +species that comprise the rest of the library. +Though +we can be confident that the aggregate set of candidate +library species is capable of explaining the INMS data, +the statistical significance of any one species is difficult +to show using such frequentist statistics under these con- +ditions. +SUPPLEMENTARY DISCUSSION +Comparison to other results +It is important to consider the entire body of statistical +evidence when drawing conclusions about which species + +14 +10 +3 +10 +1 +10 +1 +10 +3 +C3H7NO2 +Noise Floor +1 Uncertainty +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +10 +3 +10 +1 +10 +1 +10 +3 +CO2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Mass (u) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Counts +Supplementary Fig. 3. Mass spectra for alanine (C3H7NO2) +and CO2. The x-axis shows the mass of each fragmentation +product, while the y-axis quantifies the proportional abun- +dance of each fragment scaled to the calculated mixing ratio +for alanine (note the log-scale). All spectral features (green +bars) are masked by the associated count uncertainty at each +mass channel (shaded blue region). Both species have a sin- +gle dominant peak at 44 u that extends above the noise floor +(dashed black line). +0.00 +0.25 +0.50 +0.75 +1.00 +C2H6O2 +C4H10 +10 +20 +30 +40 +50 +60 +70 +0.00 +0.25 +0.50 +0.75 +1.00 +C3H8O +10 +20 +30 +40 +50 +60 +70 +C5H12 +Mass (u) +Counts +Supplementary Fig. 4. Mass spectra for representative alco- +hols and species with 43 u fragments. Blue bars (left) show +representative alcohols, C2H6O2 and C3H8O, with ρ = 0.91. +Red bars (right) show representative species with strong 43 u +signatures, C4H10 and C5H12, with ρ = 0.95. +are detected in the plume. The model-averaged mixing +ratios, the minimum AICc model, the individual species +probabilities, and the relative model likelihoods can all be +used to assess the confidence of each detection. The more +heuristics that point towards the detection of a given +species, the more confident one may be that said species +is truly present in the plume. +The minimum AICc model consists of 11 species. One +interpretation of this result is to classify all 11 species +as conclusive detections. +However, more parsimonious +models exist that explain the INMS data nearly as well. +A more conservative approach would be to interpret only +the species that comprise the best-fitting, least complex +model with λ > 1/e as essential to the fit. We favor an +even more nuanced approach and suggest using a tiered +Supplementary Table 2. Results of hypothesis testing on the +INMS data set. High dimensionality and collinearity prevent all +but the most prominent spectral features from achieving statis- +tical significance. +Species +t-Statistic p-Value +H2O +378 +3.2 × 10−79 +CO2 +3.9 +1.9 × 10−4 +All others < 1.1 +> 0.13 +hierarchy of confidence based on the holistic analysis of +the main text. +Below, we contextualize these results +within the collection of previous studies on the composi- +tion of Enceladus and other icy bodies. +In the main text, we present an upper limit on the +NH3 mixing ratio that is consistent with previous analy- +ses of the slow flyby INMS spectra [4, 31] as well as those +obtained during the E2 [6] and E3 [5] flybys. In their +supplementary material, Waite et al. [4] argue, based on +the slow flyby data, that the residual left at mass 16 by +fitting H2O, CH4, and CO2 alone unambiguously indi- +cates the presence of NH3. For the spectrum analyzed +in the main text of this work, we find that this residual +is equal to ∼880 counts, corresponding to ∼1.1σ at 16 u +(see Fig. 4a and Methods of the main text). However for +independent Gaussian uncertainties, it is expected that +∼16% of all mass channels would be undercounted by +> 1σ due to random chance alone (see also Extended +Data Fig. +1 of the main text). +As such, the residual +at 16 u is not enough to imply the unambiguous detec- +tion of NH3. +Of course, this conclusion is dependent +on the estimation of count uncertainties in INMS data, +for which multiple different procedures have been pro- +posed [4, 5, 18, 19]. Still, other instruments including +the Cassini Visible and Infrared Mapping Spectrometer +(VIMS), Ultraviolet Imaging Spectrograph (UVIS), and +Cosmic Dust Analyzer (CDA) have failed to find con- +clusive evidence of NH3 at Enceladus and are unable to +corroborate its presence in the plume [31, 65, 70]. Tele- +scopic data have suggested the presence of NH3 or NH3- +hydrate [71–74], though these observations are also not +definitive [75]. Consequently, additional studies that fur- +ther constrain the INMS count rate at mass channel 16 +beyond the typical 30% uncertainty suggested in ref. [4] +are required before NH3 can be confirmed in the plume. +As discussed in the main text, suggestive evidence for +nitrogen at Enceladus in the form of HCN has been previ- +ously reported based on other INMS data sets. HCN has +also been suggested to help explain yet unresolved sig- +natures in Cassini CDA spectra of ice grains in Saturn’s +E-ring [7], though could not be definitively identified by +CDA as a cation species due to its high ionization poten- +tial [10]. As such, the HCN mixing ratio determined in +this work is the first definitive detection of nitrile chem- +istry at Enceladus. +Our analysis shows that CH2O and C2H2 are also + +15 +present in the plume with concentrations exceeding 100 +ppm. As for HCN, both species have been detected in +comets [35] and circumstantial evidence for their exis- +tence at Enceladus has been previously suggested based +on other Cassini flybys. Both CH2O and C2H2 were sus- +pected by Waite et al. [5] during the high-velocity E3 +and E5 flybys, though correlations between abundance +and spacecraft velocity suggest that impact fragmenta- +tion may have been responsible. Only upper limits for +both species were reported on a reanalysis of the slower +E2 flyby [5]. The presence of CH2O would also help ex- +plain features at 28-30 u seen in CDA ice grain spectra +[31], which are consistent with the INMS data. +Concerning the higher mass organics, we find that +trace amounts (∼40 ppm) of C3H6 are present in the +plume as well. C3H6 is present on Titan [19] and was +produced as a fragmentation product during the E3 and +E5 flybys of Enceladus [5]. Although not initially identi- +fied in the E2 flyby data [6], reanalysis suggests that it +may have been present [5]—although it was not identified +by Waite et al. [4, 31] as an intrinsic plume constituent. +Our analysis shows that C3H6 is by far the most likely +explanation for the majority of counts in the 37-42 u re- +gion of the slow flyby INMS spectrum (Fig. 4d of the +main text) and provides the first conclusive evidence for +native C3 organics in the plume. +The O2 mixing ratio reported in the main text is con- +sistent with the limit on native mass 32 counts imposed +by Waite et al. [4] after correcting for the surface pro- +cessing of H2O in the INMS instrument antechamber. +Interestingly, the source of O2 at Enceladus presents a +few challenges. Unlike Europa, where charged particle +bombardment of the surface is known to drive radioly- +sis of water and other elements to O2, H2O2, SO2− +4 , and +other oxidants [50, 76–78], the radiation flux near Ence- +ladus is considerably lower [79], and evidence for oxidants +on the surface is lacking, despite proposals of such radi- +olytic chemistry [80]. Even with moderate levels of sur- +face radiolysis, a key problem would be the efficiency of +oxidant production at the high temperatures observed in +the South Polar Terrain [81, 82]. Here we do not pro- +pose a solution for this issue but rather note that our +results are consistent with the presence of O2, be it from +surface radiolysis and subsequent delivery to the ocean, +or via other production mechanisms. Notably, Waite et +al. [4] found that radiolysis of H2O due to radioactive +isotopes in Enceladus’ core could also produce O2 at the +abundance reported here. +Evidence for 40Ar stems from the large ∼2.4σ resid- +ual at 40 u, which is the only mass channel with strong +enough signal to significantly influence the calculation of +its mixing ratio. +This signal does have strong overlap +with the cracking pattern of C3H4 (ρ = 0.75), but the +abundance of this species is limited by the larger contri- +bution of C3H6 at neighboring mass channels. Though +not identified in a previous analysis of the slow flybys +[4], 40Ar was detected during the E3 and E5 flybys and +may indicate significant water-rock interactions and the +leaching of salts within Enceladus [5, 54]. CH3CN also +has a reasonably high correlation with 40Ar (ρ = 0.47) +and could potentially contribute to the observed signal at +mass 40. Although there is no prior published evidence +for CH3CN at Enceladus, its presence would not be unex- +pected based on the evidence for HCN and hydrocarbons +such as C3H6 (see Discussion in the main text). +Native alcohols have not been previously identified in +the plume, but CH3OH has been suggested as a possible +fragmentation product based on high velocity INMS and +ice grain spectra [12]. CH3OH has also been observed via +ground-based methods in the vicinity of Enceladus and +could be produced through chemical processing of CH4 +in the nearby gas cloud [83] or by the partial combustion +of endogenous CH4 within the ocean. Both CH3OH and +C2H6O2 have been detected in relatively high abundance +in several comets [84, 85]. +Our results also provide evidence for an ambiguous +species with a strong 43 u signal. +Ambiguity at this +mass channel was previously reported based on the E5 +flyby of Enceladus [5]. One explanation of this signal is +C2H6N2. Although this species has an additional large +peak at 15 u, this feature is masked by the much larger +contribution from CH4 in the INMS data. +The corre- +lated ice grain features at 15 and 43 u seen in “Type II” +CDA spectra [10] might be explained by fragmentation of +C2H6N2 into CH+ +3 and CH3N+ +2 . Alternative explanations +including acetyl group-bearing species such as C3H6O or +C4H6O2 are also possible. +Sulfur compounds have not been definitively identi- +fied at Enceladus, though a past detection of H2S based +on the E5 flyby [5] supports our finding that it may be +present. H2S would be expected if there is active serpen- +tinization taking place on the ocean floor. +For the remaining species listed in Table 1 of the main +text, prior evidence for their existence in the plume is +lacking. +Phosphorous compounds have not been pre- +viously reported in the plume, though PH3 may have +been observed in the coma of comet 67P/Churyumov- +Gerasimenko [86]. C3H5Cl has also not been identified +at Enceladus, but Cl has been detected by CDA as NaCl +and KCl salts residing in plume ice grains [8, 9]. Lastly, +although there is no strong evidence for alanine at Ence- +ladus (see the Supplementary Results section above), we +note that alanine and other amino acids are abundant in +carbonaceous chondrites [69]. +Comparison to other methodologies +In +order +to +adequately +account +for +the +high- +dimensionality and (approximate) collinearity of the +INMS plume dataset, it is necessary to perform a type +of variable selection that constrains the parameter space + +16 +of possible model fits. Such variable selection techniques +trade off a small increase in model bias for a significant +reduction in model variability [13, 17]. +The resulting +models are far less likely to overfit noisy features in the +training data and tend to be significantly more accurate +in predicting future observations [13]. Moreover, variable +selection reduces the impact of collinearity by identify- +ing which model components are better at explaining the +observed data and discarding those that are superfluous. +Such a process then allows for the evaluation of individ- +ual model components without the confounding presence +of their collinear counterparts. +In the main text, we outlined two variable selection +procedures: an exhaustive best subset selection for mod- +els with fewer than 15 species and a forward stepwise +selection algorithm for more complex models. +Other +common algorithms such as ridge regression [87] and +the Least Absolute Shrinkage and Selection Operator +(LASSO) are frequently applied in a wide variety of ma- +chine learning and model validation contexts [13, 17, 88– +90]. These methods seek to reduce the influence of ex- +traneous parameters via L2 or L1 regularization, respec- +tively. Multi-model averaging is similar to L1 regular- +ization in that it allows for explicit dimensionality re- +duction, whereas L2 regularization does not. This prop- +erty leads to high model interpretability, which is of ut- +most importance when performing compositional analy- +ses. Other heuristics besides the AICc can be used to +select models for averaging, but these alternative statis- +tics are not based on minimizing information loss and +are therefore not well-suited for model selection when +the structure of the unknown, true distribution is poorly +constrained. +Furthermore, model inference using the +AICc has been shown to asymptotically approximate re- +sults based on cross-validation (another broadly accepted +model validation technique) while requiring much fewer +computational resources [27, 91, 92]. +The first few studies of INMS data collected at Ence- +ladus produced landmark results, including the charac- +terization of major plume constituents and the discovery +of molecular H2 as a potential indicator of hydrothermal +activity [4–6]. These papers (and related works published +throughout the duration of the Cassini mission) also doc- +umented a detailed description of the INMS instrument +response under varying spacecraft conditions and laid the +groundwork for follow-up studies focused solely on com- +positional analyses. However, early studies of the Ence- +ladus plume—though foundational—may not have been +well-suited to resolve minor species ambiguities for var- +ious reasons. In order to facilitate comparison with our +methodology, we briefly describe the spectral deconvo- +lution procedure developed by the authors of ref. [19] +that has been implemented in various other works (e.g., +refs. [20–22]). +In their procedure, the authors first determine the con- +tributions from major species through a visual analysis of +prominent spectral features. Mixing ratios for the major +species are estimated from the base peaks of each species, +assuming they contribute 100% of the measured counts +at these mass channels. Minor species are then identified +sequentially by subtracting their contributions from the +total spectrum to produce a residual spectrum. For small +portions of the spectrum where a few candidate species +exhibit overlaying signatures, species are fit based on a +custom-defined fit statistic using a grid search algorithm. +Iterative minimization of the fit statistic is achieved nu- +merically by sweeping through various mixing ratios at +increasingly finer resolution. For species that share the +same base peak (e.g., N2 and C2H4), the fit statistic is +manipulated to exclude this mass channel. The high com- +putational intensity of the grid search algorithm prohibits +fitting more than four species at a time. +We believe that the methodology of ref. [19] described +above, though useful and effective, may not be optimal +for identifying minor species in the INMS data. The order +in which species are subtracted from the initial spectrum +could potentially influence the outcome of the analysis. +Although it is true that a bias-variance tradeoff can be +useful for combatting high dimensionality, the described +procedure is not amenable to quantitative assessments of +inter-model uncertainty. Indeed, the authors note that +subjectivity of their analysis is a valid concern. Further- +more, the practice of fitting individual species to small +portions of the spectrum neglects potential contributions +from complex compounds with cracking patterns that +span a large mass range. Moreover, grid searches that +consider only a few species at a time may not be able to +reliably identify minor species when the spectral library +is highly collinear. In this regime, covariances between +mixing ratios become strongly model dependent (see Fig. +2 of the main text), and multi-model inference based on +model-averaged parameters is warranted. Additionally, +to our knowledge, the fit statistic described in ref. [19] is +not a standard metric, and we suspect that the use of dif- +ferent fit statistics for different model parameters could +lead to difficulties in interpreting the results. Lastly, the +authors’ procedure does not employ dimensionality re- +duction or account for the possibility of over-fitting to +noise. +By contrast, our approach quantitatively addresses +both inter- and intra-model uncertainty in the spectral +decomposition of INMS data. +While previous studies, +such as those presented in refs. [31, 33], have concluded +that minor species identification requires a higher res- +olution mass spectrometer, we have presented a math- +ematical framework capable of discriminating between +previously ambiguous species. The heuristics used in this +analysis are based on maximum likelihood estimation and +relative entropy minimization—foundational principles of +statistical inference and information theory. +Nevertheless, this study is not without limitations. A +major challenge for any compositional analysis of INMS + +17 +data stems from the large number of candidate plume +species. The chemistry of the ocean and ice shell could +include hundreds to thousands of unique compounds that +contribute to the observed INMS spectrum. +Our ap- +proach using the AICc is based on the principle of par- +simony in that the least-complex, best-fitting model is +favored over similarly performing models of higher com- +plexity. Although this is a fruitful approach to developing +conservative models of plume composition, nature does +not necessarily reflect this ideal. Future investigations +using higher resolution mass spectrometers with larger +mass ranges will shed light on the full extent of chemical +diversity within the plume and the ocean beneath. +A number of interesting follow-up studies could be con- +ducted to validate the results presented in this work. A +strong approach would be to treat each of the slow flybys +(E14, E17, and E18) as individual data sets, as opposed +to averaging them together. +Machine learning models +could then be trained on one data set and evaluated on +another. A sort of round-robin procedure could be used +to estimate the uncertainty associated with training on +a particular data set. Such a methodology would elimi- +nate the need for heuristic statistics such as the AICc in +favor of actual independent test set performance. This +implementation would, however, require correcting for +instrument artifacts in each of the individual Enceladus +flybys. +∗ jonahpeter@g.harvard.edu +[1] M. K. Dougherty, K. K. Khurana, F. M. Neubauer, C. T. +Russell, J. Saur, J. S. Leisner, and M. E. Burton, Identi- +fication of a Dynamic Atmosphere at Enceladus with the +Cassini Magnetometer, Science 311, 1406 (2006). +[2] C. C. Porco, P. Helfenstein, P. C. Thomas, A. P. Ingersoll, +J. Wisdom, R. West, G. Neukum, T. Denk, R. Wagner, +T. Roatsch, S. Kieffer, E. Turtle, A. McEwen, T. V. John- +son, J. Rathbun, J. Veverka, D. Wilson, J. Perry, J. Spi- +tale, A. Brahic, J. A. Burns, A. D. DelGenio, L. Dones, +C. D. Murray, and S. Squyres, Cassini Observes the Ac- +tive South Pole of Enceladus, Science 311, 1393 (2006). +[3] C. J. Hansen, +L. W. Esposito, +A. I. F. Stewart, +B. Meinke, B. Wallis, J. E. Colwell, A. R. Hendrix, +K. Larsen, W. Pryor, and F. Tian, Water vapour jets +inside the plume of gas leaving Enceladus, Nature 456, +477 (2008). +[4] J. H. Waite, C. R. Glein, R. S. Perryman, B. D. Teolis, +B. A. Magee, G. Miller, J. Grimes, M. E. Perry, K. E. +Miller, A. Bouquet, J. I. Lunine, T. Brockwell, and S. J. +Bolton, Cassini finds molecular hydrogen in the Ence- +ladus plume: Evidence for hydrothermal processes, Sci- +ence 356, 155 (2017). +[5] J. H. Waite Jr, W. S. Lewis, B. A. Magee, J. I. Lunine, +W. B. McKinnon, C. R. Glein, O. Mousis, D. T. Young, +T. Brockwell, J. Westlake, M.-J. Nguyen, B. D. Teolis, +H. B. Niemann, R. L. McNutt Jr, M. Perry, and W.-H. Ip, +Liquid water on Enceladus from observations of ammonia +and 40Ar in the plume, Nature 460, 487 (2009). +[6] J. H. Waite, M. R. Combi, W.-H. Ip, T. E. Cravens, R. L. +McNutt, W. Kasprzak, R. Yelle, J. Luhmann, H. Nie- +mann, D. Gell, B. Magee, G. Fletcher, J. Lunine, and +W.-L. Tseng, Cassini Ion and Neutral Mass Spectrome- +ter: Enceladus Plume Composition and Structure, Sci- +ence 311, 1419 (2006). +[7] J. K. Hillier, S. F. Green, N. McBride, J. P. Schwanethal, +F. +Postberg, +R. +Srama, +S. +Kempf, +G. +Moragas- +Klostermeyer, J. A. M. McDonnell, and E. Grun, The +composition of Saturn’s E ring, Monthly Notices of the +Royal Astronomical Society 377, 1588 (2007). +[8] F. Postberg, J. Schmidt, J. Hillier, S. Kempf, and +R. Srama, A salt-water reservoir as the source of a com- +positionally stratified plume on Enceladus., Nature 474, +620 (2011). +[9] F. Postberg, S. Kempf, J. Schmidt, N. Brilliantov, +A. Beinsen, B. Abel, U. Buck, and R. Srama, Sodium +salts in E-ring ice grains from an ocean below the surface +of Enceladus, Nature 459, 1098 (2009). +[10] F. Postberg, S. Kempf, J. K. Hillier, R. Srama, S. F. +Green, N. McBride, and E. Gr¨un, The E-ring in the vicin- +ity of Enceladus: II. Probing the moon’s interior—The +composition of E-ring particles, Icarus 193, 438 (2008). +[11] J.-P. Combe, T. B. McCord, D. L. Matson, T. V. John- +son, A. G. Davies, F. Scipioni, and F. Tosi, Nature, dis- +tribution and origin of CO2 on Enceladus, Icarus 317, +491 (2019). +[12] F. +Postberg, +N. +Khawaja, +B. +Abel, +G. +Choblet, +C. R. Glein, M. S. Gudipati, B. L. Henderson, H.-W. +Hsu, S. Kempf, F. Klenner, G. Moragas-Klostermeyer, +B. Magee, L. N¨olle, M. Perry, R. Reviol, J. Schmidt, +R. Srama, F. Stolz, G. Tobie, M. Trieloff, and J. H. Waite, +Macromolecular organic compounds from the depths of +Enceladus, Nature 558, 564 (2018). +[13] G. James, D. Witten, T. Hastie, and R. Tibshirani, eds., +An introduction to statistical learning: with applications +in R, Springer texts in statistics No. 103 (Springer, New +York, 2013). +[14] B. Chandrasekaran and A. Jain, Quantization Complex- +ity and Independent Measurements, IEEE Transactions +on Computers C-23, 102 (1974). +[15] G. Hughes, On the mean accuracy of statistical pattern +recognizers, IEEE Transactions on Information Theory +14, 55 (1968). +[16] G. V. Trunk, A Problem of Dimensionality: A Simple +Example, IEEE Transactions on Pattern Analysis and +Machine Intelligence PAMI-1, 306 (1979). +[17] I. Guyon and A. Elisseeff, An introduction to variable and +feature selection., Journal of Machine Learning Research +3, 1157 (2003). +[18] J. Cui, R. V. Yelle, V. Vuitton, J. H. Waite, W. T. +Kasprzak, D. A. Gell, H. B. Niemann, I. C. F. M¨uller- +Wodarg, N. Borggren, G. G. Fletcher, E. L. Patrick, +E. Raaen, and B. A. Magee, Analysis of Titan’s neutral +upper atmosphere from Cassini Ion Neutral Mass Spec- +trometer measurements, Icarus 200, 581 (2009). +[19] B. A. Magee, J. H. Waite, K. E. Mandt, J. West- +lake, J. Bell, and D. A. Gell, INMS-derived composi- +tion of Titan’s upper atmosphere: Analysis methods and +model comparison, Planetary and Space Science 57, 1895 +(2009), publisher: Elsevier. +[20] J. H. Waite, H. Niemann, R. V. Yelle, W. T. Kasprzak, +T. E. Cravens, J. G. Luhmann, R. L. McNutt, W.- +H. Ip, D. Gell, V. De La Haye, I. M¨uller-Wordag, + +18 +B. Magee, N. Borggren, S. Ledvina, G. Fletcher, E. Wal- +ter, R. Miller, S. Scherer, R. Thorpe, J. Xu, B. Block, +and K. Arnett, Ion Neutral Mass Spectrometer Results +from the First Flyby of Titan, Science 308, 982 (2005). +[21] J. H. Waite, D. T. Young, T. E. Cravens, A. J. Coates, +F. J. Crary, B. Magee, and J. Westlake, The Process of +Tholin Formation in Titan’s Upper Atmosphere, Science +316, 870 (2007). +[22] J. H. Waite, R. S. Perryman, M. E. Perry, K. E. Miller, +J. Bell, T. E. Cravens, C. R. Glein, J. Grimes, M. Hed- +man, J. Cuzzi, T. Brockwell, B. Teolis, L. Moore, D. G. +Mitchell, A. Persoon, W. S. Kurth, J.-E. Wahlund, +M. Morooka, L. Z. Hadid, S. Chocron, J. Walker, +A. Nagy, R. Yelle, S. Ledvina, R. Johnson, W. Tseng, +O. J. Tucker, and W.-H. Ip, Chemical interactions be- +tween Saturn’s atmosphere and its rings, Science 362, +eaat2382 (2018). +[23] H. Akaike, Information Theory and an Extension of the +Maximum Likelihood Principle., in Proceedings of the +2nd International Symposium on Information Theory, +edited by B. N. Petrov and F. Csaki (Akademiai Kiado, +Budapest, 1973) pp. 267–281. +[24] C. M. Hurvich and C.-L. Tsai, Regression and time se- +ries model selection in small samples, Biometrika 76, 297 +(1989). +[25] Y. Sakamoto, M. Ishiguro, and G. Kitagawa, Akaike in- +formation criterion statistics (KTK Scientific Publishers, +Tokyo, 1986). +[26] N. Sugiura, Further analysts of the data by akaike’ s in- +formation criterion and the finite corrections, Communi- +cations in Statistics - Theory and Methods 7, 13 (1978), +publisher: Taylor & Francis. +[27] K. P. Burnham and D. R. Anderson, eds., Model Selec- +tion and Multimodel Inference (Springer New York, New +York, NY, 2004). +[28] H. Akaike, On the Likelihood of a Time Series Model, +The Statistician 27, 217 (1978). +[29] S. Kullback and R. A. Leibler, On Information and Suf- +ficiency, The Annals of Mathematical Statistics 22, 79 +(1951). +[30] J. B. Johnson and K. S. Omland, Model selection in ecol- +ogy and evolution, Trends in Ecology & Evolution 19, +101 (2004). +[31] F. Postberg, R. N. Clark, C. J. Hansen, A. J. Coates, +C. M. Dalle Ore, F. Scipioni, M. M. Hedman, and J. H. +Waite, Plume and Surface Composition of Enceladus, +in Enceladus and the Icy Moons of Saturn, edited by +P. Schenk (University of Arizona Press, Tucson, 2018) +pp. 129–162. +[32] H. T. Smith, M. Shappirio, R. E. Johnson, D. Reisen- +feld, E. C. Sittler, F. J. Crary, D. J. McComas, and +D. T. Young, Enceladus: A potential source of ammonia +products and molecular nitrogen for Saturn’s magneto- +sphere: SATURN’S MAGNETOSPHERIC NITROGEN +SOURCES, Journal of Geophysical Research: +Space +Physics 113, 10.1029/2008JA013352 (2008). +[33] W. B. McKinnon, J. H. Waite, O. Mousis, J. I. Lunine, +and M. Y. Zolotov, The Mysterious Origin of Enceladus: +A Compositional Perspective, in Enceladus and the Icy +Moons of Saturn (The University of Arizona Press, 2018). +[34] D. Bockel´ee-Morvan, J. Crovisier, M. J. Mumma, and +H. A. Weaver, Comets II, edited by M. Festou, H. U. +Keller, and H. A. Weaver (University of Arizona Press, +Tucson, 2004). +[35] M. J. Mumma and S. B. Charnley, The Chemical Compo- +sition of Comets—Emerging Taxonomies and Natal Her- +itage, Annual Review of Astronomy and Astrophysics 49, +471 (2011), publisher: Annual Reviews. +[36] R. L. Newburn, M. Neugebauer, and J. Rahe, eds., +Comets in the Post-Halley Era: In Part Based on Re- +views Presented at the 121st Colloquium of the Interna- +tional Astronomical Union, Held in Bamberg, Germany, +April 24–28, 1989, Astrophysics and Space Science Li- +brary, Vol. 167 (Springer Netherlands, Dordrecht, 1991). +[37] A. Affholder, F. Guyot, B. Sauterey, R. Ferri`ere, and +S. Mazevet, Bayesian analysis of Enceladus’s plume data +to assess methanogenesis, Nature Astronomy 5, 805 +(2021). +[38] T. M. Hoehler, Implications of H2/CO2 disequilibrium +for life on Enceladus, Nature Astronomy 6, 3 (2022). +[39] M. J. Russell and W. Nitschke, Methane: Fuel or Exhaust +at the Emergence of Life?, Astrobiology 17, 1053 (2017). +[40] M. Y. Zolotov, A model for low-temperature biogeochem- +istry of sulfur, carbon, and iron on Europa, Journal of +Geophysical Research 109, E06003 (2004). +[41] S. L. Miller, The mechanism of synthesis of amino acids +by electric discharges, Biochimica et Biophysica Acta 23, +480 (1957). +[42] L. E. Orgel, Prebiotic Chemistry and the Origin of +the RNA World, Critical Reviews in Biochemistry and +Molecular Biology 39, 99 (2004), publisher: Taylor & +Francis. +[43] J. Or´o and A. P. Kimball, Synthesis of purines under +possible primitive earth conditions. I. Adenine from hy- +drogen cyanide, Archives of Biochemistry and Biophysics +94, 217 (1961). +[44] B. H. Patel, C. Percivalle, D. J. Ritson, C. D. Duffy, +and J. D. Sutherland, Common origins of RNA, protein +and lipid precursors in a cyanosulfidic protometabolism, +Nature Chemistry 7, 301 (2015). +[45] R. Breslow, On the mechanism of the formose reaction, +Tetrahedron Letters 1, 22 (1959). +[46] H. J. Cleaves II, The prebiotic geochemistry of formalde- +hyde, Precambrian Research 164, 111 (2008). +[47] S. Miyakawa, H. J. Cleaves, and S. L. Miller, THE COLD +ORIGIN OF LIFE: A. IMPLICATIONS BASED ON +THE HYDROLYTIC STABILITIES OF HYDROGEN +CYANIDE AND FORMAMIDE, Origins of Life and Evo- +lution of the Biosphere 32, 195 (2002). +[48] S. L. Miller and L. E. Orgel, The origins of life on the +earth, Concepts of modern biology series (Prentice-Hall, +Englewood Cliffs, N.J, 1974). +[49] M. Levy, S. L. Miller, K. Brinton, and J. L. Bada, Pre- +biotic Synthesis of Adenine and Amino Acids Under +Europa-like Conditions, Icarus 145, 609 (2000). +[50] K. P. Hand, On the physics and chemistry of the ice shell +and sub-surface ocean of Europa, Ph.D. thesis, Stanford +University (2007). +[51] K. P. Hand, R. W. Carlson, and A. I. Tsapin, Labora- +tory Analysis Of Water, Hydrocarbon And Ammonia Ice +Mixtures Exposed To High-energy Electron Irradiation, +in Bulletin of the American Astronomical Society, Vol. 38 +(2006) p. 606. +[52] K. P. Hand, C. F. Chyba, J. C. Priscu, R. W. Carlson, +and K. H. Nealson, Astrobiology and the Potential for +Life on Europa, in Europa, edited by R. T. Pappalardo, +W. B. McKinnon, and K. K. Khurana (University of Ari- +zona Press, Tucson, 2009). + +19 +[53] H.-W. +Hsu, +F. +Postberg, +Y. +Sekine, +T. +Shibuya, +S. +Kempf, +M. +Hor´anyi, +A. +Juh´asz, +N. +Altobelli, +K. Suzuki, Y. Masaki, T. Kuwatani, S. Tachibana, S.- +i. Sirono, G. Moragas-Klostermeyer, and R. Srama, On- +going hydrothermal activities within Enceladus, Nature +519, 207 (2015). +[54] C. R. Glein, F. Postberg, and S. D. Vance, The Geo- +chemistry of Enceladus: Composition and Controls, in +Enceladus and the Icy Moons of Saturn (The University +of Arizona Press, 2018). +[55] C. P. McKay, A. Davila, C. R. Glein, K. Hand, and A. M. +Stockton, Enceladus Astrobiology, Habitability, and the +Origin of Life, in Enceladus and the Icy Moons of Saturn +(The University of Arizona Press, Tucson, 2018). +[56] M. Y. Zolotov, G. Tobie, F. Postberg, B. Magee, J. H. +Waite, and L. Esposito, Chemical and phase composi- +tion of Enceladus: Insights from Cassini data, in Eu- +ropean Planetary Science Conference, edited by E. Ab- +stracts (EPSC-DPS2011-1330, 2011) p. 6. +[57] E. Bl¨ochl, M. Keller, G. Wachtersh¨auser, and K. O. +Stetter, Reactions depending on iron sulfide and link- +ing geochemistry with biochemistry., Proceedings of the +National Academy of Sciences 89, 8117 (1992). +[58] G. W¨achtersh¨auser, Before enzymes and templates: the- +ory of surface metabolism, Microbiological Reviews 52, +452 (1988), publisher: American Society for Microbiol- +ogy. +[59] S. A. Benner, +E. A. Bell, +E. Biondi, +R. Brasser, +T. Carell, H. Kim, S. J. Mojzsis, A. Omran, M. A. +Pasek, and D. Trail, When Did Life Likely Emerge on +Earth in an RNA-First Process?, ChemSystemsChem 2, +10.1002/syst.201900035 (2020). +[60] D. D. Sasselov, J. P. Grotzinger, and J. D. Sutherland, +The origin of life as a planetary phenomenon, Science +Advances 6, eaax3419 (2020). +[61] M. W. Powner, B. Gerland, and J. D. Sutherland, Syn- +thesis of activated pyrimidine ribonucleotides in prebiot- +ically plausible conditions, Nature 459, 239 (2009). +[62] J. H. Waite, W. S. Lewis, W. T. Kasprzak, V. G. Ani- +cich, B. P. Block, T. E. Cravens, G. G. Fletcher, W.-H. +Ip, J. G. Luhmann, R. L. Mcnutt, H. B. Niemann, J. K. +Parejko, J. E. Richards, R. L. Thorpe, E. M. Walter, and +R. V. Yelle, The Cassini Ion and Neutral Mass Spectrom- +eter (INMS) Investigation, Space Science Reviews 114, +113 (2004). +[63] K. Miller, J. Waite, R. Perryman, M. Perry, A. Bouquet, +B. Magee, B. Bolton, T. Brockwell, M. Hedman, and +C. Glein, Cassini INMS constraints on the composition +and latitudinal fractionation of Saturn ring rain material, +Icarus 339, 113595 (2020). +[64] C. J. Hansen, E. L., S. A. I. F., C. J., H. A., P. W., +S. D., and W. R., Enceladus’ Water Vapor Plume, Science +311, 1422 (2006), publisher: American Association for +the Advancement of Science. +[65] R. H. Brown, C. N. Clark, B. J. Buratti, D. P. Cruik- +shank, B. J. W., M. R. M. E., B. J., N. S., M. T., B. K. +H., B. G., C. F., C. P., C. M., C. A., D. P., F. V., J. R., +L. Y., M. D. L., M. T. B., N. R. M., N. P. D., S. B., and +S. C., Composition and Physical Properties of Enceladus’ +Surface, Science 311, 1425 (2006), publisher: American +Association for the Advancement of Science. +[66] B. D. Teolis, H. B. Niemann, J. H. Waite, D. A. Gell, +R. S. Perryman, W. T. Kasprzak, K. E. Mandt, R. V. +Yelle, A. Y. Lee, F. J. Pelletier, G. P. Miller, D. T. Young, +J. M. Bell, B. A. Magee, E. L. Patrick, J. Grimes, G. G. +Fletcher, and V. Vuitton, A Revised Sensitivity Model for +Cassini INMS: Results at Titan, Space Science Reviews +190, 47 (2015). +[67] W. L. Fitch and A. D. Sauter, Calculation of relative +electron impact total ionization cross sections for organic +molecules, Analytical Chemistry 55, 832 (1983). +[68] V. Graves, B. Cooper, and J. Tennyson, Calculated elec- +tron impact ionisation fragmentation patterns, Journal +of Physics B: Atomic, Molecular and Optical Physics 54, +235203 (2021). +[69] D. P. Glavin, A. S. Burton, J. E. Elsila, J. C. Aponte, +and J. P. Dworkin, The Search for Chiral Asymmetry as +a Potential Biosignature in our Solar System, Chemical +Reviews 120, 4660 (2020). +[70] C. +Hansen, +L. +Esposito, +J. +Colwell, +A. +Hendrix, +G. Portyankina, A. Stewart, and R. West, The composi- +tion and structure of Enceladus’ plume from the complete +set of Cassini UVIS occultation observations, Icarus 344, +113461 (2020). +[71] M. Zastrow, J. T. Clarke, A. R. Hendrix, and K. S. Noll, +UV spectrum of Enceladus, Icarus 220, 29 (2012). +[72] W. Grundy, Near-Infrared Spectra of Icy Outer Solar +System Surfaces: +Remote Determination of H2O Ice +Temperatures, Icarus 142, 536 (1999). +[73] J. P. Emery, D. M. Burr, D. P. Cruikshank, R. H. +Brown, +and +J. +B. +Dalton, +Near-infrared +(0.8–4.0 +$\mathsf{\mu}$m) spectroscopy of Mimas, Enceladus, +Tethys, and Rhea, Astronomy & Astrophysics 435, 353 +(2005). +[74] A. J. Verbiscer, +D. E. Peterson, +M. F. Skrutskie, +M. Cushing, P. Helfenstein, M. J. Nelson, J. Smith, +and J. C. Wilson, Near-infrared spectra of the leading +and trailing hemispheres of Enceladus, Icarus 182, 211 +(2006). +[75] D. Cruikshank, T. Owen, C. Ore, T. Geballe, T. Roush, +C. Debergh, S. Sandford, F. Poulet, G. Benedix, and +J. Emery, A spectroscopic study of the surfaces of Sat- +urn’s large satellites: HO ice, tholins, and minor con- +stituents, Icarus 175, 268 (2005). +[76] K. P. Hand and M. E. Brown, KECK II OBSERVA- +TIONS OF HEMISPHERICAL DIFFERENCES IN H 2 +O 2 ON EUROPA, The Astrophysical Journal 766, L21 +(2013). +[77] J. +R. +Spencer +and +W. +M. +Calvin, +Condensed +O[TINF]2[/TINF] on Europa and Callisto, The Astro- +nomical Journal 124, 3400 (2002). +[78] S. K. Trumbo, M. E. Brown, and K. P. Hand, H 2 O 2 +within Chaos Terrain on Europa’s Leading Hemisphere, +The Astronomical Journal 158, 127 (2019). +[79] C. Paranicas, E. Roussos, N. Krupp, P. Kollmann, +A. Hendrix, T. Cassidy, R. Johnson, P. Schenk, G. Jones, +J. Carbary, D. Mitchell, and K. Dialynas, Energetic +charged particle weathering of Saturn’s inner satellites, +Planetary and Space Science 61, 60 (2012). +[80] C. Ray, C. R. Glein, J. H. Waite, B. Teolis, T. Hoehler, +J. A. Huber, J. Lunine, and F. Postberg, Oxidation pro- +cesses diversify the metabolic menu on Enceladus, Icarus +364, 114248 (2021). +[81] K. P. Hand and R. W. Carlson, H2O2 production by +high-energy electrons on icy satellites as a function of +surface temperature and electron flux, Icarus 215, 226 +(2011). +[82] J. R. Spencer, J. C. Pearl, M. Segura, F. M. Flasar, + +20 +A. Mamoutkine, P. Romani, B. J. Buratti, A. R. Hen- +drix, L. J. Spilker, and R. M. C. Lopes, Cassini Encoun- +ters Enceladus: Background and the Discovery of a South +Polar Hot Spot, Science 311, 1401 (2006). +[83] E. Drabek-Maunder, J. Greaves, H. J. Fraser, D. L. +Clements, and L.-N. Alconcel, Ground-based detection of +a cloud of methanol from Enceladus: when is a biomarker +not a biomarker?, International Journal of Astrobiology +18, 25 (2019), publisher: Cambridge University Press. +[84] N. Biver, D. Bockel´ee-Morvan, V. Debout, J. Crovisier, +J. Boissier, D. C. Lis, N. Dello Russo, R. Moreno, +P. Colom, G. Paubert, R. Vervack, and H. A. Weaver, +Complex organic molecules in comets C/2012 F6 (Lem- +mon) and C/2013 R1 (Lovejoy): detection of ethylene +glycol and formamide, Astronomy & Astrophysics 566, +L5 (2014). +[85] J. Crovisier, D. Bockel´ee-Morvan, N. Biver, P. Colom, +D. Despois, and D. C. Lis, Ethylene glycol in comet +C/1995 O1 (Hale-Bopp), Astronomy & Astrophysics +418, L35 (2004). +[86] K. Altwegg, H. Balsiger, A. Bar-Nun, J.-J. Berthelier, +A. Bieler, P. Bochsler, C. Briois, U. Calmonte, M. R. +Combi, H. Cottin, J. De Keyser, F. Dhooghe, B. Fiethe, +S. A. Fuselier, S. Gasc, T. I. Gombosi, K. C. Hansen, +M. Haessig, A. J¨ackel, E. Kopp, A. Korth, L. Le Roy, +U. Mall, B. Marty, O. Mousis, T. Owen, H. R`eme, +M. Rubin, T. S´emon, C.-Y. Tzou, J. Hunter Waite, +and P. Wurz, Prebiotic chemicals—amino acid and +phosphorus—in the coma of comet 67P/Churyumov- +Gerasimenko, Science Advances 2, e1600285 (2016). +[87] A. E. Hoerl and R. W. Kennard, Ridge Regression: Bi- +ased Estimation for Nonorthogonal Problems, Techno- +metrics 12, 55 (1970). +[88] R. R. Hocking, A Biometrics Invited Paper. The Analysis +and Selection of Variables in Linear Regression, Biomet- +rics 32, 1 (1976), publisher: [Wiley, International Bio- +metric Society]. +[89] F. Santosa and W. W. Symes, Linear Inversion of Band- +Limited Reflection Seismograms, SIAM Journal on Scien- +tific and Statistical Computing 7, 1307 (1986), publisher: +Society for Industrial and Applied Mathematics. +[90] R. Tibshirani, Regression Shrinkage and Selection via the +Lasso, Journal of the Royal Statistical Society. Series B +(Methodological) 58, 267 (1996), publisher: [Royal Sta- +tistical Society, Wiley]. +[91] M. Stone, An Asymptotic Equivalence of Choice of Model +by Cross-Validation and Akaike’s Criterion, Journal of +the Royal Statistical Society: Series B (Methodological) +39, 44 (1977). +[92] P. Stoica, P. Eykhoff, P. Janssen, and T. S¨oderstr¨om, +Model-structure selection by cross-validation, Interna- +tional Journal of Control 43, 1841 (1986). + diff --git a/x9E4T4oBgHgl3EQfxg23/content/tmp_files/load_file.txt b/x9E4T4oBgHgl3EQfxg23/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cd5d37b6100c768a369e094151e3c0188707a45b --- /dev/null +++ b/x9E4T4oBgHgl3EQfxg23/content/tmp_files/load_file.txt @@ -0,0 +1,1877 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf,len=1876 +page_content='Detection of HCN and diverse redox chemistry in the plume of Enceladus Jonah S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Peter,1, 2, ∗ Tom A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Nordheim,1 and Kevin P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hand1 1Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA 2Biophysics Program, Harvard University, Boston, Massachusetts 02115, USA The Cassini spacecraft discovered that Saturn’s moon Enceladus possesses a series of jets erupting from its South Polar Terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Previous studies of in situ data collected by Cassini’s Ion and Neutral Mass Spectrometer (INMS) have identified H2O, CO2, CH4, H2, and NH3 within the plume of ejected material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Identification of minor species in the plume remains an ongoing challenge, owing to the large number of possible combinations that can be used to fit the INMS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Here, we present the discovery of several new compounds of strong importance to the habitability of Enceladus, including HCN, CH2O, C2H2, and C3H6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Our analyses of the low velocity INMS data coupled with our detailed statistical framework enable discriminating between previously ambiguous species in the plume by alleviating the effects of high-dimensional model fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Together with plausible mineralogical catalysts and redox gradients derived from surface radiolysis, these compounds could potentially support extant microbial communities or drive complex organic synthesis leading to the origin of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Shortly after its arrival in the Saturn system, the Cassini spacecraft discovered intense plume activity at the mid-sized moon, Enceladus [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Cassini in situ and remote sensing observations have confirmed that the plume consists primarily of H2O gas [3–6] as well as H2O-ice grains that feed Saturn’s E-ring [7–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' CO2 has also been detected in both the gaseous phase within the plume itself [4–6] and as a condensate in plume de- posits on Enceladus’ surface [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In situ measurements of the plume’s neutral gas component made by Cassini’s Ion and Neutral Mass Spectrometer (INMS) further in- dicate the presence of CH4, H2, and NH3 within Ence- ladus’ subsurface ocean [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Although early publications identified several additional species within the plume [5], more recent work suggests that many of these compounds resulted from incidental high velocity impact fragmenta- tion of larger molecules within the instrument antecham- ber [4, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Analyses of plume material sampled during the lower velocity flybys of Enceladus (for which this frag- mentation was less significant) do imply the existence of additional plume species, however no study to date has been able to verify the identity of any other intrinsic com- pounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Difficulty in resolving minor plume constituents stems from the large number of plausible compounds relative to the low mass resolution of INMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' When training statis- tical models in this high dimensional regime, simple re- gression techniques tend to form overly complex models that produce specious results [13–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Models of INMS spectra suffer from an additional complexity in that the signals produced by individual molecules are not neces- sarily linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' As such, there may be mul- tiple different combinations of species that appear to fit the data equally well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The resulting large correlations between model components generally reduce model per- formance and can mask the importance of any particular component by limiting statistical power [13, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Studies that leave these issues unaddressed are likely to encounter model ambiguities that preclude reliable statistical infer- ence about the plume’s composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In this work, we seek to resolve the apparent composi- tional ambiguity of Enceladus’ plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' By characterizing the information content of the average low velocity spec- trum obtained during the E14, E17, and E18 flybys, we determine constraints on the number of species that can reliably be extracted from the INMS dataset under opti- mal spacecraft conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' We then use relative entropy minimization to assess the likelihood of tens of billions of potential models and show that multi-model inference al- lows for the identification of several new compounds not previously confirmed at Enceladus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Our results indicate the presence of a rich, chemically diverse environment that could support complex organic synthesis and possi- bly even the origin of life (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' CONSTRAINTS ON THE COMPLEXITY OF PLUME MODELS Deconvolving the overlapping signals of each species in the plume requires comparing features in the INMS flight data to a library of known mass spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Consequently, species in the plume cannot be identified unless explicitly included within the models used for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Previ- ous studies have constructed model INMS spectra using a variety of methods, including singular value decom- position [18] and the application of custom-defined fit statistics [19–22] (see Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In all cases, these fitting procedures seek to minimize the training error associated with the residual counts be- tween the INMS spectrum and the reconstructed model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Crucially, however, the training error is not an ap- propriate metric for evaluating model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In fact, it is a well-known concept in statistical modeling that the training error will continue to decrease with the inclusion of additional model components, regardless of arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='05259v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='EP] 12 Jan 2023 2 0 + 4 4 + 3 + 2 + 1 3 2 1 CH2O, C2H6N2 CH4 CO2 HCN C3H6, CH3OH C2H2, C2H6O2 Oxidation state Plume species H2 O2 H2 HCN O2 40Ar C3H6 CH2O C2H2 43 u fragments Alcohols a b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' New compounds identified in the Enceladus plume indicate a potentially habitable environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' (a) Jets emanating from ice fissures in Enceladus’ South Polar Terrain feed a plume of ejected material containing organic molecules with varying oxidation states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Electron bombardment of the surface might help facilitate the production of oxidants and prebiotic feedstock molecules observed in the plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' These compounds could potentially support biologically-mediated redox metabolisms or polymerize to form nucleic and amino acid precursors leading to the origin of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' (b) The average oxidation state of carbon for organic compounds confirmed or suspected in the plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Plume-derived H2 and O2 could act as strong reducing and oxidizing agents, respectively, and may be responsible for the diverse redox chemistry seen at Enceladus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' whether those components are genuinely related to the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Instead, it is necessary to approximate how each model would perform on a set of independent observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' This is the preferred method of model val- idation when additional data sets are unavailable for ex- plicit model testing [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Analysis techniques that utilize only the training error risk developing overly complex models and claiming false species detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Here, we evaluate model performance using the small sample bias-corrected Akaike Information Crite- rion (AICc) [23–27] which estimates the relative entropy between a given model and the unknown, true distribu- tion that produced the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' We construct each candidate model as a linear combination of end-member spectra representing the different cracking patterns of in- dividual molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A model M is given by, M : ˆyi = d � k=1 βkxk,i (1) where ˆyi is the total modeled counts in mass channel i, xk,i is the cracking pattern of species k at mass channel i, βk ≥ 0 is the regression coefficient for species k, and d is the total number of species in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The observed INMS counts at each mass channel, yi, are treated as independent data points with unequal Gaussian uncer- tainties, σi (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The AICc for each model is then given by, AICc = 2d − 2 ln[L(M0|y)] + 2d(d + 1) n − d − 1 (2) where n is the total number of mass channels and L(M|y) is the model’s likelihood function, L(M|y) = n � j=1 � 1 σj √ 2π � exp � n � i=1 �yi − ˆyi σi �2� (3) evaluated at the maximum likelihood estimate, M0, ob- tained by optimizing the set of {βk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' (2) is a correction factor that penalizes overly complex models when the sample size is small (n/d ≲ 40) [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The model with the minimium AICc (AICcmin) asymptotically approximates the model with the lowest Kullback-Leibler information loss relative to the observed data [27–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Whereas standard maximum likelihood estimation seeks to minimize the training error quanti- fied by the variance-weighted sum of squared residuals, �n i=1((yi − ˆyi)/σi)2, minimization of the AICc accounts for the bias introduced by estimating the regression co- efficients via the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The relative likelihood, or evidence ratio, of each model follows as [30], λ = exp {−(AICc − AICcmin)/2} (4) and can be used to compare models of differing complex- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Generally, models with λ < 1/e exhibit little to no 3889QQQ3 more redundancy less redundancy underfit overfit a b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Model performance as a function of model complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' (a) Maximum relative likelihoods across all models with d species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Blue circles indicate best-fit (highest λ) models identified via an exhaustive search for d < 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Orange circles represent models constructed using a benchmark forward modeling procedure extending to d = 50 (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' There is good agreement between the statistics obtained with forward modeling and the exhaustive search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Models with λ > 1/e (dashed line) exhibit strong predictive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' All models with d < 10 underfit the data, while those with d > 13 are overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' (b) The maximum magnitude of pairwise correlations rij = cov(βiβj)/ � var(βi)var(βj) between regression coefficients in the best-fitting models at each value of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Blue and orange circles represent the same models as in panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Correlations follow a logistic curve demonstrating model redundancy for d > 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The inflection point occurs within the optimal performance interval d ∈ [10, 13] characterized by λ > 1/e (green shaded area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' predictive power while those above this threshold form a family of most probable models suitable for multi-model inference [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' To capture the full range of possible plume con- stituents, we composed a large spectral library contain- ing the most recently published list of plausible INMS- detectable plume species [31] as well as several addi- tional compounds found in organic synthesis and labo- ratory experiments simulating icy satellites (see Meth- ods and Extended Data Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Using this library, we performed an exhaustive search up to d = 14 of tens of billions of potential models for the plume’s composi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 2a demonstrates that optimal model perfor- mance is achieved with only 10-13 species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The maxi- mum relative likelihood across all models peaks sharply at 11 species and drops precipitously for more complex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A benchmark forward modeling procedure ex- tending to d = 50 confirms this trend for large d (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Models with greater than 13 species dras- tically overfit the INMS data and incorrectly extract false signal from statistical noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Large correlations rij = cov(βiβj)/ � var(βi)var(βj) between regression co- efficients in these models indicate the erroneous inclusion of redundant species with similar mass spectra (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' This leads to overfitting and poor model performance, despite a monotonic decrease in the training error with increasing complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' By contrast, models with fewer than 10 species exhibit low correlations between model parameters, indicating the presence of additional features within the INMS spectrum that have not yet been fit by a candidate species (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', underfitting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Optimal perfor- mance occurs near the inflection point after all major data features have been explained but before redundant species are incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' NEW SPECIES DETECTED IN THE PLUME The most recently published list of neutral gas species confirmed in the plume consists only of H2O, CO2, CH4, H2, and NH3 [4, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' According to that work, no other compounds could be definitively detected by INMS dur- ing the low velocity flybys due to model ambiguities at low mixing ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Here, we account for those ambigu- ities via a multi-model averaging procedure that treats the minimum Kullback-Leibler divergence as a statistical random variable and weights each model by its probabil- ity of minimizing this information loss (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Whereas previous studies have relied on ad-hoc assess- ments of model ambiguity, the procedure here is rooted in fundamental concepts of information theory and explic- itly incorporates uncertainties resulting from the model- ing process into the standard error (SE) of each mixing ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In addition to H2O, CO2, and CH4, we find signifi- cant evidence—beyond 2 SE precision—for HCN, CH2O, C2H2, and C3H6 in the plume (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Our results are agnostic to the presence of H2 which requires analysis of additional INMS data not examined here (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' We also demonstrate strong evidence for 40Ar and O2 at 1 SE precision, as well as the probable detection of an alcohol (likely C2H6O2 or CH3OH), and an unidenti- 4 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Volume mixing ratios for the Enceladus plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Probabilities and mixing ratios are shown for all species with upper limits above the estimated INMS noise floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Values are calculated using a multi-model averaging procedure that incorporates uncertainties due to model ambiguities (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Mixing ratios are given as mean +/- SE and are scaled to incorporate the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='9% H2 number abundance reported in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [4] (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Upper limits (< 3 SE) are reported for species with mean values below their corresponding SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The minimal model is comprised of “confirmed” species with > 2 SE precision and represents the most conservative model of the plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Species listed in brackets are included in the alcohol mixing ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Species listed in parentheses are included in the 43 u fragment mixing ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Evidence Species Probabilityd Mixing Ratio (%)d Previous Limit (%)e Confirmed H2O 1 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='3 96 − 99 CO2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='8 CH4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='3 HCN > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='01 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='2 C2H2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='025 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='01 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='2 CH2O > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='026 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='01 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='2 C3H6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0041 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0020 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='01 H2 a − − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='4 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='9f Strong 40Ar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0014 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0009 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='01 Alcoholsb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='81 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='021 − [C2H6O2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='38 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='021 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='01 [CH3OH] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='31 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0041 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='01 43 u fragmentsc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='81 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0016 − (C3H6O) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='27 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0015 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='01 (C2H6N2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='26 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0016 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='01 O2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0027 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0020 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='01, < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='004f Moderate H2S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='44 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0031 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='01 PH3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='36 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0025 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='01 Poor C3H5Cl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='17 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0012 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='01 CH3CN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='15 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0028 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='01 NH3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='10 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='4 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='3 C3H7NO2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='08 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0014 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='01 a Determination of the H2 mixing ratio requires analysis of additional INMS data not examined here (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' b Other alcohols such as C3H8O, C2H6O, or C4H10O might also contribute to this mixing ratio (see Supplementary Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' c Other mass 43 fragments produced by C4H10, C4H6O2, C4H9N, C5H9N, C8H18, C5H12, or C2H4O2 might also contribute to this mixing ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' d Denotes values calculated in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' e Denotes values presented in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [31] unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' f Denotes values presented in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' fied 43 u fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Because the spectrum analyzed here is free of the major fragmentation signatures produced during the high velocity flybys, these species are most likely intrinsic to the plume itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In general, our mixing ratios are in excellent agreement with the most recently published limits [4, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The reconstructed model fit is plotted alongside the INMS spectrum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Interestingly, our results suggest that NH3 is not re- quired to fit the INMS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Although our upper limit of < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='58% is consistent with the lower end of the range reported by Waite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [4, 31], we conclude that high inter-model uncertainty stemming from large contribu- tions of CH4 and H2O at overlapping mass channels precludes the firm detection of NH3 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Some amount of NH3 is likely required to explain the pres- ence of nitrogen-bearing ions observed in Saturn’s mag- netosphere [32], but a stricter bound on the mass 16 count rate is needed before a percent-level concentration of NH3 from Enceladus can be presumed (see Supple- mentary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Notably, it has been shown that even a significantly smaller concentration of NH3 in the plume could still source the observed nitrogen abundance on Titan [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' We find that nitrogen is, however, definitively present at Enceladus in the form of HCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Previous studies have been unable to resolve the HCN abundance due to con- founding signals from fragmentation products at mass 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In their analysis of the E2 flyby, Waite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [6] quote an upper limit of < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='5%, whereas upper limits of < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='58% and < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='74% are given by Waite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [5], based on the high velocity E3 and E5 flybys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' For the faster flybys in particular, the authors note that the ele- vated count rate at 28 u introduces a model ambiguity at masses 27 and 28 (N2+HCN versus C2H4) that precludes the identification of HCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Here we correct the counts at mass 28 based on the INMS Open Source Neutral Beam (OSNB) data [4] to reveal that HCN is required at 27 u (see Methods, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 4b, and Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The mixing ratio reported here (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='04%) is consistent with that observed in cometary comae [34–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Average low velocity INMS spectrum and recon- structed model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The black silhouette shows the 12-47 u range of the average INMS spectrum obtained during the E14, E17, and E18 flybys adapted from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [12] with cor- rection for minor artifacts (see Methods and Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Counts at all other mass channels are at or below the estimated noise floor (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Error bars show 1σ Gaussian uncertainty in the observed count rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Red circles indicate the model fit based on the mixing ratios in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' We also report the first strong evidence for native O2 in the plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In their analysis of the low velocity fly- bys, Waite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [4] noted that mass 32 exhibits an el- evated count rate, likely in part due to surface process- ing between H2O and the Ti instrument antechamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Accounting for this instrument effect, they estimated a corrected mass 32 signal equal to (100/45) × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='004% = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0089% of the counts measured at mass 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In account- ing for additional species, our statistical analysis yields an excess at mass channel 32 that is best explained by an O2 mixing ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0027%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The value of mass 32 counts used in this work (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 4c) is well within the limit imposed by Waite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', and thus we argue that it is a true measure of native O2 from Enceladus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Our analysis also indicates moderate (Table 1) evi- dence for either H2S or PH3 in the plume, with probabil- ities of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='44 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='36, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' This conclusion stems primarily from masses 33 and 34, where the omission of these species leaves a small residual signal that cannot be explained by other library spectra (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' However, this signal alone is not enough to unambiguously con- clude their presence within the INMS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Both species improve the model fit when added to the minimal model (“confirmed” species in Table 1), but not when additional higher likelihood species are included (“strong” species in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Our results are therefore only suggestive of a 34 u compound in the plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Given the profound astro- biological implications of finding sulfur or phosphorous compounds at Enceladus, mass spectra for alternative 34 u species (such as H2O2) should be characterized exper- imentally and investigated in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' a b c d Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Contributions of individual species to the model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Black silhouettes show different mass ranges for the average low velocity INMS spectrum plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Error bars show 1σ Gaussian uncertainty in the observed count rates and dashed lines denote the INMS noise floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' (a) Blue cir- cles show the total contribution of H2O+CO2+CH4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Green circles show the contribution of NH3, which is not required for satisfactory model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' (b) Model contributions from CO2, HCN, CH2O, and C2H2 (magenta, cyan, orange, and purple circles, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' (c) Navy circles show the contribution from O2 added to the total contribution from alcohols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Pink circles show the tentatively proposed contri- bution of H2S+PH3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Low signal-to-noise ratios at these mass channels preclude the firm identification of H2S or PH3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' (d) Brown and indigo circles show contributions from C3H6 and 40Ar, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Teal circles show the total contribution from 43 u fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' DISCUSSION In aggregate, the results presented here indicate that Enceladus is host to a multiphasic and compositionally diverse chemical environment that is consistent with a habitable subsurface ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The new species identified in this work also suggest that this ocean may contain the necessary building blocks required to synthesize com- pounds important to the origin of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The detection of CH2O, O2, and an alcohol is par- ticularly interesting as these species potentially imply a diverse redox environment within the ocean (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Past measurements of CH4 and H2 in the plume [4–6] have supported the hypothesis that Enceladus may be hydrothermally active and could be a source of biologi- cally useful reductants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In particular, methanogenesis via the reduction of CO2 has been proposed as a potential pathway that could support extant microbial communi- ties near the sea floor [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' However, without addi- tional oxidants, reductants in the ocean would be of lit- tle biochemical utility, as no electron transfer mechanism beyond methanogenesis would be available to yield a neg- 6 ative change in Gibbs free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The detection of O2 and partially oxidized carbon compounds may solve this problem, as they provide a multitude of highly exergonic redox pathways that could help power life in Enceladus’ subsurface ocean (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [39, 40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Our results also provide the first conclusive evidence for HCN, CH2O, C2H2, and C3H6 in the plume, which, in addition to their significance for habitability, are particu- larly intriguing for their relevance to prebiotic chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' HCN polymerization is implicated in a number of poten- tial pathways for the formation of nucleobases and amino acids [41–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The autocatalysis of CH2O to form simple sugars and RNA precursors is also well-documented [45], as is its production on the early Earth [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Although these reactions might be limited within a dilute subsur- face ocean, concentrated conditions favorable for poly- merization could be achieved within the ice shell via eu- tectic freezing [42, 47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Indeed, Levy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [49] demon- strated the production of alanine, glycine, aspartic acid, and adenine from HCN and NH3 under conditions similar to those of icy satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Repeated freezing and thawing caused by the cycling of material between the ice shell and the warmer interior fissures of the plume vents may provide conditions favorable for organic synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In accordance with this scenario, there does exist ev- idence that the plume contains at least some material that has not been diluted by the ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Liquid water fa- cilitates the rapid hydrolysis of HCN into CH3NO, which subsequently decays into CH2O2 and NH3 [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The si- multaneous presence of ppt amounts of HCN and absence of CH3NO and CH2O2 suggests that the plume may be sourced, in part, by solid-phase material residing in the ice shell [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' HCN at Enceladus could be primordial in nature or produced via the radiolysis of nitrogen-bearing surface ices by magnetospheric electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Laboratory ex- periments of hydrocarbon-rich ices simulating the tem- perature and surface radiolysis conditions of Enceladus support the evidence for cyano group-containing species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hand [50] and Hand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [51] found that the warming of H2O+NH3+C3H6 ice films from 70 K to 300 K after irradiation by 10 keV electrons produces HCN and nitrile species, including CH3CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Irradiation of H2O+CO2 ice in the presence of hydrocarbons can also produce CH2O and alcohols [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Thus, the detection of HCN, CH2O, and an alcohol (as well as O2 and possibly CH3CN) in the plume could potentially be explained by the aerosoliza- tion of radiolytically processed material on or near the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Additional evidence from organic-rich ice grains detected in the plume further suggests the presence of solid-phase organic films that exist above the water ta- ble and are transported to the surface via the plume vents [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The macromolecular nature of these com- pounds, some in excess of 200 u, might be evidence of ongoing synthetic chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Accumulation and isolation of buoyant organic compounds at the cold ice-water in- terface between the ocean and ice shell may also promote longer lifetimes and inhibit hydrolysis [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Of course, the availability of organic compounds at Enceladus to support or facilitate the origin of life likely depends strongly on the geochemistry of the subsurface ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Although the mineral composition of the ocean floor is unknown, the simultaneous detections of SiO2 particles by CDA [53] and gaseous H2 by INMS [4] indi- cate the presence of a complex hydrothermal environ- ment [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Similarities between the inferred tempera- ture and pH of Enceladus’ ocean and those of the Lost City hydrothermal vents point towards serpentinization reactions as a possible source for the observed H2 abun- dance [4, 55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Such an explanation would also be consistent with the tentative evidence for H2S presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' If ferrous iron is present at Enceladus (as it is at the Lost City sites) along with H2S, sufficient reducing power may be available for a metabolic pathway to abio- genesis [42, 57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Laboratory evidence suggests that confirmation of C3H6 in the plume might allow for the formation of vesicle-type structures at Enceladus, which could then shelter the burgeoning proto-metabolism [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' An alternative scenario for complex organic synthesis could be realized through the photochemical processing of ocean material ejected by the plume onto the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' HCN might be sequestered as ferrocyanide by sodium or potassium salts, both of which have been found in plume ice grains [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Ferrocyanide is an important feedstock molecule for the cyanosulfidic prebiotic chem- istry paradigm, which relies on reductive homologation of HCN to form the sugars needed for ribonucleotide as- sembly [44, 59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' If there exists sufficient UV photon processing of organic material on Enceladus’ surface, the selective production of RNA and amino acid precursors from plume-derived HCN, CH2O, C2H2, (and possibly H2S or PH3) together with plausible mineralogical cata- lysts might occur [44, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' This surface-based production could then feed back into the plume or ocean via down- ward transport through the ice shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Whether this type of chemistry is efficient under Enceladus-like conditions could be explored in future experimental studies, while more detailed examination of Enceladus’ oceanic mate- rial will require future robotic missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' METHODS Instrument overview The Cassini Ion and Neutral Mass Spectrometer (INMS) was a quadrupole mass spectrometer designed primarily for neutral gas analysis [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' When INMS was operating in its Closed Source Neutral (CSN) mode, neu- tral molecules were accumulated in the instrument an- techamber before being directed towards a 70 eV electron source for subsequent ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Detections of ionized molecules at the instrument target were then recorded 7 sequentially at individual mass channels corresponding to mass-to-charge ratios (m/z), where z is typically as- sumed to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The INMS mass channels ranged from 1 to 99 u at a resolution of 1 u, excluding 9, 10, and 11 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' When neutral molecules enter the ionization region of a mass spectrometer such as INMS, emitted electrons from the electron source both ionize incoming parent molecules and produce dissociated, ionized fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' For a given electron energy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', 70 eV), each incoming parent molecule fragments according to a specific crack- ing pattern that describes the relative proportions of the dissociated products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' As such, a mass spectrum obtained from a single molecular species will exhibit counts at mass channels pertaining to the molecular weight of the ionized parent molecule, as well as those of each of the ionized dissociation products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' INMS spectra therefore consist of a combination of overlapping spectral features resulting from each parent molecule’s cracking pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' INMS was not able to detect parent molecules or frag- ments with masses above the maximum mass of 99 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' However, larger molecules may have impacted the instru- ment walls and fragmented into smaller molecules that were within the detectable mass range [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The rate of these impact fragmentation events depends heavily on the kinetic energy of the spacecraft [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' As such, spec- tra generated from faster flybys (> 10-15 km/s) exhibit a suite of spectral features (associated with impact frag- ments of higher mass molecules) that are not observed at lower speeds [4, 12, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' As a result, measurements obtained during the lower velocity flybys of Enceladus provide the best opportunity to study the intrinsic plume composition in the absence of fragmentation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Data set selection During the E14, E17, and E18 flybys of Enceladus, Cassini made a sequence of three low velocity (< 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='5 km/s) passages through the plume along nearly identi- cal trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' These flybys, referred to as the “slow” flybys [4], produced the most consistent INMS data with which to analyze the plume’s intrinsic neutral gas com- position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Here, we use the averaged CSN spectrum ob- tained across these three flybys, as done by Waite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' This averaged slow flyby spectrum is identi- cal to that presented in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [12], with minor differences described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In their supplementary material, Waite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [4] deduce that the majority of counts at mass 28 are attributable to fragmentation products (possibly of CO2) that are not re- flective of the plume’s intrinsic composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Fragments from higher mass hydrocarbon species likely also con- tribute to this mass channel [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Measurements taken during the E21 flyby using the INMS Open Source Neu- tral Beam (OSNB) mode place an upper limit on intrinsic mass 28 counts at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='05% of the signal at mass 18 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' As this study is focused on identifying native species in the plume, we account for this effect by correcting the counts at mass 28 accordingly (Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In order to quantify variability in the INMS data be- tween flybys, we assume a 30% Gaussian uncertainty on all mass channels, similar to that described by Waite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' in their own analysis of the slow flyby spectrum [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' However, given the strong signal at mass 18, we use a standard Poisson uncertainty at this mass channel, equal to the square root of the number of counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' This choice is warranted given that mass 18 is the anchor mass used to standardize the E14, E17, and E18 flybys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Moreover, mass channels beyond 46 u exhibit significantly lower signal-to-noise ratios than lower mass channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' To avoid erroneously fitting to these data points, we implement a larger fractional uncertainty on peaks that lie below the count levels at these mass channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' This larger uncer- tainty is equal to the typical count value measured at these mass channels (2 counts) and serves as the global noise floor for the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' When INMS was operating in the CSN mode, the vast majority of counts at 1 and 2 u arose due to interactions between gas phase H2O molecules and the walls of the instrument antechamber [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Proper assessment of these mass channels requires a careful analysis of data from the instrument’s OSNB mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' This problem has been explored in detail by Waite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [4] who attribute ∼98% of the total signal at 1 and 2 u to H2 (at a mixing ratio of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='9%) and H2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The cracking pattern for H2 does not contribute to any other mass channels and is therefore independent from the deconvolution of the remainder of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Furthermore, the mixing ratio of H2O is predominantly dictated by high leverage points at masses 18 and 17 and not significantly affected by these lower mass channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' For these reasons, we chose to adopt the H2 mixing ratio of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [4] and exclude masses 1 and 2 from our analysis (Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Spectral library As discussed in the main text, species that are present within the INMS spectrum cannot be identified unless explicitly included within a reference library used for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' However, large spectral libraries suffer from high dimensionality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' the inclusion of too many library species leads to models that are unnecessarily complex and prone to overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' This phenomenon is similar to how any univariate relationship can be fit with zero residual error using a sufficiently high order polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Although overfitting can be ameliorated by using bias correcting heuristics such as the AICc, chemically im- plausible species may still be incorporated into the final model if they are included within the initial library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' As an additional hindrance, the inclusion of many (possibly collinear) model parameters reduces statistical power and 8 raises the computational intensity of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' It is therefore desirable to constrain the library of candidate species as much as possible based on prior knowledge and the results of previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Our library consists of a carefully chosen set of 50 compounds, including the most recent list of INMS- detectable candidate plume species presented in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' All library species and their reasons for inclusion are pre- sented in Extended Data Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Because all counts ob- served above 46 u during the slow flybys are below the estimated noise floor, no species with base peaks above this threshold were included in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The spectra in our library are 70 eV electron ionization mass spectra collected from the INMS Refurbished Engineering Unit (REU) and the National Institute of Standards and Tech- nology (NIST) online database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' We adopt the method used in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [19, 63] to prioritize the REU spectra when available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' REU spectra were obtained from the most recent INMS calibration file provided on the Planetary Data System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' All spectra were analyzed at 1 u resolu- tion and matched to the mass detection range of INMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' All spectra were base peak normalized to ensure a uni- form method of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Benchmarking model complexity In deducing limits on INMS model complexity, we con- structed all possible combinations of plume species up to d = 14 from a candidate library of 50 plausible com- pounds (Extended Data Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' To reduce the com- putational intensity associated with this analysis, we as- sumed H2O, CO2, and CH4 to be present in each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H2O and CO2 have been verified at Enceladus by inde- pendent Cassini instruments [64, 65], and CH4 is among the most consistent INMS observations, having been de- tected on every flyby that sampled the plume [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' This resulted in a total of �14 d=3 47!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' (d−3)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='(50−d)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='4×1010 mod- els for direct comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In order to efficiently sample the parameter space of more complex models, we im- plemented a forward modeling stepwise selection proce- dure [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Starting from the three-component model of H2O+CO2+CH4, the next most complex model was con- structed by including an additional species that, when added to the model, resulted in the greatest decrease in the variance-weighted sum of squared residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' This selection process was repeated—sequentially incorporat- ing new species based on their influence on the sum of squared residuals—to produce a set of nested models, each containing one more species than the last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Notably, the sum of squared residuals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', the training error) was only used to compare models of equal complexity and therefore selects the same species as would comparisons based on the AICc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 2 of the main text, the results of this forward modeling algorithm closely re- semble those of the exhaustive model search for d < 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Extrapolation of these results to more complex models confirms a monotonic decrease in relative model likeli- hood for d > 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Multi-model inference In order to determine mixing ratios that properly ac- count for the ambiguity induced by model degeneracy, we computed model-averaged regression coefficients based on the relative likelihood of each model, ¯βk = R � j=1 wjβk,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' (5) Here, βk,j is the regression coefficient for species k in model j, R is the total number of most probable mod- els (those with λ > 1/e), and wj = λj/ �R m λm is the Akaike weight [27] of each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Because the observed INMS spectrum consists of sample data taken from the unknown population distribution of plume contents, the minimum AICc model is a statistical random variable that estimates the actual (also unknown) model that minimizes the Kullback-Leibler divergence with the true distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Therefore, the Akaike weight may be inter- preted as the probability that a given model minimizes this information loss between the model and the un- known, true distribution from which the observed INMS data were sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The probability that each species is present in the true minimum AICc model follows accord- ingly as, Pk = R � j=1 wjΘ(βk,j) (6) where Θ(βk,j) is the Heaviside step-function that equals 1 when βk,j > 0 and is zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' We then com- puted uncertainties in the model parameters using the unconditional standard error estimator [27], SE(¯βk) = R � j=1 wj � var(βk,j) + (βk,j − ¯βk)2 (7) where var(βk,j) is the variance of the regression coeffi- cient conditional on model Mj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Here, var(βk,j) charac- terizes the intra-model uncertainty associated with maxi- mum likelihood estimation, whereas the term (βk,j − ¯βk)2 quantifies inter-model variability due to the presence of additional high likelihood models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Importantly, the re- sults presented in this work incorporate the relative prob- abilities of each model and account for ambiguities in the model selection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 9 Conversion to mixing ratios The formula for converting INMS counts to ambient densities is described in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [66] and given by, nk = �T0 Ta �1/2 1 Dk Xk sk (8) where Xk is the count rate of species k (measured at the base peak), sk is the INMS sensitivity, Dk is the ram enhancement factor, Ta is the ambient temperature, and T0 is room temperature (293 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' When available, INMS sensitivities were obtained from refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [18, 19, 63] and otherwise estimated based on the electron impact ionization cross section procedure of Fitch and Sauter (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 2 of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [67]) adapted to INMS data in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [19, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' For one species (PH3), cross section data was taken from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' As done in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [63], a 30% uncertainty was implemented on all sensitivities estimated from NIST spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The count rate is related to the model-averaged re- gression coefficient of Equation (5) by Xk = ¯βkc0 where c0 is the standardized (species-independent) base peak count rate of each library spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The mixing ratio for each species mk (scaled to include the H2 mixing ra- tio mH2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='9% given in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [4]) is then, mk = (100 − mH2) nk � l nl = (100 − mH2) ¯βk Dksk � l Dlsl ¯βl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' (9) At suprathermal spacecraft speed and small ram angle (conditions valid during the E14, E17, and E18 flybys), the ram enhancement factor is approximately [4, 66], Dk ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='7u � 2πµk kBTa (10) for spacecraft speed u and molecular mass µk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The final expression for the mixing ratio of each species is there- fore, mk = (100 − mH2) ¯βk √µksk � l √µlsl ¯βl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' (11) Future updates to INMS sensitivity coefficients or ram enhancement factors may change the relative mixing ra- tios presented here but will not affect which species have been identified—or the evidence for their detection—as these depend only on the set of {¯βk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hunter Waite and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Brian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Magee for their help on interpreting INMS instrument effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' thanks Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Masao Sako and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kiri L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Wagstaff for useful discussions and statistical insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' also thanks Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Dimitar D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Sasselov for helpful discussions on prebiotic chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' All authors acknowl- edge the support of the Cassini Data Analysis Program (NNN13D466T) and the Jet Propulsion Laboratory, Cal- ifornia Institute of Technology, under a contract with NASA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' was also supported by an appointment to the NASA Postdoctoral Fellowship Program at the Jet Propulsion Laboratory administered by Oak Ridge Associated Universities and Universities Space Research Association through a contract with NASA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' also acknowledges support from the NASA Astrobiology Pro- gram (80NSSC19K1427) and the Europa Lander Pre- Project, managed by the Jet Propulsion Laboratory, Cal- ifornia Institute of Technology, under a contract with NASA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 10 Extended Data Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Complete list of species included in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Those taken from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [31] comprise the most recently published list of INMS-detectable species in the plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' REU: INMS Refurbished Engineering Unit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' NIST: National Institute of Standards and Technology online database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Species Name Mass (u) Source sk sk Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Reason/Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' CH4 Methane 16 REU 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='01×104 [63] [4, 31] NH3 Ammonia 17 REU 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='77×104 [63] [4, 31] H2O Water 18 REU 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='34×104 [63] [4, 31] C2H2 Acetylene 26 REU 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='81×104 [63] [6, 31] HCN Hydrogen Cyanide 27 REU 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='20×104 [63] [5, 31] C2H4 Ethylene 28 REU 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='21×104 [63] [4, 31] CO Carbon Monoxide 28 REU 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='60×104 [63] [4, 31] N2 Nitrogen 28 REU 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='29×104 [63] [4, 31] CH2O Formaldehyde 30 NIST 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='21×104 [63] [5, 31] NO Nitric Oxide 30 NIST 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='28×104 − [31] C2H6 Ethane 30 REU 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='94×104 [63] [5, 31] CH5N Methylamine 31 NIST 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='07×104 [63] [31] CH3OH Methanol 32 NIST 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='17×104 [63] [4, 31] H2S Hydrogen Sulfide 34 NIST 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='38×104 [63] [5, 31] PH3 Phospine 34 NIST 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='98×104 − [31] O2 Oxygen 36 NIST 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='03×104 [63] [4, 31] 36Ar Argon 36 36 REU 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='87×104 [18] [4, 31] C3H4 Propyne 40 REU 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='19×104 [63] [5, 31] 40Ar Argon 40 40 REU 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='29×104 [19] [5, 31] CH3CN Acetonitrile 41 REU 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='28×104 [63] [31, 50] C2H2O Ketene 42 NIST 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='37×104 [63] [31] C3H6 Propene 42 NIST 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='06×104 [63] [5, 31] CH2N2 Cyanamide 42 NIST 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='55×104 − Organic synthesis [59, 61] C2H4O Acetaldehyde 44 NIST 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='42×104 [63] [5, 31] C3H8 Propane 44 REU 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='11×104 [63] [6, 31] CO2 Carbon Dioxide 44 REU 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='09×104 [63] [4, 31] C2H7N Ethylamine 45 NIST 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='71×104 [63] [31] CH3NO Formamide 45 NIST 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='22×104 [63] HCN decomposition [47] C2H6O Ethanol 46 NIST 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='62×104 [63] [5, 31] CH2O2 Formic Acid 46 NIST 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='94×104 [63] HCN decomposition [47] C4H8 1-Butene 56 NIST 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='73×104 [63] [5, 31] C2H6N2 Azomethane 58 NIST 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='46×104 − [31] C3H6O Acetone 58 NIST 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='70×104 [63] [5, 31] C4H10 Isobutane 58 NIST 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='24×105 [63] [5, 31] C2H4O2 Acetic Acid 60 NIST 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='51×104 [63] [5, 31] C3H8O 1-Propanol 60 NIST 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='80×105 [63] [5, 31] C2H7NO Monoethanolamine 61 NIST 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='67×103 − [31] C2H6O2 1,2-Ethanediol 62 NIST 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='77×105 [63] [5, 31] C5H10 Cyclopentane 70 NIST 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='39×104 [63] [31] C4H9N Pyrrolidine 71 NIST 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='18×103 − [31] C5H12 Pentane 72 NIST 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='53×104 [63] [5, 31] C4H10O 1-Butanol 74 NIST 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='14×105 − [31] C2H5NO2 Glycine 75 NIST 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='14×105 [63] [5, 31] C3H5Cl Allyl Chloride 76 NIST 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='63×104 − [31] C5H9N Butyl Isocyanide 83 NIST 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='34×104 − Hydrocarbon irradiation [50] C4H6O2 2,3-Butanedione 86 NIST 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='91×104 [63] [5, 31] C3H7NO2 Alanine 89 NIST 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='73×103 − [31] C8H18 Octane 114 NIST 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='65×103 − [31] C6H12N4 Methenamine 140 NIST 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='86×103 − [31] C6H14N2O2 Lysine 146 NIST 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='34×103 − Amino acid [69] 11 a b c H2 correction for 28 u fragments Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Dataset corrections and minimal model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' (a) The black silhouette shows the full mass range of the INMS spectrum used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Shaded gray bars indicate count values that differ from the spectrum presented in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [12] (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The noise floor (dashed line) is estimated from the count rates of noisy mass channels > 46 u as ϵ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Blue circles show the results of fitting the minimal model consisting of the “confirmed” species in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' (b) Scatterplot of the standardized residuals produced by fitting the minimal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' There is no discernable pattern amongst the residuals or evidence of heteroscedasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' (c) Histogram of the standardized residuals (blue bars) compared to a reference Gaussian distribution with zero mean (black curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The residuals show good agreement with the Gaussian distribution, indicating a robust model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 12 SUPPLEMENTARY RESULTS Analysis of pairwise collinearity As discussed in the main text, compositional ambigui- ties within INMS spectra arise due to the large number of candidate species combined with the relatively low mass resolution of the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In other words, models of the plume’s composition contain many possible model components with comparatively few data points avail- able to constrain them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' This difficulty is accentuated when there exist combinations of multiple species that can reproduce the signal produced by another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' If this collinearity is exact, discriminating between such singu- lar species in the composite INMS spectrum is impossi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In practice, even approximately collinear mass spec- tra can present a significant obstacle when interpreting data with a finite mass resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' These issues have been encountered in previous studies [5, 6, 31] and have greatly hindered efforts to identify trace compounds in the plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Here, we quantify the extent of pairwise collinearity amongst library spectra by computing the correlation matrix for each species’ cracking pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' We use ρ to de- note correlation coefficients of cracking patterns, in con- trast to r (defined in the main text), which signifies cor- relations between regression coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Whereas r is a model dependent quantity (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 2 of the main text), ρ is a static property of the spectral library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 2 demonstrates that large positive correlations are common and manifest between molecules with similar cracking patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A correlation coefficient between two species of ρ = 1 would imply identical mass spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Pairs of species with correlation coefficients close to 1 are approximately collinear and may still be indis- tinguishable in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' By contrast, pairs of species with correlation coefficients close to 0 lack a consistent pattern of strong overlapping features in their cracking patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Values near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='5 represent the intermediate case where two species share certain features but also possess additional large mass peaks that are not shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' NH3 and CH4, for example, have a correlation coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='47, owing to their shared peak at mass 16 and unshared peaks at masses 17 (NH3) and 15 (CH4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Of course, large negative values are also possible, though they are effec- tively absent from the spectral library due to the inher- ent structural similarities between organic compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Large negative values would signify that one species has significant mass peaks predominantly at mass channels where another species does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' For compounds with cracking patterns dominated by only a few major peaks, sharing as few as one of these peaks can lead to high correlation coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A notable example is CO2 and alanine (C3H7NO2), which exhibit a correlation coefficient of ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Based on the results of the main text, it is tempting to view the presence of alanine in 13 of 157 high likelihood models as the first ten- tative evidence for amino acids at Enceladus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' However, at such low concentrations, alanine is virtually indistin- guishable from CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Although the cracking pattern for alanine contains counts at many different mass channels, the peak at 44 u is by far the dominant feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' As a re- sult, this mass channel acts as a high leverage point and drags up correlations between alanine and other species with large peaks at 44 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' This effect is particularly pro- nounced when there are no other major peaks present, as is the case for CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 3 shows the mean contribution of alanine to the INMS spectrum cal- culated based on the multi-model averaging procedure presented in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' An equal amount of CO2 is shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' All peaks are well within the as- sociated 1σ uncertainty for each mass channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The base peak is the only feature that extends above the noise floor and mimics the signal for CO2 at similar concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' This ambiguity precludes the detection of trace amounts of alanine in the plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Similar effects underlie the ambiguities amongst the al- cohols and mass 43 fragments discussed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Large positive correlations amongst the alcohols (as high as ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='91) manifest via high count rates at 31 u, corre- sponding to the hydroxymethyl group CH2OH+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Species with prominent 43 u fragments exhibit correlations as high as ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='95 and could be attributable to a number of different structures (see Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 4 and the Supplementary Discussion section below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Ambiguities of this nature might have important impli- cations for the detection of amino acids on future space- craft missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The high pairwise correlation between CO2 and alanine suggests that alanine likely cannot be independently detected at Enceladus using a 1 u resolu- tion mass spectrometer (such as INMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Instead, an inde- pendent measurement of the CO2 mixing ratio with pre- cision at least as great as the instrument used to detect alanine would be necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A similar argument would apply to glycine (C2H5NO2) in an environment with high NO abundance (ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='99) due to the large peak at mass 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Ramifications for hypothesis testing Extensive amounts of collinearity between model com- ponents can have profound consequences on statistical in- ference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Although one might expect traditional hypothe- sis testing to identify important model parameters, high- dimensionality and collinearity of the spectral library limit statistical power and prevent individual regression coefficients from achieving statistical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Sup- plementary Table 2 demonstrates this phenomenon for the low velocity INMS data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' An F-test for overall sig- nificance indicates that the spectral library,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' in aggregate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='36Ar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='40Ar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H2O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H4O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H4O2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H5NO2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H6N2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H6O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H6O2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H7N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H7NO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C3H4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C3H5Cl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C3H6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C3H6O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C3H7NO2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C3H8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C3H8O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C4H10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C4H10O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C4H6O2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C4H8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C4H9N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C5H10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C5H12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C5H9N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C6H12N4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C6H14N2O2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C8H18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CH2N2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CH2O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CH2O2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CH3CN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CH3NO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CH3OH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CH4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CH5N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CO2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='H2O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='H2S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='HCN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='N2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='NH3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='NO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='O2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='40Ar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H2O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H4O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H4O2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H5NO2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H6N2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H6O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H6O2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H7N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C2H7NO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C3H4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C3H5Cl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C3H6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C3H6O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C3H7NO2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C3H8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C3H8O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C4H10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C4H10O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C4H6O2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C4H8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C4H9N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C5H10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C5H12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C5H9N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C6H12N4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C6H14N2O2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='C8H18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CH2N2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CH2O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CH2O2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CH3CN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CH3NO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CH3OH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CH4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CH5N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='CO2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='H2O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='H2S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='HCN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='N2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='NH3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='NO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='O2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='PH3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0 Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Correlation matrix for the spectral library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The correlation coefficient (−1 ≤ ρ ≤ 1) measures the extent of collinearity between two cracking patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Large positive values (brighter colors) indicate high levels of collinearity, whereas smaller values signify less collinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Large negative values (darker colors) are essentially absent, indicating that most collinear species are positively correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' does indeed better explain the INMS data than does the “null model,” which consists of fitting only the regression coefficient for H2O (F = 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' p = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='1 × 10−13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' However, one-tailed t-tests indicate that only the two strongest spectral features, H2O and CO2, are individually sta- tistically significant (p = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='2 × 10−79 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='9 × 10−4, respectively) in the presence of the entire spectral li- brary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' These species account for the majority of counts at masses 18, 28, and 44 and produce signals that are large enough to stand out amongst the majority of collinear species that comprise the rest of the library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Though we can be confident that the aggregate set of candidate library species is capable of explaining the INMS data, the statistical significance of any one species is difficult to show using such frequentist statistics under these con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' SUPPLEMENTARY DISCUSSION Comparison to other results It is important to consider the entire body of statistical evidence when drawing conclusions about which species 14 10 3 10 1 10 1 10 3 C3H7NO2 Noise Floor 1 Uncertainty 0 10 20 30 40 50 60 70 80 90 100 10 3 10 1 10 1 10 3 CO2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0 Mass (u) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0 Counts Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Mass spectra for alanine (C3H7NO2) and CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The x-axis shows the mass of each fragmentation product, while the y-axis quantifies the proportional abun- dance of each fragment scaled to the calculated mixing ratio for alanine (note the log-scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' All spectral features (green bars) are masked by the associated count uncertainty at each mass channel (shaded blue region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Both species have a sin- gle dominant peak at 44 u that extends above the noise floor (dashed black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='00 C2H6O2 C4H10 10 20 30 40 50 60 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='00 C3H8O 10 20 30 40 50 60 70 C5H12 Mass (u) Counts Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Mass spectra for representative alco- hols and species with 43 u fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Blue bars (left) show representative alcohols, C2H6O2 and C3H8O, with ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Red bars (right) show representative species with strong 43 u signatures, C4H10 and C5H12, with ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' are detected in the plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The model-averaged mixing ratios, the minimum AICc model, the individual species probabilities, and the relative model likelihoods can all be used to assess the confidence of each detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The more heuristics that point towards the detection of a given species, the more confident one may be that said species is truly present in the plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The minimum AICc model consists of 11 species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' One interpretation of this result is to classify all 11 species as conclusive detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' However, more parsimonious models exist that explain the INMS data nearly as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A more conservative approach would be to interpret only the species that comprise the best-fitting, least complex model with λ > 1/e as essential to the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' We favor an even more nuanced approach and suggest using a tiered Supplementary Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Results of hypothesis testing on the INMS data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' High dimensionality and collinearity prevent all but the most prominent spectral features from achieving statis- tical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Species t-Statistic p-Value H2O 378 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='2 × 10−79 CO2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='9 × 10−4 All others < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='13 hierarchy of confidence based on the holistic analysis of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Below, we contextualize these results within the collection of previous studies on the composi- tion of Enceladus and other icy bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In the main text, we present an upper limit on the NH3 mixing ratio that is consistent with previous analy- ses of the slow flyby INMS spectra [4, 31] as well as those obtained during the E2 [6] and E3 [5] flybys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In their supplementary material, Waite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [4] argue, based on the slow flyby data, that the residual left at mass 16 by fitting H2O, CH4, and CO2 alone unambiguously indi- cates the presence of NH3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' For the spectrum analyzed in the main text of this work, we find that this residual is equal to ∼880 counts, corresponding to ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='1σ at 16 u (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 4a and Methods of the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' However for independent Gaussian uncertainties, it is expected that ∼16% of all mass channels would be undercounted by > 1σ due to random chance alone (see also Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 1 of the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' As such, the residual at 16 u is not enough to imply the unambiguous detec- tion of NH3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Of course, this conclusion is dependent on the estimation of count uncertainties in INMS data, for which multiple different procedures have been pro- posed [4, 5, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Still, other instruments including the Cassini Visible and Infrared Mapping Spectrometer (VIMS), Ultraviolet Imaging Spectrograph (UVIS), and Cosmic Dust Analyzer (CDA) have failed to find con- clusive evidence of NH3 at Enceladus and are unable to corroborate its presence in the plume [31, 65, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Tele- scopic data have suggested the presence of NH3 or NH3- hydrate [71–74], though these observations are also not definitive [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Consequently, additional studies that fur- ther constrain the INMS count rate at mass channel 16 beyond the typical 30% uncertainty suggested in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [4] are required before NH3 can be confirmed in the plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' As discussed in the main text, suggestive evidence for nitrogen at Enceladus in the form of HCN has been previ- ously reported based on other INMS data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' HCN has also been suggested to help explain yet unresolved sig- natures in Cassini CDA spectra of ice grains in Saturn’s E-ring [7], though could not be definitively identified by CDA as a cation species due to its high ionization poten- tial [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' As such, the HCN mixing ratio determined in this work is the first definitive detection of nitrile chem- istry at Enceladus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Our analysis shows that CH2O and C2H2 are also 15 present in the plume with concentrations exceeding 100 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' As for HCN, both species have been detected in comets [35] and circumstantial evidence for their exis- tence at Enceladus has been previously suggested based on other Cassini flybys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Both CH2O and C2H2 were sus- pected by Waite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [5] during the high-velocity E3 and E5 flybys, though correlations between abundance and spacecraft velocity suggest that impact fragmenta- tion may have been responsible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Only upper limits for both species were reported on a reanalysis of the slower E2 flyby [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The presence of CH2O would also help ex- plain features at 28-30 u seen in CDA ice grain spectra [31], which are consistent with the INMS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Concerning the higher mass organics, we find that trace amounts (∼40 ppm) of C3H6 are present in the plume as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' C3H6 is present on Titan [19] and was produced as a fragmentation product during the E3 and E5 flybys of Enceladus [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Although not initially identi- fied in the E2 flyby data [6], reanalysis suggests that it may have been present [5]—although it was not identified by Waite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [4, 31] as an intrinsic plume constituent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Our analysis shows that C3H6 is by far the most likely explanation for the majority of counts in the 37-42 u re- gion of the slow flyby INMS spectrum (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 4d of the main text) and provides the first conclusive evidence for native C3 organics in the plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The O2 mixing ratio reported in the main text is con- sistent with the limit on native mass 32 counts imposed by Waite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [4] after correcting for the surface pro- cessing of H2O in the INMS instrument antechamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Interestingly, the source of O2 at Enceladus presents a few challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Unlike Europa, where charged particle bombardment of the surface is known to drive radioly- sis of water and other elements to O2, H2O2, SO2− 4 , and other oxidants [50, 76–78], the radiation flux near Ence- ladus is considerably lower [79], and evidence for oxidants on the surface is lacking, despite proposals of such radi- olytic chemistry [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Even with moderate levels of sur- face radiolysis, a key problem would be the efficiency of oxidant production at the high temperatures observed in the South Polar Terrain [81, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Here we do not pro- pose a solution for this issue but rather note that our results are consistent with the presence of O2, be it from surface radiolysis and subsequent delivery to the ocean, or via other production mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Notably, Waite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [4] found that radiolysis of H2O due to radioactive isotopes in Enceladus’ core could also produce O2 at the abundance reported here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Evidence for 40Ar stems from the large ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='4σ resid- ual at 40 u, which is the only mass channel with strong enough signal to significantly influence the calculation of its mixing ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' This signal does have strong overlap with the cracking pattern of C3H4 (ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='75), but the abundance of this species is limited by the larger contri- bution of C3H6 at neighboring mass channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Though not identified in a previous analysis of the slow flybys [4], 40Ar was detected during the E3 and E5 flybys and may indicate significant water-rock interactions and the leaching of salts within Enceladus [5, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' CH3CN also has a reasonably high correlation with 40Ar (ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='47) and could potentially contribute to the observed signal at mass 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Although there is no prior published evidence for CH3CN at Enceladus, its presence would not be unex- pected based on the evidence for HCN and hydrocarbons such as C3H6 (see Discussion in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Native alcohols have not been previously identified in the plume, but CH3OH has been suggested as a possible fragmentation product based on high velocity INMS and ice grain spectra [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' CH3OH has also been observed via ground-based methods in the vicinity of Enceladus and could be produced through chemical processing of CH4 in the nearby gas cloud [83] or by the partial combustion of endogenous CH4 within the ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Both CH3OH and C2H6O2 have been detected in relatively high abundance in several comets [84, 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Our results also provide evidence for an ambiguous species with a strong 43 u signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Ambiguity at this mass channel was previously reported based on the E5 flyby of Enceladus [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' One explanation of this signal is C2H6N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Although this species has an additional large peak at 15 u, this feature is masked by the much larger contribution from CH4 in the INMS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The corre- lated ice grain features at 15 and 43 u seen in “Type II” CDA spectra [10] might be explained by fragmentation of C2H6N2 into CH+ 3 and CH3N+ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Alternative explanations including acetyl group-bearing species such as C3H6O or C4H6O2 are also possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Sulfur compounds have not been definitively identi- fied at Enceladus, though a past detection of H2S based on the E5 flyby [5] supports our finding that it may be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H2S would be expected if there is active serpen- tinization taking place on the ocean floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' For the remaining species listed in Table 1 of the main text, prior evidence for their existence in the plume is lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Phosphorous compounds have not been pre- viously reported in the plume, though PH3 may have been observed in the coma of comet 67P/Churyumov- Gerasimenko [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' C3H5Cl has also not been identified at Enceladus, but Cl has been detected by CDA as NaCl and KCl salts residing in plume ice grains [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Lastly, although there is no strong evidence for alanine at Ence- ladus (see the Supplementary Results section above), we note that alanine and other amino acids are abundant in carbonaceous chondrites [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Comparison to other methodologies In order to adequately account for the high- dimensionality and (approximate) collinearity of the INMS plume dataset, it is necessary to perform a type of variable selection that constrains the parameter space 16 of possible model fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Such variable selection techniques trade off a small increase in model bias for a significant reduction in model variability [13, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The resulting models are far less likely to overfit noisy features in the training data and tend to be significantly more accurate in predicting future observations [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Moreover, variable selection reduces the impact of collinearity by identify- ing which model components are better at explaining the observed data and discarding those that are superfluous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Such a process then allows for the evaluation of individ- ual model components without the confounding presence of their collinear counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In the main text, we outlined two variable selection procedures: an exhaustive best subset selection for mod- els with fewer than 15 species and a forward stepwise selection algorithm for more complex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Other common algorithms such as ridge regression [87] and the Least Absolute Shrinkage and Selection Operator (LASSO) are frequently applied in a wide variety of ma- chine learning and model validation contexts [13, 17, 88– 90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' These methods seek to reduce the influence of ex- traneous parameters via L2 or L1 regularization, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Multi-model averaging is similar to L1 regular- ization in that it allows for explicit dimensionality re- duction, whereas L2 regularization does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' This prop- erty leads to high model interpretability, which is of ut- most importance when performing compositional analy- ses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Other heuristics besides the AICc can be used to select models for averaging, but these alternative statis- tics are not based on minimizing information loss and are therefore not well-suited for model selection when the structure of the unknown, true distribution is poorly constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Furthermore, model inference using the AICc has been shown to asymptotically approximate re- sults based on cross-validation (another broadly accepted model validation technique) while requiring much fewer computational resources [27, 91, 92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The first few studies of INMS data collected at Ence- ladus produced landmark results, including the charac- terization of major plume constituents and the discovery of molecular H2 as a potential indicator of hydrothermal activity [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' These papers (and related works published throughout the duration of the Cassini mission) also doc- umented a detailed description of the INMS instrument response under varying spacecraft conditions and laid the groundwork for follow-up studies focused solely on com- positional analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' However, early studies of the Ence- ladus plume—though foundational—may not have been well-suited to resolve minor species ambiguities for var- ious reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In order to facilitate comparison with our methodology, we briefly describe the spectral deconvo- lution procedure developed by the authors of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [19] that has been implemented in various other works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [20–22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In their procedure, the authors first determine the con- tributions from major species through a visual analysis of prominent spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Mixing ratios for the major species are estimated from the base peaks of each species, assuming they contribute 100% of the measured counts at these mass channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Minor species are then identified sequentially by subtracting their contributions from the total spectrum to produce a residual spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' For small portions of the spectrum where a few candidate species exhibit overlaying signatures, species are fit based on a custom-defined fit statistic using a grid search algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Iterative minimization of the fit statistic is achieved nu- merically by sweeping through various mixing ratios at increasingly finer resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' For species that share the same base peak (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', N2 and C2H4), the fit statistic is manipulated to exclude this mass channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The high com- putational intensity of the grid search algorithm prohibits fitting more than four species at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' We believe that the methodology of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [19] described above, though useful and effective, may not be optimal for identifying minor species in the INMS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The order in which species are subtracted from the initial spectrum could potentially influence the outcome of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Although it is true that a bias-variance tradeoff can be useful for combatting high dimensionality, the described procedure is not amenable to quantitative assessments of inter-model uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Indeed, the authors note that subjectivity of their analysis is a valid concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Further- more, the practice of fitting individual species to small portions of the spectrum neglects potential contributions from complex compounds with cracking patterns that span a large mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Moreover, grid searches that consider only a few species at a time may not be able to reliably identify minor species when the spectral library is highly collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' In this regime, covariances between mixing ratios become strongly model dependent (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 2 of the main text), and multi-model inference based on model-averaged parameters is warranted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Additionally, to our knowledge, the fit statistic described in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [19] is not a standard metric, and we suspect that the use of dif- ferent fit statistics for different model parameters could lead to difficulties in interpreting the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Lastly, the authors’ procedure does not employ dimensionality re- duction or account for the possibility of over-fitting to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' By contrast, our approach quantitatively addresses both inter- and intra-model uncertainty in the spectral decomposition of INMS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' While previous studies, such as those presented in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [31, 33], have concluded that minor species identification requires a higher res- olution mass spectrometer, we have presented a math- ematical framework capable of discriminating between previously ambiguous species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The heuristics used in this analysis are based on maximum likelihood estimation and relative entropy minimization—foundational principles of statistical inference and information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Nevertheless, this study is not without limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A major challenge for any compositional analysis of INMS 17 data stems from the large number of candidate plume species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The chemistry of the ocean and ice shell could include hundreds to thousands of unique compounds that contribute to the observed INMS spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Our ap- proach using the AICc is based on the principle of par- simony in that the least-complex, best-fitting model is favored over similarly performing models of higher com- plexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Although this is a fruitful approach to developing conservative models of plume composition, nature does not necessarily reflect this ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Future investigations using higher resolution mass spectrometers with larger mass ranges will shed light on the full extent of chemical diversity within the plume and the ocean beneath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A number of interesting follow-up studies could be con- ducted to validate the results presented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A strong approach would be to treat each of the slow flybys (E14, E17, and E18) as individual data sets, as opposed to averaging them together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Machine learning models could then be trained on one data set and evaluated on another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A sort of round-robin procedure could be used to estimate the uncertainty associated with training on a particular data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Such a methodology would elimi- nate the need for heuristic statistics such as the AICc in favor of actual independent test set performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' This implementation would, however, require correcting for instrument artifacts in each of the individual Enceladus flybys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' ∗ jonahpeter@g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='edu [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Dougherty, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Khurana, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Neubauer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Russell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Saur, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Leisner, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Burton, Identi- fication of a Dynamic Atmosphere at Enceladus with the Cassini Magnetometer, Science 311, 1406 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Porco, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Helfenstein, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Thomas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Ingersoll, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Wisdom, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' West, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Neukum, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Denk, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Wagner, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Roatsch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kieffer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Turtle, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' McEwen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' John- son, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Rathbun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Veverka, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Wilson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Perry, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Spi- tale, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Brahic, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Burns, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' DelGenio, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Dones, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Murray, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Squyres, Cassini Observes the Ac- tive South Pole of Enceladus, Science 311, 1393 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hansen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Esposito, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Stewart, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Meinke, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Wallis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Colwell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hendrix, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Larsen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Pryor, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Tian, Water vapour jets inside the plume of gas leaving Enceladus, Nature 456, 477 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Waite, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Glein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Perryman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Teolis, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Magee, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Miller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Grimes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Perry, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Miller, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Bouquet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Lunine, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Brockwell, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Bolton, Cassini finds molecular hydrogen in the Ence- ladus plume: Evidence for hydrothermal processes, Sci- ence 356, 155 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Waite Jr, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Lewis, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Magee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Lunine, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' McKinnon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Glein, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Mousis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Young, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Brockwell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Westlake, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Nguyen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Teolis, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Niemann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' McNutt Jr, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Perry, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Ip, Liquid water on Enceladus from observations of ammonia and 40Ar in the plume, Nature 460, 487 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Waite, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Combi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Ip, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Cravens, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' McNutt, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kasprzak, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Yelle, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Luhmann, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Nie- mann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Gell, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Magee, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Fletcher, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Lunine, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Tseng, Cassini Ion and Neutral Mass Spectrome- ter: Enceladus Plume Composition and Structure, Sci- ence 311, 1419 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hillier, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Green, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' McBride, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Schwanethal, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Postberg, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Srama, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kempf, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Moragas- Klostermeyer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' McDonnell, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Grun, The composition of Saturn’s E ring, Monthly Notices of the Royal Astronomical Society 377, 1588 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [8] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Postberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Schmidt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hillier, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kempf, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Srama, A salt-water reservoir as the source of a com- positionally stratified plume on Enceladus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', Nature 474, 620 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [9] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Postberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kempf, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Schmidt, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Brilliantov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Beinsen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Abel, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Buck, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Srama, Sodium salts in E-ring ice grains from an ocean below the surface of Enceladus, Nature 459, 1098 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [10] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Postberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kempf, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hillier, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Srama, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Green, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' McBride, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Gr¨un, The E-ring in the vicin- ity of Enceladus: II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Probing the moon’s interior—The composition of E-ring particles, Icarus 193, 438 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Combe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' McCord, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Matson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' John- son, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Davies, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Scipioni, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Tosi, Nature, dis- tribution and origin of CO2 on Enceladus, Icarus 317, 491 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [12] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Postberg, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Khawaja, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Abel, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Choblet, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Glein, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Gudipati, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Henderson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hsu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kempf, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Klenner, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Moragas-Klostermeyer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Magee, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' N¨olle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Perry, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Reviol, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Schmidt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Srama, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Stolz, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Tobie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Trieloff, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Waite, Macromolecular organic compounds from the depths of Enceladus, Nature 558, 564 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [13] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' James, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Witten, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hastie, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Tibshirani, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', An introduction to statistical learning: with applications in R, Springer texts in statistics No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 103 (Springer, New York, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [14] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Chandrasekaran and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Jain, Quantization Complex- ity and Independent Measurements, IEEE Transactions on Computers C-23, 102 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [15] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hughes, On the mean accuracy of statistical pattern recognizers, IEEE Transactions on Information Theory 14, 55 (1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [16] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Trunk, A Problem of Dimensionality: A Simple Example, IEEE Transactions on Pattern Analysis and Machine Intelligence PAMI-1, 306 (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [17] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Guyon and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Elisseeff, An introduction to variable and feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', Journal of Machine Learning Research 3, 1157 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Cui, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Yelle, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Vuitton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Waite, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kasprzak, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Gell, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Niemann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' M¨uller- Wodarg, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Borggren, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Fletcher, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Patrick, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Raaen, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Magee, Analysis of Titan’s neutral upper atmosphere from Cassini Ion Neutral Mass Spec- trometer measurements, Icarus 200, 581 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [19] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Magee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Waite, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Mandt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' West- lake, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Bell, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Gell, INMS-derived composi- tion of Titan’s upper atmosphere: Analysis methods and model comparison, Planetary and Space Science 57, 1895 (2009), publisher: Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Waite, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Niemann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Yelle, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kasprzak, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Cravens, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Luhmann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' McNutt, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='- H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Ip, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Gell, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' De La Haye, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' M¨uller-Wordag, 18 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Magee, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Borggren, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Ledvina, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Fletcher, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Wal- ter, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Miller, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Scherer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Thorpe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Block, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Arnett, Ion Neutral Mass Spectrometer Results from the First Flyby of Titan, Science 308, 982 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Waite, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Young, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Cravens, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Coates, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Crary, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Magee, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Westlake, The Process of Tholin Formation in Titan’s Upper Atmosphere, Science 316, 870 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [22] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Waite, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Perryman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Perry, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Miller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Bell, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Cravens, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Glein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Grimes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hed- man, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Cuzzi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Brockwell, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Teolis, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Moore, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Mitchell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Persoon, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kurth, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Wahlund, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Morooka, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hadid, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Chocron, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Walker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Nagy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Yelle, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Ledvina, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Johnson, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Tseng, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Tucker, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Ip, Chemical interactions be- tween Saturn’s atmosphere and its rings, Science 362, eaat2382 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [23] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Akaike, Information Theory and an Extension of the Maximum Likelihood Principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', in Proceedings of the 2nd International Symposium on Information Theory, edited by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Petrov and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Csaki (Akademiai Kiado, Budapest, 1973) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 267–281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [24] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hurvich and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Tsai, Regression and time se- ries model selection in small samples, Biometrika 76, 297 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [25] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Sakamoto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Ishiguro, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kitagawa, Akaike in- formation criterion statistics (KTK Scientific Publishers, Tokyo, 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [26] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Sugiura, Further analysts of the data by akaike’ s in- formation criterion and the finite corrections, Communi- cations in Statistics - Theory and Methods 7, 13 (1978), publisher: Taylor & Francis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [27] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Burnham and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Anderson, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', Model Selec- tion and Multimodel Inference (Springer New York, New York, NY, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [28] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Akaike, On the Likelihood of a Time Series Model, The Statistician 27, 217 (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [29] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kullback and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Leibler, On Information and Suf- ficiency, The Annals of Mathematical Statistics 22, 79 (1951).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [30] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Johnson and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Omland, Model selection in ecol- ogy and evolution, Trends in Ecology & Evolution 19, 101 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [31] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Postberg, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Clark, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hansen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Coates, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Dalle Ore, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Scipioni, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hedman, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Waite, Plume and Surface Composition of Enceladus, in Enceladus and the Icy Moons of Saturn, edited by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Schenk (University of Arizona Press, Tucson, 2018) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 129–162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [32] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Smith, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Shappirio, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Johnson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Reisen- feld, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Sittler, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Crary, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' McComas, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Young, Enceladus: A potential source of ammonia products and molecular nitrogen for Saturn’s magneto- sphere: SATURN’S MAGNETOSPHERIC NITROGEN SOURCES, Journal of Geophysical Research: Space Physics 113, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='1029/2008JA013352 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [33] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' McKinnon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Waite, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Mousis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Lunine, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Zolotov, The Mysterious Origin of Enceladus: A Compositional Perspective, in Enceladus and the Icy Moons of Saturn (The University of Arizona Press, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [34] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Bockel´ee-Morvan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Crovisier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Mumma, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Weaver, Comets II, edited by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Festou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Keller, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Weaver (University of Arizona Press, Tucson, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Mumma and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Charnley, The Chemical Compo- sition of Comets—Emerging Taxonomies and Natal Her- itage, Annual Review of Astronomy and Astrophysics 49, 471 (2011), publisher: Annual Reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [36] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Newburn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Neugebauer, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Rahe, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', Comets in the Post-Halley Era: In Part Based on Re- views Presented at the 121st Colloquium of the Interna- tional Astronomical Union, Held in Bamberg, Germany, April 24–28, 1989, Astrophysics and Space Science Li- brary, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 167 (Springer Netherlands, Dordrecht, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [37] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Affholder, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Guyot, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Sauterey, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Ferri`ere, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Mazevet, Bayesian analysis of Enceladus’s plume data to assess methanogenesis, Nature Astronomy 5, 805 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [38] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hoehler, Implications of H2/CO2 disequilibrium for life on Enceladus, Nature Astronomy 6, 3 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [39] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Russell and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Nitschke, Methane: Fuel or Exhaust at the Emergence of Life?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', Astrobiology 17, 1053 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [40] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Zolotov, A model for low-temperature biogeochem- istry of sulfur, carbon, and iron on Europa, Journal of Geophysical Research 109, E06003 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [41] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Miller, The mechanism of synthesis of amino acids by electric discharges, Biochimica et Biophysica Acta 23, 480 (1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [42] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Orgel, Prebiotic Chemistry and the Origin of the RNA World, Critical Reviews in Biochemistry and Molecular Biology 39, 99 (2004), publisher: Taylor & Francis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [43] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Or´o and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kimball, Synthesis of purines under possible primitive earth conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Adenine from hy- drogen cyanide, Archives of Biochemistry and Biophysics 94, 217 (1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [44] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Patel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Percivalle, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Ritson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Duffy, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Sutherland, Common origins of RNA, protein and lipid precursors in a cyanosulfidic protometabolism, Nature Chemistry 7, 301 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [45] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Breslow, On the mechanism of the formose reaction, Tetrahedron Letters 1, 22 (1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [46] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Cleaves II, The prebiotic geochemistry of formalde- hyde, Precambrian Research 164, 111 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [47] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Miyakawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Cleaves, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Miller, THE COLD ORIGIN OF LIFE: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' IMPLICATIONS BASED ON THE HYDROLYTIC STABILITIES OF HYDROGEN CYANIDE AND FORMAMIDE, Origins of Life and Evo- lution of the Biosphere 32, 195 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [48] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Miller and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Orgel, The origins of life on the earth, Concepts of modern biology series (Prentice-Hall, Englewood Cliffs, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='J, 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [49] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Levy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Miller, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Brinton, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Bada, Pre- biotic Synthesis of Adenine and Amino Acids Under Europa-like Conditions, Icarus 145, 609 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [50] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hand, On the physics and chemistry of the ice shell and sub-surface ocean of Europa, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' thesis, Stanford University (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [51] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hand, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Carlson, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Tsapin, Labora- tory Analysis Of Water, Hydrocarbon And Ammonia Ice Mixtures Exposed To High-energy Electron Irradiation, in Bulletin of the American Astronomical Society, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 38 (2006) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [52] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hand, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Chyba, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Priscu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Carlson, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Nealson, Astrobiology and the Potential for Life on Europa, in Europa, edited by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Pappalardo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' McKinnon, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Khurana (University of Ari- zona Press, Tucson, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 19 [53] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hsu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Postberg, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Sekine, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Shibuya, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kempf, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hor´anyi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Juh´asz, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Altobelli, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Suzuki, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Masaki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kuwatani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Tachibana, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='- i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Sirono, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Moragas-Klostermeyer, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Srama, On- going hydrothermal activities within Enceladus, Nature 519, 207 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [54] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Glein, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Postberg, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Vance, The Geo- chemistry of Enceladus: Composition and Controls, in Enceladus and the Icy Moons of Saturn (The University of Arizona Press, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [55] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' McKay, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Davila, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Glein, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hand, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Stockton, Enceladus Astrobiology, Habitability, and the Origin of Life, in Enceladus and the Icy Moons of Saturn (The University of Arizona Press, Tucson, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [56] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Zolotov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Tobie, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Postberg, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Magee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Waite, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Esposito, Chemical and phase composi- tion of Enceladus: Insights from Cassini data, in Eu- ropean Planetary Science Conference, edited by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Ab- stracts (EPSC-DPS2011-1330, 2011) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [57] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Bl¨ochl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Keller, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Wachtersh¨auser, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Stetter, Reactions depending on iron sulfide and link- ing geochemistry with biochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', Proceedings of the National Academy of Sciences 89, 8117 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [58] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' W¨achtersh¨auser, Before enzymes and templates: the- ory of surface metabolism, Microbiological Reviews 52, 452 (1988), publisher: American Society for Microbiol- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [59] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Benner, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Bell, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Biondi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Brasser, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Carell, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Mojzsis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Omran, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Pasek, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Trail, When Did Life Likely Emerge on Earth in an RNA-First Process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', ChemSystemsChem 2, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='1002/syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='201900035 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [60] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Sasselov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Grotzinger, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Sutherland, The origin of life as a planetary phenomenon, Science Advances 6, eaax3419 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [61] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Powner, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Gerland, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Sutherland, Syn- thesis of activated pyrimidine ribonucleotides in prebiot- ically plausible conditions, Nature 459, 239 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [62] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Waite, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Lewis, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kasprzak, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Ani- cich, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Block, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Cravens, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Fletcher, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Ip, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Luhmann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Mcnutt, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Niemann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Parejko, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Richards, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Thorpe, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Walter, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Yelle, The Cassini Ion and Neutral Mass Spectrom- eter (INMS) Investigation, Space Science Reviews 114, 113 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [63] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Miller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Waite, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Perryman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Perry, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Bouquet, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Magee, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Bolton, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Brockwell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hedman, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Glein, Cassini INMS constraints on the composition and latitudinal fractionation of Saturn ring rain material, Icarus 339, 113595 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [64] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hansen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', Enceladus’ Water Vapor Plume, Science 311, 1422 (2006), publisher: American Association for the Advancement of Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [65] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Brown, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Clark, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Buratti, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Cruik- shank, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', Composition and Physical Properties of Enceladus’ Surface, Science 311, 1425 (2006), publisher: American Association for the Advancement of Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [66] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Teolis, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Niemann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Waite, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Gell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Perryman, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kasprzak, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Mandt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Yelle, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Lee, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Pelletier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Miller, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Young, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Bell, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Magee, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Patrick, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Grimes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Fletcher, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Vuitton, A Revised Sensitivity Model for Cassini INMS: Results at Titan, Space Science Reviews 190, 47 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [67] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Fitch and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Sauter, Calculation of relative electron impact total ionization cross sections for organic molecules, Analytical Chemistry 55, 832 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [68] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Graves, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Cooper, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Tennyson, Calculated elec- tron impact ionisation fragmentation patterns, Journal of Physics B: Atomic, Molecular and Optical Physics 54, 235203 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [69] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Glavin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Burton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Elsila, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Aponte, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Dworkin, The Search for Chiral Asymmetry as a Potential Biosignature in our Solar System, Chemical Reviews 120, 4660 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [70] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hansen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Esposito, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Colwell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hendrix, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Portyankina, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Stewart, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' West, The composi- tion and structure of Enceladus’ plume from the complete set of Cassini UVIS occultation observations, Icarus 344, 113461 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [71] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Zastrow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Clarke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hendrix, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Noll, UV spectrum of Enceladus, Icarus 220, 29 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [72] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Grundy, Near-Infrared Spectra of Icy Outer Solar System Surfaces: Remote Determination of H2O Ice Temperatures, Icarus 142, 536 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [73] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Emery, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Burr, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Cruikshank, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Brown, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Dalton, Near-infrared (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='8–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='0 $\\mathsf{\\mu}$m) spectroscopy of Mimas, Enceladus, Tethys, and Rhea, Astronomy & Astrophysics 435, 353 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [74] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Verbiscer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Peterson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Skrutskie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Cushing, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Helfenstein, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Nelson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Smith, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Wilson, Near-infrared spectra of the leading and trailing hemispheres of Enceladus, Icarus 182, 211 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [75] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Cruikshank, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Owen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Ore, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Geballe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Roush, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Debergh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Sandford, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Poulet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Benedix, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Emery, A spectroscopic study of the surfaces of Sat- urn’s large satellites: HO ice, tholins, and minor con- stituents, Icarus 175, 268 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [76] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hand and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Brown, KECK II OBSERVA- TIONS OF HEMISPHERICAL DIFFERENCES IN H 2 O 2 ON EUROPA, The Astrophysical Journal 766, L21 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [77] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Spencer and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Calvin, Condensed O[TINF]2[/TINF] on Europa and Callisto, The Astro- nomical Journal 124, 3400 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [78] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Trumbo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Brown, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hand, H 2 O 2 within Chaos Terrain on Europa’s Leading Hemisphere, The Astronomical Journal 158, 127 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [79] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Paranicas, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Roussos, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Krupp, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kollmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hendrix, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Cassidy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Johnson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Schenk, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Jones, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Carbary, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Mitchell, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Dialynas, Energetic charged particle weathering of Saturn’s inner satellites, Planetary and Space Science 61, 60 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [80] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Ray, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Glein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Waite, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Teolis, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hoehler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Huber, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Lunine, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Postberg, Oxidation pro- cesses diversify the metabolic menu on Enceladus, Icarus 364, 114248 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [81] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hand and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Carlson, H2O2 production by high-energy electrons on icy satellites as a function of surface temperature and electron flux, Icarus 215, 226 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [82] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Spencer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Pearl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Segura, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Flasar, 20 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Mamoutkine, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Romani, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Buratti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hen- drix, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Spilker, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Lopes, Cassini Encoun- ters Enceladus: Background and the Discovery of a South Polar Hot Spot, Science 311, 1401 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [83] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Drabek-Maunder, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Greaves, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Fraser, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Clements, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Alconcel, Ground-based detection of a cloud of methanol from Enceladus: when is a biomarker not a biomarker?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=', International Journal of Astrobiology 18, 25 (2019), publisher: Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [84] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Biver, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Bockel´ee-Morvan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Debout, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Crovisier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Boissier, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Lis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Dello Russo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Moreno, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Colom, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Paubert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Vervack, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Weaver, Complex organic molecules in comets C/2012 F6 (Lem- mon) and C/2013 R1 (Lovejoy): detection of ethylene glycol and formamide, Astronomy & Astrophysics 566, L5 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [85] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Crovisier, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Bockel´ee-Morvan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Biver, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Colom, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Despois, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Lis, Ethylene glycol in comet C/1995 O1 (Hale-Bopp), Astronomy & Astrophysics 418, L35 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [86] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Altwegg, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Balsiger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Bar-Nun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Berthelier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Bieler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Bochsler, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Briois, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Calmonte, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Combi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Cottin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' De Keyser, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Dhooghe, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Fiethe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Fuselier, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Gasc, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Gombosi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hansen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Haessig, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' J¨ackel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kopp, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Korth, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Le Roy, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Mall, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Marty, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Mousis, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Owen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R`eme, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Rubin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' S´emon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Tzou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hunter Waite, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Wurz, Prebiotic chemicals—amino acid and phosphorus—in the coma of comet 67P/Churyumov- Gerasimenko, Science Advances 2, e1600285 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [87] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hoerl and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Kennard, Ridge Regression: Bi- ased Estimation for Nonorthogonal Problems, Techno- metrics 12, 55 (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [88] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Hocking, A Biometrics Invited Paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' The Analysis and Selection of Variables in Linear Regression, Biomet- rics 32, 1 (1976), publisher: [Wiley, International Bio- metric Society].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [89] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Santosa and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Symes, Linear Inversion of Band- Limited Reflection Seismograms, SIAM Journal on Scien- tific and Statistical Computing 7, 1307 (1986), publisher: Society for Industrial and Applied Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [90] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Tibshirani, Regression Shrinkage and Selection via the Lasso, Journal of the Royal Statistical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Series B (Methodological) 58, 267 (1996), publisher: [Royal Sta- tistical Society, Wiley].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [91] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Stone, An Asymptotic Equivalence of Choice of Model by Cross-Validation and Akaike’s Criterion, Journal of the Royal Statistical Society: Series B (Methodological) 39, 44 (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' [92] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Stoica, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Eykhoff, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' Janssen, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} +page_content=' S¨oderstr¨om, Model-structure selection by cross-validation, Interna- tional Journal of Control 43, 1841 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfxg23/content/2301.05259v1.pdf'} diff --git a/x9FQT4oBgHgl3EQfxTb9/content/tmp_files/2301.13405v1.pdf.txt b/x9FQT4oBgHgl3EQfxTb9/content/tmp_files/2301.13405v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a4360bf704cba4374d695ba2518ca91158b8ea23 --- /dev/null +++ b/x9FQT4oBgHgl3EQfxTb9/content/tmp_files/2301.13405v1.pdf.txt @@ -0,0 +1,3270 @@ +IPMU23-0002, KEK-TH-2494 +Novel loop-diagrammatic approach to QCD θ +parameter and application to the left-right model +Junji Hisano,a,b,c Teppei Kitahara,b,d,e,f Naohiro Osamura,a and Atsuyuki Yamadaa +aDepartment of Physics, Nagoya University, Furo-cho Chikusa-ku, Nagoya 464-8602 Japan +bKobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, +Furo-cho Chikusa-ku, Nagoya 464-8602 Japan +cKavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa 277-8584, Japan +dInstitute for Advanced Research, Nagoya University, Furo-cho Chikusa-ku, Nagoya 464-8601, +Japan +eKEK Theory Center, IPNS, KEK, Tsukuba 305–0801, Japan +fCAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy +of Sciences, Beijing 100190, China +E-mail: hisano@eken.phys.nagoya-u.ac.jp, teppeik@kmi.nagoya-u.ac.jp, +osamura.naohiro.j2@s.mail.nagoya-u.ac.jp, +yamada.atsuyuki.k3@s.mail.nagoya-u.ac.jp +Abstract: +When the QCD axion is absent in full theory, the strong CP problem has +to be explained by an additional mechanism, e.g., the left-right symmetry. Even though +tree-level QCD ¯θ parameter is restricted by the mechanism, radiative corrections to ¯θ are +mostly generated, which leads to a dangerous neutron electric dipole moment (EDM). The +ordinary method for calculating the radiative ¯θ utilizes an equation ¯θ = arg det mloop +q +based +on the chiral rotations of complex quark masses. In this paper, we point out that when full +theory includes extra heavy quarks, the ordinary method is unsettled for the extra quark +contributions and does not contain its full radiative corrections. We formulate a novel +method to calculate the radiative corrections to ¯θ through a direct loop-diagrammatic +approach, which should be more robust than the ordinary one. +As an application, we +investigate the radiative ¯θ in the minimal left-right symmetric model. We first confirm a +seminal result that two-loop level radiative ¯θ completely vanishes (corresponding to one- +loop corrections to the quark mass matrices). Furthermore, we estimate the size of a non- +vanishing radiative ¯θ at three-loop level. It is found that the resultant induced neutron +EDM is comparable to the current experimental bound, and the expected size is restricted +by the perturbative unitarity bound in the minimal left-right model. +Keywords: CP violation, Electric Dipole Moments, Left-Right Models +arXiv:2301.13405v1 [hep-ph] 31 Jan 2023 + +Contents +1 +Introduction +1 +2 +Loop-diagrammatic evaluation of QCD θ parameter +3 +2.1 +Operator Schwinger method +4 +2.2 +Fock-Schwinger gauge method +5 +3 +The minimal left-right symmetric model +8 +3.1 +Model +8 +3.2 +Parametrization of Yukawa coupling constants +10 +3.3 +Quark EDMs +12 +4 +Confirmation of vanishing QCD θ parameter in two-loop order +12 +5 +Non-vanishing contribution to QCD θ parameter in three-loop order +19 +5.1 +Leading contribution: PDF of δθduu +20 +5.2 +Case for the GIM by universal vector-like mass +23 +6 +Conclusions and discussion +24 +A Loop functions +26 +A.1 I(1;3) and I(3;1) +27 +A.2 I(2;2) +28 +1 +Introduction +The QCD θ term is known as a topological one and is P- and T-odd, and then CP-odd +under the CPT invariance. Because it is identical to the total derivative, it never locally +affects physics at the classical level (as long as the momentum conservation holds), while +its effect occurs only via nonperturbative processes [1–5]. It is known that this interaction +induces the neutron electric dipole moment (EDM) [6, 7]. Measurement of the neutron +EDM by the nEDM collaboration has set the severe upper bound: |dn|exp < 1.8×10−26 e cm +(90% CL) [8]. Using the latest lattice result dn = −0.00148(34) ¯θ e fm [9]#1 and assuming +that ¯θ is the only source of CP violation, one obtains the upper bound on the angle, +|¯θ| ≲ 1.2 × 10−10 +(90% CL) , +(1.1) +#1This is the first statistically significant result using the lattice QCD calculation. +– 1 – + +where ¯θ is the physical CP-violating angle in the QCD Lagrangian which will be defined +explicitly in the next section. +Although the experimental bound requires that ¯θ must be around zero, such a CP- +violating phase is not restricted in the Standard Model (SM). In fact, the CP-violating +phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix [10, 11] is O(1); δCKM ≃ 66◦ = +1.2 rad [12] (in the standard parameterization [13, 14]). If there is no trick in the full theory, +¯θ ≪ 1, or equivalently ¯θ ≪ δCKM = O(1), requires a fine-tuning at O(10−10) level. This is +known as the strong CP problem. +The massless up quark could be a solution to the strong CP problem if the observed +hadron masses are explained by the nonperturbative effect [15–19]. However, some lattice +studies ruled out this solution [20]. +Thus, the strong CP problem would suggest that +the SM has to be extended to suppress ¯θ without the fine-tuning. Axion is the simplest +solution to the strong CP problem [21–23], though it suffers from another fine-tuning in +the quantum gravity sector (axion quality problem) [24–26]. +Alternatively, one may resolve the strong CP problem by extended parity symmetry +[27–30].#2 In such scenarios, the extended parity involves the left-right (LR) gauge sym- +metry. The parity symmetry forbids the bare ¯θ parameter, while O(1) of δCKM is allowed. +It is known that even though the bare ¯θ parameter is strictly forbidden by the parity sym- +metry, radiative correction to ¯θ is regenerated since the parity symmetry must be softly +broken in nature. Eventually, one has to consider the experimental bound on the model +from the neutron EDM measurements in Eq. (1.1) through the radiatively regenerated ¯θ. +The ordinary method for calculating the radiatively generated ¯θ parameter, adopted +in many papers, utilizes +¯θ = arg det mloop +u ++ arg det mloop +d +. +(1.2) +Here, mloop +u,d are the up- and down-type quark mass matrices including the radiative correc- +tions. This relation is based on the chiral rotations for the complex quark masses and an +anomalous divergence of the axial-vector current, known as the Adler-Bell-Jackiw anomaly +[34, 35]. Or, it is also derived using the path-integral formalism referred to as the Fujikawa +method [36]. +The ordinary method is simple though it may not be accurate. For instance, within +the SM, even if one sets the bare ¯θ parameter to be zero, it is radiatively produced via the +CP-violating phase in the CKM matrix. It is shown with the above method in Ref. [37] +that the contribution via the radiative corrections to quark masses is of O(G2 +F α3 +s) (GF is +the Fermi coupling constant and αs is the QCD coupling constant). On the other hand, +the direct loop calculation of the correction to ¯θ shows that it is derived at four-loop +order (O(G2 +F αs)) [38]. It is consistent with the fact that the EDMs (and also the chromo- +EDMs) of quarks are induced at three-loop order (O(G2 +F αs)). In fact, the ordinary method +corresponds to the diagrams where the external gluons are attached in the same fermion +line in loop diagrams contributing to ¯θ. It implies that the leading ¯θ in the SM [38] comes +from diagrams with external gluons attached to different fermion lines. +#2Other possibilities are spontaneous CP violation referred to as the Nelson-Barr mechanism [31–33]. +– 2 – + +In this paper, we formulate a novel approach to evaluate the radiative corrections to +¯θ through a direct loop-diagrammatic calculation, which should be more robust than the +ordinary one. In Ref. [38], the external gluon field is introduced to calculate the correction +to ¯θ from the CKM matrix, while details of the technique are not written. We introduce +the Fock-Swinger gauge method to directly calculate the radiative corrections to ¯θ under +the gluon field-strength background. +As an application, we investigate the radiative ¯θ in the minimal LR symmetric model +[29, 30], in which the bare and one-loop level ¯θ parameters are strictly forbidden by the +LR symmetry. Although the extra heavy quarks whose Yukawa interactions violate CP +symmetry are introduced, the CP-violating Yukawa interactions do not contribute to the +¯θ parameters at one-loop level. Furthermore, it is known that two-loop level ¯θ parame- +ter also vanishes, which corresponds to one-loop corrections to the quark masses in the +ordinary method [29, 39, 40]. However, the ordinary method is unsettled for the extra +quark contributions to ¯θ and indeed does not contain its full radiative corrections, like the +SM calculations [38]. We first confirm this seminal result by using the proposed method. +While new type diagrams contribute to ¯θ at two-loop level, the sum of the diagrams still +gives no contribution to ¯θ. Next, we estimate the size of a non-vanishing radiative ¯θ at +three-loop level. It will be found that the resultant induced neutron EDM is comparable to +the current experimental bound. We will also investigate a relation between the radiative +three-loop level ¯θ and the perturbative unitarity bounds of the Yukawa couplings in the +minimal LR symmetric model. +This paper is organized as follows. In Sec. 2, we discuss methods of direct calculation +of the loop diagrams contributing to ¯θ. We show that the operator Schwinger method and +the Fock-Schwinger gauge method are applicable, though the latter method has a merit +to extend the calculation to the higher-loop diagrams. In Sec. 3, the minimal LR sym- +metric model is briefly summarized. We also derive the parameterization by the physical +parameters based on the seesaw mechanism in the LR symmetric model. We confirm that +the two-loop level radiative ¯θ vanishes by using the proposed method in Sec. 4. In Sec. 5, +we investigate the numerical size of the non-vanishing radiative ¯θ and compare both ex- +perimental (from neutron EDM) and theoretical bounds (from the perturbative unitarity +bound). Section 6 is devoted to conclusions and discussion. Details of the loop calculations +are given in the Appendix. +2 +Loop-diagrammatic evaluation of QCD θ parameter +In the QCD Lagrangian, imaginary parts of the quark masses and the QCD θ term are +P-odd and T-odd interactions that are not restricted from the SU(3)C gauge symmetry, +L/P , /T = − +� +q=all +Im(mq)¯qiγ5q + θG +αs +8πGa +µν ˜Gaµν , +(2.1) +where mq stands for the complex quark masses with mq ≡ |mq| exp(iθq), Ga +µν is the gluon +field-strength tensor, ˜Gaµν ≡ 1 +2ϵµνρσGa +ρσ with ϵ0123 = +1, and αs = g2 +s/(4π) is the SU(3)C +– 3 – + +coupling constant. It is well-known that the axial rotation of quarks +q → q′ = exp +� +− i +2θqγ5 +� +q , +¯q → ¯q′ = ¯q exp +� +− i +2θqγ5 +� +, +(2.2) +turns off the imaginary part of the quark masses and generates an additional QCD θ term, +L/P , /T = ¯θ αs +8πGa +µν ˜Gaµν , +(2.3) +where +¯θ ≡ θG − +� +q +θq , +(2.4) +is a physical ¯θ parameter, if all quarks are massive. This is derived using the path-integral +formalism referred to as the Fujikawa method [36] or with the Adler-Bell-Jackiw anomaly +[34, 35, 41, 42] +∂µ(¯qγµγ5q) = 2i Re(mq)¯qγ5q − 2 Im(mq)¯qq − αs +4πGa +µν ˜Gaµν . +(2.5) +The contribution of the quark mass phase to the QCD θ term should be able to be +directly evaluated by loop-diagrammatically integrating out quarks, not via the Adler-Bell- +Jackiw anomaly or transformation of measure in the path integral (the Fujikawa method). +However, it does not generate the QCD θ term if the momenta in the diagrams are con- +served. It is because the θ term is equivalent to total-derivative and the total momentum +has to be zero. Thus, we have to abandon the momentum conservation or equivalently the +translation invariance in order to evaluate the QCD θ term with the loop-diagrammatic +calculation. +It can be realized by introducing the gluon field strength background. In this section, +we evaluate the QCD θ term with two different methods, 1) the operator Schwinger method +and 2) the Fock-Schwinger gauge method. We show that they produce consistent results +with Eq. (2.4). +2.1 +Operator Schwinger method +First, we consider the operator Schwinger method.#3 The effective action ∆S induced by +the integration of quarks at one-loop level is given by the log-determinant (or trace-log) of +the Dirac operator. Now we introduce the complex mass parameters for quarks. In this +case, the effective action is given as +∆S = −iTr log +��� +���/P − (m∗ +qPL + mqPR) +��� +��� , +(2.6) +where Pµ = iDµ, Dµ ≡ ∂µ + igsT aGa +µ is the QCD covariant derivative, and PL/R = +(1 ∓ γ5)/2. In this method, the following basic commutation relation is used, +[Pµ, Pν] = igsT aGa +µν(≡ igsGµν) , +(2.7) +#3See Ref. [43] for the review about the operator Schwinger method. +– 4 – + +since the gluon field-strength tensor appears from it. Then, the derivative of ∆S over the +fermion mass mq for PR is +d +dmq +∆S = iTr +� +1 +/P − (m∗qPL + mqPR)PR +� += iTr +� +1 +P 2 − |mq|2 + i +2gsσµνGµν m∗ +qPR +� +. +(2.8) +Since the Levi-Civita tensor appears from the trace of a product of four γ’s and γ5, the +second order of i +2gsσµνGµν leads to the G ˜G term as +d +dmq +∆S ⊃iTr +� +1 +P 2 − |mq|2 +� +− i +2gsσµνGµν +� +1 +P 2 − |mq|2 +� +− i +2gsσρσGρσ +� +1 +P 2 − |mq|2 m∗ +qPR +� += +� +d4x +� +d4p +(2π)4 +g2 +s +2 +−m∗ +q +(p2 − |mq|2)3 Ga +µν ˜Gaµν + · · · +⊃ +� +d4x +i +32π2 +g2 +s +2 +1 +mq +Ga +µν ˜Gaµν , +(2.9) +where Tr(T aT b) = (1/2)δab is used. Here, P 2 is replaced by −∂2, and it is integrated in +momentum space. Similarly, d∆S/dm∗ +q leads to the G ˜G term. By Integrating d∆S/dmq +with mq and d∆S/dm∗ +q with m∗ +q, we get +∆L = −i +32π2 +g2 +s +2 log m∗ +q +mq +Ga +µν ˜Gaµν += − θq +αs +8πGa +µν ˜Gaµν . +(2.10) +Now we obtain the contribution from a quark with complex mass to the QCD θ term by +integrating out the quark, which is consistent with the axial rotation (2.3). +2.2 +Fock-Schwinger gauge method +In the previous section, we showed that the operator Schwinger method enables us to +derive the physical QCD θ term by the diagrammatic evaluation. However, the operator +Schwinger method is not suitable to calculate effective operators induced at higher loops +because it is the method to obtain an effective action by integrating fermions out with the +log-determinant of the Dirac operator. Here alternatively, we introduce the Fock-Schwinger +gauge method, which is more applicable to diagrammatic calculation.#4 +The Fock-Schwinger gauge is to take such a gauge +(xµ − xµ +0)Ga +µ(x) = 0 , +(2.11) +which violates the translation symmetry. Because of breaking the translation symmetry, +we can derive perturbatively the QCD θ term as will be shown below. While the Fock- +Schwinger gauge violates the translation symmetry, the physical observables do not depend +#4See Ref. [43] for the review about the Fock-Schwinger gauge method. +– 5 – + +ADxHichVLbhMxFL3p8Cjl0RY2SGxGREFlE5yqAtRVRSUEu +7QlbaU2RJ6Jm46L8ZOQhNP4AfAIlVK7FAfAYbfoBFPwGxLBKbLnrsuAiIUmzN+Pre849rWXhoFUjB2VJpwLFy9dnrwydfXa9RvTM7M312XSzXzR8JMwyTY9LkUYxKhAhWKzTQTPJCseHtLev4Rk9kMkjiF2qQimbEO3GwE/hcwfUy384i93lUuHNR69X91kyZVZk +Z7qhRs0aZ7Kgns6U6bVObEvKpSxEJiknBDomTxNyiGjFK4WtSDl8GKzBxQVNAatgvca/D38b6y65wDwGKsLUGV3wCHBw4Pbw72C3Zb0x9rqNPw+dIT4MnC7VGHf2Cd2zL6yz+w7OxnLlRsOrXaA1RtiRdqafnt7d/URFWrfo36lzNinZwNq01gPbUePQp/CG+9+bd8 +driaiW/xw7ZD+g/YEfsC04Q9376H1fE6odz9JwpUbgBfQ8SnBWwao9C9iI9wPRhc1SsmnvqwJOCQ6+j+PG1pO3wLnZt28kYdt/0IDJcMSI5/MPcwryF5h+eM3ROZUQKy8IR0z3sIxIgJzW5A/QlQzaHBmZxqvTuTpT4CwF3nTt3xc8aqzPV2sPqwsrC+WlJ/Z1T9Iduk +tzqPWIlugZ1amBKhm9pwM6dJ46oSOd7jB1omQxt+iv4eyfAv70CM=Im(mq) +ADt3ichVLbhMxFL3p8GjLoy1s +kNhEREGsglNVPLqYMyaUkbqUSRZ+IkVufF2EkIo/wAbEFdsAKJBeIz2PADLPoJiGWR2LDg2HEREKXYmvH1vfece+xrPw2l0owdFRa8M2fPnV9cWr5w8dLldW1K7sqGWSBaARJmGRNnysRylg0tNShaKaZ4JEf +ij3/4KGJ7w1FpmQSP9bjVLQi3otlVwZcw1VvtldLrMLsKM4aVWeUyI1aslao0RPqUEIBDSgiQTFp2CFxUpj7VCVGKXwtyuHLYEkbFzShZWA1rGf4j+DvYO1TEZh7QEWYJmMAHgEODtwB/j3s9p03xt5UVZY/gI4Q +XwbuIpXZF/aBHbP7CP7yn7O5coth1E7xupPsSJtr7y4tvPjv6gIq1H9G3WqZk1dnM1oldCeWo85RTDFD58fHu9sbpfzm+wd+wb9b9kR+4QTxMPvwfu62H5zip4TJRo3YO5BgbMVuPRyN6k25gBbI6KFXtPXhS +cJh1Fj+/lnId7mPXcZ2MY9sDyLFSOSwz/Nndi30PrDc4LOqYTIxLFwxEwPR4hI5KQ2d4y+ZMjm0KAc03x1JtdkCpxlgjd/fcFzxq765XqncpGfaO09cC97kW6TjfoFmrdpS16RDVqoIqgl/SKXnv3vbX9fr +T1IWCw1ylv4b39BdazctCX +ADuXichVLbhMxFL3p8Cjl0RY2SGwioiBWwYMqQGVTwYZl2pC2Uokiz8RN3MxLYychjPIDSGzp +ghVILBCfwYfYNFPQCyLxIYFx46LgCjF1oyv73n3GNfB1klWbsqLTgnTl7vzihaWLly5fWV5Zvbqt0kEei +maYRm+G3AlIpmIpY6ErtZLngcRGIn6D828Z2hyJVMk6d6nIlWzLuJ3Jch13A1+m2/vVJhNWZHedbwnVEhN+ +rpaqlOz6hDKYU0oJgEJaRhR8RJYe6RT4wy+FpUwJfDkjYuaEJLwGpYz/Efwd/B2qMyMA+AijFNxgA8AhwcuD7 ++Xez2nDfB3lRVlj+EjghfDu4yVdkX9oEds8/sI/vKfs7lKiyHUTvGkyxImsv7ze+PFfVIzVqP6NOlWzpn2 +czWiV0J5ZjzlFOMUPXxweN9a3qsUt9o59g/637Ih9wgmS4fw/abYenOKnhMlGjdg7kGBswpW49HIXqc7mCFs +jo1e09deDJwmHUWP7+Wch3uYdxnUxgj2wPYsuVIFLAP82d2LfQ+sNzgi6ogsjEsXDETA9HiEjkZDZ3jL7ky +ObQoBzTfHUm12QKnGWCN+3/+4Jnje27Nf9ebW1zrbLxyL3uRbpBN+k2at2nDXpCdWqiSpde0Ws69B563Ot5B9 +PUhZLDXKO/hqd+Acdhy/k=k1 +ADuXichVLbhMxFL3p8Cjl0RY2SGwio +iBWwakqQGVT0Q3LtCFtpRJFnombmMxLYychjPIDSN3SRVdFYoH4Db8AIt+AmJZJDYsOHZcBEQptmZ8fe895x72k9DqTRjJ4U578LFS5fnryxcvXb9xuLS8s1tlfSzQDSCJEyXZ8rEcpYNLTUodhNM8EjPxQ7fm/DxHcGIlM +yiZ/rUSqaEe/Ecl8GXMNV7VWkslVmF2FKeNqjNK5EYtWS7U6AW1KaGA+hSRoJg07JA4Kcw9qhKjFL4m5fBlsKSNCxrTArAa1iv8h/C3sXapCMxjoCJMk9EHjwAHB6Hfwe7PeNsTdVleUPoCPEl4G7SGX2hX1gp+wz+8i+s +p8zuXLYdSOsPoTrEhbi29u13/8FxVhNap/o87VrGkfZzNaJbSn1mNOEUzwg9eHp/W1rXJ+j71j36D/mJ2wTzhBPgevN8UW0fn6DlTonED5h4UOMtgNR6N7DV6gBnA5qhYsfUgScFh1mn8bNrKdfhLnZt18kY9tD2ILJcMSI +5/JPcsX0LzT8Z+icSoiMHQtHzPRwiIhETmpzR+hLhmwODcoxzVZnck2mwFnGeNPVf1/wtLG9Uqk+rKxurpbWn7rXPU936C7dR61HtE7PqEYNVOnQAb2lQ+Jx72u93KSOldwmFv01/DUL8q0y/o=k2 +ADvnichVK7bhNBFL3O8gjhkQcNEo2FZ +USDmY0iQKks0lA6CU4sBcvaXY+dkfelnbEdZ+UfoKRJEZpEokB8Bg0/QJFPQJRBoqHgzHiCAMthRrtz5957zpyZe/0FIxdlaYc65cvXZ9/sbCzVu37ywuLa/syKSfBbweJGSNXxP8lDEvK6ECnkjzbgX+SHf9XsbOr474Jk +USfxKjVLejLxuLDoi8BRcjce9lotvtbVUYhVmRnHacK1RIjtqyXKhRq+pTQkF1KeIOMWkYIfkcTcI5cYpfA1KYcvgyVMnNOYFoBVsA7wH8LfxrpPRWCeAxVh6ow+eDg4POB6+Hex27PeGHt9qjT8AXSE+DJwF6nMvrAP7Jx9Z +h/ZV/ZzJlduOLTaEVZ/guVpa/HNve0f/0VFWLXq36hLNSvq4G5aq4D21Hj0LYIJfnB4dL69vlXOH7JT9g36T9gZ+4QbxIPvwftNvnV8iZ4LJQovoN9BgrMVu1RyF6nJ5gBbA8nVsw7deFJwaHXafzs6St8D52bVvJGPbQ1CA +yXDEiOfyT3LHpheYfngt0TiVExpbFQ0zXcIiIQE5qckeoS4ZsDxqkZqtTufqTI67jNHT7r8dPG3srFbcp5W1zbVS9YXt7nm6Tw/oEc56RlV6STWqm859S8f0zqk6HSdyknqXMFi7tJfwzn4BQEnzYE=�k1 � k2 +Figure 1: Feynman amplitude iΠq +X for the loop-diagrammatic evaluation of the ¯θ param- +eter. +on it. +In the below argument, we take gauge dependence parameter x0 as x0 = 0 for +simplicity. +The gluon field Ga +µ can be expanded under this gauge around x = 0 and it is given +with the gluon field-strength tensor at x = 0, Ga +µν(0), as [43] +Ga +µ(x) = 1 +2xνGa +νµ(0) + · · · += +� +d4ke−ik·x +� +− i +2Ga +νµ(0) ∂ +∂kν +δ(4)(k) +� ++ · · · . +(2.12) +Here, the discarded terms are covariant derivatives of the background gluon field-strength +tensor, which are irrelevant to the calculation of the QCD θ term. We can systematically +evaluate the interaction of the propagating quarks with the background gluon field-strength +tensor in this gauge fixing. However, we found that the effective gluon operators such +as the QCD θ term cannot be evaluated from the simple quark bubble diagrams. The +background gluon fields bring momenta, k in Eq. (2.12), which are taken to be zero in +the last of calculation due to δ(4)(k). Thus, the quark momentum is not constant due to +interaction with the background field and the quark line cannot be closed without violating +momentum conservation. +In order to fix this problem, we introduce an auxiliary (dimensionless) background +field X, and it is coupled to the CP-odd quark mass terms as +L/P , /T = − +� +q=all +Im(mq)¯qiγ5q +⇒ − +� +q=all +Im(mq) (¯qiγ5q) X . +(2.13) +We evaluate the leading contribution of Im(mq) to the QCD θ term in perturbative way, +assuming Im(mq) ≪ Re(mq). The field X is taken to be 1 in the last of calculation.#5 +The radiative QCD θ term comes from a bubble diagram. The Feynman diagram in +Fig. 1 shows the leading contribution, which is realized by integrating the delta function +#5This technique has been applied for evaluation of the Weinberg operator in the QCD [44]. +– 6 – + +in Eq. (2.12) as +iΠq +X = − +� +d4k1d4k2 +d4p +(2π)4 ˜X(−k1 − k2) +× Tr +� +(−igsγµT a) +� +− i +2Ga +ρµ(0) +∂ +∂k1;ρ +δ(4)(k1) +� +i +/p + /k1 − Re(mq)Im(mq)γ5 +× +i +/p − /k2 − Re(mq)(−igsγνT b) +� +− i +2Gb +σν(0) +∂ +∂k2;σ +δ(4)(k2) +� +i +/p − Re(mq) +� +, +(2.14) +where ˜X(k) = +� d4xeik·xX(x) and p is the loop momentum. We followed the Feynman rules +under the Fock-Schwinger gauge, which includes the gluon field Ga +µ expressed as Eq. (2.12) +and the modified CP-odd quark mass term in Eq. (2.13). Until integration of the delta +functions, two independent momenta k1, k2 flow into the vertices with the background field- +strength tensors Ga +ρµ and Gb +σν, respectively, and the artificial background field X brings a +momentum −k1 − k2 (see Fig. 1). After some calculation, we get +iΠq +X = i Im(mq)g2 +s +8 Ga +ρµ(0)Ga +σν(0) +� +d4p +(2π)4 +4iRe(mq)ϵµνρσ +[p2 − (Re(mq))2]3 ˜X(0) += iαs +8π +� +−Im(mq) +Re(mq) +� +Ga +µν(0) ˜Gaµν(0) +≃ −iθq +αs +8πGa +µν(0) ˜Gaµν(0) . +(2.15) +Here, we take ˜X(0) = 1. +After integrating the quark q out in the full theory, ∆L = Πq +X is obtained in the +effective action of the gluon. Eventually, one can derive the QCD θ term in the Fock- +Schwinger gauge method. This result is consistent with that of the chiral rotation, q → q′ = +exp(− i +2θqγ5) q, in Eq. (2.3) and also the operator Schwinger method in Eq. (2.10). Hence, +we reached a clarification of the equivalence among the Fock-Schwinger gauge method, +the operator Schwinger method, and the ordinary chiral rotation, and we noticed that the +Fock-Schwinger gauge method is more intuitive than the operator Schwinger method for +higher-loop order calculations. +It might be concerned that the diagrammatic evaluations of the light quark contribu- +tion to the QCD θ term is not justified from the viewpoint of perturbation, since the loop +momentum around the quark mass dominates the integrals in Eqs. (2.9) and (2.15). It +might be healthy to evaluate the light quark mass phases above the ΛQCD scale and derive +the QCD θ parameter by the chiral rotation. However, since the diagrammatic evaluations +are consistent with those of the chiral rotation, we may forget the problem in practical +cases. +In this paper, to evaluate the QCD θ term diagrammatically, we will use the Fock- +Schwinger gauge method. Note that the auxiliary background field X should be attached +to any perturbative interactions, but we suppress them in the following calculations for the +sake of clarity. +– 7 – + +SU(3)C +SU(2)L +SU(2)R +U(1)B−L +U(1)Y +Qi +L ≡ (ui +L, di +L)T +□ +□ +1 +1/6 +(1/6, 1/6) +Qi +R ≡ (ui +R, di +R)T +□ +1 +□ +1/6 +(2/3, −1/3) +H +1 +□ +1 +1/2 +(1/2, 1/2) +H′ +1 +1 +□ +1/2 +(1, 0) +Ua +L +□ +1 +1 +2/3 +2/3 +Ua +R +□ +1 +1 +2/3 +2/3 +Da +L +□ +1 +1 +−1/3 +−1/3 +Da +R +□ +1 +1 +−1/3 +−1/3 +Table 1: The matter contents and their gauge charges in the minimal LR symmetric +model, where U(1)Y = T R +3 + U(1)B−L. The indices i and a represent the flavors for the +doublet and singlet quarks, respectively. +3 +The minimal left-right symmetric model +3.1 +Model +From this section, we introduce the minimal LR symmetric model that can solve the +strong CP problem. The LR symmetry, which is formed by introducing a new SU(2)R +gauge symmetry, with spatial parity symmetry is motivated to forbid the QCD ¯θ term at +tree level. In particular, we focus on the minimal LR symmetric model, which embeds the +SU(2)L singlet right-handed quarks, uR and dR, to the SU(2)R doublets, QR ≡ (uR, dR)T . +Furthermore, a SU(2)R doublet Higgs, H′, and three flavors of the up-type and down-type +vector-like quarks, UL, UR, DL and DR, have to be introduced. The matter contents are +listed in Table 1. +To solve the strong CP problem, the spatial parity symmetry has to be extended to +symmetrize the left-handed and right-handed sectors as well as the SU(2)L and SU(2)R +gauge bosons, +⃗x ←→ −⃗x , +Wµ ←→ W ′µ , +(3.1) +QL, UL, DL, H ←→ QR, UR, DR, H′ , +while the other gauge bosons are invariant. The spontaneous violation of the extended +parity symmetry, SU(2)R ×U(1)B−L → U(1)Y , is caused by the vacuum expectation value +(VEV) of H′ 0, ⟨H′⟩ = (0, v′). After the symmetry breaking, the U(1)Y gauge symmetry +is generated with the gauge charge of U(1)Y = T R +3 ++ U(1)B−L. +The SU(2)R gauge +bosons absorb the Nambu-Goldstone (NG) bosons in the doublet H′ (ϕ′ + and ϕ′ 0) to +– 8 – + +become massive states (W ′ + and Z′). The physical neutral Higgs boson associated with +this symmetry breaking is denoted as h′. Then, the SU(2)L × U(1)Y gauge symmetry is +broken to U(1)EM by the VEV of H0, ⟨H⟩ = (0, v). The W + and Z bosons absorb the NG +bosons in the doublet H (ϕ+ and ϕ0), and the physical (SM) neutral Higgs boson with +this symmetry breaking is denoted as h.#6 +The resultant parity violation in nature comes from v′ ̸= v. Namely, we assume that +soft parity breaking terms are contained in the H and H′ Higgs potentials which lead to +v′ ≫ v ̸= 0. These two VEVs can be chosen as real and positive without loss of generality. +Since the extended parity is a discrete symmetry, its spontaneous breaking leads to the +formation of the domain walls, which dominate the energy density in the Universe. This +domain wall problem can be naturally solved by the Planck suppressed higher-dimensional +operators, which explicitly violate the parity symmetry [40]. +The Yukawa interactions and Dirac mass terms for the vector-like quarks are repre- +sented as +−LY =Qi +Lxia +u Ua +R ˜H + Qi +Rxia +u Ua +L ˜H′ + Ma +uUa +LUa +R ++ Qi +Lxia +d Da +RH + Qi +Rxia +d Da +LH′ + Ma +d Da +LDa +R + h.c. , +(3.2) +where i = 1–3 is a flavor index for SU(2)L/R doublets, a = 1–3 is that for the singlets +(vector-like quarks), and ˜H(′) = ϵH(′)∗ (ϵ12 = 1). The LR symmetry requires that the +Yukawa xia +u/d in the first two terms (in both lines) must be the same complex matrices, +and the Dirac mass terms Mu and Md must be Hermitian. The Dirac mass terms Ma +u and +Ma +d in Eq. (3.2) are diagonalized to real and positive eigenvalues by the field redefinitions +of the vector-like quarks.#7 The SM quark masses are realized by the seesaw mechanism +such as higher dimensional operators induced by integrating out the vector-like quarks.#8 +Before discussing the mass matrices in detail in the next section, let us count on the +number of physical CP phases in this model. The Yukawa couplings xia +u/d are 3×3 complex +matrices. Nine real parameters are removed from the Yukawa matrices by field redefinition +of Qi +L/R as Qi +L/R → UijQj +L/R with a unitary matrix U. Furthermore, phase redefinition of +Ua +L/R and Da +L/R removes five phases in the total. A remaining phase rotation corresponds +to the baryon number conservation, and it does not change xu and xd. Thus, xu and xd +have a total of 22 physical real parameters. We parametrize these 22 parameters as +xu = Φ†(θd3, θd8) VQ Φ(θu3, θu8) ¯xuVU , +xd = ¯xdVD , +(3.3) +#6The h–h′ and Z–Z′ mixings are induced in the model at tree level, though the mixings are suppressed +by v/v′ and (v/v′)2, respectively [29]. Since we take v/v′ → 0 in the calculation of the ¯θ parameter, they +are ignored. +#7One can also consider non-Hermitian vector-like quark mass matrices which correspond to soft parity +breaking terms [29]. However, such contributions produce large quark EDM and radiative ¯θ, and thus they +are severely constrained from the EDM bounds [40, 45]. +#8One can also extend the lepton sector that is insensitive to the QCD ¯θ term. If one considers SU(5)L × +SU(5)R grand unification [46, 47], vector-like neutral leptons are absent and the neutrinos keep massless +at the tree level. Interestingly, suitable Dirac neutrino masses are generated from the two-loop radiative +corrections [48] with predicting a nonzero ∆Neff [49]. +Furthermore, an O(10) keV sterile neutrino dark +matter with the leptogenesis mechanism can be incorporated [50, 51]. +– 9 – + +with +Φ(θ3, θ8) ≡ exp(iτ3θ3) exp(iτ8θ8) , +(3.4) +where τ3 and τ8 are the third and eighth Gell-Mann matrices. Here, ¯xu/d are real diagonal +matrices and VQ, VU, and VD are CKM-like unitary matrices which have three rotation +angles and one CP-violating phase. It is found that there are seven CP-violating phases +in this model (θu3/u8, θd3/d8, and three phases in VQ/U/D). When the Dirac masses are +assumed to be universal such as Ma +u = Mu and Ma +d = Md for a = 1–3, the parameters +θu3/u8, θd3/d8, VU/D become unphysical and only VQ remains physical. +In this paper, we assume that v′ <∼ Ma +q and will utilize expansions by v′/M a +q . This +inequality is motivated because the seesaw mechanism may explain the SM fermion mass +hierarchy naturally. On the other hand, if v′ ≫ Ma +q , a copy of the SM fermions has a mass +spectrum similar to the SM fermions, which spread over five orders of magnitude. A new +naturalness problem might appear in such a model, but the QCD ¯θ term is suppressed by +Ma +q /v′ since only the CKM phase survives in a limit of Ma +q → 0. In the following, we will +consider the case of v′ <∼ Ma +q . +3.2 +Parametrization of Yukawa coupling constants +In this section, we show the quark mass matrices and define the mass eigenstates. In the +mass matrices the Yukawa coupling constants, xu and xd, appear, and we have to determine +them from the observed quark masses and CKM matrix in order to evaluate the radiative +¯θ parameter. We give the parameterization of the Yukawa coupling constants assuming +the SM quark masses are given by the seesaw mechanism with v′ ≲ Ma +q . +From Eq. (3.2), the quark mass matrices in the flavor eigenstates are given as +−LM = +� +ui +L, Ua +L +� � +0 +xib +u v +x†aj +u v′ Ma +uδab +� � +uj +R +Ub +R +� ++ +� +d +i +L, Da +L +� � +0 +xib +d v +x†aj +d v′ Ma +d δab +� � +dj +R +Db +R +� ++ h.c. +≡ Up +LM(0)pq +u +Uq +R + Dp +LM(0)pq +d +Dq +R + h.c. , +(3.5) +where Up +L/R and Dp +L/R (p, q = 1, · · · , 6) are the up- and down-type flavor eigenstates, and +v ≃ 174.1 GeV. Here, Ma +u/d are real diagonal, while xia +u/d are complex matrices. It is obvious +that arg det M(0) +u/d = 0. Then, the 6×6 fermion mass matrices are diagonalized by bi-unitary +matrices, VqL and VqR, as +M(0)pq +q += V †pP +qL +¯ +MP +q V Pq +qR +for q = u and d , +(3.6) +with diagonal mass matrices ¯ +Mq. The mass eigenstates, UP +ML/R and DP +ML/R for P = 1–6, +are given as +UP +ML = V Pp +uL Up +L , +UP +MR = V Pp +uR Up +R , +DP +ML = V Pp +dL Dp +L , +DP +MR = V Pp +dR Dp +R . +(3.7) +It is difficult to reconstruct the model parameters from the experimental data in gen- +eral. Here we assume that v′ <∼ Ma +q and we take leading terms in the expansion of v′/M a +q +– 10 – + +for the quark mass eigenvalues. In this expansion, the SM quark masses are given by the +following seesaw relation +V †iI +q +mI +q +v V Ij +q += xia +q +v′ +Maq +x†aj +q +, +(3.8) +with a 3 × 3 unitary matrix Vq and I = 1–3, while the heavy quark masses are given by +Ma +q , for q = u and d. +Now let us rewrite Eq. (3.8) as +I3 = +� √v +√mq +Vqxq +√ +v′ +�Mq +� � √ +v′ +�Mq +x† +qV † +q +√v +√mq +� +≡ UqU† +q , +(3.9) +where I3 is a unit matrix in the three-dimensional space and we ignore the indices of +matrices.#9 Thus, the Yukawa matrices xq are given with a unitary matrix Uq by +xq = V † +q +√mq +√v Uq +�Mq +√ +v′ . +(3.10) +According to the previous section, one can remove some unphysical parameters in Uq and +Vq by the field redefinitions. Then, we get +xu = V † +CKM +√mu +√v Φ(θu3, θu8)VU +√Mu +√ +v′ , +xd = +√md +√v Φ(θd3, θd8)VD +√Md +√ +v′ , +(3.11) +where VCKM(≡ VuV † +d ) corresponds to the CKM matrix in the SM. Here, VU/D are CKM- +like unitary matrices with three mixing angles and one CP-violating phase, though they +are different from those in Eq. (3.3). Now we have seven physical CP-violating phases +(θu3/u8, θd3/d8, and three phases in VU/D and VCKM), which is consistent with our previous +counting, and all phases can be O(1) under the extended parity symmetry. +Since we assume that v′ <∼ Ma +q , the 6×6 diagonalization matrices in Eq. (3.7) are given +of leading terms in the expansion of v′/M a +q as +VuL ≃ +� +−VCKM VCKMxu v +Mu +v +Mu x† +u +I3 +� +, +VuR ≃ +� +VCKM −VCKMxu v′ +Mu +v′ +Mu x† +u +I3 +� +, +VdL ≃ +� +−I3 xd v +Md +v +Md x† +d +I3 +� +, +VdR ≃ +� +I3 +−xd v′ +Md +v′ +Md x† +d +I3 +� +, +(3.12) +where xq are given by Eq. (3.11). Here, the diagonal eigenvalue matrices ¯ +Mq are given as, +¯ +MP +q = diag(mI +q, Ma +q ) +for q = u and d . +(3.13) +#9A similar parameterization technique, referred to as the Casas-Ibarra parameterization, is applied in +the minimal seesaw model [52, 53]. +– 11 – + +3.3 +Quark EDMs +Before discussing the radiative ¯θ parameter in the minimal LR symmetric model, let us +comment on contributions to quark (chromo) EDMs. As long as the vector-like mass ma- +trices are Hermitian, the quark EDMs vanish completely at one-loop level. The W (′)± +contribution at one-loop level vanishes trivially since the chirality is conserved in the dia- +grams. On the other hand, the one-loop quark EDM contributions from neutral Higgses +and Z(′) may have a chirality flip in the diagrams. Nevertheless, they also vanish because +the extended parity symmetry restricts the product of two vertices to be strictly real, as +shown in Ref. [40]. +4 +Confirmation of vanishing QCD θ parameter in two-loop +order +The parity symmetry is spontaneously broken by the ⟨H′⟩ (≫ ⟨H⟩) in the LR symmetric +model. Since arg det M(0) +u/d = 0 holds, fermion one-loop contributions to the QCD ¯θ term +remain zero. However, it is expected that fermion-loop diagrams at a higher than one-loop +level would generate it. In this section, we show fermion two-loop contributions to the +QCD ¯θ term still vanish. +Integrating out quarks, the following higher-dimensional operators are expected to be +generated, +Leff = +� +n=1 +Cn +|H′|2n − |H|2n +M2n +q +αs +8πGa +µν ˜Gaµν , +(4.1) +with Mq as the scale of vector-like quark masses, and the QCD ¯θ term are induced by the +spontaneous parity symmetry breaking ⟨H′⟩ (≫ ⟨H⟩), +¯θ = +� +n=1 +Cn +� +⟨H′⟩ +Mq +�2n +. +(4.2) +We evaluate the Wilson coefficients of the operators Cn in the following. +First, we consider the contributions to the QCD ¯θ term at two-loop level coming from +an exchange of the W ′ ± boson. The two-loop fermion bubble diagrams mediated by the +W ′ ± boson under the gluon field-strength background conserve chirality in the fermion +line, and then it is proportional to +V Pi +dRV †iQ +uR V Qj +uR V †jP +dR f +� +( ¯ +MP +d )2, ( ¯ +MQ +u )2, m2 +W ′ +� +, +(4.3) +where i and j run 1–3 as the flavor index for the SU(2)R doublet, while P and Q run 1–6 +for the quark mass eigenstates. Here, a two-loop function f[( ¯ +MP +d )2, ( ¯ +MQ +u )2, m2 +W ′] is a real +function. Since +� +V Pi +dRV †iQ +uR V Qj +uR V †jP +dR +�∗ = V Pj +dR V †jQ +uR V Qi +uRV †iP +dR +and it corresponds to Eq. (4.3) +by an exchange of i ↔ j, the contribution is real so that it does not generate the QCD ¯θ +term. +– 12 – + +ADAnichVI7T9xAGBxMeIbHQZpIaU45gahOexEKiAopFDSReB0gwelk+xbOwi/svSNgXUeVP0CRJ +iBRECpEm46GP5Cn4BoIhEpTYqMfQbxELCWd2fn2xnPrtfwbStUQly0aK2v2to7Oru6X/f09vVnBgYXQ68WmL +JoerYXLBt6KG3LlUVlKVsu+4HUHcOWS8bGp7i+VJdBaHnugtr2ZcnR1rzTJ1Raq06uiqaup2VGyUP5czOZE +XScs+BoU5JC2GS/zG6uowIOJGhxIuFDENnSEfFZQgIBProSIXEBkJXWJBrqpVURf2G+Rr3CsIkvNOFUOn3hFj +T6SHjp1G+zXOVtJWZfz+Kth4m8yh803oHcWQ+KXOBLX4lwci0vx70mvKPGI025zNJpa6Zf7v76d/uiyuEYp7 +5VPZtZY17i7NazO4nTLwLs6mv7+xdz0/MDUXD4kBcMf+uBn3IFb/2Mezsq5b8/kCdNTr3JWSU/XJd5KzsVJ +krqsROSbaxvJ/yndYW7UEXKsNGIXonCwvwGCx+yBc+5kdnR3OTU+nl6MQ7vMcIfcYwiWnMoMgkm9jDd+xru +9oP7UQ7bS7VWlLNG9xr2s/qtytHw=UM +ADAnichVI7T9xAGBxMCI+QcJAGiebECUR12kMoICokKGiQeB0gwelk+xbOr9i7x0P6zq/AEKG +kCigFQRbo0+QMU/IQoTSQaFJk7DMoCQpZy7uz8+2MZ9dr+LYVKiFu2rT2Fx0vO7u6e171vn7Tl+kfWAu9em +DKounZXrBh6KG0LVcWlaVsueEHUncMW64btdm4vt6QWh57qra92XJ0Xdca9sydUWqtOXoqmrqdjTXLC+UMzm +RF0nLPgWFOSQtkUv8wNbqMCDiTocSLhQxDZ0hHw2UYCAT6EiFxAZCV1iSZ6qFVEe+x3yVc4VpGlZoqh0+8o +k4fSQ+duhr7Hc42U9blP5qmPibzGHzDeidxYi4FhfiVnwVH8U38fOfXlHiEafd52i0tNIv930YXLn/r8rhGK +d+VD2bWGbe4uzWszuJ0y8C7Olbxwc3a5ML49Eo+JMfGf+U3EjvnAHbuPOPF+Sy8fP5AnTU69yVklP1yXeTc7F +SZK6rETkW2ubyf8p/cY8qCPkWGnGLrwShb8vwFOwNp4vMtPLE3kZubSy9GFIQxjD6TmME8FlFkvc4wglOt +UPtUvukXbWam2p5i3+aNrnX373rQ4=DM +ADAnichVI7T9xAGBxMCI+QcJAGiebECU +R12kMoICokKGiQeB0gwelk+xbOr9i7x0P6zq/AEKGkCigFQRbo0+QMU/IQoTSQaFJk7DMoCQpZy7uz8+2MZ9dr+LYVKiFu2rT2Fx0vO7u6e171vn7Tl+kfWAu9emDKounZXrBh6KG0LVcWlaVsueEHUncMW64btdm4vt6Q +Wh57qra92XJ0Xdca9sydUWqtOXoqmrqdjTXLC+UMzmRF0nLPgWFOSQtkUv8wNbqMCDiTocSLhQxDZ0hHw2UYCAT6EiFxAZCV1iSZ6qFVEe+x3yVc4VpGlZoqh0+8ok4fSQ+duhr7Hc42U9blP5qmPibzGHzDeidxYi4Fhfi +VnwVH8U38fOfXlHiEafd52i0tNIv930YXLn/r8rhGKd+VD2bWGbe4uzWszuJ0y8C7Olbxwc3a5ML49Eo+JMfGf+U3EjvnAHbuPOPF+Sy8fP5AnTU69yVklP1yXeTc7FSZK6rETkW2ubyf8p/cY8qCPkWGnGLrwShb8vwFOwNp +4vMtPLE3kZubSy9GFIQxjD6TmME8FlFkvc4wglOtUPtUvukXbWam2p5i3+aNrnX373rQ4=DM +ADAnichVI7T9xAGBxMeIbHQZpIaU45ga +hOexEKiAopFDSReB0gwelk+xbOwi/svSNgXUeVP0CRJiBRECpEm46GP5Cn4BoIhEpTYqMfQbxELCWd2fn2xnPrtfwbStUQly0aK2v2to7Oru6X/f09vVnBgYXQ68WmLJoerYXLBt6KG3LlUVlKVsu+4HUHcOWS8bGp7i+VJdBa +Hnugtr2ZcnR1rzTJ1Raq06uiqaup2VGyUP5czOZEXScs+BoU5JC2GS/zG6uowIOJGhxIuFDENnSEfFZQgIBProSIXEBkJXWJBrqpVURf2G+Rr3CsIkvNOFUOn3hFjT6SHjp1G+zXOVtJWZfz+Kth4m8yh803oHcWQ+KXOBLX +4lwci0vx70mvKPGI025zNJpa6Zf7v76d/uiyuEYp75VPZtZY17i7NazO4nTLwLs6mv7+xdz0/MDUXD4kBcMf+uBn3IFb/2Mezsq5b8/kCdNTr3JWSU/XJd5KzsVJkrqsROSbaxvJ/yndYW7UEXKsNGIXonCwvwGCx+yB +c+5kdnR3OTU+nl6MQ7vMcIfcYwiWnMoMgkm9jDd+xru9oP7UQ7bS7VWlLNG9xr2s/qtytHw=UM +ADAnichVI7T9xAGBxMeIbHQZpIaU45ga +hOexEKiAopFDSReB0gwelk+xbOwi/svSNgXUeVP0CRJiBRECpEm46GP5Cn4BoIhEpTYqMfQbxELCWd2fn2xnPrtfwbStUQly0aK2v2to7Oru6X/f09vVnBgYXQ68WmLJoerYXLBt6KG3LlUVlKVsu+4HUHcOWS8bGp7i+VJdBa +Hnugtr2ZcnR1rzTJ1Raq06uiqaup2VGyUP5czOZEXScs+BoU5JC2GS/zG6uowIOJGhxIuFDENnSEfFZQgIBProSIXEBkJXWJBrqpVURf2G+Rr3CsIkvNOFUOn3hFjT6SHjp1G+zXOVtJWZfz+Kth4m8yh803oHcWQ+KXOBLX +4lwci0vx70mvKPGI025zNJpa6Zf7v76d/uiyuEYp75VPZtZY17i7NazO4nTLwLs6mv7+xdz0/MDUXD4kBcMf+uBn3IFb/2Mezsq5b8/kCdNTr3JWSU/XJd5KzsVJkrqsROSbaxvJ/yndYW7UEXKsNGIXonCwvwGCx+yB +c+5kdnR3OTU+nl6MQ7vMcIfcYwiWnMoMgkm9jDd+xru9oP7UQ7bS7VWlLNG9xr2s/qtytHw=UM +ADAnichVI7T9xAGBxMCI+QcJAGiebECU +R12kMoICokKGiQeB0gwelk+xbOr9i7x0P6zq/AEKGkCigFQRbo0+QMU/IQoTSQaFJk7DMoCQpZy7uz8+2MZ9dr+LYVKiFu2rT2Fx0vO7u6e171vn7Tl+kfWAu9emDKounZXrBh6KG0LVcWlaVsueEHUncMW64btdm4vt6Q +Wh57qra92XJ0Xdca9sydUWqtOXoqmrqdjTXLC+UMzmRF0nLPgWFOSQtkUv8wNbqMCDiTocSLhQxDZ0hHw2UYCAT6EiFxAZCV1iSZ6qFVEe+x3yVc4VpGlZoqh0+8ok4fSQ+duhr7Hc42U9blP5qmPibzGHzDeidxYi4Fhfi +VnwVH8U38fOfXlHiEafd52i0tNIv930YXLn/r8rhGKd+VD2bWGbe4uzWszuJ0y8C7Olbxwc3a5ML49Eo+JMfGf+U3EjvnAHbuPOPF+Sy8fP5AnTU69yVklP1yXeTc7FSZK6rETkW2ubyf8p/cY8qCPkWGnGLrwShb8vwFOwNp +4vMtPLE3kZubSy9GFIQxjD6TmME8FlFkvc4wglOtUPtUvukXbWam2p5i3+aNrnX373rQ4=DM +ADzHichVK7bhNBFL3O8gjhESdQINFYWEahwIxRFCqCBoq5CQ4iRQba3Y9s +UfZx2h3bOstqXgByhoAIki8Bk0/ABFPgFRBomGgjPjDQIshxntzp17zn3zNxlS8TzdhRYcY5c +/bc+dkLcxcvXb4yX1xY3EqifuyJhf5Ubzj8kT4MhQNLbUvdlQseOD6Ytvdf2Ti2wMRJzIKn+qR +Eq2Ad0O5Jz2u4WoXrzUHPFY9+SxdaqpYBuJ26U7WLpZldlRmjRquVGmfNSjhUKdmtShiDzqU0C +CQtKwfeKUYO5SjRgp+FqUwhfDkjYuKM5YDWs5/gP4e9g7VEJmAdABZgmow8eAQ4O3D7+Xex2c2+ +IvamaWH4POnx8MbhLVGFf2CE7Zp/ZR/aV/ZzKlVoOo3aE1R1jhWrPv7y+eO/qACrUf0bdapmTX +s4m9EqoV1ZjzmFN8YPDl4db65uVNJb7B37Bv1v2RH7hBOEg+/e+3Wx8foUPSdKNG7A3EMCzgpYj +Ucje5XuYnqwOSpW7T14VHgMOskfnqtJO9wD7tO3skQ9tD2ILBcISIp/OPczL6F1h+eE3RKZUSy +nIUjZno4REQiR9ncEfoSI5tDQ5IzTVdnck2mwFnMm679+4Inja171dr96vL6cntYf6Z+kG3aQl +1FqhNXpMdWqgygG9oUP64DxtJM62Th1pBjrtJfw3nxC2w30yw= +'(0)� +ADzHichVK7bhNBFL3O8gjh +ESdQINFYWEahwIxRFCqCBoq5CQ4iRQba3Y9sUfZx2h3bOstqXgByhoAIki8Bk0/ABFPgFRBomGgjPjDQIshxntzp17zn3zNxlS8TzdhRYcY5c/bc+dkLcxcvXb4yX1xY3EqifuyJhf5Ubzj8k +T4MhQNLbUvdlQseOD6Ytvdf2Ti2wMRJzIKn+qREq2Ad0O5Jz2u4WoXrzUHPFY9+SxdaqpYBuJ26U7WLpZldlRmjRquVGmfNSjhUKdmtShiDzqU0CQtKwfeKUYO5SjRgp+FqUwhfDkjYuKM5YDWs5 +/gP4e9g7VEJmAdABZgmow8eAQ4O3D7+Xex2c2+IvamaWH4POnx8MbhLVGFf2CE7Zp/ZR/aV/ZzKlVoOo3aE1R1jhWrPv7y+eO/qACrUf0bdapmTXs4m9EqoV1ZjzmFN8YPDl4db65uVNJb7B37Bv1v +2RH7hBOEg+/e+3Wx8foUPSdKNG7A3EMCzgpYjUcje5XuYnqwOSpW7T14VHgMOskfnqtJO9wD7tO3skQ9tD2ILBcISIp/OPczL6F1h+eE3RKZUSynIUjZno4REQiR9ncEfoSI5tDQ5IzTVdnck2mwF +nMm679+4Inja171dr96vL6cntYf6Z+kG3aQl1FqhNXpMdWqgygG9oUP64DxtJM62Th1pBjrtJfw3nxC2w30yw= +'(0)� +ADzHichVK7bhNBFL3O8gjh +ESdQINFYWEahwIxRFCqCBoq5CQ4iRQba3Y9sUfZx2h3bOstqXgByhoAIki8Bk0/ABFPgFRBomGgjPjDQIshxntzp17zn3zNxlS8TzdhRYcY5c/bc+dkLcxcvXb4yX1xY3EqifuyJhf5Ubzj8k +T4MhQNLbUvdlQseOD6Ytvdf2Ti2wMRJzIKn+qREq2Ad0O5Jz2u4WoXrzUHPFY9+SxdaqpYBuJ26U7WLpZldlRmjRquVGmfNSjhUKdmtShiDzqU0CQtKwfeKUYO5SjRgp+FqUwhfDkjYuKM5YDWs5 +/gP4e9g7VEJmAdABZgmow8eAQ4O3D7+Xex2c2+IvamaWH4POnx8MbhLVGFf2CE7Zp/ZR/aV/ZzKlVoOo3aE1R1jhWrPv7y+eO/qACrUf0bdapmTXs4m9EqoV1ZjzmFN8YPDl4db65uVNJb7B37Bv1v +2RH7hBOEg+/e+3Wx8foUPSdKNG7A3EMCzgpYjUcje5XuYnqwOSpW7T14VHgMOskfnqtJO9wD7tO3skQ9tD2ILBcISIp/OPczL6F1h+eE3RKZUSynIUjZno4REQiR9ncEfoSI5tDQ5IzTVdnck2mwF +nMm679+4Inja171dr96vL6cntYf6Z+kG3aQl1FqhNXpMdWqgygG9oUP64DxtJM62Th1pBjrtJfw3nxC2w30yw= +'(0)� +(a) diagram A +(b) diagram B +(c) diagram C +Figure 2: The charged NG boson contributions to the QCD ¯θ term at two-loop level. +The reason why the exchange of the W ′ ± boson does not contribute to the QCD ¯θ +term at two-loop level is clear. However, the above discussion is based on the structure of +the mixing matrices in the contribution, not on the structure of Lagrangian parameters, +such as xu/d and Mu/d, and then, it is unclear what is required to generate the QCD ¯θ term +in higher-order diagrams. We make it clear by explicit calculation of the loop diagrams in +the following. +In the unitary gauge, the lowest dimension operator (n = 1) in Eq. (4.1) might come +from diagrams which include the longitudinal mode of the W ′ ± boson. +It is because +the propagator is proportional to kµkν/m2 +W ′ (kν the momentum of the W ′ ± boson), and it +could give the lowest order contribution with regard to v′ 2. The Yukawa coupling constants +xu and xd are multiplied with the Higgs VEVs in the mixing matrices as in Eq. (3.12). +In our calculation, we adopt the Rξ gauge with the Feynman-’t Hooft gauge ξ = 1 +(for SU(2)L × SU(2)R × U(1)B−L gauge), in order to avoid the messy calculation in the +unitary gauge. The lowest dimension operator (n = 1) in Eq. (4.1) could arise the charged +NG boson exchange ϕ′ ± in this gauge. The charged NG boson is absorbed by W ′ ± boson +in the Higgs mechanism, and its mass, mϕ′, is equal to the W ′ ± mass. The charged NG +boson interactions are given as +−Lϕ′± = ui +Rxia +d Da +Lϕ′+ − d +i +Rxia +u Ua +Lϕ′ − + h.c. += (xia +d V Pi +uRV ∗Qa +dL ) UP +MRDQ +MLϕ′+ − (x∗ia +u V Pa +uL V ∗Qi +dR ) UP +MLDQ +MRϕ′+ + h.c. . +(4.4) +Both the left- and right-handed quarks are coupled with the charged NG boson. We will +show that the charged NG boson diagrams at two-loop level do not contribute to the QCD +¯θ term. +Three diagrams (dubbed as diagrams A, B and C) in Fig. 2 could give contributions +to the ¯θ parameter. By using the Fock-Schwinger gauge method (for SU(3)C gauge) in +– 13 – + +Sec. 2.2, we obtain +δθ|A = +1 +8π2 Im +� +xia +u V ∗Pa +uL V Qi +dR xjb +d V Pj +uR V ∗Qb +dL +� ¯ +MP +u ¯ +MQ +d ¯I(1;3) +� +( ¯ +MP +u )2; ( ¯ +MQ +d )2; m2 +ϕ′ +� +, +(4.5) +δθ|B = +1 +8π2 Im +� +xia +u V ∗Pa +uL V Qi +dR xjb +d V Pj +uR V ∗Qb +dL +� ¯ +MP +u ¯ +MQ +d ¯I(3;1) +� +( ¯ +MP +u )2; ( ¯ +MQ +d )2; m2 +ϕ′ +� +, +(4.6) +δθ|C = +1 +8π2 Im +� +xia +u V ∗Pa +uL V Qi +dR xjb +d V Pj +uR V ∗Qb +dL +� ¯ +MP +u ¯ +MQ +d I(2;2) +� +( ¯ +MP +u )2; ( ¯ +MQ +d )2; m2 +ϕ′ +� +, +(4.7) +where the two-loop functions ¯I(1;3), ¯I(3;1), and I(2;2) are defined in Appendix A. In the +above evaluation, we pick up contributions proportional to both xu and xd, which are +also proportional to the quark masses in the mass eigenstate propagators. +The terms +proportional to xu and x∗ +u (or xd and x∗ +d) is real. +Here, we use the dimensional regularization (d = 4 − 2ϵ) for loop momentum integrals +and the partial diagrams produce UV divergence. However, the contributions from dia- +grams A and B, proportional to ¯Iϵ(1;3) and ¯Iϵ(3;1) in Eq. (A.13), respectively, vanish, so +that the correction to the ¯θ parameter is finite and scale-independent. The UV divergent +parts (1/ϵ terms) in ¯Iϵ(1;3) and ¯Iϵ(3;1) cancel out since +Im +� +xia +u V ∗Pa +uL V Qi +dR xjb +d V Pj +uR V ∗Qb +dL +� ¯ +MP +u +¯ +MQ +d += Im +� +x∗ia +u xia +u +� += 0 , +(4.8) +Im +� +xia +u V ∗Pa +uL V Qi +dR xjb +d V Pj +uR V ∗Qb +dL +� ¯ +MQ +d +¯ +MP +u += Im +� +x∗ia +d xia +d +� += 0 . +(4.9) +To derive the above equations we use the following equations, +[(M(0) +q )−1]pq = V †pP +qR ( ¯ +M−1 +q +)P V Pq +qL += +� +− 1 +vv′ (x† +q)−1 +ic Mc +q(xq)−1 +cj +1 +v′ (x† +q)−1 +ib +1 +v(xq)−1 +aj +0 +� +for q = u and d , +(4.10) +in addition to Eq. (3.6) with ( ¯ +MP +q )∗ = ¯ +MP +q . Furthermore, we observed that the following +combinations also vanish#10 +Im +� +xia +u V ∗Pa +uL V Qi +dR xjb +d V Pj +uR V ∗Qb +dL +� ¯ +MP +u +¯ +MQ +d +log ¯ +MQ +d = 0 , +(4.11) +Im +� +xia +u V ∗Pa +uL V Qi +dR xjb +d V Pj +uR V ∗Qb +dL +� ¯ +MQ +d +¯ +MP +u +log ¯ +MP +u = 0 . +(4.12) +Therefore, the second terms of ¯Iϵ(1;3) and ¯Iϵ(3;1) in Eq. (A.13) also do not affect the ¯θ +parameter. On the other hand, the loop function I(2;2) in Eq. (4.7) is UV finite. +Similar to the ϕ′ ± contribution, the contribution from the SM charged NG boson ϕ±, +absorbed into W ±, is derived by replacing L(R) with R(L), and m2 +ϕ′ with m2 +ϕ in the above +#10The factors 1/M log M (M: quark mass) in Eqs. (4.11) and (4.12) correspond to the O(ϵ) term in +Eq. (2.15) when changing d4p to ddp (d = 4 − 2ϵ). They become O(ϵ0) since 1/ϵ comes from the quark +self-energy subdiagrams in diagrams A and B. Equations (4.11) and (4.12) can be perturbatively proved +by assuming the off-diagonal terms in the quark mass matrices are small. The similar trick is also used +around Eq. (4.26). We also checked Eqs. (4.11) and (4.12) numerically. +– 14 – + +formulae. Furthermore, (−1) is multiplied since the chiralities of circulating fermions are +opposite to diagrams of Fig. 2. It means that, if one sets v = v′ corresponding to the LR +symmetric limit, those two contributions of ϕ′ ± and ϕ± cancel each other. +The diagrams A and B correspond to the one-loop correction to the fermion mass +terms. The two-loop function ¯I(3;1)(x1; x2; x3) is expressed as +¯I(3;1)(x1; x2; x3) = (16π2µ2ϵ) +� +ddp +i(2π)d +1 +(p2 − x1)3 F0(p2, x2, x3) , +(4.13) +where F0(p2, x2, x3) is a loop function of one-loop diagrams for the fermion mass correction, +F0(p2, x2, x3) = +� 1 +0 +dz log −z(1 − z)p2 + zx2 + (1 − z)x3 +Q2 +, +(4.14) +with Q2 ≡ 4πµ2e−γE and µ is the renormalization scale. (¯I(1;3)(x1; x2; x3) also has a similar +expression, see Appendix A.) ¯I(3;1)(x1; x2; x3) has an IR-singular behavior when x1 ≪ x3 +as +¯I(3;1)(x1; x2; x3) ≃ − 1 +2x1 +F0(0, x2, x3) , +(4.15) +while small x2 and x3 do not lead to IR singularities. This behavior is expected. It is +because if a fermion with real mass mf gets a constant radiative correction to the fermion +mass mf + δmf, the correction to the ¯θ parameter is given by δθ ≃ −Im(δmf)/mf, see +Eq. (2.15).#11 However, this evaluation of δθ is justified only when the correction to the +fermion mass term is independent of fermion momentum. +The IR-singular behaviors of the SM fermion masses in ¯I(3;1) and ¯I(1;3) are not physical +in δθ|A and δθ|B in Eqs. (4.5) and (4.6), and they can be removed indeed using Eq. (4.10) +#11In Eq. (2.15), the loop function and the chirality flip lead to ∼ mf/m2 +f = 1/mf, and then δθ ≃ +−Im(δmf)/mf. +– 15 – + +as +δθ|A ≃ 1 +8π2 Im +� +xia +u xjb +d +� +V ∗Aa +uL +¯ +MA +u V Aj +uR +� � +V Bi +dR +1 +¯ +MB +d +V ∗Bb +dL +�� +× +� +( ¯ +MB +d )2 ¯I(1;3) +� +( ¯ +MA +u )2; ( ¯ +MB +d )2; m2 +ϕ′ +� +− ( ¯ +MB +d )2 ¯I(1;3) +� +0; ( ¯ +MB +d )2; m2 +ϕ′ +� ++1 +2F0 +� +0, ( ¯ +MA +u )2, m2 +ϕ′ +� +− 1 +2F0 +� +0, 0, m2 +ϕ′ +�� ++ +1 +8π2 Im +� +xia +u x∗ja +u xjb +d v′ +� +V Ai +dR +1 +¯ +MA +d +V ∗Ab +dL +�� +× +� +( ¯ +MA +d )2 ¯I(1;3) +� +0; ( ¯ +MA +d )2; m2 +ϕ′ +� ++ 1 +2F0 +� +0, 0, m2 +ϕ′ +�� +, +(4.16) +δθ|B ≃ 1 +8π2 Im +� +xia +u xjb +d +� +V Aj +uR +1 +¯ +MA +u +V ∗Aa +uL +� � +V ∗Bb +dL +¯ +MB +d V Bi +dR +�� +× +� +( ¯ +MA +u )2 ¯I(3;1) +� +( ¯ +MA +u )2; ( ¯ +MB +d )2; m2 +ϕ′ +� +− ( ¯ +MA +u )2 ¯I(3;1) +� +( ¯ +MA +u )2; 0; m2 +ϕ′ +� ++1 +2F0 +� +0, ( ¯ +MB +d )2, m2 +ϕ′ +� +− 1 +2F0 +� +0, 0, m2 +ϕ′ +�� ++ +1 +8π2 Im +� +xia +u xjb +d x∗ib +d v′ +� +V Aj +uR +1 +¯ +MA +u +V +∗Aa +uL +�� +× +� +( ¯ +MA +u )2 ¯I(3;1) +� +( ¯ +MA +u )2; 0; m2 +ϕ′ +� ++ 1 +2F0 +� +0, 0, m2 +ϕ′ +�� +, +(4.17) +where A and B run 4–6 as the heavy quark mass eigenstates, see Eq. (3.13). Here, the SM +quark masses in the loop function are taken to be zero. +On the other hand, the contribution of diagram C is not associated with the correction +to the quark masses, and then it is a new type contribution to the ¯θ parameter. It is +suppressed by the heavier fermion or ϕ′ ± masses. By taking the SM quark masses to be +zero in the loop function (see Appendix A), it is given as +δθ|C ≃ +1 +8π2 Im +� +xia +u xjb +d +� +V ∗Aa +uL +¯ +MA +u V Aj +uR +� � +V ∗Bb +dL +¯ +MB +d V Bi +dR +�� +I(2;2) +� +( ¯ +MA +u )2; ( ¯ +MB +d )2; m2 +ϕ′ +� +, +(4.18) +where A and B run 4–6 as the heavy quark mass eigenstates. It is found that the diagram +C does not contain any IR-singular behavior, unlike the diagrams A and B. +When v′ <∼ Ma +q , the leading contributions of O(v′2/(Ma +q )2) are given as +δθ|A ≃ +1 +8π2 Im +� +(Aa +u)ij(Ab +d)ji� � +v′2 ¯I(1;3) +� +(Ma +u)2; (Mb +d)2; m2 +ϕ′ +� ++ +v′2 +2(Mb +d)2 F0 +� +0, (Ma +u)2, m2 +ϕ′ +�� +, +(4.19) +δθ|B ≃ +1 +8π2 Im +� +(Aa +u)ij(Ab +d)ji� � +v′2 ¯I(3;1) +� +(Ma +u)2; (Mb +d)2; m2 +ϕ′ +� ++ +v′2 +2(Mau)2 F0 +� +0, (Mb +d)2, m2 +ϕ′ +�� +, +(4.20) +δθ|C ≃ +1 +8π2 Im +� +(Aa +u)ij(Ab +d)ji� +v′2 I(2;2) +� +(Ma +u)2; (Mb +d)2; m2 +ϕ′ +� +, +(4.21) +– 16 – + +with a Hermitian matrix Aa +q, +(Aa +q)ij ≡ xia +q x∗ja +q +for q = u, d and not sum a index . +(4.22) +This can be derived from the above formulae by +(V † +qL)aA +¯ +MA +q +p2 − ( ¯ +MA +q )2 (VqR)Ai → +x†ai +q v′ +p2 − (Maq )2 , +(4.23) +(V † +qL)aA +¯ +MA +q +[p2 − ( ¯ +MA +q )2]2 (VqR)Ai → +x†ai +q v′ +[p2 − (Maq )2]2 , +(4.24) +(V † +qL)aA +¯ +MA +q +[p2 − ( ¯ +MA +q )2]3 (VqR)Ai → +x†ai +q v′ +[p2 − (Maq )2]3 . +(4.25) +It is found that these radiative corrections to the ¯θ parameter vanish. For example, δθ|A +is given as +δθ|A = Im Tr +� +Aa +uAb +d +� +f +� +(Ma +u)2, (Mb +d)2, m2 +ϕ′ +� += 1 +2Im Tr +� +[Aa +u, Ab +d] +� +f +� +(Ma +u)2, (Mb +d)2, m2 +ϕ′ +� += 0 , +(4.26) +where f[(Ma +u)2, (Mb +d)2, m2 +ϕ′] is the real function, and the Hermitian property of the matrix +Aa +q is used. The same conclusions are applicable to δθ|B and δθ|C at this order. +Now we showed that the charged NG boson contribution to the ¯θ parameter at two- +loop level vanishes at the leading order of v′ (n = 1 in Eq. (4.1)). +It comes from the +fact that the contributions are proportional to the fourth power of xu/d. We have also +checked that the contributions of the sixth power of xu/d, corresponding to O(v′4/(Ma +q )4) +contributions, also vanish. The contributions are derived from the above formulae with the +– 17 – + +mass-insertion approximation, +(V † +qL)aA +i ¯ +MA +q +p2 − ( ¯ +MA +q )2 (VqR)Ai → i +x†ai +q v′ +p2 − (Maq )2 ++i +1 +p2 − (Maq )2 (x† +qxq)abv′2 +x†bi +q v′ +p2 − (Mbq)2 , +(4.27) +(V † +qL)aA +i ¯ +MA +q +[p2 − ( ¯ +MA +q )2]2 (VqR)Ai → i +x†ai +q v′ +[p2 − (Maq )2]2 ++i +1 +p2 − (Maq )2 (x† +qxq)abv′2 +x†bi +q v′ +[p2 − (Mbq)2]2 ++i +1 +[p2 − (Maq )2]2 (x† +qxq)abv′2 +x†bi +q v′ +p2 − (Mbq)2 , +(4.28) +(V † +qL)aA +i ¯ +MA +q +[p2 − ( ¯ +MA +q )2]3 (VqR)Ai → i +x†ai +q v′ +[p2 − (Maq )2]3 ++i +1 +p2 − (Maq )2 (x† +qxq)abv′2 +x†bi +q v′ +[p2 − (Mbq)2]3 ++i +1 +[p2 − (Maq )2]2 (x† +qxq)abv′2 +x†bi +q v′ +[p2 − (Mbq)2]2 ++i +1 +[p2 − (Maq )2]3 (x† +qxq)abv′2 +x†bi +q v′ +p2 − (Mbq)2 . +(4.29) +Each first term is the aforementioned leading contribution, which vanishes (see Eq. (4.26)). +The next-to-leading contributions are +δθ|A ≃ 1 +8π2 Im +� +(Aa +u)ij(Ab +u)jk(Ac +d)ki� +× +� +v′4I(1,1;3) +� +(Ma +u)2, (Mb +u)2; (Mc +d)2; m2 +ϕ′ +� +− +v′4 +(Mc +d)2 B(1,1) +� +0, (Ma +u)2, (Mb +u)2; m2 +ϕ′ +�� ++ +1 +8π2 Im +� +(Aa +u)ij(Ab +d)jk(Ac +d)ki� +v′4 � +I(1;1,3) + I(1;2,2) + I(1;3,1) +� � +(Ma +u)2; (Mb +d)2, (Mc +d)2; m2 +ϕ′ +� +, +(4.30) +δθ|B ≃ 1 +8π2 Im +� +(Aa +u)ij(Ab +d)jk(Ac +d)ki� +× +� +v′4I(3;1,1) +� +(Ma +u)2; (Mb +d)2, (Mc +d)2; m2 +ϕ′ +� +− +v′4 +(Mau)2 B(1,1) +� +0, (Mb +d)2, (Mc +d)2; m2 +ϕ′ +�� ++ +1 +8π2 Im +� +(Aa +u)ij(Ab +u)jk(Ac +d)ki� +v′4 � +I(1,3;1) + I(2,2;1) + I(3,1;1) +� � +(Ma +u)2, (Mb +u)2; (Mc +d)2; m2 +ϕ′ +� +, +(4.31) +δθ|C ≃ 1 +8π2 Im +� +(Aa +u)ij(Ab +u)jk(Ac +d)ki� +v′4 � +I(1,2;2) + I(2,1;2) +� � +(Ma +u)2, (Mb +u)2; (Mc +d)2; m2 +ϕ′ +� ++ +1 +8π2 Im +� +(Aa +u)ij(Ab +d)jk(Ac +d)ki� +v′4 � +I(2:1,2) + I(2:2,1) +� � +(Ma +u)2; (Mb +d)2, (Mc +d)2; m2 +ϕ′ +� +. +(4.32) +– 18 – + +These loop functions, which come from the mass-insertion approximation, are given in +Appendix A. The sequences of masses connected by commas in the loop functions are +introduced by the mass insertion. +Again, it is found that these contributions are zero. For example, the first term in δθ|A +is given as +δθ|A = Im Tr +� +Aa +uAb +uAc +d +� +f +� +(Ma +u)2, (Mb +u)2; (Mc +d)2; m2 +ϕ′ +� ++Im Tr +� +Aa +uAb +dAc +d +� +g +� +(Ma +u)2; (Mb +d)2, (Mc +d)2; m2 +ϕ′ +� += 1 +2Im Tr +� +Ac +d [Aa +u, Ab +u] +� +f +� +(Ma +u)2, (Mb +u)2; (Mc +d)2; m2 +ϕ′ +� ++1 +2Im Tr +� +Aa +u [Ab +d, Ac +d] +� +g +� +(Ma +u)2; (Mb +d)2, (Mc +d)2; m2 +ϕ′ +� += 0 . +(4.33) +Here, above two real loop functions f and g are symmetric under exchanges of (Ma +u)2 ↔ +(Mb +u)2 and (Mb +d)2 ↔ (Mc +d)2, respectively. +Then, the above equation vanishes, see Ap- +pendix A. The symmetry comes from the mass-insertion approximation. Even if we include +the higher-order contributions of xq in the mass-insertion approximation, they still vanish +since the loop function is real and symmetric for the exchange of the heavy fermion masses. +Now we found that the charged NG boson does not give a contribution to the ¯θ +parameter at two-loop level. We also numerically checked this fact by using Eqs. (4.16)– +(4.18). +The W ′ ± contributions to the ¯θ parameter in the Feynman-’t Hooft gauge at two-loop +level vanish. The Yukawa coupling dependence comes from only the mixing matrices, and +then the leading contributions, which are proportional to the fourth power of xu/d at most, +vanish. The higher-order contributions, coming from mass-insertion approximation, also +vanish due to the symmetry of heavy fermion masses in loop functions. +A similar discussion is applicable for the other contribution, such as Z′, h′, and ϕ′ 0 at +two-loop level. Then, we confirmed the two-loop contribution to the ¯θ parameter vanishes +as far as Mq >∼⟨H′⟩ ≫ ⟨H⟩. +5 +Non-vanishing contribution to QCD θ parameter in +three-loop order +In the previous section, we confirmed that the QCD ¯θ term is not generated in the two- +loop level contribution, i.e., up to the fourth order of the Yukawa interaction xq. We also +found that it is valid even if one considers the higher-order contributions of xq by using +the mass-insertion approximation. In order to give non-vanishing contributions to the ¯θ +parameter, the commutation relation [Aa +q, Ab +q] must be nonzero, see Eq. (4.33). It implies +that non-vanishing contribution should be proportional to Im Tr(Aa +q′ [Ab +q, Ac +q]) for q, q′ = u +and/or d rather than Im Tr([Ab +q, Ac +q]), and the loop function has to be asymmetric under +exchange between (Mb +q)2 and (Mc +q)2. Thus, the contributions of the following form might +– 19 – + +ADwHichVJNbxMxEJ10+Sjloy1cKnGJiI4BaeqKOqpKpce05a0FW2odjduYnX6dhHSVP8C5Ug8IJA4IH4 +GF/4Ah/4ExLFIXDjw7GwREKXY2vV4Zt6bZ48DFQltGDstTHiXLl+5Onlt6vqNm7emZ2Zvb+qk4a8HiZRkm4HvuaRkLx +uhIn4tkq5HwcR3woOntj4VpenWiTyqekr3oj9lhT7IvQNXM+6z7NdlYqYD/ZmSqzC3CiOGtXcKFE+aslsoUa71KSEQup +QTJwkGdgR+aQxd6hKjBR8DcrgS2EJF+c0oClgDawX+Pfgb2JtUxGYx0DFmDajAx4ODh+4A/xb2O3kXom9raodfwgdEb4 +U3EUqsy/sAztjn9lH9pX9HMuVOQ6rto81GK52pt+Obfx47+oGKtV/Rt1oWZD+zib1SqgXTmPU4xHePTs42ltbL2X3 +2jn2D/rfslH3CWT3e/h+ja+/ukDPuRKDG7D3oMFZBqv1GQv0UPMELaPihV3Ty14FDjsOofX0vnHW5j18w7KWH3XA9 +ixyURyeAf5g7cW2j84TlHZ1RCZJCz+IjZHvYQEchRLrePvqTI9qFB50zj1dlcm8lxFvumq/+4Fjc75SfVRZWFsoLa/k +r3uS7tI9eoBai7RMq1SjOqpIOqbX9MZb8dpe4h0OUycKOeYO/TW8o1/YFs+J +v0 +ADwHichVJNbxMxEJ10+Sjloy1cKnGJiI4BaeqK +OqpKpce05a0FW2odjduYnX6dhHSVP8C5Ug8IJA4IH4GF/4Ah/4ExLFIXDjw7GwREKXY2vV4Zt6bZ48DFQltGDstTHiXLl+5Onlt6vqNm7emZ2Zvb+qk4a8HiZRkm4HvuaRkLxuhIn4tkq5HwcR3woOntj4VpenWiTyqekr3oj9lhT7IvQNX +M+6z7NdlYqYD/ZmSqzC3CiOGtXcKFE+aslsoUa71KSEQupQTJwkGdgR+aQxd6hKjBR8DcrgS2EJF+c0oClgDawX+Pfgb2JtUxGYx0DFmDajAx4ODh+4A/xb2O3kXom9raodfwgdEb4U3EUqsy/sAztjn9lH9pX9HMuVOQ6rto81GK52pt+Obfx4 +7+oGKtV/Rt1oWZD+zib1SqgXTmPU4xHePTs42ltbL2X32jn2D/rfslH3CWT3e/h+ja+/ukDPuRKDG7D3oMFZBqv1GQv0UPMELaPihV3Ty14FDjsOofX0vnHW5j18w7KWH3XA9ixyURyeAf5g7cW2j84TlHZ1RCZJCz+IjZHvYQEchRLrePvq +TI9qFB50zj1dlcm8lxFvumq/+4Fjc75SfVRZWFsoLa/kr3uS7tI9eoBai7RMq1SjOqpIOqbX9MZb8dpe4h0OUycKOeYO/TW8o1/YFs+J +v0 +ADwHichVJNbxMxEJ10+Sjloy1cKnGJiI4BaeqK +OqpKpce05a0FW2odjduYnX6dhHSVP8C5Ug8IJA4IH4GF/4Ah/4ExLFIXDjw7GwREKXY2vV4Zt6bZ48DFQltGDstTHiXLl+5Onlt6vqNm7emZ2Zvb+qk4a8HiZRkm4HvuaRkLxuhIn4tkq5HwcR3woOntj4VpenWiTyqekr3oj9lhT7IvQNX +M+6z7NdlYqYD/ZmSqzC3CiOGtXcKFE+aslsoUa71KSEQupQTJwkGdgR+aQxd6hKjBR8DcrgS2EJF+c0oClgDawX+Pfgb2JtUxGYx0DFmDajAx4ODh+4A/xb2O3kXom9raodfwgdEb4U3EUqsy/sAztjn9lH9pX9HMuVOQ6rto81GK52pt+Obfx4 +7+oGKtV/Rt1oWZD+zib1SqgXTmPU4xHePTs42ltbL2X32jn2D/rfslH3CWT3e/h+ja+/ukDPuRKDG7D3oMFZBqv1GQv0UPMELaPihV3Ty14FDjsOofX0vnHW5j18w7KWH3XA9ixyURyeAf5g7cW2j84TlHZ1RCZJCz+IjZHvYQEchRLrePvq +TI9qFB50zj1dlcm8lxFvumq/+4Fjc75SfVRZWFsoLa/kr3uS7tI9eoBai7RMq1SjOqpIOqbX9MZb8dpe4h0OUycKOeYO/TW8o1/YFs+J +v0 +ADwHichVJNbxMxEJ10+Sjloy1cKnGJiI4BaeqK +OqpKpce05a0FW2odjduYnX6dhHSVP8C5Ug8IJA4IH4GF/4Ah/4ExLFIXDjw7GwREKXY2vV4Zt6bZ48DFQltGDstTHiXLl+5Onlt6vqNm7emZ2Zvb+qk4a8HiZRkm4HvuaRkLxuhIn4tkq5HwcR3woOntj4VpenWiTyqekr3oj9lhT7IvQNX +M+6z7NdlYqYD/ZmSqzC3CiOGtXcKFE+aslsoUa71KSEQupQTJwkGdgR+aQxd6hKjBR8DcrgS2EJF+c0oClgDawX+Pfgb2JtUxGYx0DFmDajAx4ODh+4A/xb2O3kXom9raodfwgdEb4U3EUqsy/sAztjn9lH9pX9HMuVOQ6rto81GK52pt+Obfx4 +7+oGKtV/Rt1oWZD+zib1SqgXTmPU4xHePTs42ltbL2X32jn2D/rfslH3CWT3e/h+ja+/ukDPuRKDG7D3oMFZBqv1GQv0UPMELaPihV3Ty14FDjsOofX0vnHW5j18w7KWH3XA9ixyURyeAf5g7cW2j84TlHZ1RCZJCz+IjZHvYQEchRLrePvq +TI9qFB50zj1dlcm8lxFvumq/+4Fjc75SfVRZWFsoLa/kr3uS7tI9eoBai7RMq1SjOqpIOqbX9MZb8dpe4h0OUycKOeYO/TW8o1/YFs+J +v0 +ADynichVLbhMxFL3p8Cjl0bRskNhEREFsCA6qAHVwYFi7QlbaUmRJ6Jm1idh+VxkoZRdqz4ARZISCAhQHwG36ART8B +sSwSGxYcO1MERCm2Znx97z3nHvaV6FMDWOHhTnv1OkzZ+fPLZy/cPHSYnFpeStN+joQjSAJE73j81SEMhYNI0odpQWPJDse3 +vP7Dx7YHQqUzix2akRCvi3VjuyYAbuNrF5eaAa9WT7Km0jISpZvjdrHMqsyN0rRy40y5aOeLBXq1KQOJRQnyISFJOBHRKnFH +OXasRIwdeiD4NS7q4oDEtAGtgHeA/hL+DtUclYO4BFWHajD54BDg4cPv4d7Hbzb0x9rZq6vgD6AjxaXCXqMK+sA/siH1mH9lX +9nMmV+Y4rNoRVn+CFaq9+PzK5o/oiKsVvVv1ImaDe3hbFarhHblPYUwQ/ePriaHN1o5JdZ2/YN+h/zQ7ZJ5wgHnwP3q6LjZc +n6DlWYnAD9h5ScFbAaj0G2at0CzOAzVGx6u6pC48Ch12n8bNrpXmHe9h18k7GsIeuB5HjihHJ4J/kjt1baP3hOUZnVEZknLNwxG +wPh4hI5CiXO0JfNLI5NKQ502x1NtdmCpzFvunavy942ti6Xa3dqa6sr5TX7ueve56u0jW6gVp3aY0eUp0aqHJAr+gdvfcedob +edkda6QYy7TX8N79gsE+dLH +'0� +ADynichVLbhMxFL3p8Cjl0bRskNhEREFsCA6qAHVwY +Fi7QlbaUmRJ6Jm1idh+VxkoZRdqz4ARZISCAhQHwG36ART8BsSwSGxYcO1MERCm2Znx97z3nHvaV6FMDWOHhTnv1OkzZ+fPLZy/cPHSYnFpeStN+joQjSAJE73j81SEMhYNI0odpQWPJDse3vP7Dx7YHQqUzix2akRCvi3VjuyYAbuNrF5eaAa9WT7Km0jIS +pZvjdrHMqsyN0rRy40y5aOeLBXq1KQOJRQnyISFJOBHRKnFHOXasRIwdeiD4NS7q4oDEtAGtgHeA/hL+DtUclYO4BFWHajD54BDg4cPv4d7Hbzb0x9rZq6vgD6AjxaXCXqMK+sA/siH1mH9lX9nMmV+Y4rNoRVn+CFaq9+PzK5o/oiKsVvVv1ImaDe3hbFarhH +blPYUwQ/ePriaHN1o5JdZ2/YN+h/zQ7ZJ5wgHnwP3q6LjZcn6DlWYnAD9h5ScFbAaj0G2at0CzOAzVGx6u6pC48Ch12n8bNrpXmHe9h18k7GsIeuB5HjihHJ4J/kjt1baP3hOUZnVEZknLNwxGwPh4hI5CiXO0JfNLI5NKQ502x1NtdmCpzFvunavy942ti6Xa3 +dqa6sr5TX7ueve56u0jW6gVp3aY0eUp0aqHJAr+gdvfcedobedkda6QYy7TX8N79gsE+dLH +'0� +A +AD13ichVI9bxNBEB3n+AjhI05okCg4YRlRM4aRCliqChdBKcBMWOdT5v7FXuY3W3tjGnExUI0VFRUIFEgdLCL6DhD1DkJyDKINFQ8Hbv +gDLYVd3Ozsz783bnW1LT8SKscPClHXq9Jmz0+dmzl+4eGm2ODe/GYf9yOV1N/TCaLvtxNwTAa8roTy+LSPu+G2Pb7X37+n41oBHsQiDB2o +kedN3uoHYE6j4GoVrzUGTiR7YjdpyEj43Gbpgt1YsHu72b5VLEKM8MeN6q5UaJ81MK5Qo0a1KGQXOqT5wCUrA9cijG3KEqMZLwNSmBL4 +IlTJxTSjPAKliP8B/C38HaIxuYZaB8TJ3RBw8HhwPcPv5d7HZyb4C9rhobfhc6PHwRuG0qsy/sPTtin9kB+8p+TuRKDIdWO8LazrBctmZf +XNn48V+Uj1Wr/o06UbOiPZxNaxXQLo1Hn8LN8IPHr42VtbLyQ32ln2D/jfskH3CYLBd/fdGl9/fYKeYyUKN6DvIQZnGazao5C9QouYLmw +HFSvmnrwSHDodRw/uVacd7iHXSfvZAB7aHrgG64AkQT+LDc1b6H5h+cYnVAJkTRncRDTPRwiIpAjTe4IfYmQ7UBDnDNVqdzdSbHWVK86e +q/L3jc2LxVqd6uLK0tlVbv5q97mq7SdbqJWndole5Tjeqo8pQO6AN9tB5aT6xn1vMsdaqQYy7TX8N6+QtwcNd/ +'00, h0 +AD13ichVI9bxNBEB3n+AjhI05okCg4YRlRM4aRCliqChdBKcBMW +OdT5v7FXuY3W3tjGnExUI0VFRUIFEgdLCL6DhD1DkJyDKINFQ8HbvgDLYVd3Ozsz783bnW1LT8SKscPClHXq9Jmz0+dmzl+4eGm2ODe/GYf9yOV1N/TCaLvtxNwTAa8roTy+LSPu+G2Pb7X37+n41oBHsQiDB2okedN3uoHYE6j4GoVrzUGTiR7YjdpyEj43Gbpgt1YsHu72b5VLEK +M8MeN6q5UaJ81MK5Qo0a1KGQXOqT5wCUrA9cijG3KEqMZLwNSmBL4IlTJxTSjPAKliP8B/C38HaIxuYZaB8TJ3RBw8HhwPcPv5d7HZyb4C9rhobfhc6PHwRuG0qsy/sPTtin9kB+8p+TuRKDIdWO8LazrBctmZfXNn48V+Uj1Wr/o06UbOiPZxNaxXQLo1Hn8LN8IPHr42VtbLyQ32ln +2D/jfskH3CYLBd/fdGl9/fYKeYyUKN6DvIQZnGazao5C9QouYLmwHFSvmnrwSHDodRw/uVacd7iHXSfvZAB7aHrgG64AkQT+LDc1b6H5h+cYnVAJkTRncRDTPRwiIpAjTe4IfYmQ7UBDnDNVqdzdSbHWVK86eq/L3jc2LxVqd6uLK0tlVbv5q97mq7SdbqJWndole5Tjeqo8pQO6AN +9tB5aT6xn1vMsdaqQYy7TX8N6+QtwcNd/ +'00, h0 +A +ADuXichVLbhMxFL3p8GjLoy1sKrGJiIJYBQdVgMqmgk2XaULaSiWKPBM3cTsvjZ2EMoPILFtF6xAYoH4Db8AIt+AmJZJDYsOHZcBEQ +ptmZ8fe895x72k9DqTRjJ4U578LFS5fnFxavXL12fWl5ca2SvpZIJpBEibZrs+VCGUsmlrqUOymeCRH4od/Cpie8MRKZkEj/To1S0I +t6N5b4MuIar0W/X28slVmF2FKeNqjNK5EYtWSnU6Dl1KGA+hSRoJg07JA4Kcw9qhKjFL4W5fBlsKSNCxrTIrAa1gv8h/B3sPaoCMwjoCJ +Mk9EHjwAHB+4Q/y52e84bY2+qKsfQEeILwN3kcrsC/vATtln9pF9ZT9ncuWw6gdYfUnWJG2l16tNn78FxVhNap/o87VrGkfZzNaJbSn1 +mNOEUzwg5fHp431ejm/w96xb9D/lp2wTzhBPgevN8S9Tfn6DlTonED5h4UOMtgNR6N7HW6hxnA5qhYsfUhScFh1mn8bNrKdfhHnYd18k +Y9tD2ILJcMSI5/JPcsX0LrT8Z+icSoiMHQtHzPRwiIhETmpzR+hLhmwODcoxzVZnck2mwFnGeNPVf1/wtLF9v1J9UFnbWitPHGve5u0 +W26i1oPaYM2qUZNVOnSazqiY+x72edzBJnSs4zE36a3jqF1Z1zCQ=uR +ADuXichVLbhMxFL3p8GjLoy1sKrGJiIJYBQdVgMqmgk2XaULaSi +WKPBM3cTsvjZ2EMoPILFtF6xAYoH4Db8AIt+AmJZJDYsOHZcBEQptmZ8fe895x72k9DqTRjJ4U578LFS5fnFxavXL12fWl5ca2SvpZIJpBEibZrs+VCGUsmlrqUOymeCRH4od/Cpie8MRKZkEj/To1S0It6N5b4MuIar0W/X28slVmF2FKeNqjNK5EYtWSnU6Dl1KGA+hSRoJ +g07JA4Kcw9qhKjFL4W5fBlsKSNCxrTIrAa1gv8h/B3sPaoCMwjoCJMk9EHjwAHB+4Q/y52e84bY2+qKsfQEeILwN3kcrsC/vATtln9pF9ZT9ncuWw6gdYfUnWJG2l16tNn78FxVhNap/o87VrGkfZzNaJbSn1mNOEUzwg5fHp431ejm/w96xb9D/lp2wTzhBPgevN8S9Tfn6DlTon +ED5h4UOMtgNR6N7HW6hxnA5qhYsfUhScFh1mn8bNrKdfhHnYd18kY9tD2ILJcMSI5/JPcsX0LrT8Z+icSoiMHQtHzPRwiIhETmpzR+hLhmwODcoxzVZnck2mwFnGeNPVf1/wtLF9v1J9UFnbWitPHGve5u0W26i1oPaYM2qUZNVOnSazqiY+x72edzBJnSs4zE36a3jqF1Z1zC +Q=uR +A +ADuXichVI9b1MxFL3pK7SUj7awILFEREFMwUFVQWpYGFgSBvSVipR5PfiJqbvS89O0vQpfwCJlQ5MIDEgfgZL/wBDfwJiLBILA8eOi4A +oxdZ7vr73nOPfe2noVSasZPCjDd74eLc/KWFy1euXltcWr6+pZJeFohGkIRJtuNzJUIZi4aWOhQ7aSZ45Idi29/YuLbfZEpmcTP9TAVz +Yh3YrknA67hqjdaz1pLJVZhdhQnjaozSuRGLVku1OgFtSmhgHoUkaCYNOyQOCnMXaoSoxS+JuXwZbCkjQsa0QKwGtYB/gP421i7VATmIVA +RpsnogUeAgwO3j38Hu13njbE3VZXlD6AjxJeBu0hl9oV9ZKfsmH1iX9nPqVy5TBqh1j9MVakrcVXN+s/ouKsBrVv1Hnata0h7MZrRLaU ++sxpwjG+P7h0Wl9bOc32Hv2Tfof8dO2GecIO5/Dz5siM235+g5U6JxA+YeFDjLYDUejew1uocZwOaoWLH31IEnBYdZJ/HTaynX4S52bdf +JGPbA9iCyXDEiOfzj3JF9C80/PGfonEqIjBwLR8z0cICIRE5qc4foS4ZsDg3KMU1XZ3JNpsBZRnjT1X9f8KSxdb9SXa2sbKyU1h+71z1Pt ++g23UWtB7ROT6lGDVTp0Gt6Q0feI497Xe/lOHWm4DA36K/hqV/X1Mv+UL +ADuXichVLbhMxFL3p8GjLoy1sKrGJiIJYBQdVgMqmgk2XaULaSiWKPBM3cTsvjZ2EMoPILFtF6xAYoH4Db8AIt+AmJZJDYs +OHZcBEQptmZ8fe895x72k9DqTRjJ4U578LFS5fnFxavXL12fWl5ca2SvpZIJpBEibZrs+VCGUsmlrqUOymeCRH4od/Cpie8M +RKZkEj/To1S0It6N5b4MuIar0WnX28slVmF2FKeNqjNK5EYtWSnU6Dl1KGA+hSRoJg07JA4Kcw9qhKjFL4W5fBlsKSNCxrTIrAa1 +gv8h/B3sPaoCMwjoCJMk9EHjwAHB+4Q/y52e84bY2+qKsfQEeILwN3kcrsC/vATtln9pF9ZT9ncuWw6gdYfUnWJG2l16tNn78Fx +VhNap/o87VrGkfZzNaJbSn1mNOEUzwg5fHp431ejm/w96xb9D/lp2wTzhBPgevN8S9Tfn6DlTonED5h4UOMtgNR6N7HW6hxnA5q +hYsfUhScFh1mn8bNrKdfhHnYd18kY9tD2ILJcMSI5/JPcsX0LrT8Z+icSoiMHQtHzPRwiIhETmpzR+hLhmwODcoxzVZnck2mwFn +GeNPVf1/wtLF9v1J9UFnbWitPHGve5u0W26i1oPaYM2qUZNVOnSazqiY+x72edzBJnSs4zE36a3jqFx3QzBM=dR +ADuXichVLbhMxFL3p8Cjl0QcbJDYRURCr4FQVoLKpgAULFmlD2kolijwTNzGdl8ZO0j +DKD1Tqli5YgcQC8Rls+AEW/QTEskhsWHDsuAiIUmzN+Pre849rWfhlJpxo4LM9658xcuzl6au3zl6rX5hcWlTZX0skA0giRMsm2fKxHKWDS01KHYTjPBIz8UW/7eYxPf6otMySR+roepaEa8E8tdGXANV/1J61lrocQqzI7ipF1RoncqCWLhRq9oDYlFCPIhIUk4YdEieFuUNVYpTC16QcvgyWtHFBI5oDVsPax38Afxtrl +4rAPAqwjQZPfAIcHDg9vDvYLfjvDH2pqy/AF0hPgycBepzL6wD+yEfWYf2Vf2cypXbjmM2iFWf4wVaWv+4Eb9x39REVaj+jfqTM2adnE2o1VCe2o95hTBGN9/dXRSX90o57fZO/YN+t+yY/YJ4j734P362LjzRl6TpVo3IC5BwXOMliNRyN7le5iBrA5KlbsPXgScFh1kn89FrKdbiLXdt1MoY9sD2ILFeMSA7/OHdk30L +zD8pOqcSIiPHwhEzPRwgIpGT2twh+pIhm0ODckzT1ZlckylwlhHedPXfFzxpbC5XqvcqK+srpbVH7nXP0k26RXdQ6z6t0VOqUQNVOnRIr+nIe+hxr+u9HKfOFBzmOv01PULny/L7Q=DL +A +ADuXichVI9b1MxFL3pK7SUj7awILFEREFMwUFVQWpYGFgSBvSVipR5PfiJqbvS89O0vQpfwCJlQ5MIDEgfgZL/wBDfwJiLBILA8eOi4A +oxdZ7vr73nOPfe2noVSasZPCjDd74eLc/KWFy1euXltcWr6+pZJeFohGkIRJtuNzJUIZi4aWOhQ7aSZ45Idi29/YuLbfZEpmcTP9TAVz +Yh3YrknA67hqjdaz1pLJVZhdhQnjaozSuRGLVku1OgFtSmhgHoUkaCYNOyQOCnMXaoSoxS+JuXwZbCkjQsa0QKwGtYB/gP421i7VATmIVA +RpsnogUeAgwO3j38Hu13njbE3VZXlD6AjxJeBu0hl9oV9ZKfsmH1iX9nPqVy5TBqh1j9MVakrcVXN+s/ouKsBrVv1Hnata0h7MZrRLaU ++sxpwjG+P7h0Wl9bOc32Hv2Tfof8dO2GecIO5/Dz5siM235+g5U6JxA+YeFDjLYDUejew1uocZwOaoWLH31IEnBYdZJ/HTaynX4S52bdf +JGPbA9iCyXDEiOfzj3JF9C80/PGfonEqIjBwLR8z0cICIRE5qc4foS4ZsDg3KMU1XZ3JNpsBZRnjT1X9f8KSxdb9SXa2sbKyU1h+71z1Pt ++g23UWtB7ROT6lGDVTp0Gt6Q0feI497Xe/lOHWm4DA36K/hqV/X1Mv+UL +ADuXichVI9b1MxFL3pK7SUj7awILFEREFMwUFVQWpYGFgSBvSVi +pR5PfiJqbvS89O0vQpfwCJlQ5MIDEgfgZL/wBDfwJiLBILA8eOi4AoxdZ7vr73nOPfe2noVSasZPCjDd74eLc/KWFy1euXltcWr6+pZJeFohGkIRJtuNzJUIZi4aWOhQ7aSZ45Idi29/YuLbfZEpmcTP9TAVzYh3YrknA67hqjdaz1pLJVZhdhQnjaozSuRGLVku1OgFtSmhgHoUka +CYNOyQOCnMXaoSoxS+JuXwZbCkjQsa0QKwGtYB/gP421i7VATmIVARpsnogUeAgwO3j38Hu13njbE3VZXlD6AjxJeBu0hl9oV9ZKfsmH1iX9nPqVy5TBqh1j9MVakrcVXN+s/ouKsBrVv1Hnata0h7MZrRLaU+sxpwjG+P7h0Wl9bOc32Hv2Tfof8dO2GecIO5/Dz5siM235+g5U6 +JxA+YeFDjLYDUejew1uocZwOaoWLH31IEnBYdZJ/HTaynX4S52bdfJGPbA9iCyXDEiOfzj3JF9C80/PGfonEqIjBwLR8z0cICIRE5qc4foS4ZsDg3KMU1XZ3JNpsBZRnjT1X9f8KSxdb9SXa2sbKyU1h+71z1Pt+g23UWtB7ROT6lGDVTp0Gt6Q0feI497Xe/lOHWm4DA36K/hqV/X1M +v+UL +ADuXichVI9b1MxFL3pK7SUj7awILFEREFMwUFVQWpYGFgSBvSVi +pR5PfiJqbvS89O0vQpfwCJlQ5MIDEgfgZL/wBDfwJiLBILA8eOi4AoxdZ7vr73nOPfe2noVSasZPCjDd74eLc/KWFy1euXltcWr6+pZJeFohGkIRJtuNzJUIZi4aWOhQ7aSZ45Idi29/YuLbfZEpmcTP9TAVzYh3YrknA67hqjdaz1pLJVZhdhQnjaozSuRGLVku1OgFtSmhgHoUka +CYNOyQOCnMXaoSoxS+JuXwZbCkjQsa0QKwGtYB/gP421i7VATmIVARpsnogUeAgwO3j38Hu13njbE3VZXlD6AjxJeBu0hl9oV9ZKfsmH1iX9nPqVy5TBqh1j9MVakrcVXN+s/ouKsBrVv1Hnata0h7MZrRLaU+sxpwjG+P7h0Wl9bOc32Hv2Tfof8dO2GecIO5/Dz5siM235+g5U6 +JxA+YeFDjLYDUejew1uocZwOaoWLH31IEnBYdZJ/HTaynX4S52bdfJGPbA9iCyXDEiOfzj3JF9C80/PGfonEqIjBwLR8z0cICIRE5qc4foS4ZsDg3KMU1XZ3JNpsBZRnjT1X9f8KSxdb9SXa2sbKyU1h+71z1Pt+g23UWtB7ROT6lGDVTp0Gt6Q0feI497Xe/lOHWm4DA36K/hqV/X1M +v+UL +ADuXichVLbhMxFL3p8GjLoy1sKrGJiIJYBQdVgMqmgk2Xa +ULaSiWKPBM3cTsvjZ2EMoPILFtF6xAYoH4Db8AIt+AmJZJDYsOHZcBEQptmZ8fe895x72k9DqTRjJ4U578LFS5fnFxavXL12fWl5ca2SvpZIJpBEibZrs+VCGUsmlrqUOymeCRH4od/Cpie8MRKZkEj/To1S0It6N5b4MuIar0WnX28slVmF2FKeNqjNK5EYtWSn +U6Dl1KGA+hSRoJg07JA4Kcw9qhKjFL4W5fBlsKSNCxrTIrAa1gv8h/B3sPaoCMwjoCJMk9EHjwAHB+4Q/y52e84bY2+qKsfQEeILwN3kcrsC/vATtln9pF9ZT9ncuWw6gdYfUnWJG2l16tNn78FxVhNap/o87VrGkfZzNaJbSn1mNOEUzwg5fHp431ejm/w96xb9D/ +lp2wTzhBPgevN8S9Tfn6DlTonED5h4UOMtgNR6N7HW6hxnA5qhYsfUhScFh1mn8bNrKdfhHnYd18kY9tD2ILJcMSI5/JPcsX0LrT8Z+icSoiMHQtHzPRwiIhETmpzR+hLhmwODcoxzVZnck2mwFnGeNPVf1/wtLF9v1J9UFnbWitPHGve5u0W26i1oPaYM2qUZNVO +nSazqiY+x72edzBJnSs4zE36a3jqFx3QzBM=dR +ADuXichVLbhMxFL3p8GjLoy1sKrGJiIJYBQdVgMqmgk2Xa +ULaSiWKPBM3cTsvjZ2EMoPILFtF6xAYoH4Db8AIt+AmJZJDYsOHZcBEQptmZ8fe895x72k9DqTRjJ4U578LFS5fnFxavXL12fWl5ca2SvpZIJpBEibZrs+VCGUsmlrqUOymeCRH4od/Cpie8MRKZkEj/To1S0It6N5b4MuIar0WnX28slVmF2FKeNqjNK5EYtWSn +U6Dl1KGA+hSRoJg07JA4Kcw9qhKjFL4W5fBlsKSNCxrTIrAa1gv8h/B3sPaoCMwjoCJMk9EHjwAHB+4Q/y52e84bY2+qKsfQEeILwN3kcrsC/vATtln9pF9ZT9ncuWw6gdYfUnWJG2l16tNn78FxVhNap/o87VrGkfZzNaJbSn1mNOEUzwg5fHp431ejm/w96xb9D/ +lp2wTzhBPgevN8S9Tfn6DlTonED5h4UOMtgNR6N7HW6hxnA5qhYsfUhScFh1mn8bNrKdfhHnYd18kY9tD2ILJcMSI5/JPcsX0LrT8Z+icSoiMHQtHzPRwiIhETmpzR+hLhmwODcoxzVZnck2mwFnGeNPVf1/wtLF9v1J9UFnbWitPHGve5u0W26i1oPaYM2qUZNVO +nSazqiY+x72edzBJnSs4zE36a3jqFx3QzBM=dR +ADuXichVLbhMxFL3p8GjLoy1sKrGJiIJYBQdVgMqmgk2Xa +ULaSiWKPBM3cTsvjZ2EMoPILFtF6xAYoH4Db8AIt+AmJZJDYsOHZcBEQptmZ8fe895x72k9DqTRjJ4U578LFS5fnFxavXL12fWl5ca2SvpZIJpBEibZrs+VCGUsmlrqUOymeCRH4od/Cpie8MRKZkEj/To1S0It6N5b4MuIar0WnX28slVmF2FKeNqjNK5EYtWSn +U6Dl1KGA+hSRoJg07JA4Kcw9qhKjFL4W5fBlsKSNCxrTIrAa1gv8h/B3sPaoCMwjoCJMk9EHjwAHB+4Q/y52e84bY2+qKsfQEeILwN3kcrsC/vATtln9pF9ZT9ncuWw6gdYfUnWJG2l16tNn78FxVhNap/o87VrGkfZzNaJbSn1mNOEUzwg5fHp431ejm/w96xb9D/ +lp2wTzhBPgevN8S9Tfn6DlTonED5h4UOMtgNR6N7HW6hxnA5qhYsfUhScFh1mn8bNrKdfhHnYd18kY9tD2ILJcMSI5/JPcsX0LrT8Z+icSoiMHQtHzPRwiIhETmpzR+hLhmwODcoxzVZnck2mwFnGeNPVf1/wtLF9v1J9UFnbWitPHGve5u0W26i1oPaYM2qUZNVO +nSazqiY+x72edzBJnSs4zE36a3jqFx3QzBM=dR +ADuXichVLbhMxFL3p8Cjl0QcbJDYRURCr4FQVoLKpgAULFmlD2kolijwTNzGdl8ZO0j +DKD1Tqli5YgcQC8Rls+AEW/QTEskhsWHDsuAiIUmzN+Pre849rWfhlJpxo4LM9658xcuzl6au3zl6rX5hcWlTZX0skA0giRMsm2fKxHKWDS01KHYTjPBIz8UW/7eYxPf6otMySR+roepaEa8E8tdGXANV/1J61lrocQqzI7ipF1RoncqCWLhRq9oDYlFCPIhIUk4YdEieFuUNVYpTC16QcvgyWtHFBI5oDVsPax38Afxtrl +4rAPAqwjQZPfAIcHDg9vDvYLfjvDH2pqy/AF0hPgycBepzL6wD+yEfWYf2Vf2cypXbjmM2iFWf4wVaWv+4Eb9x39REVaj+jfqTM2adnE2o1VCe2o95hTBGN9/dXRSX90o57fZO/YN+t+yY/YJ4j734P362LjzRl6TpVo3IC5BwXOMliNRyN7le5iBrA5KlbsPXgScFh1kn89FrKdbiLXdt1MoY9sD2ILFeMSA7/OHdk30L +zD8pOqcSIiPHwhEzPRwgIpGT2twh+pIhm0ODckzT1ZlckylwlhHedPXfFzxpbC5XqvcqK+srpbVH7nXP0k26RXdQ6z6t0VOqUQNVOnRIr+nIe+hxr+u9HKfOFBzmOv01PULny/L7Q=DL +Figure 3: Diagrams that contribute to Eq. (5.2) and have asymmetric structure in the +loop function. The neutral scalar propagator corresponds to the neutral NG boson ϕ′ 0 and +the neutral Higgs boson h′. +be leading if they are non-vanishing in the three-loop order, +δθuuu ≈ +1 +(16π2)2 +v′2 +� +M2 Im Tr +� +Aa +u [Ab +u, Ac +u] +� +fabc +uuu , +(5.1) +δθduu ≈ +1 +(16π2)2 +v′2 +� +M2 Im Tr +� +Aa +d [Ab +u, Ac +u] +� +fabc +duu , +(5.2) +where � +M is the heaviest quark mass in the loop diagrams. Here, fabc +uuu is a dimensionless +three-loop function which is totally antisymmetric under permutation of (Ma +u)2, (Mb +u)2 and +(Mc +u)2, while fabc +duu is antisymmetric under permutation of (Mb +u)2 and (Mc +u)2. Although the +other types such as Im Tr(Aa +u [Ab +d, Ac +d]) and Im Tr(Aa +d [Ab +d, Ac +d]) can also contribute to the +¯θ parameter, they are suppressed by the SM down-type quark masses, so we do not take +them into account in this paper. +We found that δθuuu in Eq. (5.1) is not generated from diagrams in the three-loop +order in the minimal LR model. The corresponding three-loop diagrams do not have an +asymmetric structure for three Dirac fermion masses when the internal scalar lines respect +the U(1)B−L. If the neutral scalar lines break the U(1)B−L (or the scalar lines pick v′ +using the four-point Higgs interaction), the diagrams may have the asymmetric structure. +However, in the case, the contribution the ¯θ parameter is proportional to v′4/ � +M4 (n = 2 +in Eq. (4.1)), not v′2/ � +M2. +This situation is not changed even in the four-loop order. +Thus, we conclude that δθuuu is not the leading contribution and the largest non-vanishing +contribution to the ¯θ parameter comes from δθduu in Eq. (5.2) in the minimal LR symmetric +model. +5.1 +Leading contribution: PDF of δθduu +In this section, we estimate the size of δθduu in Eq. (5.2). We find that diagrams in Fig. 3 +would provide the antisymmetric three-loop function and produce the non-vanishing δθduu. +When one considers a universal down-type vector-like quark mass M1 +d = M2 +d = M3 +d ≡ +� +Md for simplicity, Φ(θd3, θd8)VD in the down-type Yukawa xd in Eq. (3.11) become un- +physical parameters, because these are removed by changing the basis of DL/R. Then, the +– 20 – + +M3 +u/M 1 +u = 10−3 +M3 +u/M 1 +u = 10−2 +max(xu) < 4π +42.1% (58.9%) +32.2% (46.2%) +max(xu) < +√ +4π +3.65% (10.6%) +38.8% (55.9%) +Table 2: Ratios of excluded parameter regions for the parameter sets in Fig. 4 from the +current neutron EDM measurement, under an assumption of ¯f13 +duu = 1. The numbers in +parentheses are those in Fig. 5 under an assumption of ¯f13 +duu = ¯f23 +duu = 1. The parameter +sets are restricted by the perturbative bound of the Yukawa coupling xu as 4π or +√ +4π. +Here, � +M = � +Md = M1 +u = M2 +u. +contribution from δθduu is simplified as +δθduu ≈ +1 +(16π2)2 +v′2 +� +M2 +� +MdMb +uMc +u +v′3 +mi +dmk +u +� +mj +umlu +v3 +× Im Tr +� +V †ij +CKMV ′jb +U V ′†bk +U +V ′kc +U V ′†cl +U +V li +CKM − (b ↔ c) +� ˜fbc +duu +≈ +4 +(16π2)2 +v′2 +� +M2 +� +MdMb +uMc +u +v′3 +mbm +3 +2 +t +√mc +v3 +Im +� +V †33 +CKMV ′3b +U V ′†b3 +U +V ′3c +U V ′†c2 +U +V 23 +CKM +� ˜fbc +duu , (5.3) +where a three-loop function ˜fbc +duu is antisymmetric under the permutation of b and c, and +˜fbc +duu = f1bc +duu = f2bc +duu = f3bc +duu for the universal down-type Dirac quark mass. Here, V ′ +U ≡ +Φ(θu3, θu8)VU. A term proportional to mbm2 +t (corresponding to i = 3 and j = k = l = 3) +vanishes by Im(V †33 +CKMV ′3b +U V ′†b3 +U +V ′3c +U V ′†c3 +U +V 33 +CKM) = 0. Although the above contribution (i = +3, j = k = 3 and l = 2) is suppressed by V 23 +CKM ≃ 0.04, we found that it can provide a +larger contribution than a term proportional to mbmtmc (i = 3, j = l = 3 and k = 2). +We considered the benchmark points where � +M = M1 +d = M2 +d = M3 +d = M1 +u = M2 +u = +103M3 +u and � +M = M1 +d = M2 +d = M3 +d = M1 +u = M2 +u = 102M3 +u. Both benchmark points have +the degenerate down-type vector-like quark masses and the partially degenerate up-type +masses. This is because we would like to focus on the case that the hierarchy in the SM +down-type quark masses is explained by not one in the down-type vector-like quark masses +Ma +d but one in the components of the Yukawa coupling xia +d +in the seesaw mechanism +Eq. (3.8). +This is motivated by the fact that the SM down-type quark masses have a +moderate hierarchy compared to the up-type ones. +We estimate the size of the three-loop function as v′2/ � +M2 ˜fbc +duu, where � +M is the heaviest +quark mass. It is naively expected that when a loop function is made up of O(1) mass-ratio +parameter, its size is maximized. In this case a mass-ratio m2 +ϕ′/(M3 +u)2 can become O(1), so +that the three-loop function would be maximized when b or c = 3. Therefore, the leading +contributions in Eq. (5.3) would be dominated by (b, c) = (1, 3) and (2, 3). +In Figs. 4a and 4b, we show the probability density functions (PDFs) of the absolute +value of δθduu in Eq. (5.2) with b = 1 and c = 3 (also included b = 3 and c = 1), by +the solid lines. The total integral of the solid lines is 1 in all plots. When M1 +u ≫ M2 +u +(reflecting the fact that mu ≪ mc), this contribution would be dominant in the non- +– 21 – + +-13 +-12 +-11 +-10 +-9 +-8 +10-4 +0.001 +0.010 +0.100 +1 +Log10 |δθduu/f duu| +Probability Density +(a) M 1 +u = 103M 3 +u +-13 +-12 +-11 +-10 +-9 +-8 +10-4 +0.001 +0.010 +0.100 +1 +Log10 |δθduu/f duu| +Probability Density +(b) M 1 +u = 102M 3 +u +Figure 4: The probability density functions (PDFs) of |δθduu| in Eq. (5.2) with only b = 1 +and c = 3 normalized by a three-loop function | ¯f13 +duu| (solid line). We take � +M = � +Md = +M1 +u = 103M3 +u in (a) and � +M = � +Md = M1 +u = 102M3 +u in (b). After imposing the perturbative +unitarity bound of max(xu) < 4π ( +√ +4π), the PDFs are expected by the dashed (dotted) +lines. The areas of dashed and dotted lines are not normalized by one (see text). Note +that the solid and dashed lines overlap in (b). If ¯f13 +duu = 1, the right area of the vertical +line is excluded at 90% CL by the neutron EDM measurement (1.1). +-13 +-12 +-11 +-10 +-9 +-8 +10-4 +0.001 +0.010 +0.100 +1 +Log10 |δθduu/f duu| +Probability Density +(a) M 1 +u = M 2 +u = 103M 3 +u +-13 +-12 +-11 +-10 +-9 +-8 +10-4 +0.001 +0.010 +0.100 +1 +Log10 |δθduu/f duu| +Probability Density +(b) M 1 +u = M 2 +u = 102M 3 +u +Figure 5: The PDFs of |δθduu| in Eq. (5.2) with b = 1, 2 and c = 3 normalized by +| ¯f13 +duu| = | ¯f23 +duu| (solid line). We take � +M = � +Md = M1 +u = M2 +u = 103M3 +u in (a) and � +M = +� +Md = M1 +u = M2 +u = 102M3 +u in (b). After imposing the perturbative unitarity bound of +max(xu) < 4π ( +√ +4π), the PDFs are expected by the dashed (dotted) lines. Note that the +solid and dashed lines overlap in (b). If ¯f13 +duu = ¯f23 +duu = 1, the right area of the vertical line +is excluded at 90% CL by the neutron EDM measurement (1.1). +vanishing δθduu, see Eq. (5.3). In the PDFs, the δθduu is normalized by | ¯f13 +duu| where we +define ¯f13 +duu = ˜f13 +duu − ˜f31 +duu = 2 ˜f13 +duu. In Fig. 4a and 4b, we take � +M = � +Md = M1 +u = 103M3 +u +and � +M = � +Md = M1 +u = 102M3 +u, respectively, with v′ = M3 +u, by solid lines, whose hierarchy +is concerned with the top quark masses in the SM and the seesaw mechanism in Eq. (3.8). +– 22 – + +In the PDFs, the three mixing angles and one CP-violating phase in the VU matrix and +two additional CP-violating phases in Φ(θu3, θu8) are taken random values between [0, 2π]. +Although the radiative ¯θ parameter should be renormalization scale invariant, in order +to investigate the perturbative unitarity bounds for the Yukawa interactions, we used the +SM (running) quark masses at µ = 1 TeV.#12 Furthermore, by assuming ¯f13 +duu = 1, the +vertical line in the figures stands for the experimental upper bound from the neutron EDM +measurement in Eq. (1.1) and the right area is excluded at 90% CL. +In the same figures, we find that the Yukawa coupling constants xu are larger than +O(1), depending on the mixing angle and CP-violating phase parameters. Therefore, we +impose the largest eigenvalue in xu Yukawa matrix to be smaller than 4π or +√ +4π for +the dashed or dotted lines, respectively, as the perturbative unitarity bound. In order to +display the reduction in statistics as a result of setting the perturbative bound, we do not +normalize the areas of dashed and dotted lines by 1. +The ratios of the excluded parameter regions in each of the PDFs are shown in Table 2 +under an assumption of ¯f13 +duu = 1. We found that some fractions of parameter regions have +already been excluded even if the perturbative bound is imposed. In the case of the pertur- +bative bound of 4π to the eigenvalues of xu matrix, about 30–40% of the whole parameter +region is already excluded. On the other hand, in the case of +√ +4π, the dependence of +M3 +u appears obviously. In particular, only 3.65% is excluded in the case of M1 +u = 103M3 +u. +These results show that this model is sensitive to the current bound from the neutron EDM +experiments and has a possibility to be explored by future improvement of the experiments. +Moreover, in Figs. 5a and 5b, we show the PDFs of the absolute value of δθduu with +(b, c) = (1, 3) plus (2, 3) (also included (3, 1) and (3, 2)), with assuming M1 +u = M2 +u which +should be a somewhat aggressive parameter choice in light of mu ≪ mc. We observed that +the PDFs in Fig. 5 are slightly larger than the PDFs in Fig. 4. The ratios of excluded +parameter regions by the neutron EDM measurement are shown in Table 2 at the numbers +in parentheses. +5.2 +Case for the GIM by universal vector-like mass +Next, we investigate a case that all Dirac quark masses are degenerate as � +M = Ma +u = Ma +d . +In this case, the GIM-like mechanism occurs and then the CKM matrix becomes a +unique source of the CP-violating phase. +As a consequence, the radiative ¯θ parame- +ter would be significantly suppressed, and it is expected as the minimum value of the +¯θ parameter in the minimal LR symmetric model. +The induced ¯θ parameter should +be proportional to the Jarlskog invariant of the CKM matrix JCKM, which is given by +Im(V ij +CKMV †jk +CKMV kl +CKMV †li +CKM) = JCKM +� +m,n ϵikmϵjln [55] and JCKM = (3.08+0.15 +−0.13)×10−5 [14]. +#12We take mt = 143 GeV, mb = 2.41 GeV, mc = 528 MeV, and ms = 45.4 MeV at µ = 1 TeV [54]. +– 23 – + +If the ¯θ parameter is induced at three-loop level, the contribution would be given as +δθminimum ≈ +1 +(16π2)2 +v′8 +� +M8 Im Tr +� +A2 +uA2 +dAuAd +� +fCKM += +1 +(16π2)2 +v′2 +� +M2 JCKMfCKM +× 1 +v6 (mt − mc) (mt − mu) (mc − mu) (mb − ms) (mb − md) (ms − md) , +≃ 1 × 10−19 v′2 +� +M2 fCKM , +(5.4) +where fCKM is a dimensionless loop function of O(1). It is assumed that in the three-loop +diagrams, two scalar bosons are exchanged inside the fermion loop, and the other scalar +lines are replaced by v′. The above contribution is proportional to v′8. The perturbativity +of the top Yukawa requires � +M ≃ v′. Then, δθminimum is expected as at most 10−19 or +smaller when all vector-like quark masses are degenerate. This size is comparable to or +smaller than the CKM phase contribution to the ¯θ parameter in the SM at four-loop level, +evaluated in Ref. [38] (though the top quark had been assumed to be lighter than the +W boson). This is because the quark mass suppression of the contribution in the SM is +milder than Eq. (5.4). In addition, the neutron EDM induced by the CKM phase via long- +distance hadronic contributions in the SM [56, 57] is much larger than by the contribution +to the ¯θ parameter in the above benchmark point. Then, a more precise evaluation of the +¯θ parameter in the benchmark point is too academic and beyond our scope. +6 +Conclusions and discussion +When the QCD axion is absent, one has to solve the strong CP problem by an additional +discrete symmetry. The extended parity with the LR gauge symmetry can solve it with +generating the SM as the low-energy theory. However, it is known that the radiative ¯θ +parameter is induced from the soft symmetry breaking of the parity. In this paper, we +first formulated a novel method of direct loop-diagrammatic calculation of the radiative ¯θ +parameter by using Fock-Schwinger gauge. This approach should be more robust than the +the ordinary calculation method based on the chiral rotations. By using the Fock-Schwinger +gauge method, we confirmed a seminal result that two-loop level ¯θ vanishes completely in +the minimal LR symmetric model. +Furthermore, we estimated the size of the leading contributions to the non-vanishing +radiative ¯θ parameter at three-loop level. We derive the parameterization by the physical +parameters based on the seesaw mechanism in the LR symmetric model, and we obtained +the probability density functions of the radiative ¯θ parameter by varying all free but physical +parameters. Here, we also investigated the impact of the perturbative unitarity conditions +for the LR symmetric Yukawa matrices. It is found that the resultant ¯θ parameters are +partially excluded by the current neutron EDM bound. It implies that this model has a +possibility to be explored by future improvement of the experiments. One should note that +the minimal LR symmetric model predicts that all hadronic EDMs are dominated by the +– 24 – + +radiative ¯θ parameter. Therefore, this model can predict a distinctive and non-vanishing +correlation between the neutron, proton, nuclei (2H, 3He), diamagnetic atoms (Hg, Ra), +paramagnetic atoms and molecules (YbF, HfF, ThO) EDMs [45, 58, 59]. +The large mass-scale difference between v and v′ can be explained by a Higgs parity +mechanism that predicts λSM(µ = v′) ≃ 0 and v′ = O(1010) GeV [39, 60–62]. Since the +above estimation of the radiative ¯θ parameter is insensitive to the mass scale itself of the +right-handed sector, one could predict the neutron EDM for the case of v′ = O(1010) GeV. +We also comment on the radiative ¯θ parameter in the spontaneous CP violation +(Nelson-Barr) model [31–33], which is another model that can explain the strong CP prob- +lem by the discrete symmetry. In Ref. [63], the radiative ¯θ parameter has been investigated +in detail. It is found that although the reducible ¯θ, which comes from quartic couplings of +scalar fields responsible for the spontaneous CP violation with the SM Higgs one, is induced +at two-loop level, it can be numerically neglected if the couplings are small enough. On +the other hand, the irreducible ¯θ, which is related to the CKM phase, is induced at three- +loop level and is safely below the current experimental bounds. The loop-diagrammatic +approach we proposed would provide a more robust estimation if possible. +It would be an interesting direction to investigate correlations between the radiative +¯θ parameter and other (flavor) observables in the minimal LR symmetric model: The +lepton flavor universality violation in B → D(∗)lν (R(D(∗)) anomaly) (the recent review +[64, 65]), the Cabibbo angle anomaly (the recent review [66]) and the W-boson mass +anomaly [67] could be explained [68–70]. +Furthermore, the investigation of a correla- +tion with electroweak-like baryogenesis in the right-handed sector should be an attractive +prospect [71, 72]. +Acknowledgments +We would like to acknowledge Keisuke Harigaya for a collaboration at the early stage. This +work is supported by the JSPS Grant-in-Aid for Scientific Research Grant No. 20H01895 +(J.H.), No. 21K03572 (J.H.) and for Early-Career Scientists Grant No. 19K14706 (T.K.). +The work of J.H. is also supported by World Premier International Research Center Ini- +tiative (WPI Initiative), MEXT, Japan. This work is also supported by JSPS Core-to- +Core Program Grant No. JPJSCCA20200002. This work was financially supported by JST +SPRING, Grant Number JPMJSP2125. The author (A.Y.) would like to take this oppor- +tunity to thank the “Interdisciplinary Frontier Next-Generation Researcher Program of the +Tokai Higher Education and Research System.” +– 25 – + +A +Loop functions +We define the loop functions in this section. The two-loop functions used in this paper are +given as +I(n1,··· ;m1,··· )(x1, · · · ; x2, · · · ; x3) += (16π2µ2ϵ)2 +� +ddp +(2π)d +ddq +(2π)d +1 +[p2 − x1]n1 · · · [q2 − x2]m1 · · · [(p + q)2 − x3] , +(A.1) +where the dimensional regularization is used on d = 4 − 2ϵ dimension and µ is the renor- +malization scale. The functions can also be derived by derivative or finite difference of +I(x1; x2; x3) (≡ I(1;1)(x1; x2; x3)) as +I(n;m)(x1; x2; x3) = +1 +(n − 1)!(m − 1)! +dn−1 +dxn−1 +1 +dm−1 +dxm−1 +2 +I(x1; x2; x3) , +(A.2) +I(1,1;1)(x1, x′ +1; x2; x3) = +1 +x1 − x′ +1 +�I(x1; x2; x3) − I(x′ +1; x2; x3) +� , +(A.3) +where +I(x1; x2; x3) ≡ I(1;1)(x1; x2; x3) += (16π2µ2ϵ)2 +� ddpddq +(2π)2d +1 +[p2 − x1][q2 − x2][(p + q)2 − x3] . +(A.4) +The explicit form of I(x1; x2; x3) is given as +I(x1; x2; x3) = ¯Iϵ(x1; x2; x3) + ¯I(x1; x2; x3) , +(A.5) +with the UV divergent part +¯Iϵ(x1; x2; x3) = − +� +i=1,2,3 +xi +� +1 +2ϵ2 − 1 +ϵ +� +log xi +Q2 − 3 +2 +� ++ +� +1 + π2 +12 − log xi +Q2 + 1 +2 log2 xi +Q2 +�� +, +(A.6) +while the finite part +¯I(x1; x2; x3) = −1 +2 +� +(−x1 + x2 + x3) log x2 +Q2 log x3 +Q2 + (x1 − x2 + x3) log x1 +Q2 log x3 +Q2 ++ (x1 + x2 − x3) log x1 +Q2 log x2 +Q2 − 4 +� +x1 log x1 +Q2 + x2 log x2 +Q2 + x3 log x3 +Q2 +� ++ 5(x1 + x2 + x3) + ξ(x1, x2, x3) +� +, +(A.7) +ξ(x1, x2, x3) = R +� +2 log +�x3 + x1 − x2 − R +2x3 +� +log +�x3 − x1 + x2 − R +2x3 +� +− log x1 +x3 +log x2 +x3 +−2Li2 +�x3 + x1 − x2 − R +2x3 +� +− 2Li2 +�x3 − x1 + x2 − R +2x3 +� ++ π2 +3 +� +, +(A.8) +where R = +� +x2 +1 + x2 +2 + x3 +3 − 2x1x2 − 2x2x3 − 2x3x1 and Q2 ≡ 4πµ2e−γE [73–75]. Note +that although the last terms of ¯Iϵ are UV finite, they do not affect any physical quantity +[76]. These terms are suppressed by O(ϵ) at one-loop level, while they are uplifted to O(ϵ0) +at two-loop level. +– 26 – + +A.1 +I(1;3) and I(3;1) +The two-loop function I(3;1) is also given by the two-point one-loop function B0 as +I(3;1)(x1; x2; x3) = −(16π2µ2ϵ) +� +ddp +i(2π)d +1 +(p2 − x1)3 B0(p2, x2, x3) , +(A.9) +where +B0(p2, x2, x3) ≡ (16π2µ2ϵ) +� +ddq +i(2π)d +1 +[q2 − x2][(p + q)2 − x3] += 1 +ϵ − F0(p2, x2, x3) , +(A.10) +F0(p2, x2, x3) = +� 1 +0 +dz log −z(1 − z)p2 + zx2 + (1 − z)x3 +Q2 +. +(A.11) +Similarly, ¯I(1;3) is obtained by replacements of x1 ↔ x2 in the above equations of ¯I(3;1). +The finite part of I(3;1) (= ¯I(3;1)) is given as +¯I(3;1)(x1; x2; x3) = (16π2µ2ϵ) +� +ddp +i(2π)d +1 +(p2 − x1)3 F0(p2, x2, x3) , +(A.12) +while the divergent part is +¯Iϵ(3;1)(x1; x2; x3) = +1 +2x1 +�1 +ϵ − log x1 +Q2 +� +. +(A.13) +In the case of x1 ≪ x3, ¯I(3;1) is given as +¯I(3;1)(x1; x2; x3) ≃ − 1 +2x1 +F0(0, x2, x3) , +(A.14) +where +F0(0, x2, x3) = +� 1 +0 +dz log zx2 + (1 − z)x3 +Q2 += −x2 − x3 − x2 log(x2/Q2) + x3 log(x3/Q2) +x2 − x3 +. +(A.15) +On the other hand, in the case of x2 ≪ x3, ¯I(3;1) is given as +¯I(3;1)(x1; x2; x3) ≃ +1 +2x1 +x1 − x3 − x1 log(x1/Q2) + x3 log(x3/Q2) +x1 − x3 +. +(A.16) +Furthermore, we define the loop function at one-loop level as +B(n1,··· )(p2, x1, · · · ; x3) = (16π2µ2ϵ) +� +ddq +i(2π)d +1 +[q2 − x1]n1 · · · [(p + q)2 − x3] . (A.17) +– 27 – + +A.2 +I(2;2) +The two-loop function I(2;2) does not contain the UV divergence. +I(2;2)(x1; x2; x3) is a +symmetric function in variables x1 and x2. In the case of x1, x2 ≪ x3, I(2;2) is given as +I(2;2)(x1; x2; x3) ≃ 1 +x3 +� +π2 +3 + log x1x2 +x2 +3 ++ log x1 +x3 +log x2 +x3 +� +. +(A.18) +In the case of x1 ≪ x2 < x3, we find +I(2;2)(x1; x2; x3) ≃ +x3 +(x3 − x2)2 +� +π2 +3 − x2 +x3 +log x1 +x2 ++ log x1x2 +x2 +3 ++ log x1 +x3 +log x2 +x3 +−2 +� +log x2 +x3 +log +� +1 − x2 +x3 +� ++ Li2 +�x2 +x3 +��� +, +(A.19) +while for x1 ≪ x3 < x2 +I(2;2)(x1; x2; x3) ≃ +x3 +(x3 − x2)2 +� +−x2 +x3 +log x1 +x2 ++ log x1x2 +x2 +3 ++ log x1 +x3 +log x2 +x3 ++ 2Li2 +� +1 − x2 +x3 +�� +. +(A.20) +References +[1] A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Y. S. Tyupkin, “Pseudoparticle +Solutions of the Yang-Mills Equations,” Phys. Lett. B 59 (1975) 85–87. +[2] G. ’t Hooft, “Symmetry Breaking Through Bell-Jackiw Anomalies,” Phys. Rev. Lett. 37 +(1976) 8–11. +[3] C. G. Callan, Jr., R. F. Dashen, and D. J. Gross, “The Structure of the Gauge Theory +Vacuum,” Phys. Lett. B 63 (1976) 334–340. +[4] R. Jackiw and C. Rebbi, “Vacuum Periodicity in a Yang-Mills Quantum Theory,” Phys. Rev. +Lett. 37 (1976) 172–175. +[5] E. Witten, “Dyons of Charge e theta/2 pi,” Phys. Lett. B 86 (1979) 283–287. +[6] V. Baluni, “CP Violating Effects in QCD,” Phys. Rev. D 19 (1979) 2227–2230. +[7] R. J. Crewther, P. Di Vecchia, G. Veneziano, and E. Witten, “Chiral Estimate of the Electric +Dipole Moment of the Neutron in Quantum Chromodynamics,” Phys. Lett. B 88 (1979) 123. +[Erratum: Phys.Lett.B 91, 487 (1980)]. +[8] C. Abel et al., “Measurement of the Permanent Electric Dipole Moment of the Neutron,” +Phys. Rev. Lett. 124 (2020) 081803 [arXiv:2001.11966]. +[9] J. Liang, et al., “Nucleon Electric Dipole Moment from the θ Term with Lattice Chiral +Fermions.” arXiv:2301.04331. +[10] N. Cabibbo, “Unitary Symmetry and Leptonic Decays,” Phys. Rev. Lett. 10 (1963) 531–533. +[11] M. Kobayashi and T. Maskawa, “CP Violation in the Renormalizable Theory of Weak +Interaction,” Prog. Theor. Phys. 49 (1973) 652–657. +[12] CKMfitter Group Collaboration, “CP violation and the CKM matrix: Assessing the +impact of the asymmetric B factories,” Eur. Phys. J. C 41 (2005) 1–131 [hep-ph/0406184]. +updated results and plots available at: http://ckmfitter.in2p3.fr. +– 28 – + +[13] L.-L. Chau and W.-Y. Keung, “Comments on the Parametrization of the +Kobayashi-Maskawa Matrix,” Phys. Rev. Lett. 53 (1984) 1802. +[14] Particle Data Group Collaboration, “Review of Particle Physics,” PTEP 2022 (2022) +083C01. +[15] H. Georgi and I. N. McArthur, “INSTANTONS AND THE mu QUARK MASS.”. +[16] D. B. Kaplan and A. V. Manohar, “Current Mass Ratios of the Light Quarks,” Phys. Rev. +Lett. 56 (1986) 2004. +[17] K. Choi, C. W. Kim, and W. K. Sze, “Mass Renormalization by Instantons and the Strong +CP Problem,” Phys. Rev. Lett. 61 (1988) 794. +[18] T. Banks, Y. Nir, and N. Seiberg, “Missing (up) mass, accidental anomalous symmetries, +and the strong CP problem,” in 2nd IFT Workshop on Yukawa Couplings and the Origins of +Mass, pp. 26–41. 1994. hep-ph/9403203. +[19] M. Srednicki, “Comment on ”Ambiguities in the up-quark mass”,” Phys. Rev. Lett. 95 +(2005) 059101 [hep-ph/0503051]. +[20] C. Alexandrou, et al., “Ruling Out the Massless Up-Quark Solution to the Strong CP +CP +CP +Problem by Computing the Topological Mass Contribution with Lattice QCD,” Phys. Rev. +Lett. 125 (2020) 232001 [arXiv:2002.07802]. +[21] R. D. Peccei and H. R. Quinn, “CP Conservation in the Presence of Instantons,” Phys. Rev. +Lett. 38 (1977) 1440–1443. +[22] S. Weinberg, “A New Light Boson?” Phys. Rev. Lett. 40 (1978) 223–226. +[23] F. Wilczek, “Problem of Strong P and T Invariance in the Presence of Instantons,” Phys. +Rev. Lett. 40 (1978) 279–282. +[24] M. Kamionkowski and J. March-Russell, “Planck scale physics and the Peccei-Quinn +mechanism,” Phys. Lett. B 282 (1992) 137–141 [hep-th/9202003]. +[25] R. Holman, et al., “Solutions to the strong CP problem in a world with gravity,” Phys. Lett. +B 282 (1992) 132–136 [hep-ph/9203206]. +[26] S. M. Barr and D. Seckel, “Planck scale corrections to axion models,” Phys. Rev. D 46 +(1992) 539–549. +[27] M. A. B. Beg and H. S. Tsao, “Strong P, T Noninvariances in a Superweak Theory,” Phys. +Rev. Lett. 41 (1978) 278. +[28] R. N. Mohapatra and G. Senjanovic, “Natural Suppression of Strong p and t Noninvariance,” +Phys. Lett. B 79 (1978) 283–286. +[29] K. S. Babu and R. N. Mohapatra, “A Solution to the Strong CP Problem Without an +Axion,” Phys. Rev. D 41 (1990) 1286. +[30] S. M. Barr, D. Chang, and G. Senjanovic, “Strong CP problem and parity,” Phys. Rev. Lett. +67 (1991) 2765–2768. +[31] A. E. Nelson, “Naturally Weak CP Violation,” Phys. Lett. B 136 (1984) 387–391. +[32] S. M. Barr, “Solving the Strong CP Problem Without the Peccei-Quinn Symmetry,” Phys. +Rev. Lett. 53 (1984) 329. +[33] S. M. Barr, “A Natural Class of Nonpeccei-quinn Models,” Phys. Rev. D 30 (1984) 1805. +– 29 – + +[34] S. L. Adler, “Axial vector vertex in spinor electrodynamics,” Phys. Rev. 177 (1969) +2426–2438. +[35] J. S. Bell and R. Jackiw, “A PCAC puzzle: π0 → γγ in the σ model,” Nuovo Cim. A 60 +(1969) 47–61. +[36] K. Fujikawa, “Path Integral Measure for Gauge Invariant Fermion Theories,” Phys. Rev. +Lett. 42 (1979) 1195–1198. +[37] J. R. Ellis and M. K. Gaillard, “Strong and Weak CP Violation,” Nucl. Phys. B 150 (1979) +141–162. +[38] I. B. Khriplovich, “Quark Electric Dipole Moment and Induced θ Term in the +Kobayashi-Maskawa Model,” Phys. Lett. B 173 (1986) 193–196. +[39] L. J. Hall and K. Harigaya, “Implications of Higgs Discovery for the Strong CP Problem and +Unification,” JHEP 10 (2018) 130 [arXiv:1803.08119]. +[40] N. Craig, I. Garcia Garcia, G. Koszegi, and A. McCune, “P not PQ,” JHEP 09 (2021) 130 +[arXiv:2012.13416]. +[41] S. L. Adler and W. A. Bardeen, “Absence of higher order corrections in the anomalous axial +vector divergence equation,” Phys. Rev. 182 (1969) 1517–1536. +[42] G. ’t Hooft and M. J. G. Veltman, “Regularization and Renormalization of Gauge Fields,” +Nucl. Phys. B 44 (1972) 189–213. +[43] V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, “Calculations in +External Fields in Quantum Chromodynamics. Technical Review,” Fortsch. Phys. 32 (1984) +585. +[44] T. Abe, J. Hisano, and R. Nagai, “Model independent evaluation of the Wilson coefficient of +the Weinberg operator in QCD,” JHEP 03 (2018) 175 [arXiv:1712.09503]. [Erratum: +JHEP 09, 020 (2018)]. +[45] J. de Vries, P. Draper, and H. H. Patel, “Do Minimal Parity Solutions to the Strong CP +Problem Work?” arXiv:2109.01630. +[46] A. Davidson and K. C. Wali, “Universal Seesaw Mechanism?” Phys. Rev. Lett. 59 (1987) +393. +[47] C.-H. Lee and R. N. Mohapatra, “Vector-Like Quarks and Leptons, SU(5) ⊗ SU(5) Grand +Unification, and Proton Decay,” JHEP 02 (2017) 080 [arXiv:1611.05478]. +[48] K. S. Babu and X. G. He, “Dirac Neutrino Masses as Two-Loop Radiative Corrections,” +Mod. Phys. Lett. A 4 (1989) 61. +[49] K. S. Babu, X.-G. He, M. Su, and A. Thapa, “Naturally light Dirac and pseudo-Dirac +neutrinos from left-right symmetry,” JHEP 08 (2022) 140 [arXiv:2205.09127]. +[50] J. A. Dror, D. Dunsky, L. J. Hall, and K. Harigaya, “Sterile Neutrino Dark Matter in +Left-Right Theories,” JHEP 07 (2020) 168 [arXiv:2004.09511]. +[51] D. Dunsky, L. J. Hall, and K. Harigaya, “Sterile Neutrino Dark Matter and Leptogenesis in +Left-Right Higgs Parity,” JHEP 01 (2021) 125 [arXiv:2007.12711]. +[52] J. A. Casas and A. Ibarra, “Oscillating neutrinos and µ → e, γ,” Nucl. Phys. B 618 (2001) +171–204 [hep-ph/0103065]. +– 30 – + +[53] J. R. Ellis, J. Hisano, M. Raidal, and Y. Shimizu, “A New parametrization of the seesaw +mechanism and applications in supersymmetric models,” Phys. Rev. D 66 (2002) 115013 +[hep-ph/0206110]. +[54] K. G. Chetyrkin, J. H. Kuhn, and M. Steinhauser, “RunDec: A Mathematica package for +running and decoupling of the strong coupling and quark masses,” Comput. Phys. Commun. +133 (2000) 43–65 [hep-ph/0004189]. +[55] C. Jarlskog, “Commutator of the Quark Mass Matrices in the Standard Electroweak Model +and a Measure of Maximal CP Nonconservation,” Phys. Rev. Lett. 55 (1985) 1039. +[56] S. Dar, “The Neutron EDM in the SM: A Review.” hep-ph/0008248. +[57] T. Mannel and N. Uraltsev, “Loop-Less Electric Dipole Moment of the Nucleon in the +Standard Model,” Phys. Rev. D 85 (2012) 096002 [arXiv:1202.6270]. +[58] W. Dekens, et al., “Unraveling models of CP violation through electric dipole moments of +light nuclei,” JHEP 07 (2014) 069 [arXiv:1404.6082]. +[59] J. de Vries, P. Draper, K. Fuyuto, J. Kozaczuk, and D. Sutherland, “Indirect Signs of the +Peccei-Quinn Mechanism,” Phys. Rev. D 99 (2019) 015042 [arXiv:1809.10143]. +[60] D. Dunsky, L. J. Hall, and K. Harigaya, “Higgs Parity, Strong CP, and Dark Matter,” JHEP +07 (2019) 016 [arXiv:1902.07726]. +[61] L. J. Hall and K. Harigaya, “Higgs Parity Grand Unification,” JHEP 11 (2019) 033 +[arXiv:1905.12722]. +[62] D. Dunsky, L. J. Hall, and K. Harigaya, “Dark Matter, Dark Radiation and Gravitational +Waves from Mirror Higgs Parity,” JHEP 02 (2020) 078 [arXiv:1908.02756]. +[63] A. Valenti and L. Vecchi, “The CKM phase and θ in Nelson-Barr models,” JHEP 07 (2021) +203 [arXiv:2105.09122]. +[64] HFLAV Collaboration, “Averages of b-hadron, c-hadron, and τ-lepton properties as of +2021.” arXiv:2206.07501. +[65] S. Iguro, T. Kitahara, and R. Watanabe, “Global fit to b → cτν anomalies 2022 +mid-autumn.” arXiv:2210.10751. +[66] A. Crivellin, M. Kirk, T. Kitahara, and F. Mescia, “Global Fit of Modified Quark Couplings +to EW Gauge Bosons and Vector-Like Quarks in Light of the Cabibbo Angle Anomaly.” +arXiv:2212.06862. +[67] CDF Collaboration, “High-precision measurement of the W boson mass with the CDF II +detector,” Science 376 (2022) 170–176. +[68] K. S. Babu, B. Dutta, and R. N. Mohapatra, “A theory of R(D∗, D) anomaly with +right-handed currents,” JHEP 01 (2019) 168 [arXiv:1811.04496]. +[69] K. S. Babu and R. Dcruz, “Resolving W Boson Mass Shift and CKM Unitarity Violation in +Left-Right Symmetric Models with Universal Seesaw.” arXiv:2212.09697. +[70] R. Dcruz, “Flavor Physics Constraints on Left-Right Symmetric Models with Universal +Seesaw.” arXiv:2301.10786. +[71] K. Fujikura, K. Harigaya, Y. Nakai, and I. R. Wang, “Electroweak-like baryogenesis with +new chiral matter,” JHEP 07 (2021) 224 [arXiv:2103.05005]. [Erratum: JHEP 12, 192 +(2021), Erratum: JHEP 1, 156 (2022), Erratum: JHEP 01, 156 (2022)]. +– 31 – + +[72] K. Harigaya and I. R. Wang, “Baryogenesis in a Parity Solution to the Strong CP Problem.” +arXiv:2210.16207. +[73] C. Ford, I. Jack, and D. R. T. Jones, “The Standard model effective potential at two loops,” +Nucl. Phys. B 387 (1992) 373–390 [hep-ph/0111190]. [Erratum: Nucl.Phys.B 504, 551–552 +(1997)]. +[74] J. R. Espinosa and R.-J. Zhang, “Complete two loop dominant corrections to the mass of the +lightest CP even Higgs boson in the minimal supersymmetric standard model,” Nucl. Phys. +B 586 (2000) 3–38 [hep-ph/0003246]. +[75] S. P. Martin, “Two Loop Effective Potential for a General Renormalizable Theory and Softly +Broken Supersymmetry,” Phys. Rev. D 65 (2002) 116003 [hep-ph/0111209]. +[76] S. P. Martin and H. H. Patel, “Two-loop effective potential for generalized gauge fixing,” +Phys. Rev. D 98 (2018) 076008 [arXiv:1808.07615]. +– 32 – + diff --git a/x9FQT4oBgHgl3EQfxTb9/content/tmp_files/load_file.txt b/x9FQT4oBgHgl3EQfxTb9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b52d41cda9de7c2c0cabcd9c6a1d02c112bc2c2f --- /dev/null +++ b/x9FQT4oBgHgl3EQfxTb9/content/tmp_files/load_file.txt @@ -0,0 +1,1642 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf,len=1641 +page_content='IPMU23-0002,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' KEK-TH-2494 Novel loop-diagrammatic approach to QCD θ parameter and application to the left-right model Junji Hisano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='c Teppei Kitahara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='f Naohiro Osamura,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='a and Atsuyuki Yamadaa aDepartment of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Nagoya University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Furo-cho Chikusa-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Nagoya 464-8602 Japan bKobayashi-Maskawa Institute for the Origin of Particles and the Universe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Nagoya University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Furo-cho Chikusa-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Nagoya 464-8602 Japan cKavli IPMU (WPI),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' UTIAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Kashiwa 277-8584,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Japan dInstitute for Advanced Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Nagoya University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Furo-cho Chikusa-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Nagoya 464-8601,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Japan eKEK Theory Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' IPNS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' KEK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Tsukuba 305–0801,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Japan fCAS Key Laboratory of Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Institute of Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' China E-mail: hisano@eken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='nagoya-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='jp, teppeik@kmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='nagoya-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='jp, osamura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='naohiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='j2@s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='nagoya-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='jp, yamada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='atsuyuki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='k3@s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='nagoya-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='jp Abstract: When the QCD axion is absent in full theory, the strong CP problem has to be explained by an additional mechanism, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=', the left-right symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Even though tree-level QCD ¯θ parameter is restricted by the mechanism, radiative corrections to ¯θ are mostly generated, which leads to a dangerous neutron electric dipole moment (EDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The ordinary method for calculating the radiative ¯θ utilizes an equation ¯θ = arg det mloop q based on the chiral rotations of complex quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In this paper, we point out that when full theory includes extra heavy quarks, the ordinary method is unsettled for the extra quark contributions and does not contain its full radiative corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' We formulate a novel method to calculate the radiative corrections to ¯θ through a direct loop-diagrammatic approach, which should be more robust than the ordinary one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' As an application, we investigate the radiative ¯θ in the minimal left-right symmetric model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' We first confirm a seminal result that two-loop level radiative ¯θ completely vanishes (corresponding to one- loop corrections to the quark mass matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Furthermore, we estimate the size of a non- vanishing radiative ¯θ at three-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It is found that the resultant induced neutron EDM is comparable to the current experimental bound, and the expected size is restricted by the perturbative unitarity bound in the minimal left-right model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Keywords: CP violation, Electric Dipole Moments, Left-Right Models arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='13405v1 [hep-ph] 31 Jan 2023 Contents 1 Introduction 1 2 Loop-diagrammatic evaluation of QCD θ parameter 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1 Operator Schwinger method 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2 Fock-Schwinger gauge method 5 3 The minimal left-right symmetric model 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1 Model 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2 Parametrization of Yukawa coupling constants 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3 Quark EDMs 12 4 Confirmation of vanishing QCD θ parameter in two-loop order 12 5 Non-vanishing contribution to QCD θ parameter in three-loop order 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1 Leading contribution: PDF of δθduu 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2 Case for the GIM by universal vector-like mass 23 6 Conclusions and discussion 24 A Loop functions 26 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1 I(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3) and I(3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) 27 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2 I(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2) 28 1 Introduction The QCD θ term is known as a topological one and is P- and T-odd, and then CP-odd under the CPT invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Because it is identical to the total derivative, it never locally affects physics at the classical level (as long as the momentum conservation holds), while its effect occurs only via nonperturbative processes [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It is known that this interaction induces the neutron electric dipole moment (EDM) [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Measurement of the neutron EDM by the nEDM collaboration has set the severe upper bound: |dn|exp < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='8×10−26 e cm (90% CL) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Using the latest lattice result dn = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='00148(34) ¯θ e fm [9]#1 and assuming that ¯θ is the only source of CP violation, one obtains the upper bound on the angle, |¯θ| ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2 × 10−10 (90% CL) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) #1This is the first statistically significant result using the lattice QCD calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' – 1 – where ¯θ is the physical CP-violating angle in the QCD Lagrangian which will be defined explicitly in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Although the experimental bound requires that ¯θ must be around zero, such a CP- violating phase is not restricted in the Standard Model (SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In fact, the CP-violating phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix [10, 11] is O(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' δCKM ≃ 66◦ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2 rad [12] (in the standard parameterization [13, 14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' If there is no trick in the full theory, ¯θ ≪ 1, or equivalently ¯θ ≪ δCKM = O(1), requires a fine-tuning at O(10−10) level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' This is known as the strong CP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The massless up quark could be a solution to the strong CP problem if the observed hadron masses are explained by the nonperturbative effect [15–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' However, some lattice studies ruled out this solution [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Thus, the strong CP problem would suggest that the SM has to be extended to suppress ¯θ without the fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Axion is the simplest solution to the strong CP problem [21–23], though it suffers from another fine-tuning in the quantum gravity sector (axion quality problem) [24–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Alternatively, one may resolve the strong CP problem by extended parity symmetry [27–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='#2 In such scenarios, the extended parity involves the left-right (LR) gauge sym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The parity symmetry forbids the bare ¯θ parameter, while O(1) of δCKM is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It is known that even though the bare ¯θ parameter is strictly forbidden by the parity sym- metry, radiative correction to ¯θ is regenerated since the parity symmetry must be softly broken in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Eventually, one has to consider the experimental bound on the model from the neutron EDM measurements in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) through the radiatively regenerated ¯θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The ordinary method for calculating the radiatively generated ¯θ parameter, adopted in many papers, utilizes ¯θ = arg det mloop u + arg det mloop d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2) Here, mloop u,d are the up- and down-type quark mass matrices including the radiative correc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' This relation is based on the chiral rotations for the complex quark masses and an anomalous divergence of the axial-vector current, known as the Adler-Bell-Jackiw anomaly [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Or, it is also derived using the path-integral formalism referred to as the Fujikawa method [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The ordinary method is simple though it may not be accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' For instance, within the SM, even if one sets the bare ¯θ parameter to be zero, it is radiatively produced via the CP-violating phase in the CKM matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It is shown with the above method in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' [37] that the contribution via the radiative corrections to quark masses is of O(G2 F α3 s) (GF is the Fermi coupling constant and αs is the QCD coupling constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' On the other hand, the direct loop calculation of the correction to ¯θ shows that it is derived at four-loop order (O(G2 F αs)) [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It is consistent with the fact that the EDMs (and also the chromo- EDMs) of quarks are induced at three-loop order (O(G2 F αs)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In fact, the ordinary method corresponds to the diagrams where the external gluons are attached in the same fermion line in loop diagrams contributing to ¯θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It implies that the leading ¯θ in the SM [38] comes from diagrams with external gluons attached to different fermion lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' #2Other possibilities are spontaneous CP violation referred to as the Nelson-Barr mechanism [31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' – 2 – In this paper, we formulate a novel approach to evaluate the radiative corrections to ¯θ through a direct loop-diagrammatic calculation, which should be more robust than the ordinary one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' [38], the external gluon field is introduced to calculate the correction to ¯θ from the CKM matrix, while details of the technique are not written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' We introduce the Fock-Swinger gauge method to directly calculate the radiative corrections to ¯θ under the gluon field-strength background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' As an application, we investigate the radiative ¯θ in the minimal LR symmetric model [29, 30], in which the bare and one-loop level ¯θ parameters are strictly forbidden by the LR symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Although the extra heavy quarks whose Yukawa interactions violate CP symmetry are introduced, the CP-violating Yukawa interactions do not contribute to the ¯θ parameters at one-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Furthermore, it is known that two-loop level ¯θ parame- ter also vanishes, which corresponds to one-loop corrections to the quark masses in the ordinary method [29, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' However, the ordinary method is unsettled for the extra quark contributions to ¯θ and indeed does not contain its full radiative corrections, like the SM calculations [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' We first confirm this seminal result by using the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' While new type diagrams contribute to ¯θ at two-loop level, the sum of the diagrams still gives no contribution to ¯θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Next, we estimate the size of a non-vanishing radiative ¯θ at three-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It will be found that the resultant induced neutron EDM is comparable to the current experimental bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' We will also investigate a relation between the radiative three-loop level ¯θ and the perturbative unitarity bounds of the Yukawa couplings in the minimal LR symmetric model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' 2, we discuss methods of direct calculation of the loop diagrams contributing to ¯θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' We show that the operator Schwinger method and the Fock-Schwinger gauge method are applicable, though the latter method has a merit to extend the calculation to the higher-loop diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' 3, the minimal LR sym- metric model is briefly summarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' We also derive the parameterization by the physical parameters based on the seesaw mechanism in the LR symmetric model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' We confirm that the two-loop level radiative ¯θ vanishes by using the proposed method in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' 5, we investigate the numerical size of the non-vanishing radiative ¯θ and compare both ex- perimental (from neutron EDM) and theoretical bounds (from the perturbative unitarity bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Section 6 is devoted to conclusions and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Details of the loop calculations are given in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' 2 Loop-diagrammatic evaluation of QCD θ parameter In the QCD Lagrangian, imaginary parts of the quark masses and the QCD θ term are P-odd and T-odd interactions that are not restricted from the SU(3)C gauge symmetry, L/P , /T = − � q=all Im(mq)¯qiγ5q + θG αs 8πGa µν ˜Gaµν , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) where mq stands for the complex quark masses with mq ≡ |mq| exp(iθq), Ga µν is the gluon field-strength tensor, ˜Gaµν ≡ 1 2ϵµνρσGa ρσ with ϵ0123 = +1, and αs = g2 s/(4π) is the SU(3)C – 3 – coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It is well-known that the axial rotation of quarks q → q′ = exp � − i 2θqγ5 � q , ¯q → ¯q′ = ¯q exp � − i 2θqγ5 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2) turns off the imaginary part of the quark masses and generates an additional QCD θ term, L/P , /T = ¯θ αs 8πGa µν ˜Gaµν , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3) where ¯θ ≡ θG − � q θq , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='4) is a physical ¯θ parameter, if all quarks are massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' This is derived using the path-integral formalism referred to as the Fujikawa method [36] or with the Adler-Bell-Jackiw anomaly [34, 35, 41, 42] ∂µ(¯qγµγ5q) = 2i Re(mq)¯qγ5q − 2 Im(mq)¯qq − αs 4πGa µν ˜Gaµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='5) The contribution of the quark mass phase to the QCD θ term should be able to be directly evaluated by loop-diagrammatically integrating out quarks, not via the Adler-Bell- Jackiw anomaly or transformation of measure in the path integral (the Fujikawa method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' However, it does not generate the QCD θ term if the momenta in the diagrams are con- served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It is because the θ term is equivalent to total-derivative and the total momentum has to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Thus, we have to abandon the momentum conservation or equivalently the translation invariance in order to evaluate the QCD θ term with the loop-diagrammatic calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It can be realized by introducing the gluon field strength background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In this section, we evaluate the QCD θ term with two different methods, 1) the operator Schwinger method and 2) the Fock-Schwinger gauge method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' We show that they produce consistent results with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1 Operator Schwinger method First, we consider the operator Schwinger method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='#3 The effective action ∆S induced by the integration of quarks at one-loop level is given by the log-determinant (or trace-log) of the Dirac operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Now we introduce the complex mass parameters for quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In this case, the effective action is given as ∆S = −iTr log ��� ���/P − (m∗ qPL + mqPR) ��� ��� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='6) where Pµ = iDµ, Dµ ≡ ∂µ + igsT aGa µ is the QCD covariant derivative, and PL/R = (1 ∓ γ5)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In this method, the following basic commutation relation is used, [Pµ, Pν] = igsT aGa µν(≡ igsGµν) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='7) #3See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' [43] for the review about the operator Schwinger method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' – 4 – since the gluon field-strength tensor appears from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Then, the derivative of ∆S over the fermion mass mq for PR is d dmq ∆S = iTr � 1 /P − (m∗qPL + mqPR)PR � = iTr � 1 P 2 − |mq|2 + i 2gsσµνGµν m∗ qPR � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='8) Since the Levi-Civita tensor appears from the trace of a product of four γ’s and γ5, the second order of i 2gsσµνGµν leads to the G ˜G term as d dmq ∆S ⊃iTr � 1 P 2 − |mq|2 � − i 2gsσµνGµν � 1 P 2 − |mq|2 � − i 2gsσρσGρσ � 1 P 2 − |mq|2 m∗ qPR � = � d4x � d4p (2π)4 g2 s 2 −m∗ q (p2 − |mq|2)3 Ga µν ˜Gaµν + · · · ⊃ � d4x i 32π2 g2 s 2 1 mq Ga µν ˜Gaµν , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='9) where Tr(T aT b) = (1/2)δab is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Here, P 2 is replaced by −∂2, and it is integrated in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Similarly, d∆S/dm∗ q leads to the G ˜G term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' By Integrating d∆S/dmq with mq and d∆S/dm∗ q with m∗ q, we get ∆L = −i 32π2 g2 s 2 log m∗ q mq Ga µν ˜Gaµν = − θq αs 8πGa µν ˜Gaµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='10) Now we obtain the contribution from a quark with complex mass to the QCD θ term by integrating out the quark, which is consistent with the axial rotation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2 Fock-Schwinger gauge method In the previous section, we showed that the operator Schwinger method enables us to derive the physical QCD θ term by the diagrammatic evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' However, the operator Schwinger method is not suitable to calculate effective operators induced at higher loops because it is the method to obtain an effective action by integrating fermions out with the log-determinant of the Dirac operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Here alternatively, we introduce the Fock-Schwinger gauge method, which is more applicable to diagrammatic calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='#4 The Fock-Schwinger gauge is to take such a gauge (xµ − xµ 0)Ga µ(x) = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='11) which violates the translation symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Because of breaking the translation symmetry, we can derive perturbatively the QCD θ term as will be shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' While the Fock- Schwinger gauge violates the translation symmetry, the physical observables do not depend #4See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' [43] for the review about the Fock-Schwinger gauge method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='– 5 – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ADxHichVLbhMxFL3p8Cjl0RY2SGxGREFlE5yqAtRVRSUEu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='7QlbaU2RJ6Jm46L8ZOQhNP4AfAIlVK7FAfAYbfoBFPwGxLBKbLnrsuAiIUmzN+Pre849rWXhoFUjB2VJpwLFy9dnrwydfXa9RvTM7M312XSzXzR8JMwyTY9LkUYxKhAhWKzTQTPJCseHtLev4Rk9kMkjiF2qQimbEO3GwE/hcwfUy384i93lUuHNR69X91kyZVZk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='Z7qhRs0aZ7Kgns6U6bVObEvKpSxEJiknBDomTxNyiGjFK4WtSDl8GKzBxQVNAatgvca/D38b6y65wDwGKsLUGV3wCHBw4Pbw72C3Zb0x9rqNPw+dIT4MnC7VGHf2Cd2zL6yz+w7OxnLlRsOrXaA1RtiRdqafnt7d/URFWrfo36lzNinZwNq01gPbUePQp/CG+9+bd8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='driaiW/xw7ZD+g/YEfsC04Q9376H1fE6odz9JwpUbgBfQ8SnBWwao9C9iI9wPRhc1SsmnvqwJOCQ6+j+PG1pO3wLnZt28kYdt/0IDJcMSI5/MPcwryF5h+eM3ROZUQKy8IR0z3sIxIgJzW5A/QlQzaHBmZxqvTuTpT4CwF3nTt3xc8aqzPV2sPqwsrC+WlJ/Z1T9Iduk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='tzqPWIlugZ1amBKhm9pwM6dJ46oSOd7jB1omQxt+iv4eyfAv70CM=Im(mq) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ADt3ichVLbhMxFL3p8GjLoy1s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='kNhEREGsglNVPLqYMyaUkbqUSRZ+IkVufF2EkIo/wAbEFdsAKJBeIz2PADLPoJiGWR2LDg2HEREKXYmvH1vfece+xrPw2l0owdFRa8M2fPnV9cWr5w8dLldW1K7sqGWSBaARJmGRNnysRylg0tNShaKaZ4JEf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ij3/4KGJ7w1FpmQSP9bjVLQi3otlVwZcw1VvtldLrMLsKM4aVWeUyI1aslao0RPqUEIBDSgiQTFp2CFxUpj7VCVGKXwtyuHLYEkbFzShZWA1rGf4j+DvYO1TEZh7QEWYJmMAHgEODtwB/j3s9p03xt5UVZY/gI4Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='XwbuIpXZF/aBHbP7CP7yn7O5coth1E7xupPsSJtr7y4tvPjv6gIq1H9G3WqZk1dnM1oldCeWo85RTDFD58fHu9sbpfzm+wd+wb9b9kR+4QTxMPvwfu62H5zip4TJRo3YO5BgbMVuPRyN6k25gBbI6KFXtPXhS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='cJh1Fj+/lnId7mPXcZ2MY9sDyLFSOSwz/Nndi30PrDc4LOqYTIxLFwxEwPR4hI5KQ2d4y+ZMjm0KAc03x1JtdkCpxlgjd/fcFzxq765XqncpGfaO09cC97kW6TjfoFmrdpS16RDVqoIqgl/SKXnv3vbX9fr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='T1IWCw1ylv4b39BdazctCX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ADuXichVLbhMxFL3p8Cjl0RY2SGwioiBWwYMqQGVTwYZl2pC2Uokiz8RN3MxLYychjPIDSGzp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ghVILBCfwYfYNFPQCyLxIYFx46LgCjF1oyv73n3GNfB1klWbsqLTgnTl7vzihaWLly5fWV5Zvbqt0kEei ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='maYRm+G3AlIpmIpY6ErtZLngcRGIn6D828Z2hyJVMk6d6nIlWzLuJ3Jch13A1+m2/vVJhNWZHedbwnVEhN+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='rpaqlOz6hDKYU0oJgEJaRhR8RJYe6RT4wy+FpUwJfDkjYuaEJLwGpYz/Efwd/B2qMyMA+AijFNxgA8AhwcuD7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='+Xez2nDfB3lRVlj+EjghfDu4yVdkX9oEds8/sI/vKfs7lKiyHUTvGkyxImsv7ze+PFfVIzVqP6NOlWzpn2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='czWiV0J5ZjzlFOMUPXxweN9a3qsUt9o59g/637Ih9wgmS4fw/abYenOKnhMlGjdg7kGBswpW49HIXqc7mCFs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='jo1e09deDJwmHUWP7+Wch3uYdxnUxgj2wPYsuVIFLAP82d2LfQ+sNzgi6ogsjEsXDETA9HiEjkZDZ3jL7ky ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ObQoBzTfHUm12QKnGWCN+3/+4Jnje27Nf9ebW1zrbLxyL3uRbpBN+k2at2nDXpCdWqiSpde0Ws69B563Ot5B9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='PUhZLDXKO/hqd+Acdhy/k=k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ADuXichVLbhMxFL3p8Cjl0RY2SGwio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='iBWwakqQGVT0Q3LtCFtpRJFnombmMxLYychjPIDSN3SRVdFYoH4Db8AIt+AmJZJDYsOHZcBEQptmZ8fe895x72k9DqTRjJ4U578LFS5fnryxcvXb9xuLS8s1tlfSzQDSCJEyXZ8rEcpYNLTUodhNM8EjPxQ7fm/DxHcGIlM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='yiZ/rUSqaEe/Ecl8GXMNV7VWkslVmF2FKeNqjNK5EYtWS7U6AW1KaGA+hSRoJg07JA4Kcw9qhKjFL4m5fBlsKSNCxrTArAa1iv8h/C3sXapCMxjoCJMk9EHjwAHB6Hfwe7PeNsTdVleUPoCPEl4G7SGX2hX1gp+wz+8i+s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='p8zuXLYdSOsPoTrEhbi29u13/8FxVhNap/o87VrGkfZzNaJbSn1mNOEUzwg9eHp/W1rXJ+j71j36D/mJ2wTzhBPgevN8UW0fn6DlTonED5h4UOMtgNR6N7DV6gBnA5qhYsfUgScFh1mn8bNrKdfhLnZt18kY9tD2ILJcMSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='5/JPcsX0LzT8Z+icSoiMHQtHzPRwiIhETmpzR+hLhmwODcoxzVZnck2mwFnGeNPVf1/wtLG9Uqk+rKxurpbWn7rXPU936C7dR61HtE7PqEYNVOnQAb2lQ+Jx72u93KSOldwmFv01/DUL8q0y/o=k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ADvnichVK7bhNBFL3O8gjhkQcNEo2FZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='USDmY0iQKks0lA6CU4sBcvaXY+dkfelnbEdZ+UfoKRJEZpEokB8Bg0/QJFPQJRBoqHgzHiCAMthRrtz5957zpyZe/0FIxdlaYc65cvXZ9/sbCzVu37ywuLa/syKSfBbweJGSNXxP8lDEvK6ECnkjzbgX+SHf9XsbOr474Jk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='USfxKjVLejLxuLDoi8BRcjce9lotvtbVUYhVmRnHacK1RIjtqyXKhRq+pTQkF1KeIOMWkYIfkcTcI5cYpfA1KYcvgyVMnNOYFoBVsA7wH8LfxrpPRWCeAxVh6ow+eDg4POB6+Hex27PeGHt9qjT8AXSE+DJwF6nMvrAP7Jx9Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='h/ZV/ZzJlduOLTaEVZ/guVpa/HNve0f/0VFWLXq36hLNSvq4G5aq4D21Hj0LYIJfnB4dL69vlXOH7JT9g36T9gZ+4QbxIPvwftNvnV8iZ4LJQovoN9BgrMVu1RyF6nJ5gBbA8nVsw7deFJwaHXafzs6St8D52bVvJGPbQ1CA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='yXDEiOfyT3LHpheYfngt0TiVExpbFQ0zXcIiIQE5qckeoS4ZsDxqkZqtTufqTI67jNHT7r8dPG3srFbcp5W1zbVS9YXt7nm6Tw/oEc56RlV6STWqm859S8f0zqk6HSdyknqXMFi7tJfwzn4BQEnzYE=�k1 � k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='Figure 1: Feynman amplitude iΠq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='X for the loop-diagrammatic evaluation of the ¯θ param- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='eter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In the below argument, we take gauge dependence parameter x0 as x0 = 0 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The gluon field Ga µ can be expanded under this gauge around x = 0 and it is given with the gluon field-strength tensor at x = 0, Ga µν(0), as [43] Ga µ(x) = 1 2xνGa νµ(0) + · · · = � d4ke−ik·x � − i 2Ga νµ(0) ∂ ∂kν δ(4)(k) � + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='12) Here, the discarded terms are covariant derivatives of the background gluon field-strength tensor, which are irrelevant to the calculation of the QCD θ term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' We can systematically evaluate the interaction of the propagating quarks with the background gluon field-strength tensor in this gauge fixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' However, we found that the effective gluon operators such as the QCD θ term cannot be evaluated from the simple quark bubble diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The background gluon fields bring momenta, k in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='12), which are taken to be zero in the last of calculation due to δ(4)(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Thus, the quark momentum is not constant due to interaction with the background field and the quark line cannot be closed without violating momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In order to fix this problem, we introduce an auxiliary (dimensionless) background field X, and it is coupled to the CP-odd quark mass terms as L/P , /T = − � q=all Im(mq)¯qiγ5q ⇒ − � q=all Im(mq) (¯qiγ5q) X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='13) We evaluate the leading contribution of Im(mq) to the QCD θ term in perturbative way, assuming Im(mq) ≪ Re(mq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The field X is taken to be 1 in the last of calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='#5 The radiative QCD θ term comes from a bubble diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The Feynman diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' 1 shows the leading contribution, which is realized by integrating the delta function #5This technique has been applied for evaluation of the Weinberg operator in the QCD [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' – 6 – in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='12) as iΠq X = − � d4k1d4k2 d4p (2π)4 ˜X(−k1 − k2) × Tr � (−igsγµT a) � − i 2Ga ρµ(0) ∂ ∂k1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ρ δ(4)(k1) � i /p + /k1 − Re(mq)Im(mq)γ5 × i /p − /k2 − Re(mq)(−igsγνT b) � − i 2Gb σν(0) ∂ ∂k2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='σ δ(4)(k2) � i /p − Re(mq) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='14) where ˜X(k) = � d4xeik·xX(x) and p is the loop momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' We followed the Feynman rules under the Fock-Schwinger gauge, which includes the gluon field Ga µ expressed as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='12) and the modified CP-odd quark mass term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Until integration of the delta functions, two independent momenta k1, k2 flow into the vertices with the background field- strength tensors Ga ρµ and Gb σν, respectively, and the artificial background field X brings a momentum −k1 − k2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' After some calculation, we get iΠq X = i Im(mq)g2 s 8 Ga ρµ(0)Ga σν(0) � d4p (2π)4 4iRe(mq)ϵµνρσ [p2 − (Re(mq))2]3 ˜X(0) = iαs 8π � −Im(mq) Re(mq) � Ga µν(0) ˜Gaµν(0) ≃ −iθq αs 8πGa µν(0) ˜Gaµν(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='15) Here, we take ˜X(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' After integrating the quark q out in the full theory, ∆L = Πq X is obtained in the effective action of the gluon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Eventually, one can derive the QCD θ term in the Fock- Schwinger gauge method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' This result is consistent with that of the chiral rotation, q → q′ = exp(− i 2θqγ5) q, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3) and also the operator Schwinger method in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Hence, we reached a clarification of the equivalence among the Fock-Schwinger gauge method, the operator Schwinger method, and the ordinary chiral rotation, and we noticed that the Fock-Schwinger gauge method is more intuitive than the operator Schwinger method for higher-loop order calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It might be concerned that the diagrammatic evaluations of the light quark contribu- tion to the QCD θ term is not justified from the viewpoint of perturbation, since the loop momentum around the quark mass dominates the integrals in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It might be healthy to evaluate the light quark mass phases above the ΛQCD scale and derive the QCD θ parameter by the chiral rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' However, since the diagrammatic evaluations are consistent with those of the chiral rotation, we may forget the problem in practical cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In this paper, to evaluate the QCD θ term diagrammatically, we will use the Fock- Schwinger gauge method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Note that the auxiliary background field X should be attached to any perturbative interactions, but we suppress them in the following calculations for the sake of clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' – 7 – SU(3)C SU(2)L SU(2)R U(1)B−L U(1)Y Qi L ≡ (ui L, di L)T □ □ 1 1/6 (1/6, 1/6) Qi R ≡ (ui R, di R)T □ 1 □ 1/6 (2/3, −1/3) H 1 □ 1 1/2 (1/2, 1/2) H′ 1 1 □ 1/2 (1, 0) Ua L □ 1 1 2/3 2/3 Ua R □ 1 1 2/3 2/3 Da L □ 1 1 −1/3 −1/3 Da R □ 1 1 −1/3 −1/3 Table 1: The matter contents and their gauge charges in the minimal LR symmetric model, where U(1)Y = T R 3 + U(1)B−L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The indices i and a represent the flavors for the doublet and singlet quarks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' 3 The minimal left-right symmetric model 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1 Model From this section, we introduce the minimal LR symmetric model that can solve the strong CP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The LR symmetry, which is formed by introducing a new SU(2)R gauge symmetry, with spatial parity symmetry is motivated to forbid the QCD ¯θ term at tree level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In particular, we focus on the minimal LR symmetric model, which embeds the SU(2)L singlet right-handed quarks, uR and dR, to the SU(2)R doublets, QR ≡ (uR, dR)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Furthermore, a SU(2)R doublet Higgs, H′, and three flavors of the up-type and down-type vector-like quarks, UL, UR, DL and DR, have to be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The matter contents are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' To solve the strong CP problem, the spatial parity symmetry has to be extended to symmetrize the left-handed and right-handed sectors as well as the SU(2)L and SU(2)R gauge bosons, ⃗x ←→ −⃗x , Wµ ←→ W ′µ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) QL, UL, DL, H ←→ QR, UR, DR, H′ , while the other gauge bosons are invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The spontaneous violation of the extended parity symmetry, SU(2)R ×U(1)B−L → U(1)Y , is caused by the vacuum expectation value (VEV) of H′ 0, ⟨H′⟩ = (0, v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' After the symmetry breaking, the U(1)Y gauge symmetry is generated with the gauge charge of U(1)Y = T R 3 + U(1)B−L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The SU(2)R gauge bosons absorb the Nambu-Goldstone (NG) bosons in the doublet H′ (ϕ′ + and ϕ′ 0) to – 8 – become massive states (W ′ + and Z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The physical neutral Higgs boson associated with this symmetry breaking is denoted as h′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Then, the SU(2)L × U(1)Y gauge symmetry is broken to U(1)EM by the VEV of H0, ⟨H⟩ = (0, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The W + and Z bosons absorb the NG bosons in the doublet H (ϕ+ and ϕ0), and the physical (SM) neutral Higgs boson with this symmetry breaking is denoted as h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='#6 The resultant parity violation in nature comes from v′ ̸= v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Namely, we assume that soft parity breaking terms are contained in the H and H′ Higgs potentials which lead to v′ ≫ v ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' These two VEVs can be chosen as real and positive without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Since the extended parity is a discrete symmetry, its spontaneous breaking leads to the formation of the domain walls, which dominate the energy density in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' This domain wall problem can be naturally solved by the Planck suppressed higher-dimensional operators, which explicitly violate the parity symmetry [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The Yukawa interactions and Dirac mass terms for the vector-like quarks are repre- sented as −LY =Qi Lxia u Ua R ˜H + Qi Rxia u Ua L ˜H′ + Ma uUa LUa R + Qi Lxia d Da RH + Qi Rxia d Da LH′ + Ma d Da LDa R + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2) where i = 1–3 is a flavor index for SU(2)L/R doublets, a = 1–3 is that for the singlets (vector-like quarks), and ˜H(′) = ϵH(′)∗ (ϵ12 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The LR symmetry requires that the Yukawa xia u/d in the first two terms (in both lines) must be the same complex matrices, and the Dirac mass terms Mu and Md must be Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The Dirac mass terms Ma u and Ma d in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2) are diagonalized to real and positive eigenvalues by the field redefinitions of the vector-like quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='#7 The SM quark masses are realized by the seesaw mechanism such as higher dimensional operators induced by integrating out the vector-like quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='#8 Before discussing the mass matrices in detail in the next section, let us count on the number of physical CP phases in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The Yukawa couplings xia u/d are 3×3 complex matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Nine real parameters are removed from the Yukawa matrices by field redefinition of Qi L/R as Qi L/R → UijQj L/R with a unitary matrix U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Furthermore, phase redefinition of Ua L/R and Da L/R removes five phases in the total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' A remaining phase rotation corresponds to the baryon number conservation, and it does not change xu and xd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Thus, xu and xd have a total of 22 physical real parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' We parametrize these 22 parameters as xu = Φ†(θd3, θd8) VQ Φ(θu3, θu8) ¯xuVU , xd = ¯xdVD , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3) #6The h–h′ and Z–Z′ mixings are induced in the model at tree level, though the mixings are suppressed by v/v′ and (v/v′)2, respectively [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Since we take v/v′ → 0 in the calculation of the ¯θ parameter, they are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' #7One can also consider non-Hermitian vector-like quark mass matrices which correspond to soft parity breaking terms [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' However, such contributions produce large quark EDM and radiative ¯θ, and thus they are severely constrained from the EDM bounds [40, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' #8One can also extend the lepton sector that is insensitive to the QCD ¯θ term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' If one considers SU(5)L × SU(5)R grand unification [46, 47], vector-like neutral leptons are absent and the neutrinos keep massless at the tree level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Interestingly, suitable Dirac neutrino masses are generated from the two-loop radiative corrections [48] with predicting a nonzero ∆Neff [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Furthermore, an O(10) keV sterile neutrino dark matter with the leptogenesis mechanism can be incorporated [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' – 9 – with Φ(θ3, θ8) ≡ exp(iτ3θ3) exp(iτ8θ8) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='4) where τ3 and τ8 are the third and eighth Gell-Mann matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Here, ¯xu/d are real diagonal matrices and VQ, VU, and VD are CKM-like unitary matrices which have three rotation angles and one CP-violating phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It is found that there are seven CP-violating phases in this model (θu3/u8, θd3/d8, and three phases in VQ/U/D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' When the Dirac masses are assumed to be universal such as Ma u = Mu and Ma d = Md for a = 1–3, the parameters θu3/u8, θd3/d8, VU/D become unphysical and only VQ remains physical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In this paper, we assume that v′ <∼ Ma q and will utilize expansions by v′/M a q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' This inequality is motivated because the seesaw mechanism may explain the SM fermion mass hierarchy naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' On the other hand, if v′ ≫ Ma q , a copy of the SM fermions has a mass spectrum similar to the SM fermions, which spread over five orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' A new naturalness problem might appear in such a model, but the QCD ¯θ term is suppressed by Ma q /v′ since only the CKM phase survives in a limit of Ma q → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In the following, we will consider the case of v′ <∼ Ma q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2 Parametrization of Yukawa coupling constants In this section, we show the quark mass matrices and define the mass eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In the mass matrices the Yukawa coupling constants, xu and xd, appear, and we have to determine them from the observed quark masses and CKM matrix in order to evaluate the radiative ¯θ parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' We give the parameterization of the Yukawa coupling constants assuming the SM quark masses are given by the seesaw mechanism with v′ ≲ Ma q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2), the quark mass matrices in the flavor eigenstates are given as −LM = � ui L, Ua L � � 0 xib u v x†aj u v′ Ma uδab � � uj R Ub R � + � d i L, Da L � � 0 xib d v x†aj d v′ Ma d δab � � dj R Db R � + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' ≡ Up LM(0)pq u Uq R + Dp LM(0)pq d Dq R + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='5) where Up L/R and Dp L/R (p, q = 1, · · · , 6) are the up- and down-type flavor eigenstates, and v ≃ 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Here, Ma u/d are real diagonal, while xia u/d are complex matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It is obvious that arg det M(0) u/d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Then, the 6×6 fermion mass matrices are diagonalized by bi-unitary matrices, VqL and VqR, as M(0)pq q = V †pP qL ¯ MP q V Pq qR for q = u and d , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='6) with diagonal mass matrices ¯ Mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The mass eigenstates, UP ML/R and DP ML/R for P = 1–6, are given as UP ML = V Pp uL Up L , UP MR = V Pp uR Up R , DP ML = V Pp dL Dp L , DP MR = V Pp dR Dp R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='7) It is difficult to reconstruct the model parameters from the experimental data in gen- eral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Here we assume that v′ <∼ Ma q and we take leading terms in the expansion of v′/M a q – 10 – for the quark mass eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In this expansion, the SM quark masses are given by the following seesaw relation V †iI q mI q v V Ij q = xia q v′ Maq x†aj q , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='8) with a 3 × 3 unitary matrix Vq and I = 1–3, while the heavy quark masses are given by Ma q , for q = u and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Now let us rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='8) as I3 = � √v √mq Vqxq √ v′ �Mq � � √ v′ �Mq x† qV † q √v √mq � ≡ UqU† q , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='9) where I3 is a unit matrix in the three-dimensional space and we ignore the indices of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='#9 Thus, the Yukawa matrices xq are given with a unitary matrix Uq by xq = V † q √mq √v Uq �Mq √ v′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='10) According to the previous section, one can remove some unphysical parameters in Uq and Vq by the field redefinitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Then, we get xu = V † CKM √mu √v Φ(θu3, θu8)VU √Mu √ v′ , xd = √md √v Φ(θd3, θd8)VD √Md √ v′ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='11) where VCKM(≡ VuV † d ) corresponds to the CKM matrix in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Here, VU/D are CKM- like unitary matrices with three mixing angles and one CP-violating phase, though they are different from those in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Now we have seven physical CP-violating phases (θu3/u8, θd3/d8, and three phases in VU/D and VCKM), which is consistent with our previous counting, and all phases can be O(1) under the extended parity symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Since we assume that v′ <∼ Ma q , the 6×6 diagonalization matrices in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='7) are given of leading terms in the expansion of v′/M a q as VuL ≃ � −VCKM VCKMxu v Mu v Mu x† u I3 � , VuR ≃ � VCKM −VCKMxu v′ Mu v′ Mu x† u I3 � , VdL ≃ � −I3 xd v Md v Md x† d I3 � , VdR ≃ � I3 −xd v′ Md v′ Md x† d I3 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='12) where xq are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Here, the diagonal eigenvalue matrices ¯ Mq are given as, ¯ MP q = diag(mI q, Ma q ) for q = u and d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='13) #9A similar parameterization technique, referred to as the Casas-Ibarra parameterization, is applied in the minimal seesaw model [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' – 11 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3 Quark EDMs Before discussing the radiative ¯θ parameter in the minimal LR symmetric model, let us comment on contributions to quark (chromo) EDMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' As long as the vector-like mass ma- trices are Hermitian, the quark EDMs vanish completely at one-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The W (′)± contribution at one-loop level vanishes trivially since the chirality is conserved in the dia- grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' On the other hand, the one-loop quark EDM contributions from neutral Higgses and Z(′) may have a chirality flip in the diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Nevertheless, they also vanish because the extended parity symmetry restricts the product of two vertices to be strictly real, as shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' 4 Confirmation of vanishing QCD θ parameter in two-loop order The parity symmetry is spontaneously broken by the ⟨H′⟩ (≫ ⟨H⟩) in the LR symmetric model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Since arg det M(0) u/d = 0 holds, fermion one-loop contributions to the QCD ¯θ term remain zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' However, it is expected that fermion-loop diagrams at a higher than one-loop level would generate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In this section, we show fermion two-loop contributions to the QCD ¯θ term still vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Integrating out quarks, the following higher-dimensional operators are expected to be generated, Leff = � n=1 Cn |H′|2n − |H|2n M2n q αs 8πGa µν ˜Gaµν , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) with Mq as the scale of vector-like quark masses, and the QCD ¯θ term are induced by the spontaneous parity symmetry breaking ⟨H′⟩ (≫ ⟨H⟩), ¯θ = � n=1 Cn � ⟨H′⟩ Mq �2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2) We evaluate the Wilson coefficients of the operators Cn in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' First, we consider the contributions to the QCD ¯θ term at two-loop level coming from an exchange of the W ′ ± boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The two-loop fermion bubble diagrams mediated by the W ′ ± boson under the gluon field-strength background conserve chirality in the fermion line, and then it is proportional to V Pi dRV †iQ uR V Qj uR V †jP dR f � ( ¯ MP d )2, ( ¯ MQ u )2, m2 W ′ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3) where i and j run 1–3 as the flavor index for the SU(2)R doublet, while P and Q run 1–6 for the quark mass eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Here, a two-loop function f[( ¯ MP d )2, ( ¯ MQ u )2, m2 W ′] is a real function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Since � V Pi dRV †iQ uR V Qj uR V †jP dR �∗ = V Pj dR V †jQ uR V Qi uRV †iP dR and it corresponds to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3) by an exchange of i ↔ j, the contribution is real so that it does not generate the QCD ¯θ term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='– 12 – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ADAnichVI7T9xAGBxMeIbHQZpIaU45gahOexEKiAopFDSReB0gwelk+xbOwi/svSNgXUeVP0CRJ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='iBRECpEm46GP5Cn4BoIhEpTYqMfQbxELCWd2fn2xnPrtfwbStUQly0aK2v2to7Oru6X/f09vVnBgYXQ68WmL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='JoerYXLBt6KG3LlUVlKVsu+4HUHcOWS8bGp7i+VJdBaHnugtr2ZcnR1rzTJ1Raq06uiqaup2VGyUP5czOZE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='XScs+BoU5JC2GS/zG6uowIOJGhxIuFDENnSEfFZQgIBProSIXEBkJXWJBrqpVURf2G+Rr3CsIkvNOFUOn3hFj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='T6SHjp1G+zXOVtJWZfz+Kth4m8yh803oHcWQ+KXOBLX4lwci0vx70mvKPGI025zNJpa6Zf7v76d/uiyuEYp7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='5VPZtZY17i7NazO4nTLwLs6mv7+xdz0/MDUXD4kBcMf+uBn3IFb/2Mezsq5b8/kCdNTr3JWSU/XJd5KzsVJ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='krqsROSbaxvJ/yndYW7UEXKsNGIXonCwvwGCx+yBc+5kdnR3OTU+nl6MQ7vMcIfcYwiWnMoMgkm9jDd+xru ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='9oP7UQ7bS7VWlLNG9xr2s/qtytHw=UM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ADAnichVI7T9xAGBxMCI+QcJAGiebECUR12kMoICokKGiQeB0gwelk+xbOr9i7x0P6zq/AEKG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='kCigFQRbo0+QMU/IQoTSQaFJk7DMoCQpZy7uz8+2MZ9dr+LYVKiFu2rT2Fx0vO7u6e171vn7Tl+kfWAu9em ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='DKounZXrBh6KG0LVcWlaVsueEHUncMW64btdm4vt6QWh57qra92XJ0Xdca9sydUWqtOXoqmrqdjTXLC+UMzm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='RF0nLPgWFOSQtkUv8wNbqMCDiTocSLhQxDZ0hHw2UYCAT6EiFxAZCV1iSZ6qFVEe+x3yVc4VpGlZoqh0+8o ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='k4fSQ+duhr7Hc42U9blP5qmPibzGHzDeidxYi4FhfiVnwVH8U38fOfXlHiEafd52i0tNIv930YXLn/r8rhGK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='d+VD2bWGbe4uzWszuJ0y8C7Olbxwc3a5ML49Eo+JMfGf+U3EjvnAHbuPOPF+Sy8fP5AnTU69yVklP1yXeTc7F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='SZK6rETkW2ubyf8p/cY8qCPkWGnGLrwShb8vwFOwNp4vMtPLE3kZubSy9GFIQxjD6TmME8FlFkvc4wglOt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='UPtUvukXbWam2p5i3+aNrnX373rQ4=DM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ADAnichVI7T9xAGBxMCI+QcJAGiebECU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='R12kMoICokKGiQeB0gwelk+xbOr9i7x0P6zq/AEKGkCigFQRbo0+QMU/IQoTSQaFJk7DMoCQpZy7uz8+2MZ9dr+LYVKiFu2rT2Fx0vO7u6e171vn7Tl+kfWAu9emDKounZXrBh6KG0LVcWlaVsueEHUncMW64btdm4vt6Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='Wh57qra92XJ0Xdca9sydUWqtOXoqmrqdjTXLC+UMzmRF0nLPgWFOSQtkUv8wNbqMCDiTocSLhQxDZ0hHw2UYCAT6EiFxAZCV1iSZ6qFVEe+x3yVc4VpGlZoqh0+8ok4fSQ+duhr7Hc42U9blP5qmPibzGHzDeidxYi4Fhfi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='VnwVH8U38fOfXlHiEafd52i0tNIv930YXLn/r8rhGKd+VD2bWGbe4uzWszuJ0y8C7Olbxwc3a5ML49Eo+JMfGf+U3EjvnAHbuPOPF+Sy8fP5AnTU69yVklP1yXeTc7FSZK6rETkW2ubyf8p/cY8qCPkWGnGLrwShb8vwFOwNp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='4vMtPLE3kZubSy9GFIQxjD6TmME8FlFkvc4wglOtUPtUvukXbWam2p5i3+aNrnX373rQ4=DM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ADAnichVI7T9xAGBxMeIbHQZpIaU45ga ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='hOexEKiAopFDSReB0gwelk+xbOwi/svSNgXUeVP0CRJiBRECpEm46GP5Cn4BoIhEpTYqMfQbxELCWd2fn2xnPrtfwbStUQly0aK2v2to7Oru6X/f09vVnBgYXQ68WmLJoerYXLBt6KG3LlUVlKVsu+4HUHcOWS8bGp7i+VJdBa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='Hnugtr2ZcnR1rzTJ1Raq06uiqaup2VGyUP5czOZEXScs+BoU5JC2GS/zG6uowIOJGhxIuFDENnSEfFZQgIBProSIXEBkJXWJBrqpVURf2G+Rr3CsIkvNOFUOn3hFjT6SHjp1G+zXOVtJWZfz+Kth4m8yh803oHcWQ+KXOBLX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='4lwci0vx70mvKPGI025zNJpa6Zf7v76d/uiyuEYp75VPZtZY17i7NazO4nTLwLs6mv7+xdz0/MDUXD4kBcMf+uBn3IFb/2Mezsq5b8/kCdNTr3JWSU/XJd5KzsVJkrqsROSbaxvJ/yndYW7UEXKsNGIXonCwvwGCx+yB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='c+5kdnR3OTU+nl6MQ7vMcIfcYwiWnMoMgkm9jDd+xru9oP7UQ7bS7VWlLNG9xr2s/qtytHw=UM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ADAnichVI7T9xAGBxMeIbHQZpIaU45ga ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='hOexEKiAopFDSReB0gwelk+xbOwi/svSNgXUeVP0CRJiBRECpEm46GP5Cn4BoIhEpTYqMfQbxELCWd2fn2xnPrtfwbStUQly0aK2v2to7Oru6X/f09vVnBgYXQ68WmLJoerYXLBt6KG3LlUVlKVsu+4HUHcOWS8bGp7i+VJdBa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='Hnugtr2ZcnR1rzTJ1Raq06uiqaup2VGyUP5czOZEXScs+BoU5JC2GS/zG6uowIOJGhxIuFDENnSEfFZQgIBProSIXEBkJXWJBrqpVURf2G+Rr3CsIkvNOFUOn3hFjT6SHjp1G+zXOVtJWZfz+Kth4m8yh803oHcWQ+KXOBLX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='4lwci0vx70mvKPGI025zNJpa6Zf7v76d/uiyuEYp75VPZtZY17i7NazO4nTLwLs6mv7+xdz0/MDUXD4kBcMf+uBn3IFb/2Mezsq5b8/kCdNTr3JWSU/XJd5KzsVJkrqsROSbaxvJ/yndYW7UEXKsNGIXonCwvwGCx+yB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='c+5kdnR3OTU+nl6MQ7vMcIfcYwiWnMoMgkm9jDd+xru9oP7UQ7bS7VWlLNG9xr2s/qtytHw=UM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ADAnichVI7T9xAGBxMCI+QcJAGiebECU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='R12kMoICokKGiQeB0gwelk+xbOr9i7x0P6zq/AEKGkCigFQRbo0+QMU/IQoTSQaFJk7DMoCQpZy7uz8+2MZ9dr+LYVKiFu2rT2Fx0vO7u6e171vn7Tl+kfWAu9emDKounZXrBh6KG0LVcWlaVsueEHUncMW64btdm4vt6Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='Wh57qra92XJ0Xdca9sydUWqtOXoqmrqdjTXLC+UMzmRF0nLPgWFOSQtkUv8wNbqMCDiTocSLhQxDZ0hHw2UYCAT6EiFxAZCV1iSZ6qFVEe+x3yVc4VpGlZoqh0+8ok4fSQ+duhr7Hc42U9blP5qmPibzGHzDeidxYi4Fhfi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='VnwVH8U38fOfXlHiEafd52i0tNIv930YXLn/r8rhGKd+VD2bWGbe4uzWszuJ0y8C7Olbxwc3a5ML49Eo+JMfGf+U3EjvnAHbuPOPF+Sy8fP5AnTU69yVklP1yXeTc7FSZK6rETkW2ubyf8p/cY8qCPkWGnGLrwShb8vwFOwNp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='4vMtPLE3kZubSy9GFIQxjD6TmME8FlFkvc4wglOtUPtUvukXbWam2p5i3+aNrnX373rQ4=DM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ADzHichVK7bhNBFL3O8gjhESdQINFYWEahwIxRFCqCBoq5CQ4iRQba3Y9s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='UfZx2h3bOstqXgByhoAIki8Bk0/ABFPgFRBomGgjPjDQIshxntzp17zn3zNxlS8TzdhRYcY5c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='/bc+dkLcxcvXb4yX1xY3EqifuyJhf5Ubzj8kT4MhQNLbUvdlQseOD6Ytvdf2Ti2wMRJzIKn+qR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='Eq2Ad0O5Jz2u4WoXrzUHPFY9+SxdaqpYBuJ26U7WLpZldlRmjRquVGmfNSjhUKdmtShiDzqU0C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='CQtKwfeKUYO5SjRgp+FqUwhfDkjYuKM5YDWs5/gP4e9g7VEJmAdABZgmow8eAQ4O3D7+Xex2c2+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='IvamaWH4POnx8MbhLVGFf2CE7Zp/ZR/aV/ZzKlVoOo3aE1R1jhWrPv7y+eO/qACrUf0bdapmTX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='s4m9EqoV1ZjzmFN8YPDl4db65uVNJb7B37Bv1v2RH7hBOEg+/e+3Wx8foUPSdKNG7A3EMCzgpYj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='Ucje5XuYnqwOSpW7T14VHgMOskfnqtJO9wD7tO3skQ9tD2ILBcISIp/OPczL6F1h+eE3RKZUSy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='nIUjZno4REQiR9ncEfoSI5tDQ5IzTVdnck2mwFnMm679+4Inja171dr96vL6cntYf6Z+kG3aQl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1FqhNXpMdWqgygG9oUP64DxtJM62Th1pBjrtJfw3nxC2w30yw= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content="'(0)� " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ADzHichVK7bhNBFL3O8gjh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ESdQINFYWEahwIxRFCqCBoq5CQ4iRQba3Y9sUfZx2h3bOstqXgByhoAIki8Bk0/ABFPgFRBomGgjPjDQIshxntzp17zn3zNxlS8TzdhRYcY5c/bc+dkLcxcvXb4yX1xY3EqifuyJhf5Ubzj8k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='T4MhQNLbUvdlQseOD6Ytvdf2Ti2wMRJzIKn+qREq2Ad0O5Jz2u4WoXrzUHPFY9+SxdaqpYBuJ26U7WLpZldlRmjRquVGmfNSjhUKdmtShiDzqU0CQtKwfeKUYO5SjRgp+FqUwhfDkjYuKM5YDWs5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='/gP4e9g7VEJmAdABZgmow8eAQ4O3D7+Xex2c2+IvamaWH4POnx8MbhLVGFf2CE7Zp/ZR/aV/ZzKlVoOo3aE1R1jhWrPv7y+eO/qACrUf0bdapmTXs4m9EqoV1ZjzmFN8YPDl4db65uVNJb7B37Bv1v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2RH7hBOEg+/e+3Wx8foUPSdKNG7A3EMCzgpYjUcje5XuYnqwOSpW7T14VHgMOskfnqtJO9wD7tO3skQ9tD2ILBcISIp/OPczL6F1h+eE3RKZUSynIUjZno4REQiR9ncEfoSI5tDQ5IzTVdnck2mwF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='nMm679+4Inja171dr96vL6cntYf6Z+kG3aQl1FqhNXpMdWqgygG9oUP64DxtJM62Th1pBjrtJfw3nxC2w30yw= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content="'(0)� " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ADzHichVK7bhNBFL3O8gjh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ESdQINFYWEahwIxRFCqCBoq5CQ4iRQba3Y9sUfZx2h3bOstqXgByhoAIki8Bk0/ABFPgFRBomGgjPjDQIshxntzp17zn3zNxlS8TzdhRYcY5c/bc+dkLcxcvXb4yX1xY3EqifuyJhf5Ubzj8k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='T4MhQNLbUvdlQseOD6Ytvdf2Ti2wMRJzIKn+qREq2Ad0O5Jz2u4WoXrzUHPFY9+SxdaqpYBuJ26U7WLpZldlRmjRquVGmfNSjhUKdmtShiDzqU0CQtKwfeKUYO5SjRgp+FqUwhfDkjYuKM5YDWs5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='/gP4e9g7VEJmAdABZgmow8eAQ4O3D7+Xex2c2+IvamaWH4POnx8MbhLVGFf2CE7Zp/ZR/aV/ZzKlVoOo3aE1R1jhWrPv7y+eO/qACrUf0bdapmTXs4m9EqoV1ZjzmFN8YPDl4db65uVNJb7B37Bv1v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2RH7hBOEg+/e+3Wx8foUPSdKNG7A3EMCzgpYjUcje5XuYnqwOSpW7T14VHgMOskfnqtJO9wD7tO3skQ9tD2ILBcISIp/OPczL6F1h+eE3RKZUSynIUjZno4REQiR9ncEfoSI5tDQ5IzTVdnck2mwF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='nMm679+4Inja171dr96vL6cntYf6Z+kG3aQl1FqhNXpMdWqgygG9oUP64DxtJM62Th1pBjrtJfw3nxC2w30yw= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content="'(0)� " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='(a) diagram A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='(b) diagram B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='(c) diagram C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='Figure 2: The charged NG boson contributions to the QCD ¯θ term at two-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The reason why the exchange of the W ′ ± boson does not contribute to the QCD ¯θ term at two-loop level is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' However, the above discussion is based on the structure of the mixing matrices in the contribution, not on the structure of Lagrangian parameters, such as xu/d and Mu/d, and then, it is unclear what is required to generate the QCD ¯θ term in higher-order diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' We make it clear by explicit calculation of the loop diagrams in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In the unitary gauge, the lowest dimension operator (n = 1) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) might come from diagrams which include the longitudinal mode of the W ′ ± boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It is because the propagator is proportional to kµkν/m2 W ′ (kν the momentum of the W ′ ± boson), and it could give the lowest order contribution with regard to v′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The Yukawa coupling constants xu and xd are multiplied with the Higgs VEVs in the mixing matrices as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In our calculation, we adopt the Rξ gauge with the Feynman-’t Hooft gauge ξ = 1 (for SU(2)L × SU(2)R × U(1)B−L gauge), in order to avoid the messy calculation in the unitary gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The lowest dimension operator (n = 1) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) could arise the charged NG boson exchange ϕ′ ± in this gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The charged NG boson is absorbed by W ′ ± boson in the Higgs mechanism, and its mass, mϕ′, is equal to the W ′ ± mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The charged NG boson interactions are given as −Lϕ′± = ui Rxia d Da Lϕ′+ − d i Rxia u Ua Lϕ′ − + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' = (xia d V Pi uRV ∗Qa dL ) UP MRDQ MLϕ′+ − (x∗ia u V Pa uL V ∗Qi dR ) UP MLDQ MRϕ′+ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='4) Both the left- and right-handed quarks are coupled with the charged NG boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' We will show that the charged NG boson diagrams at two-loop level do not contribute to the QCD ¯θ term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Three diagrams (dubbed as diagrams A, B and C) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' 2 could give contributions to the ¯θ parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' By using the Fock-Schwinger gauge method (for SU(3)C gauge) in – 13 – Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2, we obtain δθ|A = 1 8π2 Im � xia u V ∗Pa uL V Qi dR xjb d V Pj uR V ∗Qb dL � ¯ MP u ¯ MQ d ¯I(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3) � ( ¯ MP u )2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' ( ¯ MQ d )2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='5) δθ|B = 1 8π2 Im � xia u V ∗Pa uL V Qi dR xjb d V Pj uR V ∗Qb dL � ¯ MP u ¯ MQ d ¯I(3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) � ( ¯ MP u )2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' ( ¯ MQ d )2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='6) δθ|C = 1 8π2 Im � xia u V ∗Pa uL V Qi dR xjb d V Pj uR V ∗Qb dL � ¯ MP u ¯ MQ d I(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2) � ( ¯ MP u )2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' ( ¯ MQ d )2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='7) where the two-loop functions ¯I(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3), ¯I(3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1), and I(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2) are defined in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In the above evaluation, we pick up contributions proportional to both xu and xd, which are also proportional to the quark masses in the mass eigenstate propagators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The terms proportional to xu and x∗ u (or xd and x∗ d) is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Here, we use the dimensional regularization (d = 4 − 2ϵ) for loop momentum integrals and the partial diagrams produce UV divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' However, the contributions from dia- grams A and B, proportional to ¯Iϵ(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3) and ¯Iϵ(3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='13), respectively, vanish, so that the correction to the ¯θ parameter is finite and scale-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The UV divergent parts (1/ϵ terms) in ¯Iϵ(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3) and ¯Iϵ(3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) cancel out since Im � xia u V ∗Pa uL V Qi dR xjb d V Pj uR V ∗Qb dL � ¯ MP u ¯ MQ d = Im � x∗ia u xia u � = 0 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='8) Im � xia u V ∗Pa uL V Qi dR xjb d V Pj uR V ∗Qb dL � ¯ MQ d ¯ MP u = Im � x∗ia d xia d � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='9) To derive the above equations we use the following equations, [(M(0) q )−1]pq = V †pP qR ( ¯ M−1 q )P V Pq qL = � − 1 vv′ (x† q)−1 ic Mc q(xq)−1 cj 1 v′ (x† q)−1 ib 1 v(xq)−1 aj 0 � for q = u and d , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='10) in addition to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='6) with ( ¯ MP q )∗ = ¯ MP q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Furthermore, we observed that the following combinations also vanish#10 Im � xia u V ∗Pa uL V Qi dR xjb d V Pj uR V ∗Qb dL � ¯ MP u ¯ MQ d log ¯ MQ d = 0 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='11) Im � xia u V ∗Pa uL V Qi dR xjb d V Pj uR V ∗Qb dL � ¯ MQ d ¯ MP u log ¯ MP u = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='12) Therefore, the second terms of ¯Iϵ(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3) and ¯Iϵ(3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='13) also do not affect the ¯θ parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' On the other hand, the loop function I(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='7) is UV finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Similar to the ϕ′ ± contribution, the contribution from the SM charged NG boson ϕ±, absorbed into W ±, is derived by replacing L(R) with R(L), and m2 ϕ′ with m2 ϕ in the above #10The factors 1/M log M (M: quark mass) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='11) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='12) correspond to the O(ϵ) term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='15) when changing d4p to ddp (d = 4 − 2ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' They become O(ϵ0) since 1/ϵ comes from the quark self-energy subdiagrams in diagrams A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='11) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='12) can be perturbatively proved by assuming the off-diagonal terms in the quark mass matrices are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The similar trick is also used around Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' We also checked Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='11) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='12) numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' – 14 – formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Furthermore, (−1) is multiplied since the chiralities of circulating fermions are opposite to diagrams of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It means that, if one sets v = v′ corresponding to the LR symmetric limit, those two contributions of ϕ′ ± and ϕ± cancel each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The diagrams A and B correspond to the one-loop correction to the fermion mass terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The two-loop function ¯I(3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1)(x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' x3) is expressed as ¯I(3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1)(x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' x3) = (16π2µ2ϵ) � ddp i(2π)d 1 (p2 − x1)3 F0(p2, x2, x3) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='13) where F0(p2, x2, x3) is a loop function of one-loop diagrams for the fermion mass correction, F0(p2, x2, x3) = � 1 0 dz log −z(1 − z)p2 + zx2 + (1 − z)x3 Q2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='14) with Q2 ≡ 4πµ2e−γE and µ is the renormalization scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (¯I(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3)(x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' x3) also has a similar expression, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=') ¯I(3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1)(x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' x3) has an IR-singular behavior when x1 ≪ x3 as ¯I(3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1)(x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' x3) ≃ − 1 2x1 F0(0, x2, x3) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='15) while small x2 and x3 do not lead to IR singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' This behavior is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It is because if a fermion with real mass mf gets a constant radiative correction to the fermion mass mf + δmf, the correction to the ¯θ parameter is given by δθ ≃ −Im(δmf)/mf, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='#11 However, this evaluation of δθ is justified only when the correction to the fermion mass term is independent of fermion momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The IR-singular behaviors of the SM fermion masses in ¯I(3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) and ¯I(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3) are not physical in δθ|A and δθ|B in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='6), and they can be removed indeed using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='10) #11In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='15), the loop function and the chirality flip lead to ∼ mf/m2 f = 1/mf, and then δθ ≃ −Im(δmf)/mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' – 15 – as δθ|A ≃ 1 8π2 Im � xia u xjb d � V ∗Aa uL ¯ MA u V Aj uR � � V Bi dR 1 ¯ MB d V ∗Bb dL �� × � ( ¯ MB d )2 ¯I(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3) � ( ¯ MA u )2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' ( ¯ MB d )2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � − ( ¯ MB d )2 ¯I(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3) � 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' ( ¯ MB d )2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � +1 2F0 � 0, ( ¯ MA u )2, m2 ϕ′ � − 1 2F0 � 0, 0, m2 ϕ′ �� + 1 8π2 Im � xia u x∗ja u xjb d v′ � V Ai dR 1 ¯ MA d V ∗Ab dL �� × � ( ¯ MA d )2 ¯I(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3) � 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' ( ¯ MA d )2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � + 1 2F0 � 0, 0, m2 ϕ′ �� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='16) δθ|B ≃ 1 8π2 Im � xia u xjb d � V Aj uR 1 ¯ MA u V ∗Aa uL � � V ∗Bb dL ¯ MB d V Bi dR �� × � ( ¯ MA u )2 ¯I(3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) � ( ¯ MA u )2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' ( ¯ MB d )2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � − ( ¯ MA u )2 ¯I(3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) � ( ¯ MA u )2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � +1 2F0 � 0, ( ¯ MB d )2, m2 ϕ′ � − 1 2F0 � 0, 0, m2 ϕ′ �� + 1 8π2 Im � xia u xjb d x∗ib d v′ � V Aj uR 1 ¯ MA u V ∗Aa uL �� × � ( ¯ MA u )2 ¯I(3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) � ( ¯ MA u )2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � + 1 2F0 � 0, 0, m2 ϕ′ �� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='17) where A and B run 4–6 as the heavy quark mass eigenstates, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Here, the SM quark masses in the loop function are taken to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' On the other hand, the contribution of diagram C is not associated with the correction to the quark masses, and then it is a new type contribution to the ¯θ parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It is suppressed by the heavier fermion or ϕ′ ± masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' By taking the SM quark masses to be zero in the loop function (see Appendix A), it is given as δθ|C ≃ 1 8π2 Im � xia u xjb d � V ∗Aa uL ¯ MA u V Aj uR � � V ∗Bb dL ¯ MB d V Bi dR �� I(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2) � ( ¯ MA u )2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' ( ¯ MB d )2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='18) where A and B run 4–6 as the heavy quark mass eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It is found that the diagram C does not contain any IR-singular behavior, unlike the diagrams A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' When v′ <∼ Ma q , the leading contributions of O(v′2/(Ma q )2) are given as δθ|A ≃ 1 8π2 Im � (Aa u)ij(Ab d)ji� � v′2 ¯I(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3) � (Ma u)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (Mb d)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � + v′2 2(Mb d)2 F0 � 0, (Ma u)2, m2 ϕ′ �� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='19) δθ|B ≃ 1 8π2 Im � (Aa u)ij(Ab d)ji� � v′2 ¯I(3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) � (Ma u)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (Mb d)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � + v′2 2(Mau)2 F0 � 0, (Mb d)2, m2 ϕ′ �� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='20) δθ|C ≃ 1 8π2 Im � (Aa u)ij(Ab d)ji� v′2 I(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2) � (Ma u)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (Mb d)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='21) – 16 – with a Hermitian matrix Aa q, (Aa q)ij ≡ xia q x∗ja q for q = u, d and not sum a index .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='22) This can be derived from the above formulae by (V † qL)aA ¯ MA q p2 − ( ¯ MA q )2 (VqR)Ai → x†ai q v′ p2 − (Maq )2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='23) (V † qL)aA ¯ MA q [p2 − ( ¯ MA q )2]2 (VqR)Ai → x†ai q v′ [p2 − (Maq )2]2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='24) (V † qL)aA ¯ MA q [p2 − ( ¯ MA q )2]3 (VqR)Ai → x†ai q v′ [p2 − (Maq )2]3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='25) It is found that these radiative corrections to the ¯θ parameter vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' For example, δθ|A is given as δθ|A = Im Tr � Aa uAb d � f � (Ma u)2, (Mb d)2, m2 ϕ′ � = 1 2Im Tr � [Aa u, Ab d] � f � (Ma u)2, (Mb d)2, m2 ϕ′ � = 0 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='26) where f[(Ma u)2, (Mb d)2, m2 ϕ′] is the real function, and the Hermitian property of the matrix Aa q is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The same conclusions are applicable to δθ|B and δθ|C at this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Now we showed that the charged NG boson contribution to the ¯θ parameter at two- loop level vanishes at the leading order of v′ (n = 1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It comes from the fact that the contributions are proportional to the fourth power of xu/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' We have also checked that the contributions of the sixth power of xu/d, corresponding to O(v′4/(Ma q )4) contributions, also vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The contributions are derived from the above formulae with the – 17 – mass-insertion approximation, (V † qL)aA i ¯ MA q p2 − ( ¯ MA q )2 (VqR)Ai → i x†ai q v′ p2 − (Maq )2 +i 1 p2 − (Maq )2 (x† qxq)abv′2 x†bi q v′ p2 − (Mbq)2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='27) (V † qL)aA i ¯ MA q [p2 − ( ¯ MA q )2]2 (VqR)Ai → i x†ai q v′ [p2 − (Maq )2]2 +i 1 p2 − (Maq )2 (x† qxq)abv′2 x†bi q v′ [p2 − (Mbq)2]2 +i 1 [p2 − (Maq )2]2 (x† qxq)abv′2 x†bi q v′ p2 − (Mbq)2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='28) (V † qL)aA i ¯ MA q [p2 − ( ¯ MA q )2]3 (VqR)Ai → i x†ai q v′ [p2 − (Maq )2]3 +i 1 p2 − (Maq )2 (x† qxq)abv′2 x†bi q v′ [p2 − (Mbq)2]3 +i 1 [p2 − (Maq )2]2 (x† qxq)abv′2 x†bi q v′ [p2 − (Mbq)2]2 +i 1 [p2 − (Maq )2]3 (x† qxq)abv′2 x†bi q v′ p2 − (Mbq)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='29) Each first term is the aforementioned leading contribution, which vanishes (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='26)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The next-to-leading contributions are δθ|A ≃ 1 8π2 Im � (Aa u)ij(Ab u)jk(Ac d)ki� × � v′4I(1,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3) � (Ma u)2, (Mb u)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (Mc d)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � − v′4 (Mc d)2 B(1,1) � 0, (Ma u)2, (Mb u)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ �� + 1 8π2 Im � (Aa u)ij(Ab d)jk(Ac d)ki� v′4 � I(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1,3) + I(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2,2) + I(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='3,1) � � (Ma u)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (Mb d)2, (Mc d)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='30) δθ|B ≃ 1 8π2 Im � (Aa u)ij(Ab d)jk(Ac d)ki� × � v′4I(3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1,1) � (Ma u)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (Mb d)2, (Mc d)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � − v′4 (Mau)2 B(1,1) � 0, (Mb d)2, (Mc d)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ �� + 1 8π2 Im � (Aa u)ij(Ab u)jk(Ac d)ki� v′4 � I(1,3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) + I(2,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) + I(3,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='1) � � (Ma u)2, (Mb u)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (Mc d)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='31) δθ|C ≃ 1 8π2 Im � (Aa u)ij(Ab u)jk(Ac d)ki� v′4 � I(1,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2) + I(2,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2) � � (Ma u)2, (Mb u)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (Mc d)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � + 1 8π2 Im � (Aa u)ij(Ab d)jk(Ac d)ki� v′4 � I(2:1,2) + I(2:2,1) � � (Ma u)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (Mb d)2, (Mc d)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='32) – 18 – These loop functions, which come from the mass-insertion approximation, are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The sequences of masses connected by commas in the loop functions are introduced by the mass insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Again, it is found that these contributions are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' For example, the first term in δθ|A is given as δθ|A = Im Tr � Aa uAb uAc d � f � (Ma u)2, (Mb u)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (Mc d)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � +Im Tr � Aa uAb dAc d � g � (Ma u)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (Mb d)2, (Mc d)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � = 1 2Im Tr � Ac d [Aa u, Ab u] � f � (Ma u)2, (Mb u)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (Mc d)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � +1 2Im Tr � Aa u [Ab d, Ac d] � g � (Ma u)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (Mb d)2, (Mc d)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' m2 ϕ′ � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='33) Here, above two real loop functions f and g are symmetric under exchanges of (Ma u)2 ↔ (Mb u)2 and (Mb d)2 ↔ (Mc d)2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Then, the above equation vanishes, see Ap- pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The symmetry comes from the mass-insertion approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Even if we include the higher-order contributions of xq in the mass-insertion approximation, they still vanish since the loop function is real and symmetric for the exchange of the heavy fermion masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Now we found that the charged NG boson does not give a contribution to the ¯θ parameter at two-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' We also numerically checked this fact by using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='16)– (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The W ′ ± contributions to the ¯θ parameter in the Feynman-’t Hooft gauge at two-loop level vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The Yukawa coupling dependence comes from only the mixing matrices, and then the leading contributions, which are proportional to the fourth power of xu/d at most, vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' The higher-order contributions, coming from mass-insertion approximation, also vanish due to the symmetry of heavy fermion masses in loop functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' A similar discussion is applicable for the other contribution, such as Z′, h′, and ϕ′ 0 at two-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Then, we confirmed the two-loop contribution to the ¯θ parameter vanishes as far as Mq >∼⟨H′⟩ ≫ ⟨H⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' 5 Non-vanishing contribution to QCD θ parameter in three-loop order In the previous section, we confirmed that the QCD ¯θ term is not generated in the two- loop level contribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=', up to the fourth order of the Yukawa interaction xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' We also found that it is valid even if one considers the higher-order contributions of xq by using the mass-insertion approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' In order to give non-vanishing contributions to the ¯θ parameter, the commutation relation [Aa q, Ab q] must be nonzero, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' It implies that non-vanishing contribution should be proportional to Im Tr(Aa q′ [Ab q, Ac q]) for q, q′ = u and/or d rather than Im Tr([Ab q, Ac q]), and the loop function has to be asymmetric under exchange between (Mb q)2 and (Mc q)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' the contributions of the following form might ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='– 19 – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ADwHichVJNbxMxEJ10+Sjloy1cKnGJiI4BaeqKOqpKpce05a0FW2odjduYnX6dhHSVP8C5Ug8IJA4IH4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='GF/4Ah/4ExLFIXDjw7GwREKXY2vV4Zt6bZ48DFQltGDstTHiXLl+5Onlt6vqNm7emZ2Zvb+qk4a8HiZRkm4HvuaRkLx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='uhIn4tkq5HwcR3woOntj4VpenWiTyqekr3oj9lhT7IvQNXM+6z7NdlYqYD/ZmSqzC3CiOGtXcKFE+aslsoUa71KSEQup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='QTJwkGdgR+aQxd6hKjBR8DcrgS2EJF+c0oClgDawX+Pfgb2JtUxGYx0DFmDajAx4ODh+4A/xb2O3kXom9raodfwgdEb4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='U3EUqsy/sAztjn9lH9pX9HMuVOQ6rto81GK52pt+Obfx47+oGKtV/Rt1oWZD+zib1SqgXTmPU4xHePTs42ltbL2X3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='2jn2D/rfslH3CWT3e/h+ja+/ukDPuRKDG7D3oMFZBqv1GQv0UPMELaPihV3Ty14FDjsOofX0vnHW5j18w7KWH3XA9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ixyURyeAf5g7cW2j84TlHZ1RCZJCz+IjZHvYQEchRLrePvqTI9qFB50zj1dlcm8lxFvumq/+4Fjc75SfVRZWFsoLa/k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='r3uS7tI9eoBai7RMq1SjOqpIOqbX9MZb8dpe4h0OUycKOeYO/TW8o1/YFs+J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='v0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ADwHichVJNbxMxEJ10+Sjloy1cKnGJiI4BaeqK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='OqpKpce05a0FW2odjduYnX6dhHSVP8C5Ug8IJA4IH4GF/4Ah/4ExLFIXDjw7GwREKXY2vV4Zt6bZ48DFQltGDstTHiXLl+5Onlt6vqNm7emZ2Zvb+qk4a8HiZRkm4HvuaRkLxuhIn4tkq5HwcR3woOntj4VpenWiTyqekr3oj9lhT7IvQNX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='M+6z7NdlYqYD/ZmSqzC3CiOGtXcKFE+aslsoUa71KSEQupQTJwkGdgR+aQxd6hKjBR8DcrgS2EJF+c0oClgDawX+Pfgb2JtUxGYx0DFmDajAx4ODh+4A/xb2O3kXom9raodfwgdEb4U3EUqsy/sAztjn9lH9pX9HMuVOQ6rto81GK52pt+Obfx4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='7+oGKtV/Rt1oWZD+zib1SqgXTmPU4xHePTs42ltbL2X32jn2D/rfslH3CWT3e/h+ja+/ukDPuRKDG7D3oMFZBqv1GQv0UPMELaPihV3Ty14FDjsOofX0vnHW5j18w7KWH3XA9ixyURyeAf5g7cW2j84TlHZ1RCZJCz+IjZHvYQEchRLrePvq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='TI9qFB50zj1dlcm8lxFvumq/+4Fjc75SfVRZWFsoLa/kr3uS7tI9eoBai7RMq1SjOqpIOqbX9MZb8dpe4h0OUycKOeYO/TW8o1/YFs+J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='v0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ADwHichVJNbxMxEJ10+Sjloy1cKnGJiI4BaeqK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='OqpKpce05a0FW2odjduYnX6dhHSVP8C5Ug8IJA4IH4GF/4Ah/4ExLFIXDjw7GwREKXY2vV4Zt6bZ48DFQltGDstTHiXLl+5Onlt6vqNm7emZ2Zvb+qk4a8HiZRkm4HvuaRkLxuhIn4tkq5HwcR3woOntj4VpenWiTyqekr3oj9lhT7IvQNX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='M+6z7NdlYqYD/ZmSqzC3CiOGtXcKFE+aslsoUa71KSEQupQTJwkGdgR+aQxd6hKjBR8DcrgS2EJF+c0oClgDawX+Pfgb2JtUxGYx0DFmDajAx4ODh+4A/xb2O3kXom9raodfwgdEb4U3EUqsy/sAztjn9lH9pX9HMuVOQ6rto81GK52pt+Obfx4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='7+oGKtV/Rt1oWZD+zib1SqgXTmPU4xHePTs42ltbL2X32jn2D/rfslH3CWT3e/h+ja+/ukDPuRKDG7D3oMFZBqv1GQv0UPMELaPihV3Ty14FDjsOofX0vnHW5j18w7KWH3XA9ixyURyeAf5g7cW2j84TlHZ1RCZJCz+IjZHvYQEchRLrePvq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='TI9qFB50zj1dlcm8lxFvumq/+4Fjc75SfVRZWFsoLa/kr3uS7tI9eoBai7RMq1SjOqpIOqbX9MZb8dpe4h0OUycKOeYO/TW8o1/YFs+J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='v0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='ADwHichVJNbxMxEJ10+Sjloy1cKnGJiI4BaeqK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='OqpKpce05a0FW2odjduYnX6dhHSVP8C5Ug8IJA4IH4GF/4Ah/4ExLFIXDjw7GwREKXY2vV4Zt6bZ48DFQltGDstTHiXLl+5Onlt6vqNm7emZ2Zvb+qk4a8HiZRkm4HvuaRkLxuhIn4tkq5HwcR3woOntj4VpenWiTyqekr3oj9lhT7IvQNX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='M+6z7NdlYqYD/ZmSqzC3CiOGtXcKFE+aslsoUa71KSEQupQTJwkGdgR+aQxd6hKjBR8DcrgS2EJF+c0oClgDawX+Pfgb2JtUxGYx0DFmDajAx4ODh+4A/xb2O3kXom9raodfwgdEb4U3EUqsy/sAztjn9lH9pX9HMuVOQ6rto81GK52pt+Obfx4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='7+oGKtV/Rt1oWZD+zib1SqgXTmPU4xHePTs42ltbL2X32jn2D/rfslH3CWT3e/h+ja+/ukDPuRKDG7D3oMFZBqv1GQv0UPMELaPihV3Ty14FDjsOofX0vnHW5j18w7KWH3XA9ixyURyeAf5g7cW2j84TlHZ1RCZJCz+IjZHvYQEchRLrePvq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='TI9qFB50zj1dlcm8lxFvumq/+4Fjc75SfVRZWFsoLa/kr3uS7tI9eoBai7RMq1SjOqpIOqbX9MZb8dpe4h0OUycKOeYO/TW8o1/YFs+J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content='v0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FQT4oBgHgl3EQfxTb9/content/2301.13405v1.pdf'} +page_content=' 0 at the contact point +(different from the usual thin film equation with h = 0 at the contact point). We show +that this fourth order quasilinear (non-degenerate) parabolic equation, together with +the so-called partial wetting condition at the contact point, is well-posed. Also the +contact point in our thin film equation can actually move, contrary to the classical thin +film equation for a droplet arising from no-slip condition. Furthermore, we show the +global stability of steady state solutions in a periodic setting. +1 +Introduction +In this paper, we investigate a particular thin film model, where a rigid solid enters a +liquid film (cf. Figure 1), leading to movement of the contact point and the formation of a +meniscus, as the initial state is out of equilibrium. This free boundary problem gives rise to +the following fourth order non-linear parabolic equation, in one dimension, +∂th + ∂x(h3∂3 +xh) = 0 +in x > Λ, t > 0, +(1.1a) +along with the boundary conditions, +h = g, +∂xh = ∂xg − k, +h∂3 +xh = −6Λ ∂tg +g2 +at x = Λ, t > 0. +(1.1b) +∗ghosh@iam.uni-bonn.de +†velazquez@iam.uni-bonn.de +1 +arXiv:2301.04181v1 [math.AP] 10 Jan 2023 + +Figure 1: General meniscus formation +Here h = h(x, t) is the height of the fluid film, Λ = Λ(t) is the contact point (which +is a free boundary in one dimension), k is the fixed contact angle (rescaled), i.e. the angle +between the fluid-solid and the fluid-air interfaces, and g = g(x, t) is the profile of the rigid +solid. +Deduction of the above system is elaborated in Section 2. +Here the situation in +one spatial dimension is studied only, although one may consider general dimension as well +(physically relevant cases correspond to d = 1 or 2). The study of fluid problems involving +free boundary evolution mostly concerns with an interface between two phases (cf. [19]), +whereas the interest of the current paper is the situation where three phases of matter meet +(typically, fluid-solid-air), generating a contact line/point. The main differences between +the model considered here with other classical triple junction model is pointed out later. +Equation (1.1a): +Let us first discuss the equation (1.1a) which can be viewed as a par- +ticular case of the general class of thin film equations +∂th + ∂x(f(h)∂3 +xh) = 0 +in {h > 0}. +(1.2) +Equations of the form (1.2) are typically obtained by lubrication approximation of the motion +of a two-dimensional viscous fluid droplet, spreading over a solid substrate by the effect of +surface tension (cf. [2]). Classical macroscopic fluid mechanics to model such phenomena +imposes the no-slip boundary condition at the fluid-solid interface, that is the velocity of +the fluid must be equal to that of the solid substrate in contact with. This corresponds to +the case f(h) = h3. However, this condition can not hold at the moving contact point/triple +junction where the solid, liquid and gas, three phases meet. It is known to give rise to a +non-integrable singularity at the contact point (cf. [11]) which further implies that boundary +of the film (i.e. the triple junction) can not move in time (no slip paradox). On other hand, +Navier slip condition (corresponds to the case f(h) = h2), proposed by Navier [14], which +allows the fluid to slip over the solid surface (characterized by a slip parameter), resolves +such paradoxical phenomena (cf. [15] for a discussion in this direction). +Note that equation (1.2) can also be viewed as a higher order version of the Porous +medium equation +∂th − ∂x(hn∂xh) = 0, +n > 0. +However, the standard techniques used for such second order degenerate parabolic equations +(such as Maximum principle) are not available for the higher order counterpart which makes +the analysis more difficult for the later (cf. [7]). +2 + +<-Solud +GasEquation (1.1b): +Next let us discuss the boundary conditions in (1.1b). The first condi- +tion gives the position of the free boundary, whereas the second one determines the slope at +the free boundary (contact angle condition). The case where the (equilibrium) contact an- +gle is given by a non-zero constant, determined by Young’s law (balance between interfacial +energies) (cf. [3]), is known as partial wetting, whereas the zero contact angle is referred as +complete wetting. This work takes into account both the possibilities. There are also notions +of dynamic/apparent contact angle (cf. [10, 16]). The third boundary condition comes from +a matching condition with the internal region (i.e. the fluid region bounded between solid +parts), while in the standard setting of a droplet, a third condition is deduced from match- +ing the velocity of the contact point. This would lead to an over-determined problem for +a fourth order operator with fixed boundary, but not for the free boundary problem where +the boundary is an unknown a priori. +State of the art. +All the literature available in the context of thin film equation for +contact angle problem consider either compactly supported initial data (droplet case) or +an unbounded support with single free boundary. Thus the standard condition h = 0 of +the vanishing fluid height at the free boundary makes the equation ∂th + ∂x(hn∂3 +xh) = 0 +singular/degenerate. +Following the pioneering work of Bernis and Friedman [1], several +results has been established for (1.2), considering different forms of f, for example in [7], +[6]-[8], [12]-[13], [17]. The above references are not claimed to be exhaustive. We also point +out the work [9] where stability of a global solution for the general contact line problem for +Stokes equation has been discussed. +The current work is concerned with a different thin film model, deduced from the no-slip +boundary condition assumption at the fluid-solid interface. The film height being h > 0 at +the contact point makes our model non-degenerate, unlike the standard thin film equation +as discussed above. The result shows well-posedness as well as the possibility of moving +contact point. Moreover, no paradox arises in spite that the contact point moves. The +reason of such behavior in this case is that the contact angle is very close to π. As pointed +out by Solonnikov in [20], the paradox gets resolved in the case of contact angle π. The main +contribution of the paper is to bring forward this particular interesting yet simple situation +involving the evolution of triple junction and the discussion of different behavior in contrast +to the known cases. This observation intrigued our curiosity to analyze the model in detail. +Structure of the paper. We first deduce a general thin film model describing the above +particular situation of meniscus formation under the classical approach in Section 2. The +arguments used in this derivation are standard. Still we write this derivation in detail, since +we did not find any suitable reference for such model. In Section 3, the local well-posedness +result is established, for the particular case of no-slip boundary condition. +The system +being non-degenerate, standard theory for fourth order parabolic quasilinear equation can +be employed. Nevertheless, obtaining suitable regularity (in time) in order to closing the +final fixed point argument is not straight-forward (cf. +Remark 3.1). +Finally we obtain +the asymptotic behavior of a solution, in periodic setting, in Section 4. +By looking at +perturbations of a steady state of (1.1a)-(1.1b), existence of a unique global solution is +obtained with the help of a priori energy estimates. As long as the initial data remains +3 + +close (in certain norm) to the stationary solution, the local solution of the thin film model +converges to the steady state for all time. +1.1 +Main results +Our first result concerns existence of a local in time solution to the following thin film +equation with moving contact point and prescribed contact angle (including complete wet- +ting case) +∂th + ∂x(h3∂3 +xh) = 0 +in x > Λ, t > 0, +h = g, +∂xh = ∂xg − k, +h ∂3 +xh = −6Λ∂tg +g2 +at x = Λ, t > 0, +h → 1 +as x → ∞, t > 0, +h = h0 +at t = 0, x > Λ0. +(1.3) +Here Λ0 = Λ(0) denotes the initial position of the contact point. +Theorem 1.1 For any initial data (h0, Λ0), satisfying compatibility conditions, there exists +a unique strong solution (h, Λ), for short time, of the free boundary problem (1.3). +Rigorous statement and proof of the above theorem is given in Section 3. +Next we +consider the system (1.3) over an interval (0, 2L) (in order to have a bounded domain), +symmetric with respect to x = L and with periodic boundary conditions. Precisely, +∂th + ∂x(h3∂3 +xh) = 0 +in x ∈ (Λ, L), t > 0, +h = g, +∂xh = ∂xg − k, +h ∂3 +xh = 0 +at x = Λ, t > 0, +∂xh = 0 +at x = L, t > 0, +h = h0 +at t = 0, x ∈ (Λ0, L), +h(x, t) = h(x + 2L, t), +g(x, t) = g(x + 2L, t), +x, t > 0. +(1.4) +The free interface h(x, t) is defined over the domain (Λ, 2L−Λ) in this case. Also we assume +here that the solid is not moving, i.e. +∂tg = 0. +Such a solution preserves mass for all +time t > 0 (cf. Section 4.1). Then, for a given volume V0, a steady state (h, Λ) of (1.4) is +characterized by +∂x(h +3∂3 +xh) = 0 in x ∈ (Λ, L), +subject to the boundary conditions +h = g, ∂xh = ∂xg − k, ∂3 +xh = 0 +at x = Λ +and +∂xh = 0 +at x = L. +We refer to (4.8) for an explicit description of the steady state. Stability of this steady state +for small perturbation is obtained as below, we refer to Section 4.3 for rigorous statement +and proof. +Theorem 1.2 For initial data close enough to the steady state of (1.4), there is a unique +global solution (h, Λ) of the problem (1.4) which converges to the steady state as t → ∞. +4 + +2 +Lubrication approximation +In this section, we elaborate the thin film approximation of the original three phase free +boundary problem in consideration. +Original model: Let us consider a horizontal, two-dimensional thin film of viscous, in- +compressible Newtonian fluid, of thickness H at the initial time and a rigid solid (locally +convex) touching the film (cf. Figure 2). As the solid enters the liquid, the contact point +moves and makes an angle (π − θ), θ > 0, between the liquid-solid and the liquid-gas in- +terface. The liquid-gas interface is described by y = h(x, t) and the liquid-solid boundary +is given by y = g(x, t). One may think of particular cases, for example g = H +� +1 − +� +t +t0 +�n� +with n > 0 where t0 is the time scale that characterises the motion of the solid mov- +ing down. +Here n = 1 corresponds to the solid moving with constant velocity, while +n = 2 corresponds to the case of constant acceleration. A particular case of wedge-shaped +solid (in the form g(x, t) = ˜h(t) + c|x|) has been mentioned in [18, Fig 6] in a context +of correct modelling approach describing the creation of contact angle. The free interface +makes an angle ˜θ with the horizontal plane (cf. Figure 2). Therefore, tan ˜θ = −∂xh and +θ = − arctan(∂xh) + arctan(∂xg). To obtain the lubrication approximation, we need to +assume that the two angles θ and ˜θ are very small and of the same order, which means that +the contact angle (π − θ) is close to π in this setting. Note that the thin film approximation +would not be valid if ˜θ is large. The contact points are given by si = (Λi(t), y). Finally, we +assume the far field condition h → H as |x| → ∞. +Let vL = (u, v) be the velocity of the fluid, with constant density ϱL, viscosity µL and +pressure pL, governed by the Navier-Stokes equations, while the solid motion is governed by +the translational velocity only vS(t), vertically downward. Let us consider the bottom part +of the fluid domain as a reference configuration, i.e. at y = 0. +Figure 2: Thin Film approximation with contact points +As we assume a symmetric configuration with respect to y-axis (for simplicity), it is +enough to analyse the situation for x > 0 only. Therefore, in the following, we call the +5 + +Solid +k=9(x,t) +Gas +9 +-h(t) +E +个 +1 +X-0 +(A)Vcontact point Λ instead of Λ1. +Further assumptions: +Let the solid g be smooth in a neighborhood of the initial contact +point Λ0. Moreover, g > 0 for all x, t ≥ 0, i.e. the solid never touches the bottom of the +fluid domain which may lead to further singularities. +In the framework of classical fluid mechanics, conditions at contact lines are considered +only at equilibrium, i.e. in the situations where the contact line is not moving with respect +to the bulk phases. Also we consider here the Navier slip condition at the fluid-solid inter- +face, for a general framework, which covers both the no-slip and the full-slip condition. We +formulate these conditions below (cf. [4, Section 2.4]). +in the liquid: +divx vL = 0, +ϱL � +∂tvL + vL · ∇xvL� += µL∆vL − ∇xpL; +(2.1) +at the liquid-gas interface: y = h(x, t), x > Λ(t): +Vn = vL · n1, +� +pG − pL� ++ 2µL � +DvL · n1 +� +· n1 + psκ = 0, +� +DvL · n1 +� +τ = 0; +(2.2) +at the liquid-solid interface {y = 0} ∪ {y = g(x, t), 0 < x < Λ(t)}: +� +vL − vS� +· n2 = 0, +2µL � +DvL · n2 +� +τ = β +� +vL +τ − vS +τ +� +; +(2.3) +at the triple junction x = Λ(t): +h = g, +arctan(∂xh) = arctan(∂xg) − θ. +(2.4) +Here ps is the (constant) surface tension, κ is the curvature of the free interface, pG is the +constant pressure of the gas and ni, i = 1, 2 are the unit normal vectors, inward with respect +to the fluid domain. We do not distinguish here between the slip-coefficients appearing at +the lower and upper fluid-solid interface and denote both of them by β only. In principle, +they might be different. We further assume for simplicity that the lower boundary y = 0 is +fixed, i.e. vS = 0 at y = 0. +Also, we must have the matching condition for the initial data, +h(x, 0) = H +at t = 0. +(2.5) +Further, due to the symmetry assumption of the considered configuration, one has, +u = 0 +on x = 0. +(2.6) +Now let us introduce the length scale as L = H +ε where ε > 0 is small. Then the usual +scaling of the variables are given by, +(x, y) = +� x +L, y +H +� +Λ = 1 +LΛ, +h = h +H , g = g +H +t = σ +Lµt, +(u, v) = +�µ +σu, µ +εσv +� +, +p = ε2L +σ p, θ = kε, +6 + +with σ = −ε3ps is the scaled surface tension. +Rescaled system: Under these scaling, we obtain the following approximated system of +equations from (2.1)-(2.4), +∂xu + ∂yv = 0, +∂2 +yu = ∂xp, +∂yp = 0, +(2.7a) +in (x, y) ∈ (0, Λ) × (0, g) ∪ (Λ, ∞) × (0, h), +∂th + u ∂xh − v = 0, +p = −∂2 +xh, +∂yu = 0, +on x > Λ, y = h, +(2.7b) +v = 0, +∂yu = β u, +on y = 0, +(2.7c) +∂xg u = v − ∂tg, +∂yu + β u = 0, +on x < Λ, y = g, +(2.7d) +h = g, +∂xh = ∂xg − k, +at x = Λ, +(2.7e) +h → 1 +as x → ∞, +(2.7f) +where the rescaled slip coefficient β = εβL +µ += O(1). The system is complemented by equa- +tions (2.5), (2.6). +Indeed, the incompressibility condition (2.1)1 remains same in the new variables, leading +to (2.7a)1. The Navier-Stokes equation (2.1)2 in the first component becomes, +ϱL σ +µL +L +µL +� +�� +� +Re +(∂tu + u∂xu + v∂yu) = +� +∂2 +xu + 1 +ε2 ∂2 +yu +� +− 1 +ε2 ∂xp. +Thus in the limit ε → 0, both the time derivative and the non-linear term vanish compared +to the viscous term when Reynolds number satisfies ε2Re ≪ 1 and one obtains (2.7a)2. +Similarly the second component of Navier-Stokes equation reduces to (2.7a)3. +Next we discuss the boundary conditions. At the liquid-gas interface y = h(x, t), x > +Λ(t), the unit normal and tangent vectors and the curvature have the following expressions, +n1 = +� +ε∂xh, −1 +� +� +(1 + ε2|∂xh|2) +, +τ = +� +1, ε∂xh +� +� +(1 + ε2|∂xh|2) +and +κ = +ε∂2 +xh +L +� +1 + ε2|∂xh|2�3/2 . +(2.8) +The kinematic boundary condition (2.2)1 reduces to the non-dimensional form (2.7b)1. Also +the rate of strain tensor becomes, in the new variables, +DvL = 1 +2 +� +� +2∂xu +∂yu + ∂xv +∂yu + ∂xv +2∂yv +� +� = 1 +2 +σ +Lµ +� +� +2∂xu +1 +ε∂yu + ε∂xv +1 +ε∂yu + ε∂xv +2∂yv +� +� . +Therefore, the normal and the tangential stress on the surface y = h(x, t) are given by, +� +DvL · n1 +� +· n1 = σ +Lµ +1 +� +1 + ε2|∂xh|2� � +−∂yu ∂xh + ∂yv + O(ε2) +� +, +(2.9) +7 + +and +� +DvL · n1 +� +· τ = σ +Lµ +1 +2 +1 +� +1 + ε2|∂xh|2� +� +−1 +ε∂yu + O(ε) +� +. +(2.10) +Then, equation (2.2)2 becomes, with the help of (2.8) and (2.9), +pG − +σ +ε2Lp + 2σ +L +1 +� +1 + ε2|∂xh|2� � +−∂yu ∂xh + ∂yv + O(ε2) +� +− σ +ε3 +ε ∂2 +xh +L +� +1 + ε2|∂xh|2�3/2 = 0. +Now with the assumption that σ = O(1), one obtains in the limit (2.7b)2. For equation +(2.2)3, one gets (2.7b)3 using the expression (2.10). +At the liquid-solid interface, with n2 = (0, 1), τ = (1, 0) at the bottom part y = 0, the +conditions (2.3) read as, +v = 0, +µL(∂yu + ∂xv) = βu +at y = 0, +which then convert into (2.7c). Similarly, at the upper fluid-solid interface, the normal and +tangent vectors being, +n2 = +(∂xg, −1) +� +1 + |∂xg|2 , +τ = +(1, ∂xg) +� +1 + |∂xg|2 , +and vS = (0, ∂tg), the boundary conditions (2.3) transform into the non-dimensional form +(2.7d). +The conditions at the contact point (2.4) transform into (2.7e). +Thin film equations: The thin film model can now be derived from (2.7a)-(2.7f), together +with the boundary conditions and initial data as, +∂th + ∂x +��h +3 + 1 +β +� +h2∂3 +xh +� += 0, +in x > Λ, +(2.11a) +h = g, +∂xh = ∂xg − k, +at x = Λ, +(2.11b) +h → 1, +as x → ∞, +(2.11c) +h = 1, +at t = 0. +(2.11d) +The deduction of the above system is explained in the following (omitting the bar here +onwards). Integrating (2.7a)2 twice gives the profile of the horizontal velocity, +u(x, y) = 1 +2∂xp y2 + A(x, t)y + B(x, t), +x > 0, +(2.12) +where the constants A, B are to be determined. The condition at the lower boundary (2.7c)2 +implies, +A = βB, +x > 0. +(2.13) +8 + +Also the condition at the free boundary (2.7b)3 yields, +A = −∂xp h, +therefore, +B = − 1 +β ∂xp h, +x > Λ. +(2.14) +Further, as the pressure is independent of y due to (2.7a)3, the horizontal velocity (2.12) +becomes, together with (2.14) and (2.7b)2, +u = 1 +2∂3 +xh +� +(2h − y)y + 2 +β h +� +, +x > Λ. +(2.15) +Next (2.7a)1 and (2.7c)1 give, +v|y=h = − +h +� +0 +∂xu dy, +x > Λ. +Thus from (2.7b)1, we obtain, +∂th + ∂x +� +h +� +0 +u dy +� += 0, +x > Λ. +Substituting the velocity profile (2.15) into the above relation finally gives the thin film +equation (2.11a). The contact line condition (2.11b) is nothing but (2.7e) and the initial +data (2.11d) follows from (2.5). +Inner region: In order to determine the dynamics of the contact point fully, we need to +prescribe sufficient conditions in the inner region x < Λ as well. To do so, the condition at +the fluid-solid interface (2.7d)2, together with the condition (2.13), for x < Λ, y = g, which +is +[∂xp g + A] + β +�1 +2∂xp g2 + A g + A +β +� += 0, +gives a relation between ∂xp and A for x ∈ (0, Λ), that is +∂xp = −2A +g . +(2.16) +Also, the incompressibility condition (2.7a)1 gives an expression for the normal velocity for +x ∈ (0, Λ), +v|y=g = − +˜h +� +0 +∂xu dy = −g +�1 +6∂2 +xp g2 + 1 +2∂xA g + 1 +β ∂xA +� +. +Thus the condition (2.7d)1 implies, for x ∈ (0, Λ), +∂xg +�1 +2∂xp g2 + A +� +g + 1 +β +�� ++ ∂tg + g +�1 +6∂2 +xp g2 + 1 +2∂xA g + 1 +β ∂xA +� += 0. +9 + +The above equation together with the relation (2.16) yields an ODE for A with rational +coefficients, +∂xA + r1(x, t) A + r2(x, t) = 0, +x < Λ, +(2.17) +where +r1(x, t) = +∂xg +� +1 +β + 1 +3g +� +g +� +1 +β + 1 +6g +� , +r2(x, t) = +∂tg +g +� +1 +β + 1 +6g +�. +Also, due to the symmetry assumption (2.6), we have the boundary condition +A(x, t) = 0 +at x = 0. +Therefore, the ODE (2.17) determines A and in turn ∂xp in the inner region x ∈ (0, Λ). +Here we assume that the horizontal velocity u is continuous, thus the relation (2.12) holds +at x = 0 as well. Moreover, the continuity of u at Λ further imposes the condition +A = h ∂3 +xh +at x = Λ. +(2.18) +This above relation (2.18), together with the thin film equations (2.11a)-(2.11d) determines +fully the dynamics and the position of the contact point. This thin film system is described +for a particular case of wedge-shaped solid, without its detailed derivation, in [5, Section +3.3]. +3 +Local well-posedness for no-slip condition +3.1 +Notations and functional settings +Let C0[0, ∞) denote the Banach space of continuous, bounded functions on [0, ∞) +and tending to 0 as x → ∞, together with the supremum norm on [0, ∞). +The space +Ck[0, ∞), k ∈ N of functions with k times continuous and bounded derivatives is endowed +with the standard norm +∥v∥Ck[0,∞) := +k +� +m=0 +∥∂mv∥C[0,∞). +We write C[0, ∞) ≡ C0[0, ∞) for the space of all continuous, bounded functions on [0, ∞). +The Banach space of ρ-Hölder continuous functions Cρ[0, ∞), Ck+ρ[0, ∞) for k ∈ N, ρ ∈ +(0, 1), is defined by +Cρ[0, ∞) := {v ∈ C[0, ∞) : [v]ρ := +sup +x,y∈[0,∞) +x̸=y +|v(x) − v(y)| +|x − y|ρ +< ∞}, +together with the norm +∥v∥Cρ[0,∞) := ∥v∥C[0,∞) + [v]ρ, +10 + +and +Ck+ρ[0, ∞) := {v ∈ Ck[0, ∞) : [∂kv]ρ < ∞}, +with the norm +∥v∥Ck+ρ[0,∞) := ∥v∥Ck[0,∞) + [∂kv]ρ. +For ρ = 1, the above notions coincide with the class of Lipschitz functions which we denote +by C1−[0, ∞). Sometimes we omit the underlying space [0, ∞) below which should not cause +any confusion. +3.2 +Reduction to a fixed domain +We assume the no-slip boundary condition at the fluid-solid interface, i.e. +1 +β = 0. We +will show the local and global well-posedness for this reduced problem. As mentioned in +the introduction, such result is in contrast with the classical thin film equation (for droplet) +with no-slip condition. +Using the time scaling t → 3t, t0 → 3t0, the thin film model (2.11a)-(2.11d) reduces to, +∂th + ∂x(h3∂3 +xh) = 0 +in x > Λ, t > 0, +h = g, +∂xh = ∂xg − k, +h ∂3 +xh = A +at x = Λ, t > 0, +h → 1 +as x → ∞, t > 0, +h = 1 +at t = 0, x > Λ0, +where A solves the ODE in (0, Λ), +∂xA + 2∂xg +g A + 6∂tg +g2 = 0, +A(0, t) = 0. +(3.1) +Solving (3.1) gives, +A(x, t) = −6x∂tg +g2 . +Thus, we obtain the system (1.3). Recall that we have g > 0 for all x, t ≥ 0 by assumption. +Without loss of generality, we assume Λ(0) = 0. Hence, for some δ > 0, there exists +Tδ > 0 such that +|Λ(t)| ≤ δ2 +and +| ˙Λ(t)| ≤ C +for t ∈ [0, Tδ]. +(3.2) +For such a δ fixed, let us now consider a cut-off function ξδ ∈ C∞ +c (R) i.e. +ξδ(s) = +� +1, +if 0 ≤ s ≤ δ, +0, +if s ≥ 2δ, +such that +|ξ′ +δ| ≤ C +δ , +and the following bijection +QΛ(x, t) = (x − Λ(t))ξδ(x − Λ(t)) + x(1 − ξδ(x − Λ(t))). +11 + +Denoting by +x = QΛ(x, t) +and +H(x, t) = h(x, t), +one can compute the derivatives in the new coordinate as, +∂th = ∂tH + ˙Λ(Λξ′ +δ − ξδ)∂xH, +∂xh = (1 − Λξ′ +δ)∂xH. +Therefore, the free boundary problem (1.3) reduces to a fixed domain as +∂tH − ˙Λ(t)ξδ ∂xH + (1 − Λξ′ +δ)4 ∂x(H3∂3 +xH) + F1 = 0 +in x > 0, +H = ψ1, +∂xH = ψ2, +∂3 +xH = ψ3 +at x = 0, +H → 1 +as x → ∞, +H(x, 0) = H0 ≡ h0(x + Λ(0)) → 1 +as x → ∞, +(3.3) +where +F1 ≡ F1( ˙Λ, ξ′ +δ, H) +with supp(F1) ⊂ [δ, 2δ], +and +ψ1(Λ, t) = g(Λ, t), +ψ2(Λ, t) = ∂xg(Λ, t) − k, +ψ3(Λ, t) = −6Λ∂tg +g3 (Λ, t). +Next we use the following transformation in order to lift the boundary conditions and the +far field condition, +H(x, t) = [ψ2(Λ, t) x + ψ3(Λ, t) x3]ξδ(x) + U(x, t) + (1 − ξδ(x)) +=: a(x, t, Λ)ξδ(x) + U(x, t) + (1 − ξδ(x)), +(3.4) +where |U(x, t)| → 0 as x → 0. Observe that a(x, t, Λ) > 0 for x > 0, t ∈ (0, Tδ) for suitably +chosen δ. Then the system (3.3) becomes, omitting the bar over x, +∂tU − ˙Λ(ψ2 ξδ + ∂xU)ξδ + (1 − Λξ′ +δ)4 ∂x((aξδ + 1 − ξδ + U)3∂3 +xU) = F2 + F3 +in x > 0, +∂xU = 0, +∂3 +xU = 0 +at x = 0, +U → 0 +as x → ∞, +U(x, 0) = U0 ≡ h0(x) − a(x, 0) +for x > 0, +(3.5) +together with +U = ψ1(Λ, t) +at x = 0, t > 0, +(3.6) +where +F2 ≡ F2( ˙Λ, ξ′ +δ, U) +with supp(F2) ⊂ [δ, 2δ], +(3.7) +and +F3 = 6(1 − Λξ′ +δ)4 ∂x((aξδ + (1 − ξδ) + U)3 ψ3 x3 ξδ) − ∂ta ξδ + ˙Λ 3x2ψ3 ξ2 +δ. +(3.8) +The above transformation is useful for splitting the full system as a Cauchy problem (3.5) +and a fixed point map (3.6). The general theory for quasilinear parabolic problem can be +applied to treat the fourth order system (3.5) which in turn give the existence of a solution +for the full problem (3.3). +12 + +Remark 3.1 As can be seen from (3.6), Λ and U must have the same regularity in time; +On the other hand, it is not obvious/immediate to obtain the same time regularity from the +parabolic system (3.5) if one uses the standard Sobolev spaces. Hence we chose to use the +Hölder spaces for the solution since it is important not to lose (trace) regularity in order to +perform the fixed point argument in (3.6). +3.3 +Proof of the main result +The idea is to write down the solution of (3.5) in terms of a Green function for the linear +problem. +Lemma 3.2 Let Λ ∈ C1+ α +4 [0, T] and U0 ∈ C4+α[0, ∞) ∩ C0[0, ∞) where α ∈ (0, 1). There +exists a fundamental solution G(x, t, y, τ) of the linear boundary value problem +∂tU ∗ + (1 − Λξ′ +δ)4 ∂x((aξδ + (1 − ξδ))3∂3 +xU ∗) + ˙Λ ∂xU ∗ ξδ = 0 +in x > 0, +∂xU ∗ = 0, +∂3 +xU ∗ = 0 +at x = 0, +U ∗ → 0 +as x → ∞, +U ∗(x, 0) = U0 +for x > 0, +(3.9) +satisfying the estimate +|∂mG(x, t, y, τ)| ≤ cm(t − τ)− m+1 +4 +exp +� +−c +� |x − y| +(t − τ)1/4 +�4/3� +, +t ∈ [0, T]. +(3.10) +Proof. +The existence of such G follows from [4, Chapter IV.2, Theorem 3.4]. +As per +the notations in [4], for our system (4.21), N = 1 = n, b = 2, r1 = 1, r2 = 3 and the +boundary conditions read as B ≡ (∂x, ∂3 +x)u|x=0 = 0. The Cauchy problem (3.9) can be +reduced to a problem with zero initial condition by considering a function (U ∗ − U0) since +U0 ∈ C4+α[0, ∞). In (3.9)1, the leading order coefficient a4(x, t) = (1−Λξ′ +δ)4 (aξδ+(1−ξδ))3 +is Hölder continuous in both t ∈ (0, T) and x ∈ (0, ∞), due to Theorem 3.5. Furthermore, +the other coefficients a3(x, t) = (1 − Λξ′ +δ)4 ∂x(aξδ + (1 − ξδ))3 and a1(x, t) = ˙Λξδ are Hölder +continuous in x as well. Therefore, all the conditions of [4, Chapter IV.2, Theorem 3.4] are +satisfied and hence the existence and estimates of a fundamental solution. +Theorem 3.3 Let U0 ∈ C4+α[0, ∞) ∩ C0[0, ∞) where α ∈ (0, 1), satisfying compatible +conditions +∂xU0 = ∂3 +xU0 = 0 +at +x = 0. +Then for any δ > 0 and Λ ∈ C1+ α +4 [0, Tδ] where Tδ is as in (3.2), there exists T0 ∈ (0, Tδ) +such that (3.5) has a unique solution U(x, t) belonging to C1+ α +4 ((0, T0), C4+α[0, ∞)). +The time interval T0 depends on the upper bounds of the Hölder constants of the coeffi- +cients of (3.5). +13 + +Proof. The solution of (3.5) can be expressed by the following integro-differential equation +U(x, t) = +t +� +0 +∞ +� +0 +G(x, t, y, τ) +� +˙Λψ2 ξ2 +δ + F2 + F3 + F4 +� +(y, τ) dy dτ + +∞ +� +0 +G(x, t, y, 0)U0(y) dy, +(3.11) +where G is the fundamental solution of the linear problem (3.9), given in Lemma 3.2, F2, F3 +are defined in (3.7), (3.8) and +F4 := −(1 − Λξ′ +δ)4 ∂x({U 3 + 3U(aξδ + 1 − ξδ)(U + (aξδ + 1 − ξδ))} ∂3 +xU). +(3.12) +Solving the above equation is standard, for example one may refer to [4, Chapter III.4, +Theorem 8.3]. +The next step is to perform a fixed point argument obtaining a (local in time) solution +for the full system (3.5)-(3.6). +Theorem 3.4 Let U0 ∈ C4+α[0, ∞) ∩ C0[0, ∞) where α ∈ (0, 1), satisfying compatible +conditions +U0 = ψ1(0, 0), +∂xU0 = ∂3 +xU0 = 0 +at x = 0. +Then for small δ > 0 and small Tδ where Tδ is as in (3.2), there exists T0 ∈ (0, Tδ) such +that (3.5)-(3.6) has a unique solution (U, Λ) ∈ C1+ α +4 ((0, T0), C4+α[0, ∞)) × C1+ α +4 [0, T0]. +The time interval T0 depends on the upper bounds of the Hölder constants of the coeffi- +cients of (3.5). +Proof. As U is given by equation (3.11), in order to satisfy the boundary condition (3.6), +one must have +ψ1(Λ, t) = +t +� +0 +∞ +� +0 +G(0, t, y, τ) +� +˙Λψ2 ξ2 +δ + F2 + F3 + F4 +� +(y, τ) dy dτ + +∞ +� +0 +G(0, t, y, 0)U0(y) dy +=: I + II + III + IV + V. +(3.13) +The first integral being the most important term to be controlled (all the other terms being +small), we try to rewrite and simplify it. Observe that the mass is conserved for the Green +function satisfying (3.9). Indeed, by denoting g(x, t) := +� ∞ +0 G(x, t, y, τ)dy and integrating +the equations in (3.9) with respect to y, we get that g satisfies the same system. Hence by +uniqueness, +∞ +� +0 +G(x, t, y, τ) dy = 1, +t ∈ [0, T]. +14 + +Therefore, +I := +t +� +0 +∞ +� +0 +G(0, t, y, τ)( ˙Λψ2 ξ2 +δ)(y, τ) dy dτ += +t +� +0 +( ˙Λψ2)(τ) +∞ +� +0 +G(0, t, y, τ) dy dτ − +t +� +0 +∞ +� +0 +G(0, t, y, τ)( ˙Λψ2)(τ)(1 − ξ2 +δ)(y) dy dτ += +t +� +0 +˙Λ(τ) ψ2(Λ(τ), τ) dτ − +t +� +0 +∞ +� +0 +G(0, t, y, τ)( ˙Λψ2)(τ)(1 − ξ2 +δ)(y) dy dτ +=: I1 + I2. +(3.14) +At this point, let us denote Ψ(Λ, t) = +� Λ +0 ψ2(y, t) dy, which gives, +d +dtΨ = +Λ +� +0 +∂tψ2(y, t) dy + ψ2(Λ(t), t) ˙Λ(t). +Plugging this in to (3.14), we get +I1 = +t +� +0 +� +�� d +dτ Ψ(Λ(τ), τ) − +Λ(τ) +� +0 +∂τψ2(y, τ) dy +� +�� dτ += [Ψ(Λ(t), t) − Ψ(0, 0)] − +t +� +0 +Λ(τ) +� +0 +∂τψ2(y, τ) dy dτ. +Using the definition of Ψ and ψ2, one further obtains, after some integration by parts with +respect to y, +I1 = +Λ +� +0 +(∂yg(y, t) − k) dy − +t +� +0 +(∂τg(Λ(τ), τ) − ∂τg(0, τ)) dτ += [(g(Λ, t) − g(0, t)) − kΛ] − +t +� +0 +∂τg(Λ(τ), τ) dτ + [g(0, t) − g(0, 0)] += g(Λ, t) − g(0, 0) − k Λ(t). +(3.15) +Therefore, putting the relations (3.13), (3.14) and (3.15) together, one gets, by the definition +of ψ1, +0 = −g(0, 0) − kΛ(t) + I2 + II + III + IV + V. +15 + +In other words, as k > 0, +Λ(t) = 1 +k [−g(0, 0) + I2 + II + III + IV + V ] . +(3.16) +Next we need to estimate the different terms of (3.16), in particular to show that the above +relation is a contraction map for Λ. +Estimate of I2: +Using the estimate (3.10) for the fundamental solution, we have +|I2| = +������ +t +� +0 +˙Λ(τ) ψ2(Λ(τ), τ) +∞ +� +0 +(ξ2 +δ(y) − 1) G(0, t, y, τ) dy dτ +������ +≤ c +t +� +0 +| ˙Λ(τ) ψ2(Λ(τ), τ)| +∞ +� +δ +(t − τ)−1/4 exp +� +−c +y +(t − τ)1/4 +� +dy dτ +≤ c +t +� +0 +| ˙Λ(τ) ψ2(Λ(τ), τ)| exp +� +−c +δ +(t − τ)1/4 +� +dτ ≤ cδ(∥ψ2∥) +t +� +0 +| ˙Λ(τ)| dτ. +The above constant cδ(∥ψ2∥) < 1 can be made by choosing δ > 0 suitably. +Estimate of II: +Recalling the definition of F2 from (3.7), as it comprises several terms +of the form ˙Λ ∂k +xU ξ′ +δ, k ∈ [0, 3], passing the derivatives onto G by integration by parts in x, +and using the estimate (3.10) for the fundamental solution, one obtains the following +|II| = +������ +t +� +0 +∞ +� +0 +G(0, t, y, τ)F2(y, τ) dy dτ +������ +≤ cδ(∥U∥C((0,T0),C[0,∞))) +t +� +0 +| ˙Λ(τ)|(t − τ)−β/4 dτ, +for some β ∈ (0, 4). As before, choosing δ > 0 suitably, the constant can be made small +cδ(∥U∥C((0,T0),C[0,∞))) < 1. +Estimate of III: +All the terms in (3.8) are small. For example, +������ +t +� +0 +∞ +� +0 +G(0, t, y, τ) [ ˙Λ 3x2ψ3 ξ2 +δ](y, τ) dy dτ +������ +≤ c δ2 +t +� +0 +| ˙Λ(τ)||ψ3(Λ(τ), τ)| +δ +� +0 +|G(0, t, y, τ)| dy dτ ≤ cδ(∥ψ3∥) ∥Λ∥C1+ α +4 [0,Tδ]. +16 + +Similarly, the other two terms can be estimated as, +������ +t +� +0 +∞ +� +0 +G(0, t, y, τ) [6(1 − Λξ′ +δ)4 ∂x((aξδ + (1 − ξδ) + U)3 ψ3 x3 ξδ) − ∂ta ξδ](y, τ) dy dτ +������ +≤ Cδ(∥ψ2∥, ∥ψ3∥, ∥U(t)∥C[0,∞)) (Tδ + ∥Λ∥C1+ α +4 [0,Tδ]). +The constant can be chosen small, for suitable δ > 0. +Estimate of IV : +By the same argument as for estimating I2 and also using the fact that +|∂3 +xU| ≤ δ for |x| small as ∂3 +xU|x=0 = 0, we get +|IV | ≤ cδ(∥U∥C((0,T0),C[0,∞))) (Tδ + ∥Λ∥C1+ α +4 [0,Tδ]). +One can prove the Lipschitz estimates for each terms in (3.16) by the same argument as +above. Note that the fundamental solution G depends on Λ in a Lipschitz way, and hence +the integral V . Therefore, the map (3.16) is contractive on C1+ α +4 [0, Tδ] which yields the +existence of the contact point Λ(t) satisfying (3.6). +Finally existence of a unique local strong solution to the thin film model (1.3) is obtained +by going back to the original variable from Theorem 3.4. Let us denote by I = [Λ, ∞) for +better readability. +Theorem 3.5 Let Λ0, h0 > 0 and h0 ∈ C4+α[Λ0, ∞), α ∈ (0, 1) with h0 → 1 as |x| → ∞ +satisfying the compatibility conditions +h0 = g, +∂xh0 = ∂xg − k, +h0 ∂3 +xh0 = −6Λ0 +∂tg +g2 +at x = Λ0, t = 0. +Also, g ∈ C1+ α +4 ((0, T0], C3(I)). Then, there exists T0 > 0 which also depends on the initial +data such that (1.3) has a unique strong solution (h, Λ), with the regularity +h ∈ C1+ α +4 ((0, T0], C4(I)), +Λ ∈ C1+ α +4 [0, T0]. +(3.17) +Furthermore, the solution depends continuously on the initial data. +4 +Stability analysis +In this section, we discuss steady state/equilibrium solutions of the model (1.3) and +asymptotic stability of the non-linear problem around a steady state. +For the standard thin film equation on an unbounded domain (with slip condition), +the conservation of mass does not satisfy usually. Therefore, with our current approach, +17 + +we consider a periodic setting, or in other words, consecutive solid objects immersed in the +fluid. Precisely, we assume that the initial thin film is spatially periodic, i.e. for some L > 0, +h0(x) = h0(x + 2L) +for all x ∈ R. +(4.1) +Note that it is not possible to consider fixed lateral boundary in order to have bounded +domain (and in turn finite mass), since lubrication approximation does not hold in that case +any more. Further, it is enough to assume that the configuration is symmetric with respect +to y = L. Thus, we consider the problem (1.4). +4.1 +Conservation of mass +When the solid is stationary i.e. +∂tg = 0, observe that (h∂3 +xh)|x=Λ = 0 in (1.4)2. +Therefore we have, from the system (1.4), +d +dt +L +� +Λ(t) +h(x, t) dx = +L +� +Λ(t) +∂th dx − ˙Λ h|x=Λ = − +L +� +Λ(t) +∂x(h3∂3 +xh) dx − ˙Λ h|x=Λ += −(h3∂3 +xh)|x=Λ − ˙Λ h|x=Λ = − ˙Λ h|x=Λ +and +d +dt +Λ(t) +� +0 +g(x, t) dx = +Λ(t) +� +0 +∂tg dx + ˙Λ g|x=Λ = ˙Λ g|x=Λ +which shows due to (1.4)2 that the mass is conserved, i.e. +d +dt +� +� +� +Λ(t) +� +0 +g(x, t) dx + +L +� +Λ(t) +h(x, t) dx +� +� +� = ˙Λ g|x=Λ − ˙Λ h|x=Λ = 0. +(4.2) +4.2 +Equilibrium solution +For a fixed volume of the fluid +V0 = +Λ(t) +� +0 +g dx + +L +� +Λ(t) +h dx, +(4.3) +an equilibrium solution (h, Λ) of (1.4) is a minimizer of the total energy E, given by +E(t) := a +Λ(t) +� +0 +|∂xg|2dx + b +L +� +Λ(t) +|∂xh|2dx + c +L +� +Λ(t) +|∂xg|2dx, +(4.4) +18 + +where a, b, c are non-negative constants with b > 0, corresponding to the energy for the +liquid-solid, liquid-gas and gas-solid interfaces respectively. Note that energy of the solid- +gas interface must be included in the total energy. +One can compute the new energy for small changes (h + δh, Λ + δΛ) of the equilibrium +state as +E + δE = a +Λ+δΛ +� +0 +|∂xg|2dx + b +L +� +Λ+δΛ +|∂x(h + δh)|2dx + c +L +� +Λ+δΛ +|∂xg|2dx += E + (a − c)δΛ|∂xg|2|x=Λ − b δΛ|∂x(h + δh)|2|x=Λ + b +L +� +Λ +(2∂xh ∂x(δh) + |∂x(δh)|2) dx. +Neglecting the small quadratic terms, one obtains, +δE = (a − c) δΛ|∂xg|2|x=Λ − b δΛ|∂xh|2|x=Λ + 2b +L +� +Λ +∂xh ∂x(δh) dx. +Similarly, the new volume becomes, +V + δV = +Λ+δΛ +� +0 +g dx + +L +� +Λ+δΛ +(h + δh) dx = V + +L +� +Λ +δh dx. +Hence, +δV = +L +� +Λ +δh dx. +Therefore, we need to find a unique Lagrange multiplier λ ∈ R such that δE = λ δV , i.e. +after integrating by parts, +(a − c) δΛ|∂xg|2|x=Λ − b(∂xg|x=Λ − k)2δΛ − 2b +L +� +Λ +∂2 +xh δh dx − 2b(∂xh δh)|x=Λ = λ +L +� +Λ +δh dx. +(4.5) +Here we used the relation (1.4)2 since (h, Λ) solves the system (1.4). Choosing δΛ = 0 and +δh having compact support, one then obtains, +− 2b +L +� +Λ +∂2 +xh δh dx = λ +L +� +Λ +δh dx +which implies +∂2 +xh = − λ +2b, +x ∈ (Λ, L). +(4.6) +19 + +Also we have the continuity relation satisfied by the perturbed solution, +(h + δh)|x=Λ+δΛ = g|x=Λ+δΛ, +which gives, neglecting small terms, +h|x=Λ + ∂xh|x=Λ δΛ + δh|x=Λ = g|x=Λ + ∂xg|x=Λ δΛ +implying, by using (1.4)2, +δh|x=Λ = (∂xg − ∂xh)|x=Λ δΛ = k δΛ. +Plugging (4.6) further in the equation (4.5), we get a relation determining the contact angle, +b k2 + |∂xg|2(a − b − c) = 0, +or, +k = +�� +1 − a − c +b +� +∂xg. +(4.7) +Here we assume the condition (b + c − a)/b > 0 so that the above relation is well defined. +Also recall that k > 0 (by construction). This is a form of Young’s condition which says +that the contact angle is given only by slope of the solid and the energies of the interfaces. +From relation (4.6) and the boundary conditions (1.4)2 together with ∂xh|x=L = 0 (due +to the symmetry assumption), we can further determine the equilibrium solution completely, +h = (∂xg − k)|x=Λ +2(Λ − L) +(x − L)2 + g|x=Λ − (∂xg − k)|x=Λ +2 +(Λ − L), +x ∈ (Λ, L), +(4.8) +and also the Lagrange multiplier, in terms of Λ, as +λ = 2b(∂xg − k) +(L − Λ) . +Finally, from the volume constraint (4.3), the contact point at equilibrium Λ can be deter- +mined. +Energy dissipation (formal): +When the solid is stationary, it holds from (1.4)2 that +h|x=Λ = g|x=Λ for t > 0 and in turn, we obtain the following relation by differentiating in +time +∂th + ˙Λ ∂xh = ˙Λ ∂xg +which implies +∂th = k ˙Λ +at x = Λ. +(4.9) +20 + +Therefore, one obtains the following energy dissipation relation from (4.4) as +d +dtE = 2b +L +� +Λ(t) +∂xh ∂xth dx − b ˙Λ|∂xh|2|x=Λ + (a − c) ˙Λ|∂xg|2|x=Λ += −2b +L +� +Λ(t) +∂xh ∂2 +x(h3∂3 +xh) dx + ˙Λ((a − c)|∂xg|2 − b|∂xh|2)|x=Λ +[using (1.4)1] += −2b +L +� +Λ(t) +h3|∂3 +xh|2 dx + 2b (∂xh ∂x(h3∂3 +xh))|x=Λ + ˙Λ((a − c)|∂xg|2 − b|∂xh|2)|x=Λ += −2b +L +� +Λ(t) +h3|∂3 +xh|2 dx + ˙Λ((a − c)|∂xg|2 − b(∂xg − k)2 − 2kb(∂xg − k))|x=Λ += −2b +L +� +Λ(t) +h3|∂3 +xh|2 dx − ˙Λ((b + c − a)|∂xg|2|x=Λ − bk2) += −2b +L +� +Λ(t) +h3|∂3 +xh|2 dx. +(4.10) +We have used relations (1.4)1, (4.9) and (1.4)2 to obtain the fourth equality whereas Young’s +condition (4.7) is used at the last line. Also the condition ∂xh|x=L = 0 (due to the symmetry +assumption) have been used in the above deduction. +Steady state: +For the stationary solution of (1.4), as the time derivative vanishes, (1.4)1 +reduces to +∂x(h3∂3 +xh) = 0 +in (Λ, L), +which means two possibilities: either the thin film is given by a constant height or, a parabola +(below we give a complete description, cf. Remark 4.1). The constant thin film corresponds +to the case where slope of the thin film is zero, i.e. ˜θ = 0 (cf. Figure ??) which means +∂xg = k at x = Λ. If ∂xg ̸= k, then the shape of the thin film is given by a parabola. This +indicates that the fluid film might rupture in finite time if the volume of the fluid is small. +Recall that the steady state of the classical thin film equation ∂th + ∂x(h2∂3 +xh) = 0 on a +bounded domain (with slip boundary condition) is indeed given by a parabola. +Remark 4.1 The above energy dissipation relation (4.10) shows that the equilibrium solu- +tion (h, Λ), given by (4.8), is also a steady state of (1.4) since it satisfies +d +dtE = 0. +21 + +Remark 4.2 From the relation (4.7) and (1.4)2, one obtains that ∂xg − ∂xh = γ∂xg at +x = Λ where γ = +� +(b + c − a)/b. Depending on γ ≥ 1 or γ < 1, the steady state has a +(locally) convex or concave profile, respectively. +4.3 +Asymptotic stability of steady state +Finally we are in a situation to study the stability of the stationary solution (h, Λ) of +(1.4), given by (4.8). To this end, we introduce the following linear transformation +x = x(L − Λ) + L(Λ − Λ) +(L − Λ) +which maps the domain (Λ, L) to the fixed domain (Λ, L). Differentiating, we get +∂xx = (L − Λ) +(L − Λ), +∂tx = − ˙Λ(L − x)(L − Λ) +(L − Λ)2 +. +In this new coordinate, let us denote H(x, t) = h(x, t). Recall that g is independent of t +here since we assume that the solid is stationary. +Now consider a small perturbation of the steady state as H = h+ϕ where |ϕ| ≪ 1. First +of all, existence of a unique solution of (1.4), for some small time T0, in the periodic setting +can be obtained as given by Theorem 3.5. Considering spaces on bounded spatial interval +(Λ, L) instead of (0, ∞) in Theorem 3.4 does not pose any extra difficulty. Therefore, ϕ +satisfies the same regularity as in Theorem 3.5, for suitable initial data. Note that ϕ has +vanishing mean due to the mass conservation property (4.2), +⟨ϕ⟩ := +L +� +Λ +ϕ dx = 0. +(4.11) +Plugging in the above expression of H, the perturbation ϕ satisfies, +∂tϕ + +�L − Λ +L − Λ +�4 +∂x((h + ϕ)3∂3 +xϕ) = ˙Λ (L − x) +(L − Λ)(∂xh + ∂xϕ) +in x ∈ (Λ, L), +ϕ = ψ1, +∂xϕ = ψ2, +∂3 +xϕ = 0 +at x = Λ, +∂xϕ = 0 +at x = L, +ϕ = ˜h0 +at t = 0, +(4.12) +where ˜h0(x) = h0(x) and +ψ1(t) = g(Λ) − g(Λ), +ψ2(t) = +�L − Λ +L − Λ +� +(∂xg(Λ) − k) − (∂xg(Λ) − k). +22 + +Also, with the help of energy dissipation relation for the equilibrium solution, +d +dt +� +��a +Λ +� +0 +|∂xg|2 dx + b +L +� +Λ +|∂xh|2 dx + c +L +� +Λ +|∂xg|2 dx +� +�� = 0, +the energy equation (4.10) results into, +E(ϕ(t)) := d +dt +� +��b +L +� +Λ +|∂xϕ|2 dx + +β2 +(L − Λ)(|∂xg|2|x=Λ − k2) + O(β3) +� +�� += −2b(1 + O(β)) +L +� +Λ +(h + ϕ)3|∂3 +xϕ|2 dx, +(4.13) +where β ≡ β(t) := (Λ(t) − Λ). +Next let us prove some a priori estimate which is essential to prove the stability result. +Lemma 4.3 Any function ϕ ∈ H3(Λ, L), having the boundary conditions ∂xϕ = (k1−k2) +k1(L−Λ)ϕ +at x = Λ and ∂xϕ = 0 at x = L where k1, k2 are some non-zero constants, and with zero +mean value ⟨ϕ⟩ = 0, satisfies the following estimates, +L +� +Λ +|∂xϕ|2 dx ≤ C +L +� +Λ +|∂3 +xϕ|2 dx, +(4.14) +and +|ϕ(Λ)|2 ≤ C +L +� +Λ +|∂3 +xϕ|2 dx. +(4.15) +Remark 4.4 Note that the boundary condition ∂xϕ = (k1−k2) +k1(L−Λ)ϕ at x = Λ is crucial for the +estimate (4.14) to hold. It is possible to construct a function (e.g. a quadratic polynomial) +with zero mean value and symmetric with respect to x = L whose first derivative does not +vanish, that is the right hand side of (4.14) vanishes while not the left hand side. On the +other hand, the only function which satisfies all the conditions stated in the above lemma is +the null function. +Proof. (i) Let us first note that since ϕ ∈ H1(Λ, L), it is absolutely continuous, in particular, +it holds that +|ϕ(Λ)|2 ≤ C +L +� +Λ +|∂xϕ|2 dx. +23 + +Thus, (4.15) is a consequence of (4.14). +(ii) We prove the inequality (4.14) by contradiction argument. Suppose that for each +m ∈ N, there exists ϕm ∈ H3(Λ, L) with the conditions +∂xϕm|x=Λ = +λ +2k2bϕm +�� +x=Λ, +∂xϕm|x=L = 0, +and +⟨ϕm⟩ = 0, +such that +L +� +Λ +|∂xϕm|2 dx ≥ m +L +� +Λ +|∂3 +xϕm|2 dx. +(4.16) +Further we may renormalize the sequence as ∥∂xϕm∥L2(Λ,L) = 1. +Relation (4.16) shows that ϕm is a bounded sequence in H3(Λ, L). In fact, ∂3 +xϕm → 0 +in L2(Λ, L). Thus, there exists a subsequence, still denoted by ϕm, and a function ϕ such +that ϕm ⇀ ϕ in H3(Λ, L). Due to the compactness of H3(Λ, L) �→ H2(Λ, L), one further +has that ϕm → ϕ strongly in H2(Λ, L). Therefore, ϕ satisfies, +∂xϕ|x=Λ = +λ +2k2bϕ +�� +x=Λ, +∂xϕ|x=L = 0, +⟨ϕ⟩ = 0, +(4.17) +and +∥∂xϕm∥L2(Λ,L) = 1. +(4.18) +On the other hand, one has from (4.16) that ∂3 +xϕ = 0 in L2(Λ, L). This implies, together +with the boundary conditions and the vanishing mean (4.17), that ϕ ≡ 0 in (Λ, L) which is +a contradiction to (4.18). +Now we can establish the asymptotic stability of the energy corresponding to (1.4). +Corollary 4.5 Let (h, Λ) be a steady state of (1.4), given by (4.8). There exists ε > 0 such +that for any initial data ˜h0 ∈ C4+α(Λ, L), α ∈ (0, 1) and Λ0 ∈ R with |Λ0 − Λ| + ∥˜h0 − +h∥C1(Λ,L) ≤ ε, a solution (h, Λ) of (1.4) satisfies, for t ∈ (0, T0), +|Λ − Λ| ≤ Ce−ωt +and +∥H − h∥H1(Λ,L) ≤ Ce−ωt +for some ω > 0. +(4.19) +Proof. The right hand side of (4.13) can be estimated as, since |ϕ| ≪ 1, +2b(1 + O(α)) +L +� +Λ +(h + ϕ)3|∂3 +xϕ|2 dx ≥ C +L +� +Λ +�h +2 +�3 +|∂3 +xϕ|2 dx ≥ C min +x h +L +� +Λ +|∂3 +xϕ|2 dx. +Combining with the estimates in Lemma 4.3, we get that +2b(1 + O(α)) +L +� +Λ +(h + ϕ)3|∂3 +xϕ|2dx ≥ CE(ϕ(t)), +(4.20) +24 + +where the energy for the perturbation E(ϕ) is defined by (4.13). Therefore (4.20) yields +that +d +dtE(ϕ) ≤ −CE(ϕ) +for +0 ≤ t ≤ T0 +which finally gives, +E(ϕ(t)) ≤ exp(−Ct)E(ϕ(0)) +for +0 ≤ t ≤ T0. +In particular, one gets for the full energy defined by (4.4), +E(t) ≤ exp(−Ct)E(0) +for +0 ≤ t ≤ T ∗. +This completes the proof. +Next, to obtain the global in time existence result, we follow the path as in Section +3.3. To begin with, some suitable estimate on fundamental solution for the linear problem +corresponding to (4.12) can be obtained as before. Here onwards, the bar over x is omitted. +Lemma 4.6 Let ˜h0 ∈ C4+α(Λ, L) for α ∈ (0, 1). +There exists a fundamental solution +G(x, t, ξ, τ) of the linear problem +∂tϕ + +�L − Λ +L − Λ +�4 +∂x(h +3∂3 +xϕ) − ˙Λ (L − x) +(L − Λ)∂xϕ = 0 +in x ∈ (Λ, L), t > 0, +∂xϕ = ψ2, +∂3 +xϕ = 0 +at x = Λ, t > 0, +∂xϕ = 0 +at x = L, t > 0, +ϕ = ˜h0 +at t = 0, x ∈ (Λ, L), +(4.21) +satisfying the estimate +|∂mG(x, t, ξ, τ)| ≤ cm(t − τ)− m+1 +4 +exp +� +−c +� |x − y| +(t − τ)1/4 +�4/3� +, +t ∈ [0, T]. +(4.22) +Proof. The result follows from [4, Chapter IV.2, Theorem 3.4], as in Lemma 3.2. As per the +notations in [4], for our system (4.21), N = 1 = n, b = 2, r1 = 1, r2 = 3 and the boundary +conditions read as B ≡ (∂x, ∂3 +x)u|x=Λ = f ≡ (ψ2, 0). In (4.21)1, the leading order coefficient +a4(x, t) = +� +L−Λ +L−Λ +�4 +h +3 is Hölder continuous in both t ∈ (0, T0) and x ∈ (0, L), since Theorem +3.5 gives that Λ ∈ C1+ α +4 [0, T0] and h is smooth. Furthermore, the lower order coefficients +a3(x, t) = +� +L−Λ +L−Λ +�4 +∂x(h +3) and a1(x, t) = ˙Λ +� +L−x +L−Λ +� +are Hölder continuous in x as well. Hence, +all the conditions of [4, Chapter IV.2, Theorem 3.4] are satisfied. +Theorem 4.7 Let (h, Λ) be a steady state of (1.4) and ˜h0 ∈ C4+α(Λ, L), α ∈ (0, 1) and Λ0 ∈ +R. There exists ε > 0 such that for any initial data satisfying |Λ0−Λ|+∥˜h0−h∥C4+α(Λ,L) ≤ ε, +a unique global solution (h, Λ) of the problem (1.4) exists, with regularity given by Theorem +3.5, and satisfies, for all t > 0, +|Λ − Λ| ≤ Ce−ωt +and +∥H − h∥C4+α(Λ,L) ≤ Ce−ωt +for some ω > 0. +25 + +Proof. Theorem 3.5 provides existence of ϕ as a unique solution of the quasilinear problem +∂tϕ + +�L − Λ +L − Λ +�4 +∂x((h + ϕ)3∂3 +xϕ) = ˙Λ (L − x) +(L − Λ)(∂xh + ∂xϕ) +in x ∈ (Λ, L), t > 0, +∂xϕ = ψ2, +∂3 +xϕ = 0 +at x = Λ, t > 0, +∂xϕ = 0 +at x = L, t > 0, +ϕ = ˜h0 +at t = 0, x ∈ (Λ, L), +(4.23) +in some interval [0, T0], which in addition matches the boundary condition ϕ|x=Λ = ψ1, and +is given by the following integro-differential form, +ϕ(x, t) = +t +� +0 +L +� +Λ +G(x, t, y, τ) [F1 + F2] (y, τ) dy dτ. +(4.24) +Here G is the fundamental solution of the linear problem given in Lemma 4.6 and +F1 = +�L − Λ +L − Λ +�4 +∂x((ϕ3 + 3h ϕ (h + ϕ)) ∂3 +xϕ), +F2 = ˙Λ (L − x) +(L − Λ) ∂xh. +Since the assumptions of the local existence result (eg. [4, Chapter III.4, Theorem 8.3]) +are satisfied, there exists a unique solution ϕ1 of the Cauchy problem ϕ1|t=T0 = ϕ(x, T0) for +equation (4.23), which is defined for t ∈ (T0, T1) where T1 > T0 and belongs to C4+α(Λ, L). +In turn, also Λ ∈ C1+ α +4 [T0, T1]. The solution can be continued in this way. +Now observe that, +I = +������� +t +� +0 +L +� +Λ +G(x, t, y, τ)F1(y, τ) dy dτ +������� += +������� +t +� +0 +�L − Λ +L − Λ +�4 +(τ) +L +� +Λ +G(x, t, y, τ)∂x((ϕ3 + 3h ϕ (h + ϕ)) ∂3 +xϕ)(y, τ) dy dτ +������� += +������� +t +� +0 +�L − Λ +L − Λ +�4 +(τ) +L +� +Λ +∂xG(x, t, y, τ)(ϕ3 + 3h ϕ (h + ϕ)) ∂3 +xϕ(y, τ) dy dτ +������� +≤ c(∥ϕ∥C((0,t),C +1 +2 (Λ,L))) +t +� +0 +�L − Λ +L − Λ +�4 +(τ) +L +� +Λ +|∂xG(x, t, y, τ)| dy dτ +≤ c(∥ϕ∥C((0,t),C +1 +2 (Λ,L))) +t +� +0 +�L − Λ +L − Λ +�4 +(τ)(t − τ)− 1 +4 dτ. +26 + +In the above we used integration by parts to pass all the derivatives over G and then use +the estimate (4.22) for m = 1 and the regularity of h and ϕ. Similarly, we have +II = +������� +t +� +0 +L +� +Λ +G(x, t, y, τ)F2(y, τ) dy dτ +������� += +������� +t +� +0 +˙Λ +(L − Λ)(τ) +L +� +Λ +G(x, t, y, τ) (L − y) ∂xh(y) dy dτ +������� +≤ c +t +� +0 +���� ˙Λ(L − Λ) +(L − Λ)(τ) +���� +L +� +Λ +|G(x, t, y, τ)| dy dτ +≤ c +t +� +0 +| ˙Λ(τ)|(L − Λ) +(L − Λ) dτ ≤ c +t +� +0 +| ˙Λ(τ)|(1 + +β +(L − Λ) + O(β2)) dτ. +Therefore, from the expression (4.24), we can conclude +∥ϕ∥C4+α(Λ,L) ≤ C +� +|β(t)| + ∥ϕ∥C +1 +2 (Λ,L) +� +≤ C +� +|Λ − Λ| + ∥ϕ∥H1(Λ,L) +� +. +Thus, the local solution can be continued up to time T = ∞ in the solution space, as long +as the initial data stays close enough. This concludes the proof. +Acknowledgements. The authors acknowledge the support of the Hausdorff Center +of Mathematics at the University of Bonn, funded by the Deutsche Forschungsgemeinschaft +(DFG) through the Collaborative Research Centre "The mathematics of emerging effects" +(CRC 1060, Project-ID 211504053). +The authors certify that they do not have any conflict of interest. +References +[1] F. Bernis and A. Friedman. Higher order nonlinear degenerate parabolic equations. Journal of +Differential Equations, 83(1):179–206, 1990. +[2] A. L. Bertozzi. The mathematics of moving contact lines in thin liquid films, 1998. +[3] P. G. de Gennes. Wetting: statics and dynamics. Rev. Mod. Phys., 57:827–863, Jul 1985. +[4] S. D. Eidel’man. +Parabolic systems. +North-Holland Publishing Co., Amsterdam-London; +Wolters-Noordhoff Publishing, Groningen, 1969. Translated from the Russian by Scripta Tech- +nica, London. +[5] A. Ghosh, B. Niethammer, and J. J. L. Velázquez. Revisiting Shikhmurzaev’s approach to the +contact line problem. Acta Applicandae Mathematicae, 181, 2022. +27 + +[6] L. Giacomelli, M. V. Gnann, H. Knüpfer, and F. Otto. Well-posedness for the Navier-slip thin- +film equation in the case of complete wetting. Journal of Differential Equations, 257(1):15–81, +2014. +[7] L. Giacomelli, H. Knüpfer, and F. Otto. Smooth zero-contact-angle solutions to a thin-film +equation around the steady state. Journal of Differential Equations, 245(6):1454–1506, 2008. +[8] M. V. Gnann and M. Petrache. The Navier-slip thin-film equation for 3d fluid films: Existence +and uniqueness. Journal of Differential Equations, 265(11):5832–5958, 2018. +[9] Y. Guo and I. Tice. Stability of contact lines in fluids: 2D Stokes flow. Archive for Rational +Mechanics and Analysis, 227(2):767–854, 2018. +[10] L. M. Hocking. +Rival contact-angle models and the spreading of drops. +Journal of Fluid +Mechanics, 239:671–681, 1992. +[11] C. Huh and L.E. Scriven. Hydrodynamic model of steady movement of a solid/liquid/fluid +contact line. Journal of Colloid and Interface Science, 35(1):85 – 101, 1971. +[12] H. Knüpfer. Well-posedness for the Navier slip thin-film equation in the case of partial wetting. +Communications on Pure and Applied Mathematics, 64(9):1263–1296, 2011. +[13] H. Knüpfer and N. Masmoudi. Darcy’s Flow with Prescribed Contact Angle: Well-Posedness +and Lubrication Approximation. Archive for Rational Mechanics and Analysis, 218:589 – 646, +2015. +[14] C. L. M. H. Navier. Mémoire sur les lois du mouvement des fluides. Mém. Acad. Sci. Inst. de +France (2), pages 389–440, 1823. +[15] A. Oron, S. H. Davis, and S. G. Bankoff. Long-scale evolution of thin liquid films. Reviews of +Modern Physics, 69:931–980, 1997. +[16] W. Ren and W. E. Derivation of continuum models for the moving contact line problem based +on thermodynamic principles. Communications in Mathematical Sciences, 9:597–606, 06 2011. +[17] C. Seis. The thin-film equation close to self-similarity. Analysis & PDE, 11(5):1303 – 1342, +2018. +[18] Y. D. Shikhmurzaev. Moving contact lines and dynamic contact angles: a ‘litmus test’ for +mathematical models, accomplishments and new challenges. The European Physical Journal +Special Topics, 229(10):1945–1977, 2020. +[19] V. A. Solonnikov. Solvability of a problem on the motion of a viscous incompressible fluid +bounded by a free surface. Mathematics of the USSR-Izvestiya, 11(6):1323–1358, dec 1977. +[20] V. A. Solonnikov. On some free boundary problems for the Navier-Stokes equations with moving +contact points and lines. Mathematische Annalen, 302:743–772, 1995. +28 + diff --git a/z9E2T4oBgHgl3EQf4ghZ/content/tmp_files/load_file.txt b/z9E2T4oBgHgl3EQf4ghZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a6964940bd36ca15d9e948c2d6d95f53bf8dbc77 --- /dev/null +++ b/z9E2T4oBgHgl3EQf4ghZ/content/tmp_files/load_file.txt @@ -0,0 +1,825 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf,len=824 +page_content='A thin film model for meniscus evolution Amrita Ghosh ∗1 and Juan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Velázquez †1 1Institute for Applied Mathematics, University of Bonn Endenicher Allee 60, 53115 Bonn, Germany Abstract In this paper, we discuss a particular model arising from sinking of a rigid solid into a thin film of fluid, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' a fluid contained between two solid surfaces and part of the fluid surface is in contact with the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The fluid is governed by Navier-Stokes equation, while the contact point, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' where the gas, liquid and solid meet, is assumed to be given by a constant, non-zero contact angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' We consider a scaling limit of the fluid thickness (lubrication approximation) and the contact angle between the fluid-solid and the fluid-gas interfaces is close to π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' This resulting model is a free boundary problem for the equation ht + (h3hxxx)x = 0, for which we have h > 0 at the contact point (different from the usual thin film equation with h = 0 at the contact point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' We show that this fourth order quasilinear (non-degenerate) parabolic equation, together with the so-called partial wetting condition at the contact point, is well-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Also the contact point in our thin film equation can actually move, contrary to the classical thin film equation for a droplet arising from no-slip condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Furthermore, we show the global stability of steady state solutions in a periodic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 1 Introduction In this paper, we investigate a particular thin film model, where a rigid solid enters a liquid film (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Figure 1), leading to movement of the contact point and the formation of a meniscus, as the initial state is out of equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' This free boundary problem gives rise to the following fourth order non-linear parabolic equation, in one dimension, ∂th + ∂x(h3∂3 xh) = 0 in x > Λ, t > 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1a) along with the boundary conditions, h = g, ∂xh = ∂xg − k, h∂3 xh = −6Λ ∂tg g2 at x = Λ, t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1b) ∗ghosh@iam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='uni-bonn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='de †velazquez@iam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='uni-bonn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='de 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='04181v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='AP] 10 Jan 2023 Figure 1: General meniscus formation Here h = h(x, t) is the height of the fluid film, Λ = Λ(t) is the contact point (which is a free boundary in one dimension), k is the fixed contact angle (rescaled), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' the angle between the fluid-solid and the fluid-air interfaces, and g = g(x, t) is the profile of the rigid solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Deduction of the above system is elaborated in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Here the situation in one spatial dimension is studied only, although one may consider general dimension as well (physically relevant cases correspond to d = 1 or 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The study of fluid problems involving free boundary evolution mostly concerns with an interface between two phases (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [19]), whereas the interest of the current paper is the situation where three phases of matter meet (typically, fluid-solid-air), generating a contact line/point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The main differences between the model considered here with other classical triple junction model is pointed out later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1a): Let us first discuss the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1a) which can be viewed as a par- ticular case of the general class of thin film equations ∂th + ∂x(f(h)∂3 xh) = 0 in {h > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2) Equations of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2) are typically obtained by lubrication approximation of the motion of a two-dimensional viscous fluid droplet, spreading over a solid substrate by the effect of surface tension (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Classical macroscopic fluid mechanics to model such phenomena imposes the no-slip boundary condition at the fluid-solid interface, that is the velocity of the fluid must be equal to that of the solid substrate in contact with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' This corresponds to the case f(h) = h3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' However, this condition can not hold at the moving contact point/triple junction where the solid, liquid and gas, three phases meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' It is known to give rise to a non-integrable singularity at the contact point (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [11]) which further implies that boundary of the film (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' the triple junction) can not move in time (no slip paradox).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' On other hand, Navier slip condition (corresponds to the case f(h) = h2), proposed by Navier [14], which allows the fluid to slip over the solid surface (characterized by a slip parameter), resolves such paradoxical phenomena (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [15] for a discussion in this direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Note that equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2) can also be viewed as a higher order version of the Porous medium equation ∂th − ∂x(hn∂xh) = 0, n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' However, the standard techniques used for such second order degenerate parabolic equations (such as Maximum principle) are not available for the higher order counterpart which makes the analysis more difficult for the later (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 2 <-Solud GasEquation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1b): Next let us discuss the boundary conditions in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The first condi- tion gives the position of the free boundary, whereas the second one determines the slope at the free boundary (contact angle condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The case where the (equilibrium) contact an- gle is given by a non-zero constant, determined by Young’s law (balance between interfacial energies) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [3]), is known as partial wetting, whereas the zero contact angle is referred as complete wetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' This work takes into account both the possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' There are also notions of dynamic/apparent contact angle (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [10, 16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The third boundary condition comes from a matching condition with the internal region (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' the fluid region bounded between solid parts), while in the standard setting of a droplet, a third condition is deduced from match- ing the velocity of the contact point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' This would lead to an over-determined problem for a fourth order operator with fixed boundary, but not for the free boundary problem where the boundary is an unknown a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' State of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' All the literature available in the context of thin film equation for contact angle problem consider either compactly supported initial data (droplet case) or an unbounded support with single free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Thus the standard condition h = 0 of the vanishing fluid height at the free boundary makes the equation ∂th + ∂x(hn∂3 xh) = 0 singular/degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Following the pioneering work of Bernis and Friedman [1], several results has been established for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2), considering different forms of f, for example in [7], [6]-[8], [12]-[13], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The above references are not claimed to be exhaustive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' We also point out the work [9] where stability of a global solution for the general contact line problem for Stokes equation has been discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The current work is concerned with a different thin film model, deduced from the no-slip boundary condition assumption at the fluid-solid interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The film height being h > 0 at the contact point makes our model non-degenerate, unlike the standard thin film equation as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The result shows well-posedness as well as the possibility of moving contact point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Moreover, no paradox arises in spite that the contact point moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The reason of such behavior in this case is that the contact angle is very close to π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' As pointed out by Solonnikov in [20], the paradox gets resolved in the case of contact angle π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The main contribution of the paper is to bring forward this particular interesting yet simple situation involving the evolution of triple junction and the discussion of different behavior in contrast to the known cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' This observation intrigued our curiosity to analyze the model in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Structure of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' We first deduce a general thin film model describing the above particular situation of meniscus formation under the classical approach in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The arguments used in this derivation are standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Still we write this derivation in detail, since we did not find any suitable reference for such model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' In Section 3, the local well-posedness result is established, for the particular case of no-slip boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The system being non-degenerate, standard theory for fourth order parabolic quasilinear equation can be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Nevertheless, obtaining suitable regularity (in time) in order to closing the final fixed point argument is not straight-forward (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Finally we obtain the asymptotic behavior of a solution, in periodic setting, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' By looking at perturbations of a steady state of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1a)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1b), existence of a unique global solution is obtained with the help of a priori energy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' As long as the initial data remains 3 close (in certain norm) to the stationary solution, the local solution of the thin film model converges to the steady state for all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1 Main results Our first result concerns existence of a local in time solution to the following thin film equation with moving contact point and prescribed contact angle (including complete wet- ting case) ∂th + ∂x(h3∂3 xh) = 0 in x > Λ, t > 0, h = g, ∂xh = ∂xg − k, h ∂3 xh = −6Λ∂tg g2 at x = Λ, t > 0, h → 1 as x → ∞, t > 0, h = h0 at t = 0, x > Λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3) Here Λ0 = Λ(0) denotes the initial position of the contact point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1 For any initial data (h0, Λ0), satisfying compatibility conditions, there exists a unique strong solution (h, Λ), for short time, of the free boundary problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Rigorous statement and proof of the above theorem is given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Next we consider the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3) over an interval (0, 2L) (in order to have a bounded domain), symmetric with respect to x = L and with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Precisely, ∂th + ∂x(h3∂3 xh) = 0 in x ∈ (Λ, L), t > 0, h = g, ∂xh = ∂xg − k, h ∂3 xh = 0 at x = Λ, t > 0, ∂xh = 0 at x = L, t > 0, h = h0 at t = 0, x ∈ (Λ0, L), h(x, t) = h(x + 2L, t), g(x, t) = g(x + 2L, t), x, t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4) The free interface h(x, t) is defined over the domain (Λ, 2L−Λ) in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Also we assume here that the solid is not moving, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' ∂tg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Such a solution preserves mass for all time t > 0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Then, for a given volume V0, a steady state (h, Λ) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4) is characterized by ∂x(h 3∂3 xh) = 0 in x ∈ (Λ, L), subject to the boundary conditions h = g, ∂xh = ∂xg − k, ∂3 xh = 0 at x = Λ and ∂xh = 0 at x = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' We refer to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='8) for an explicit description of the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Stability of this steady state for small perturbation is obtained as below, we refer to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3 for rigorous statement and proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2 For initial data close enough to the steady state of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4), there is a unique global solution (h, Λ) of the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4) which converges to the steady state as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 4 2 Lubrication approximation In this section, we elaborate the thin film approximation of the original three phase free boundary problem in consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Original model: Let us consider a horizontal, two-dimensional thin film of viscous, in- compressible Newtonian fluid, of thickness H at the initial time and a rigid solid (locally convex) touching the film (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' As the solid enters the liquid, the contact point moves and makes an angle (π − θ), θ > 0, between the liquid-solid and the liquid-gas in- terface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The liquid-gas interface is described by y = h(x, t) and the liquid-solid boundary is given by y = g(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' One may think of particular cases, for example g = H � 1 − � t t0 �n� with n > 0 where t0 is the time scale that characterises the motion of the solid mov- ing down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Here n = 1 corresponds to the solid moving with constant velocity, while n = 2 corresponds to the case of constant acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' A particular case of wedge-shaped solid (in the form g(x, t) = ˜h(t) + c|x|) has been mentioned in [18, Fig 6] in a context of correct modelling approach describing the creation of contact angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The free interface makes an angle ˜θ with the horizontal plane (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Therefore, tan ˜θ = −∂xh and θ = − arctan(∂xh) + arctan(∂xg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' To obtain the lubrication approximation, we need to assume that the two angles θ and ˜θ are very small and of the same order, which means that the contact angle (π − θ) is close to π in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Note that the thin film approximation would not be valid if ˜θ is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The contact points are given by si = (Λi(t), y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Finally, we assume the far field condition h → H as |x| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Let vL = (u, v) be the velocity of the fluid, with constant density ϱL, viscosity µL and pressure pL, governed by the Navier-Stokes equations, while the solid motion is governed by the translational velocity only vS(t), vertically downward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Let us consider the bottom part of the fluid domain as a reference configuration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' at y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Figure 2: Thin Film approximation with contact points As we assume a symmetric configuration with respect to y-axis (for simplicity), it is enough to analyse the situation for x > 0 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Therefore, in the following, we call the 5 Solid k=9(x,t) Gas 9 h(t) E 个 1 X-0 (A)Vcontact point Λ instead of Λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Further assumptions: Let the solid g be smooth in a neighborhood of the initial contact point Λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Moreover, g > 0 for all x, t ≥ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' the solid never touches the bottom of the fluid domain which may lead to further singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' In the framework of classical fluid mechanics, conditions at contact lines are considered only at equilibrium, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' in the situations where the contact line is not moving with respect to the bulk phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Also we consider here the Navier slip condition at the fluid-solid inter- face, for a general framework, which covers both the no-slip and the full-slip condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' We formulate these conditions below (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [4, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' in the liquid: divx vL = 0, ϱL � ∂tvL + vL · ∇xvL� = µL∆vL − ∇xpL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1) at the liquid-gas interface: y = h(x, t), x > Λ(t): Vn = vL · n1, � pG − pL� + 2µL � DvL · n1 � n1 + psκ = 0, � DvL · n1 � τ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2) at the liquid-solid interface {y = 0} ∪ {y = g(x, t), 0 < x < Λ(t)}: � vL − vS� n2 = 0, 2µL � DvL · n2 � τ = β � vL τ − vS τ � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3) at the triple junction x = Λ(t): h = g, arctan(∂xh) = arctan(∂xg) − θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4) Here ps is the (constant) surface tension, κ is the curvature of the free interface, pG is the constant pressure of the gas and ni, i = 1, 2 are the unit normal vectors, inward with respect to the fluid domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' We do not distinguish here between the slip-coefficients appearing at the lower and upper fluid-solid interface and denote both of them by β only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' In principle, they might be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' We further assume for simplicity that the lower boundary y = 0 is fixed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' vS = 0 at y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Also, we must have the matching condition for the initial data, h(x, 0) = H at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5) Further, due to the symmetry assumption of the considered configuration, one has, u = 0 on x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='6) Now let us introduce the length scale as L = H ε where ε > 0 is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Then the usual scaling of the variables are given by, (x, y) = � x L, y H � Λ = 1 LΛ, h = h H , g = g H t = σ Lµt, (u, v) = �µ σu, µ εσv � , p = ε2L σ p, θ = kε, 6 with σ = −ε3ps is the scaled surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Rescaled system: Under these scaling, we obtain the following approximated system of equations from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4), ∂xu + ∂yv = 0, ∂2 yu = ∂xp, ∂yp = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7a) in (x, y) ∈ (0, Λ) × (0, g) ∪ (Λ, ∞) × (0, h), ∂th + u ∂xh − v = 0, p = −∂2 xh, ∂yu = 0, on x > Λ, y = h, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7b) v = 0, ∂yu = β u, on y = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7c) ∂xg u = v − ∂tg, ∂yu + β u = 0, on x < Λ, y = g, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7d) h = g, ∂xh = ∂xg − k, at x = Λ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7e) h → 1 as x → ∞, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7f) where the rescaled slip coefficient β = εβL µ = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The system is complemented by equa- tions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Indeed, the incompressibility condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1)1 remains same in the new variables, leading to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7a)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The Navier-Stokes equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1)2 in the first component becomes, ϱL σ µL L µL � �� � Re (∂tu + u∂xu + v∂yu) = � ∂2 xu + 1 ε2 ∂2 yu � − 1 ε2 ∂xp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Thus in the limit ε → 0, both the time derivative and the non-linear term vanish compared to the viscous term when Reynolds number satisfies ε2Re ≪ 1 and one obtains (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7a)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Similarly the second component of Navier-Stokes equation reduces to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7a)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Next we discuss the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' At the liquid-gas interface y = h(x, t), x > Λ(t), the unit normal and tangent vectors and the curvature have the following expressions, n1 = � ε∂xh, −1 � � (1 + ε2|∂xh|2) , τ = � 1, ε∂xh � � (1 + ε2|∂xh|2) and κ = ε∂2 xh L � 1 + ε2|∂xh|2�3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='8) The kinematic boundary condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2)1 reduces to the non-dimensional form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7b)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Also the rate of strain tensor becomes, in the new variables, DvL = 1 2 � � 2∂xu ∂yu + ∂xv ∂yu + ∂xv 2∂yv � � = 1 2 σ Lµ � � 2∂xu 1 ε∂yu + ε∂xv 1 ε∂yu + ε∂xv 2∂yv � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Therefore, the normal and the tangential stress on the surface y = h(x, t) are given by, � DvL · n1 � n1 = σ Lµ 1 � 1 + ε2|∂xh|2� � −∂yu ∂xh + ∂yv + O(ε2) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='9) 7 and � DvL · n1 � τ = σ Lµ 1 2 1 � 1 + ε2|∂xh|2� � −1 ε∂yu + O(ε) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='10) Then, equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2)2 becomes, with the help of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='9), pG − σ ε2Lp + 2σ L 1 � 1 + ε2|∂xh|2� � −∂yu ∂xh + ∂yv + O(ε2) � − σ ε3 ε ∂2 xh L � 1 + ε2|∂xh|2�3/2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Now with the assumption that σ = O(1), one obtains in the limit (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7b)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' For equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2)3, one gets (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7b)3 using the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' At the liquid-solid interface, with n2 = (0, 1), τ = (1, 0) at the bottom part y = 0, the conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3) read as, v = 0, µL(∂yu + ∂xv) = βu at y = 0, which then convert into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Similarly, at the upper fluid-solid interface, the normal and tangent vectors being, n2 = (∂xg, −1) � 1 + |∂xg|2 , τ = (1, ∂xg) � 1 + |∂xg|2 , and vS = (0, ∂tg), the boundary conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3) transform into the non-dimensional form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The conditions at the contact point (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4) transform into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Thin film equations: The thin film model can now be derived from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7a)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7f), together with the boundary conditions and initial data as, ∂th + ∂x ��h 3 + 1 β � h2∂3 xh � = 0, in x > Λ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='11a) h = g, ∂xh = ∂xg − k, at x = Λ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='11b) h → 1, as x → ∞, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='11c) h = 1, at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='11d) The deduction of the above system is explained in the following (omitting the bar here onwards).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Integrating (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7a)2 twice gives the profile of the horizontal velocity, u(x, y) = 1 2∂xp y2 + A(x, t)y + B(x, t), x > 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='12) where the constants A, B are to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The condition at the lower boundary (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7c)2 implies, A = βB, x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='13) 8 Also the condition at the free boundary (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7b)3 yields, A = −∂xp h, therefore, B = − 1 β ∂xp h, x > Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='14) Further, as the pressure is independent of y due to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7a)3, the horizontal velocity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='12) becomes, together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='14) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7b)2, u = 1 2∂3 xh � (2h − y)y + 2 β h � , x > Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='15) Next (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7a)1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7c)1 give, v|y=h = − h � 0 ∂xu dy, x > Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Thus from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7b)1, we obtain, ∂th + ∂x � h � 0 u dy � = 0, x > Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Substituting the velocity profile (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='15) into the above relation finally gives the thin film equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='11a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The contact line condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='11b) is nothing but (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7e) and the initial data (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='11d) follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Inner region: In order to determine the dynamics of the contact point fully, we need to prescribe sufficient conditions in the inner region x < Λ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' To do so, the condition at the fluid-solid interface (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7d)2, together with the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='13), for x < Λ, y = g, which is [∂xp g + A] + β �1 2∂xp g2 + A g + A β � = 0, gives a relation between ∂xp and A for x ∈ (0, Λ), that is ∂xp = −2A g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='16) Also, the incompressibility condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7a)1 gives an expression for the normal velocity for x ∈ (0, Λ), v|y=g = − ˜h � 0 ∂xu dy = −g �1 6∂2 xp g2 + 1 2∂xA g + 1 β ∂xA � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Thus the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7d)1 implies, for x ∈ (0, Λ), ∂xg �1 2∂xp g2 + A � g + 1 β �� + ∂tg + g �1 6∂2 xp g2 + 1 2∂xA g + 1 β ∂xA � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 9 The above equation together with the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='16) yields an ODE for A with rational coefficients, ∂xA + r1(x, t) A + r2(x, t) = 0, x < Λ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='17) where r1(x, t) = ∂xg � 1 β + 1 3g � g � 1 β + 1 6g � , r2(x, t) = ∂tg g � 1 β + 1 6g �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Also, due to the symmetry assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='6), we have the boundary condition A(x, t) = 0 at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Therefore, the ODE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='17) determines A and in turn ∂xp in the inner region x ∈ (0, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Here we assume that the horizontal velocity u is continuous, thus the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='12) holds at x = 0 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Moreover, the continuity of u at Λ further imposes the condition A = h ∂3 xh at x = Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='18) This above relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='18), together with the thin film equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='11a)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='11d) determines fully the dynamics and the position of the contact point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' This thin film system is described for a particular case of wedge-shaped solid, without its detailed derivation, in [5, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 3 Local well-posedness for no-slip condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1 Notations and functional settings Let C0[0, ∞) denote the Banach space of continuous, bounded functions on [0, ∞) and tending to 0 as x → ∞, together with the supremum norm on [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The space Ck[0, ∞), k ∈ N of functions with k times continuous and bounded derivatives is endowed with the standard norm ∥v∥Ck[0,∞) := k � m=0 ∥∂mv∥C[0,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' We write C[0, ∞) ≡ C0[0, ∞) for the space of all continuous, bounded functions on [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The Banach space of ρ-Hölder continuous functions Cρ[0, ∞), Ck+ρ[0, ∞) for k ∈ N, ρ ∈ (0, 1), is defined by Cρ[0, ∞) := {v ∈ C[0, ∞) : [v]ρ := sup x,y∈[0,∞) x̸=y |v(x) − v(y)| |x − y|ρ < ∞}, together with the norm ∥v∥Cρ[0,∞) := ∥v∥C[0,∞) + [v]ρ, 10 and Ck+ρ[0, ∞) := {v ∈ Ck[0, ∞) : [∂kv]ρ < ∞}, with the norm ∥v∥Ck+ρ[0,∞) := ∥v∥Ck[0,∞) + [∂kv]ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' For ρ = 1, the above notions coincide with the class of Lipschitz functions which we denote by C1−[0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Sometimes we omit the underlying space [0, ∞) below which should not cause any confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2 Reduction to a fixed domain We assume the no-slip boundary condition at the fluid-solid interface, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 1 β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' We will show the local and global well-posedness for this reduced problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' As mentioned in the introduction, such result is in contrast with the classical thin film equation (for droplet) with no-slip condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Using the time scaling t → 3t, t0 → 3t0, the thin film model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='11a)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='11d) reduces to, ∂th + ∂x(h3∂3 xh) = 0 in x > Λ, t > 0, h = g, ∂xh = ∂xg − k, h ∂3 xh = A at x = Λ, t > 0, h → 1 as x → ∞, t > 0, h = 1 at t = 0, x > Λ0, where A solves the ODE in (0, Λ), ∂xA + 2∂xg g A + 6∂tg g2 = 0, A(0, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1) Solving (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1) gives, A(x, t) = −6x∂tg g2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Thus, we obtain the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Recall that we have g > 0 for all x, t ≥ 0 by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Without loss of generality, we assume Λ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Hence, for some δ > 0, there exists Tδ > 0 such that |Λ(t)| ≤ δ2 and | ˙Λ(t)| ≤ C for t ∈ [0, Tδ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2) For such a δ fixed, let us now consider a cut-off function ξδ ∈ C∞ c (R) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' ξδ(s) = � 1, if 0 ≤ s ≤ δ, 0, if s ≥ 2δ, such that |ξ′ δ| ≤ C δ , and the following bijection QΛ(x, t) = (x − Λ(t))ξδ(x − Λ(t)) + x(1 − ξδ(x − Λ(t))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 11 Denoting by x = QΛ(x, t) and H(x, t) = h(x, t), one can compute the derivatives in the new coordinate as, ∂th = ∂tH + ˙Λ(Λξ′ δ − ξδ)∂xH, ∂xh = (1 − Λξ′ δ)∂xH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Therefore, the free boundary problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3) reduces to a fixed domain as ∂tH − ˙Λ(t)ξδ ∂xH + (1 − Λξ′ δ)4 ∂x(H3∂3 xH) + F1 = 0 in x > 0, H = ψ1, ∂xH = ψ2, ∂3 xH = ψ3 at x = 0, H → 1 as x → ∞, H(x, 0) = H0 ≡ h0(x + Λ(0)) → 1 as x → ∞, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3) where F1 ≡ F1( ˙Λ, ξ′ δ, H) with supp(F1) ⊂ [δ, 2δ], and ψ1(Λ, t) = g(Λ, t), ψ2(Λ, t) = ∂xg(Λ, t) − k, ψ3(Λ, t) = −6Λ∂tg g3 (Λ, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Next we use the following transformation in order to lift the boundary conditions and the far field condition, H(x, t) = [ψ2(Λ, t) x + ψ3(Λ, t) x3]ξδ(x) + U(x, t) + (1 − ξδ(x)) =: a(x, t, Λ)ξδ(x) + U(x, t) + (1 − ξδ(x)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4) where |U(x, t)| → 0 as x → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Observe that a(x, t, Λ) > 0 for x > 0, t ∈ (0, Tδ) for suitably chosen δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Then the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3) becomes, omitting the bar over x, ∂tU − ˙Λ(ψ2 ξδ + ∂xU)ξδ + (1 − Λξ′ δ)4 ∂x((aξδ + 1 − ξδ + U)3∂3 xU) = F2 + F3 in x > 0, ∂xU = 0, ∂3 xU = 0 at x = 0, U → 0 as x → ∞, U(x, 0) = U0 ≡ h0(x) − a(x, 0) for x > 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5) together with U = ψ1(Λ, t) at x = 0, t > 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='6) where F2 ≡ F2( ˙Λ, ξ′ δ, U) with supp(F2) ⊂ [δ, 2δ], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7) and F3 = 6(1 − Λξ′ δ)4 ∂x((aξδ + (1 − ξδ) + U)3 ψ3 x3 ξδ) − ∂ta ξδ + ˙Λ 3x2ψ3 ξ2 δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='8) The above transformation is useful for splitting the full system as a Cauchy problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5) and a fixed point map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The general theory for quasilinear parabolic problem can be applied to treat the fourth order system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5) which in turn give the existence of a solution for the full problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 12 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1 As can be seen from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='6), Λ and U must have the same regularity in time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' On the other hand, it is not obvious/immediate to obtain the same time regularity from the parabolic system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5) if one uses the standard Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Hence we chose to use the Hölder spaces for the solution since it is important not to lose (trace) regularity in order to perform the fixed point argument in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3 Proof of the main result The idea is to write down the solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5) in terms of a Green function for the linear problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2 Let Λ ∈ C1+ α 4 [0, T] and U0 ∈ C4+α[0, ∞) ∩ C0[0, ∞) where α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' There exists a fundamental solution G(x, t, y, τ) of the linear boundary value problem ∂tU ∗ + (1 − Λξ′ δ)4 ∂x((aξδ + (1 − ξδ))3∂3 xU ∗) + ˙Λ ∂xU ∗ ξδ = 0 in x > 0, ∂xU ∗ = 0, ∂3 xU ∗ = 0 at x = 0, U ∗ → 0 as x → ∞, U ∗(x, 0) = U0 for x > 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='9) satisfying the estimate |∂mG(x, t, y, τ)| ≤ cm(t − τ)− m+1 4 exp � −c � |x − y| (t − τ)1/4 �4/3� , t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The existence of such G follows from [4, Chapter IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' As per the notations in [4], for our system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='21), N = 1 = n, b = 2, r1 = 1, r2 = 3 and the boundary conditions read as B ≡ (∂x, ∂3 x)u|x=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The Cauchy problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='9) can be reduced to a problem with zero initial condition by considering a function (U ∗ − U0) since U0 ∈ C4+α[0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' In (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='9)1, the leading order coefficient a4(x, t) = (1−Λξ′ δ)4 (aξδ+(1−ξδ))3 is Hölder continuous in both t ∈ (0, T) and x ∈ (0, ∞), due to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Furthermore, the other coefficients a3(x, t) = (1 − Λξ′ δ)4 ∂x(aξδ + (1 − ξδ))3 and a1(x, t) = ˙Λξδ are Hölder continuous in x as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Therefore, all the conditions of [4, Chapter IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4] are satisfied and hence the existence and estimates of a fundamental solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3 Let U0 ∈ C4+α[0, ∞) ∩ C0[0, ∞) where α ∈ (0, 1), satisfying compatible conditions ∂xU0 = ∂3 xU0 = 0 at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Then for any δ > 0 and Λ ∈ C1+ α 4 [0, Tδ] where Tδ is as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2), there exists T0 ∈ (0, Tδ) such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5) has a unique solution U(x, t) belonging to C1+ α 4 ((0, T0), C4+α[0, ∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The time interval T0 depends on the upper bounds of the Hölder constants of the coeffi- cients of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5) can be expressed by the following integro-differential equation U(x, t) = t � 0 ∞ � 0 G(x, t, y, τ) � ˙Λψ2 ξ2 δ + F2 + F3 + F4 � (y, τ) dy dτ + ∞ � 0 G(x, t, y, 0)U0(y) dy, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='11) where G is the fundamental solution of the linear problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='9), given in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2, F2, F3 are defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='8) and F4 := −(1 − Λξ′ δ)4 ∂x({U 3 + 3U(aξδ + 1 − ξδ)(U + (aξδ + 1 − ξδ))} ∂3 xU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='12) Solving the above equation is standard, for example one may refer to [4, Chapter III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The next step is to perform a fixed point argument obtaining a (local in time) solution for the full system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4 Let U0 ∈ C4+α[0, ∞) ∩ C0[0, ∞) where α ∈ (0, 1), satisfying compatible conditions U0 = ψ1(0, 0), ∂xU0 = ∂3 xU0 = 0 at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Then for small δ > 0 and small Tδ where Tδ is as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2), there exists T0 ∈ (0, Tδ) such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='6) has a unique solution (U, Λ) ∈ C1+ α 4 ((0, T0), C4+α[0, ∞)) × C1+ α 4 [0, T0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The time interval T0 depends on the upper bounds of the Hölder constants of the coeffi- cients of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' As U is given by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='11), in order to satisfy the boundary condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='6), one must have ψ1(Λ, t) = t � 0 ∞ � 0 G(0, t, y, τ) � ˙Λψ2 ξ2 δ + F2 + F3 + F4 � (y, τ) dy dτ + ∞ � 0 G(0, t, y, 0)U0(y) dy =: I + II + III + IV + V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='13) The first integral being the most important term to be controlled (all the other terms being small), we try to rewrite and simplify it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Observe that the mass is conserved for the Green function satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Indeed, by denoting g(x, t) := � ∞ 0 G(x, t, y, τ)dy and integrating the equations in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='9) with respect to y, we get that g satisfies the same system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Hence by uniqueness, ∞ � 0 G(x, t, y, τ) dy = 1, t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 14 Therefore, I := t � 0 ∞ � 0 G(0, t, y, τ)( ˙Λψ2 ξ2 δ)(y, τ) dy dτ = t � 0 ( ˙Λψ2)(τ) ∞ � 0 G(0, t, y, τ) dy dτ − t � 0 ∞ � 0 G(0, t, y, τ)( ˙Λψ2)(τ)(1 − ξ2 δ)(y) dy dτ = t � 0 ˙Λ(τ) ψ2(Λ(τ), τ) dτ − t � 0 ∞ � 0 G(0, t, y, τ)( ˙Λψ2)(τ)(1 − ξ2 δ)(y) dy dτ =: I1 + I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='14) At this point, let us denote Ψ(Λ, t) = � Λ 0 ψ2(y, t) dy, which gives, d dtΨ = Λ � 0 ∂tψ2(y, t) dy + ψ2(Λ(t), t) ˙Λ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Plugging this in to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='14), we get I1 = t � 0 � �� d dτ Ψ(Λ(τ), τ) − Λ(τ) � 0 ∂τψ2(y, τ) dy � �� dτ = [Ψ(Λ(t), t) − Ψ(0, 0)] − t � 0 Λ(τ) � 0 ∂τψ2(y, τ) dy dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Using the definition of Ψ and ψ2, one further obtains, after some integration by parts with respect to y, I1 = Λ � 0 (∂yg(y, t) − k) dy − t � 0 (∂τg(Λ(τ), τ) − ∂τg(0, τ)) dτ = [(g(Λ, t) − g(0, t)) − kΛ] − t � 0 ∂τg(Λ(τ), τ) dτ + [g(0, t) − g(0, 0)] = g(Λ, t) − g(0, 0) − k Λ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='15) Therefore, putting the relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='13), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='15) together, one gets, by the definition of ψ1, 0 = −g(0, 0) − kΛ(t) + I2 + II + III + IV + V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 15 In other words, as k > 0, Λ(t) = 1 k [−g(0, 0) + I2 + II + III + IV + V ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='16) Next we need to estimate the different terms of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='16), in particular to show that the above relation is a contraction map for Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Estimate of I2: Using the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='10) for the fundamental solution, we have |I2| = ������ t � 0 ˙Λ(τ) ψ2(Λ(τ), τ) ∞ � 0 (ξ2 δ(y) − 1) G(0, t, y, τ) dy dτ ������ ≤ c t � 0 | ˙Λ(τ) ψ2(Λ(τ), τ)| ∞ � δ (t − τ)−1/4 exp � −c y (t − τ)1/4 � dy dτ ≤ c t � 0 | ˙Λ(τ) ψ2(Λ(τ), τ)| exp � −c δ (t − τ)1/4 � dτ ≤ cδ(∥ψ2∥) t � 0 | ˙Λ(τ)| dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The above constant cδ(∥ψ2∥) < 1 can be made by choosing δ > 0 suitably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Estimate of II: Recalling the definition of F2 from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7), as it comprises several terms of the form ˙Λ ∂k xU ξ′ δ, k ∈ [0, 3], passing the derivatives onto G by integration by parts in x, and using the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='10) for the fundamental solution, one obtains the following |II| = ������ t � 0 ∞ � 0 G(0, t, y, τ)F2(y, τ) dy dτ ������ ≤ cδ(∥U∥C((0,T0),C[0,∞))) t � 0 | ˙Λ(τ)|(t − τ)−β/4 dτ, for some β ∈ (0, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' As before, choosing δ > 0 suitably, the constant can be made small cδ(∥U∥C((0,T0),C[0,∞))) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Estimate of III: All the terms in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='8) are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' For example, ������ t � 0 ∞ � 0 G(0, t, y, τ) [ ˙Λ 3x2ψ3 ξ2 δ](y, τ) dy dτ ������ ≤ c δ2 t � 0 | ˙Λ(τ)||ψ3(Λ(τ), τ)| δ � 0 |G(0, t, y, τ)| dy dτ ≤ cδ(∥ψ3∥) ∥Λ∥C1+ α 4 [0,Tδ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 16 Similarly, the other two terms can be estimated as, ������ t � 0 ∞ � 0 G(0, t, y, τ) [6(1 − Λξ′ δ)4 ∂x((aξδ + (1 − ξδ) + U)3 ψ3 x3 ξδ) − ∂ta ξδ](y, τ) dy dτ ������ ≤ Cδ(∥ψ2∥, ∥ψ3∥, ∥U(t)∥C[0,∞)) (Tδ + ∥Λ∥C1+ α 4 [0,Tδ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The constant can be chosen small, for suitable δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Estimate of IV : By the same argument as for estimating I2 and also using the fact that |∂3 xU| ≤ δ for |x| small as ∂3 xU|x=0 = 0, we get |IV | ≤ cδ(∥U∥C((0,T0),C[0,∞))) (Tδ + ∥Λ∥C1+ α 4 [0,Tδ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' One can prove the Lipschitz estimates for each terms in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='16) by the same argument as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Note that the fundamental solution G depends on Λ in a Lipschitz way, and hence the integral V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Therefore, the map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='16) is contractive on C1+ α 4 [0, Tδ] which yields the existence of the contact point Λ(t) satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Finally existence of a unique local strong solution to the thin film model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3) is obtained by going back to the original variable from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Let us denote by I = [Λ, ∞) for better readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5 Let Λ0, h0 > 0 and h0 ∈ C4+α[Λ0, ∞), α ∈ (0, 1) with h0 → 1 as |x| → ∞ satisfying the compatibility conditions h0 = g, ∂xh0 = ∂xg − k, h0 ∂3 xh0 = −6Λ0 ∂tg g2 at x = Λ0, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Also, g ∈ C1+ α 4 ((0, T0], C3(I)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Then, there exists T0 > 0 which also depends on the initial data such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3) has a unique strong solution (h, Λ), with the regularity h ∈ C1+ α 4 ((0, T0], C4(I)), Λ ∈ C1+ α 4 [0, T0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='17) Furthermore, the solution depends continuously on the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 4 Stability analysis In this section, we discuss steady state/equilibrium solutions of the model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3) and asymptotic stability of the non-linear problem around a steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' For the standard thin film equation on an unbounded domain (with slip condition), the conservation of mass does not satisfy usually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Therefore, with our current approach, 17 we consider a periodic setting, or in other words, consecutive solid objects immersed in the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Precisely, we assume that the initial thin film is spatially periodic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' for some L > 0, h0(x) = h0(x + 2L) for all x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1) Note that it is not possible to consider fixed lateral boundary in order to have bounded domain (and in turn finite mass), since lubrication approximation does not hold in that case any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Further, it is enough to assume that the configuration is symmetric with respect to y = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Thus, we consider the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1 Conservation of mass When the solid is stationary i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' ∂tg = 0, observe that (h∂3 xh)|x=Λ = 0 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Therefore we have, from the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4), d dt L � Λ(t) h(x, t) dx = L � Λ(t) ∂th dx − ˙Λ h|x=Λ = − L � Λ(t) ∂x(h3∂3 xh) dx − ˙Λ h|x=Λ = −(h3∂3 xh)|x=Λ − ˙Λ h|x=Λ = − ˙Λ h|x=Λ and d dt Λ(t) � 0 g(x, t) dx = Λ(t) � 0 ∂tg dx + ˙Λ g|x=Λ = ˙Λ g|x=Λ which shows due to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4)2 that the mass is conserved, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' d dt � � � Λ(t) � 0 g(x, t) dx + L � Λ(t) h(x, t) dx � � � = ˙Λ g|x=Λ − ˙Λ h|x=Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2 Equilibrium solution For a fixed volume of the fluid V0 = Λ(t) � 0 g dx + L � Λ(t) h dx, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3) an equilibrium solution (h, Λ) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4) is a minimizer of the total energy E, given by E(t) := a Λ(t) � 0 |∂xg|2dx + b L � Λ(t) |∂xh|2dx + c L � Λ(t) |∂xg|2dx, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4) 18 where a, b, c are non-negative constants with b > 0, corresponding to the energy for the liquid-solid, liquid-gas and gas-solid interfaces respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Note that energy of the solid- gas interface must be included in the total energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' One can compute the new energy for small changes (h + δh, Λ + δΛ) of the equilibrium state as E + δE = a Λ+δΛ � 0 |∂xg|2dx + b L � Λ+δΛ |∂x(h + δh)|2dx + c L � Λ+δΛ |∂xg|2dx = E + (a − c)δΛ|∂xg|2|x=Λ − b δΛ|∂x(h + δh)|2|x=Λ + b L � Λ (2∂xh ∂x(δh) + |∂x(δh)|2) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Neglecting the small quadratic terms, one obtains, δE = (a − c) δΛ|∂xg|2|x=Λ − b δΛ|∂xh|2|x=Λ + 2b L � Λ ∂xh ∂x(δh) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Similarly, the new volume becomes, V + δV = Λ+δΛ � 0 g dx + L � Λ+δΛ (h + δh) dx = V + L � Λ δh dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Hence, δV = L � Λ δh dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Therefore, we need to find a unique Lagrange multiplier λ ∈ R such that δE = λ δV , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' after integrating by parts, (a − c) δΛ|∂xg|2|x=Λ − b(∂xg|x=Λ − k)2δΛ − 2b L � Λ ∂2 xh δh dx − 2b(∂xh δh)|x=Λ = λ L � Λ δh dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5) Here we used the relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4)2 since (h, Λ) solves the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Choosing δΛ = 0 and δh having compact support, one then obtains, − 2b L � Λ ∂2 xh δh dx = λ L � Λ δh dx which implies ∂2 xh = − λ 2b, x ∈ (Λ, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='6) 19 Also we have the continuity relation satisfied by the perturbed solution, (h + δh)|x=Λ+δΛ = g|x=Λ+δΛ, which gives, neglecting small terms, h|x=Λ + ∂xh|x=Λ δΛ + δh|x=Λ = g|x=Λ + ∂xg|x=Λ δΛ implying, by using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4)2, δh|x=Λ = (∂xg − ∂xh)|x=Λ δΛ = k δΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Plugging (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='6) further in the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5), we get a relation determining the contact angle, b k2 + |∂xg|2(a − b − c) = 0, or, k = �� 1 − a − c b � ∂xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7) Here we assume the condition (b + c − a)/b > 0 so that the above relation is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Also recall that k > 0 (by construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' This is a form of Young’s condition which says that the contact angle is given only by slope of the solid and the energies of the interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' From relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='6) and the boundary conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4)2 together with ∂xh|x=L = 0 (due to the symmetry assumption), we can further determine the equilibrium solution completely, h = (∂xg − k)|x=Λ 2(Λ − L) (x − L)2 + g|x=Λ − (∂xg − k)|x=Λ 2 (Λ − L), x ∈ (Λ, L), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='8) and also the Lagrange multiplier, in terms of Λ, as λ = 2b(∂xg − k) (L − Λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Finally, from the volume constraint (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3), the contact point at equilibrium Λ can be deter- mined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Energy dissipation (formal): When the solid is stationary, it holds from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4)2 that h|x=Λ = g|x=Λ for t > 0 and in turn, we obtain the following relation by differentiating in time ∂th + ˙Λ ∂xh = ˙Λ ∂xg which implies ∂th = k ˙Λ at x = Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='9) 20 Therefore, one obtains the following energy dissipation relation from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4) as d dtE = 2b L � Λ(t) ∂xh ∂xth dx − b ˙Λ|∂xh|2|x=Λ + (a − c) ˙Λ|∂xg|2|x=Λ = −2b L � Λ(t) ∂xh ∂2 x(h3∂3 xh) dx + ˙Λ((a − c)|∂xg|2 − b|∂xh|2)|x=Λ [using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4)1] = −2b L � Λ(t) h3|∂3 xh|2 dx + 2b (∂xh ∂x(h3∂3 xh))|x=Λ + ˙Λ((a − c)|∂xg|2 − b|∂xh|2)|x=Λ = −2b L � Λ(t) h3|∂3 xh|2 dx + ˙Λ((a − c)|∂xg|2 − b(∂xg − k)2 − 2kb(∂xg − k))|x=Λ = −2b L � Λ(t) h3|∂3 xh|2 dx − ˙Λ((b + c − a)|∂xg|2|x=Λ − bk2) = −2b L � Λ(t) h3|∂3 xh|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='10) We have used relations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4)1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='9) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4)2 to obtain the fourth equality whereas Young’s condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7) is used at the last line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Also the condition ∂xh|x=L = 0 (due to the symmetry assumption) have been used in the above deduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Steady state: For the stationary solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4), as the time derivative vanishes, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4)1 reduces to ∂x(h3∂3 xh) = 0 in (Λ, L), which means two possibilities: either the thin film is given by a constant height or, a parabola (below we give a complete description, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The constant thin film corresponds to the case where slope of the thin film is zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' ˜θ = 0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Figure ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=') which means ∂xg = k at x = Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' If ∂xg ̸= k, then the shape of the thin film is given by a parabola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' This indicates that the fluid film might rupture in finite time if the volume of the fluid is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Recall that the steady state of the classical thin film equation ∂th + ∂x(h2∂3 xh) = 0 on a bounded domain (with slip boundary condition) is indeed given by a parabola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='1 The above energy dissipation relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='10) shows that the equilibrium solu- tion (h, Λ), given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='8), is also a steady state of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4) since it satisfies d dtE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 21 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2 From the relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4)2, one obtains that ∂xg − ∂xh = γ∂xg at x = Λ where γ = � (b + c − a)/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Depending on γ ≥ 1 or γ < 1, the steady state has a (locally) convex or concave profile, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3 Asymptotic stability of steady state Finally we are in a situation to study the stability of the stationary solution (h, Λ) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4), given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' To this end, we introduce the following linear transformation x = x(L − Λ) + L(Λ − Λ) (L − Λ) which maps the domain (Λ, L) to the fixed domain (Λ, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Differentiating, we get ∂xx = (L − Λ) (L − Λ), ∂tx = − ˙Λ(L − x)(L − Λ) (L − Λ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' In this new coordinate, let us denote H(x, t) = h(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Recall that g is independent of t here since we assume that the solid is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Now consider a small perturbation of the steady state as H = h+ϕ where |ϕ| ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' First of all, existence of a unique solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4), for some small time T0, in the periodic setting can be obtained as given by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Considering spaces on bounded spatial interval (Λ, L) instead of (0, ∞) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4 does not pose any extra difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Therefore, ϕ satisfies the same regularity as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5, for suitable initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Note that ϕ has vanishing mean due to the mass conservation property (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2), ⟨ϕ⟩ := L � Λ ϕ dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='11) Plugging in the above expression of H, the perturbation ϕ satisfies, ∂tϕ + �L − Λ L − Λ �4 ∂x((h + ϕ)3∂3 xϕ) = ˙Λ (L − x) (L − Λ)(∂xh + ∂xϕ) in x ∈ (Λ, L), ϕ = ψ1, ∂xϕ = ψ2, ∂3 xϕ = 0 at x = Λ, ∂xϕ = 0 at x = L, ϕ = ˜h0 at t = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='12) where ˜h0(x) = h0(x) and ψ1(t) = g(Λ) − g(Λ), ψ2(t) = �L − Λ L − Λ � (∂xg(Λ) − k) − (∂xg(Λ) − k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 22 Also, with the help of energy dissipation relation for the equilibrium solution, d dt � ��a Λ � 0 |∂xg|2 dx + b L � Λ |∂xh|2 dx + c L � Λ |∂xg|2 dx � �� = 0, the energy equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='10) results into, E(ϕ(t)) := d dt � ��b L � Λ |∂xϕ|2 dx + β2 (L − Λ)(|∂xg|2|x=Λ − k2) + O(β3) � �� = −2b(1 + O(β)) L � Λ (h + ϕ)3|∂3 xϕ|2 dx, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='13) where β ≡ β(t) := (Λ(t) − Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Next let us prove some a priori estimate which is essential to prove the stability result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3 Any function ϕ ∈ H3(Λ, L), having the boundary conditions ∂xϕ = (k1−k2) k1(L−Λ)ϕ at x = Λ and ∂xϕ = 0 at x = L where k1, k2 are some non-zero constants, and with zero mean value ⟨ϕ⟩ = 0, satisfies the following estimates, L � Λ |∂xϕ|2 dx ≤ C L � Λ |∂3 xϕ|2 dx, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='14) and |ϕ(Λ)|2 ≤ C L � Λ |∂3 xϕ|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='15) Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4 Note that the boundary condition ∂xϕ = (k1−k2) k1(L−Λ)ϕ at x = Λ is crucial for the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='14) to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' It is possible to construct a function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' a quadratic polynomial) with zero mean value and symmetric with respect to x = L whose first derivative does not vanish, that is the right hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='14) vanishes while not the left hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' On the other hand, the only function which satisfies all the conditions stated in the above lemma is the null function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (i) Let us first note that since ϕ ∈ H1(Λ, L), it is absolutely continuous, in particular, it holds that |ϕ(Λ)|2 ≤ C L � Λ |∂xϕ|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 23 Thus, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='15) is a consequence of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (ii) We prove the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='14) by contradiction argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Suppose that for each m ∈ N, there exists ϕm ∈ H3(Λ, L) with the conditions ∂xϕm|x=Λ = λ 2k2bϕm �� x=Λ, ∂xϕm|x=L = 0, and ⟨ϕm⟩ = 0, such that L � Λ |∂xϕm|2 dx ≥ m L � Λ |∂3 xϕm|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='16) Further we may renormalize the sequence as ∥∂xϕm∥L2(Λ,L) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='16) shows that ϕm is a bounded sequence in H3(Λ, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' In fact, ∂3 xϕm → 0 in L2(Λ, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Thus, there exists a subsequence, still denoted by ϕm, and a function ϕ such that ϕm ⇀ ϕ in H3(Λ, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Due to the compactness of H3(Λ, L) �→ H2(Λ, L), one further has that ϕm → ϕ strongly in H2(Λ, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Therefore, ϕ satisfies, ∂xϕ|x=Λ = λ 2k2bϕ �� x=Λ, ∂xϕ|x=L = 0, ⟨ϕ⟩ = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='17) and ∥∂xϕm∥L2(Λ,L) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='18) On the other hand, one has from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='16) that ∂3 xϕ = 0 in L2(Λ, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' This implies, together with the boundary conditions and the vanishing mean (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='17), that ϕ ≡ 0 in (Λ, L) which is a contradiction to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Now we can establish the asymptotic stability of the energy corresponding to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5 Let (h, Λ) be a steady state of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4), given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' There exists ε > 0 such that for any initial data ˜h0 ∈ C4+α(Λ, L), α ∈ (0, 1) and Λ0 ∈ R with |Λ0 − Λ| + ∥˜h0 − h∥C1(Λ,L) ≤ ε, a solution (h, Λ) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4) satisfies, for t ∈ (0, T0), |Λ − Λ| ≤ Ce−ωt and ∥H − h∥H1(Λ,L) ≤ Ce−ωt for some ω > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='19) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The right hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='13) can be estimated as, since |ϕ| ≪ 1, 2b(1 + O(α)) L � Λ (h + ϕ)3|∂3 xϕ|2 dx ≥ C L � Λ �h 2 �3 |∂3 xϕ|2 dx ≥ C min x h L � Λ |∂3 xϕ|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Combining with the estimates in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3, we get that 2b(1 + O(α)) L � Λ (h + ϕ)3|∂3 xϕ|2dx ≥ CE(ϕ(t)), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='20) 24 where the energy for the perturbation E(ϕ) is defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Therefore (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='20) yields that d dtE(ϕ) ≤ −CE(ϕ) for 0 ≤ t ≤ T0 which finally gives, E(ϕ(t)) ≤ exp(−Ct)E(ϕ(0)) for 0 ≤ t ≤ T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' In particular, one gets for the full energy defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4), E(t) ≤ exp(−Ct)E(0) for 0 ≤ t ≤ T ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Next, to obtain the global in time existence result, we follow the path as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' To begin with, some suitable estimate on fundamental solution for the linear problem corresponding to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='12) can be obtained as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Here onwards, the bar over x is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='6 Let ˜h0 ∈ C4+α(Λ, L) for α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' There exists a fundamental solution G(x, t, ξ, τ) of the linear problem ∂tϕ + �L − Λ L − Λ �4 ∂x(h 3∂3 xϕ) − ˙Λ (L − x) (L − Λ)∂xϕ = 0 in x ∈ (Λ, L), t > 0, ∂xϕ = ψ2, ∂3 xϕ = 0 at x = Λ, t > 0, ∂xϕ = 0 at x = L, t > 0, ϕ = ˜h0 at t = 0, x ∈ (Λ, L), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='21) satisfying the estimate |∂mG(x, t, ξ, τ)| ≤ cm(t − τ)− m+1 4 exp � −c � |x − y| (t − τ)1/4 �4/3� , t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='22) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The result follows from [4, Chapter IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4], as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' As per the notations in [4], for our system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='21), N = 1 = n, b = 2, r1 = 1, r2 = 3 and the boundary conditions read as B ≡ (∂x, ∂3 x)u|x=Λ = f ≡ (ψ2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' In (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='21)1, the leading order coefficient a4(x, t) = � L−Λ L−Λ �4 h 3 is Hölder continuous in both t ∈ (0, T0) and x ∈ (0, L), since Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5 gives that Λ ∈ C1+ α 4 [0, T0] and h is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Furthermore, the lower order coefficients a3(x, t) = � L−Λ L−Λ �4 ∂x(h 3) and a1(x, t) = ˙Λ � L−x L−Λ � are Hölder continuous in x as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Hence, all the conditions of [4, Chapter IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4] are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='7 Let (h, Λ) be a steady state of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4) and ˜h0 ∈ C4+α(Λ, L), α ∈ (0, 1) and Λ0 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' There exists ε > 0 such that for any initial data satisfying |Λ0−Λ|+∥˜h0−h∥C4+α(Λ,L) ≤ ε, a unique global solution (h, Λ) of the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4) exists, with regularity given by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5, and satisfies, for all t > 0, |Λ − Λ| ≤ Ce−ωt and ∥H − h∥C4+α(Λ,L) ≤ Ce−ωt for some ω > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='5 provides existence of ϕ as a unique solution of the quasilinear problem ∂tϕ + �L − Λ L − Λ �4 ∂x((h + ϕ)3∂3 xϕ) = ˙Λ (L − x) (L − Λ)(∂xh + ∂xϕ) in x ∈ (Λ, L), t > 0, ∂xϕ = ψ2, ∂3 xϕ = 0 at x = Λ, t > 0, ∂xϕ = 0 at x = L, t > 0, ϕ = ˜h0 at t = 0, x ∈ (Λ, L), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='23) in some interval [0, T0], which in addition matches the boundary condition ϕ|x=Λ = ψ1, and is given by the following integro-differential form, ϕ(x, t) = t � 0 L � Λ G(x, t, y, τ) [F1 + F2] (y, τ) dy dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='24) Here G is the fundamental solution of the linear problem given in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='6 and F1 = �L − Λ L − Λ �4 ∂x((ϕ3 + 3h ϕ (h + ϕ)) ∂3 xϕ), F2 = ˙Λ (L − x) (L − Λ) ∂xh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Since the assumptions of the local existence result (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [4, Chapter III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='4, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='3]) are satisfied, there exists a unique solution ϕ1 of the Cauchy problem ϕ1|t=T0 = ϕ(x, T0) for equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='23), which is defined for t ∈ (T0, T1) where T1 > T0 and belongs to C4+α(Λ, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' In turn, also Λ ∈ C1+ α 4 [T0, T1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The solution can be continued in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Now observe that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' I = ������� t � 0 L � Λ G(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' τ)F1(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' τ) dy dτ ������� = ������� t � 0 �L − Λ L − Λ �4 (τ) L � Λ G(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' τ)∂x((ϕ3 + 3h ϕ (h + ϕ)) ∂3 xϕ)(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' τ) dy dτ ������� = ������� t � 0 �L − Λ L − Λ �4 (τ) L � Λ ∂xG(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' τ)(ϕ3 + 3h ϕ (h + ϕ)) ∂3 xϕ(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' τ) dy dτ ������� ≤ c(∥ϕ∥C((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='C 1 2 (Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='L))) t � 0 �L − Λ L − Λ �4 (τ) L � Λ |∂xG(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' τ)| dy dτ ≤ c(∥ϕ∥C((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='C 1 2 (Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='L))) t � 0 �L − Λ L − Λ �4 (τ)(t − τ)− 1 4 dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 26 In the above we used integration by parts to pass all the derivatives over G and then use the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='22) for m = 1 and the regularity of h and ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Similarly, we have II = ������� t � 0 L � Λ G(x, t, y, τ)F2(y, τ) dy dτ ������� = ������� t � 0 ˙Λ (L − Λ)(τ) L � Λ G(x, t, y, τ) (L − y) ∂xh(y) dy dτ ������� ≤ c t � 0 ���� ˙Λ(L − Λ) (L − Λ)(τ) ���� L � Λ |G(x, t, y, τ)| dy dτ ≤ c t � 0 | ˙Λ(τ)|(L − Λ) (L − Λ) dτ ≤ c t � 0 | ˙Λ(τ)|(1 + β (L − Λ) + O(β2)) dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Therefore, from the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='24), we can conclude ∥ϕ∥C4+α(Λ,L) ≤ C � |β(t)| + ∥ϕ∥C 1 2 (Λ,L) � ≤ C � |Λ − Λ| + ∥ϕ∥H1(Λ,L) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Thus, the local solution can be continued up to time T = ∞ in the solution space, as long as the initial data stays close enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The authors acknowledge the support of the Hausdorff Center of Mathematics at the University of Bonn, funded by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Centre "The mathematics of emerging effects" (CRC 1060, Project-ID 211504053).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The authors certify that they do not have any conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Bernis and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Friedman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Higher order nonlinear degenerate parabolic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Journal of Differential Equations, 83(1):179–206, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Bertozzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The mathematics of moving contact lines in thin liquid films, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' de Gennes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Wetting: statics and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=', 57:827–863, Jul 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Eidel’man.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Parabolic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' North-Holland Publishing Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=', Amsterdam-London;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Wolters-Noordhoff Publishing, Groningen, 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Translated from the Russian by Scripta Tech- nica, London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Ghosh, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Niethammer, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Velázquez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Revisiting Shikhmurzaev’s approach to the contact line problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Acta Applicandae Mathematicae, 181, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 27 [6] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Giacomelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Gnann, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Knüpfer, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Otto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Well-posedness for the Navier-slip thin- film equation in the case of complete wetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Journal of Differential Equations, 257(1):15–81, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [7] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Giacomelli, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Knüpfer, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Otto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Smooth zero-contact-angle solutions to a thin-film equation around the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Journal of Differential Equations, 245(6):1454–1506, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Gnann and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Petrache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The Navier-slip thin-film equation for 3d fluid films: Existence and uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Journal of Differential Equations, 265(11):5832–5958, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [9] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Guo and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Stability of contact lines in fluids: 2D Stokes flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Archive for Rational Mechanics and Analysis, 227(2):767–854, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [10] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Hocking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Rival contact-angle models and the spreading of drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Journal of Fluid Mechanics, 239:671–681, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [11] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Huh and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Scriven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Hydrodynamic model of steady movement of a solid/liquid/fluid contact line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Journal of Colloid and Interface Science, 35(1):85 – 101, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [12] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Knüpfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Well-posedness for the Navier slip thin-film equation in the case of partial wetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Communications on Pure and Applied Mathematics, 64(9):1263–1296, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [13] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Knüpfer and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Masmoudi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Darcy’s Flow with Prescribed Contact Angle: Well-Posedness and Lubrication Approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Archive for Rational Mechanics and Analysis, 218:589 – 646, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [14] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Navier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Mémoire sur les lois du mouvement des fluides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Mém.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' de France (2), pages 389–440, 1823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Oron, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Davis, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Bankoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Long-scale evolution of thin liquid films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Reviews of Modern Physics, 69:931–980, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [16] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Ren and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Derivation of continuum models for the moving contact line problem based on thermodynamic principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Communications in Mathematical Sciences, 9:597–606, 06 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [17] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Seis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The thin-film equation close to self-similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Analysis & PDE, 11(5):1303 – 1342, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [18] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Shikhmurzaev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Moving contact lines and dynamic contact angles: a ‘litmus test’ for mathematical models, accomplishments and new challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' The European Physical Journal Special Topics, 229(10):1945–1977, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [19] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Solonnikov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Solvability of a problem on the motion of a viscous incompressible fluid bounded by a free surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Mathematics of the USSR-Izvestiya, 11(6):1323–1358, dec 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' [20] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Solonnikov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' On some free boundary problems for the Navier-Stokes equations with moving contact points and lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' Mathematische Annalen, 302:743–772, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} +page_content=' 28' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQf4ghZ/content/2301.04181v1.pdf'} diff --git a/zdE3T4oBgHgl3EQfmgot/content/2301.04616v1.pdf b/zdE3T4oBgHgl3EQfmgot/content/2301.04616v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b22ee1b325335d82755efb4f1407e41707b610d5 --- /dev/null +++ b/zdE3T4oBgHgl3EQfmgot/content/2301.04616v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:37d2722e8e2dd732da44323ecaf920babb96000634ba86b7ccb6585277f6ccd8 +size 982322 diff --git a/zdE3T4oBgHgl3EQfmgot/vector_store/index.pkl b/zdE3T4oBgHgl3EQfmgot/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..f7dddfb893418858346937a23d995742b4792d0e --- /dev/null +++ b/zdE3T4oBgHgl3EQfmgot/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:224e70fa79d826df4d6f02f1336710451b45c9c8510c826293c8c6bb712707f8 +size 147017 diff --git a/ztFJT4oBgHgl3EQfjCwq/vector_store/index.faiss b/ztFJT4oBgHgl3EQfjCwq/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..74a72e930165ab3b83c0ed863ed3ddd5f0a68992 --- /dev/null +++ b/ztFJT4oBgHgl3EQfjCwq/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:145c54fe14ca0354e709268fa6f7281b75c3e66cc67ec41eb42693d61c5a706b +size 4980781